Subgroups — H(3,𝔽₃)×#546 #13154621

Order 1728.

1837 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₃)×#123	C₂	1	2	yes	maximal, complemented
5	H(3,𝔽₃)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,𝔽₃)×#121	C₂	1	2	yes	maximal, complemented
8	H(3,𝔽₃)×#120	C₂	1	2	yes	maximal, complemented
9	C₄×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₄×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
12	C₃²×#546	C₃	1	4	yes	maximal, complemented
13	C₂⁴×H(3,𝔽₃)	C₂²	1	2	yes	—
14	#41×H(3,𝔽₃)	—	2	4	no	—
15	C₂×Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₄×H(3,𝔽₃)	—	2	4	no	—
17	C₂×D₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
18	C₂×D₄×H(3,𝔽₃)	C₂²	1	2	yes	—
19	C₂×D₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
20	C₂×D₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
21	C₂×D₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
22	C₂×D₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
23	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
24	C₂²×C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
25	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
27	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
29	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
30	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	—
32	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
33	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
34	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
35	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
36	C₂²⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	complemented
37	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃²×#41	C₂×C₃	1	4	yes	complemented
39	C₂²×C₃²×D₄	C₂×C₃	1	4	yes	complemented
40	C₃²×#124	C₂×C₃	1	4	yes	complemented
41	C₃²×#123	C₂×C₃	1	8	yes	complemented
42	C₃²×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
43	C₃²×#121	C₂×C₃	1	4	yes	complemented
44	C₃²×#121	C₂×C₃	1	8	yes	complemented
45	C₃²×#120	C₂×C₃	1	8	yes	complemented
46	C₃²×C₄×D₄	C₂×C₃	1	4	yes	complemented
47	C₃²×C₄×D₄	C₂×C₃	1	4	yes	complemented
48	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
49	C₂³×H(3,𝔽₃)	—	2	4	no	—
50	C₂³×H(3,𝔽₃)	—	2	4	no	—
51	C₂³×H(3,𝔽₃)	—	4	2	no	—
52	C₂³×H(3,𝔽₃)	C₂³	1	2	yes	—
53	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
54	C₂³×H(3,𝔽₃)	C₂³	1	2	yes	—
55	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
56	Q₈×H(3,𝔽₃)	—	2	2	no	—
57	D₄×H(3,𝔽₃)	—	2	4	no	—
58	D₄×H(3,𝔽₃)	—	2	4	no	—
59	D₄×H(3,𝔽₃)	—	2	2	no	—
60	D₄×H(3,𝔽₃)	—	2	2	no	—
61	D₄×H(3,𝔽₃)	—	4	2	no	—
62	D₄×H(3,𝔽₃)	D₄	1	4	yes	complemented
63	C₂×C₄×H(3,𝔽₃)	—	2	4	no	—
64	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
65	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
66	C₂×C₄×H(3,𝔽₃)	—	4	2	no	—
67	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
68	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
69	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
70	C₂×C₄×H(3,𝔽₃)	C₂³	1	2	yes	—
71	C₂×C₄×H(3,𝔽₃)	C₂³	1	2	yes	—
72	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
73	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
74	C₃×#546	—	3	4	no	—
75	C₃×#546	C₃²	1	1	yes	characteristic
76	C₂⁴×C₃²	C₂²×C₃	1	8	yes	—
77	C₃²×#41	—	2	16	no	—
78	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	complemented
79	C₂×C₃²×D₄	—	2	16	no	—
80	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	—
81	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
82	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
83	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
84	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	complemented
85	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	complemented
86	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
87	C₂²×C₃²×C₄	C₂²×C₃	1	8	yes	complemented
88	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	complemented
89	C₂²×C₃²×C₄	C₂²×C₃	1	8	yes	complemented
90	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
91	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
92	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
93	C₃²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
94	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
95	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
96	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
97	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
98	C₃²×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
99	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
100	C₃²×C₄²	C₂²×C₃	1	4	yes	complemented
101	C₂²×H(3,𝔽₃)	—	2	1	no	—
102	C₂²×H(3,𝔽₃)	—	2	2	no	—
103	C₂²×H(3,𝔽₃)	—	2	2	no	—
104	C₂²×H(3,𝔽₃)	—	2	1	no	—
105	C₂²×H(3,𝔽₃)	—	2	1	no	—
106	C₂²×H(3,𝔽₃)	—	2	2	no	—
107	C₂²×H(3,𝔽₃)	—	2	1	no	—
108	C₂²×H(3,𝔽₃)	—	2	2	no	—
109	C₂²×H(3,𝔽₃)	—	2	1	no	—
110	C₂²×H(3,𝔽₃)	—	2	2	no	—
111	C₂²×H(3,𝔽₃)	—	4	8	no	—
112	C₂²×H(3,𝔽₃)	C₂×D₄	1	4	yes	—
113	C₂²×H(3,𝔽₃)	#41	1	2	yes	complemented
114	C₂²×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic
115	C₄×H(3,𝔽₃)	—	2	1	no	—
116	C₄×H(3,𝔽₃)	—	2	1	no	—
117	C₄×H(3,𝔽₃)	—	2	2	no	—
118	C₄×H(3,𝔽₃)	—	2	1	no	—
119	C₄×H(3,𝔽₃)	—	2	1	no	—
120	C₄×H(3,𝔽₃)	—	2	2	no	—
121	C₄×H(3,𝔽₃)	C₂×D₄	1	2	yes	—
122	C₂×C₃×#41	—	3	4	no	—
123	C₂×C₃×#41	C₂×C₃²	1	1	yes	characteristic
124	C₂²×C₃×D₄	—	3	4	no	—
125	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
126	C₃×#124	—	3	4	no	—
127	C₃×#124	C₂×C₃²	1	1	yes	characteristic
128	C₃×#123	—	3	8	no	—
129	C₃×#123	C₂×C₃²	1	2	yes	—
130	C₃×C₂²⋊Q₈	—	3	4	no	—
131	C₃×C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic
132	C₃×#121	—	3	4	no	—
133	C₃×#121	—	3	8	no	—
134	C₃×#121	C₂×C₃²	1	1	yes	characteristic
135	C₃×#121	C₂×C₃²	1	2	yes	—
136	C₃×#120	—	3	8	no	—
137	C₃×#120	C₂×C₃²	1	2	yes	—
138	C₃×C₄×D₄	—	3	4	no	—
139	C₃×C₄×D₄	—	3	4	no	—
140	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
141	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
142	C₂×C₃×C₂²⋊C₄	—	3	8	no	—
143	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	2	yes	—
144	C₂³×C₃²	—	2	16	no	—
145	C₂³×C₃²	—	2	16	no	—
146	C₂³×C₃²	—	4	8	no	—
147	C₂³×C₃²	C₂³×C₃	1	8	yes	—
148	C₂³×C₃²	C₂³×C₃	1	4	yes	—
149	C₂³×C₃²	C₂³×C₃	1	8	yes	—
150	C₂³×C₃²	C₂³×C₃	1	4	yes	—
151	C₃²×Q₈	—	2	8	no	—
152	C₃²×D₄	—	2	16	no	—
153	C₃²×D₄	—	2	16	no	—
154	C₃²×D₄	—	2	8	no	—
155	C₃²×D₄	—	2	8	no	—
156	C₃²×D₄	—	4	8	no	—
157	C₃²×D₄	C₃×D₄	1	16	yes	complemented
158	C₂×C₃²×C₄	—	2	16	no	—
159	C₂×C₃²×C₄	—	2	8	no	—
160	C₂×C₃²×C₄	—	2	8	no	—
161	C₂×C₃²×C₄	—	4	8	no	—
162	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
163	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
164	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
165	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
166	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
167	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
168	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
169	#546	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×H(3,𝔽₃)	—	2	2	no	—
171	C₂×H(3,𝔽₃)	—	2	4	no	—
172	C₂×H(3,𝔽₃)	—	4	1	no	—
173	C₂×H(3,𝔽₃)	—	4	2	no	—
174	C₂×H(3,𝔽₃)	C₂²×D₄	1	1	yes	characteristic
175	C₂×H(3,𝔽₃)	C₂×#41	1	1	yes	characteristic
176	C₂×H(3,𝔽₃)	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂⁴×C₃	—	3	8	no	—
178	C₂⁴×C₃	C₂²×C₃²	1	2	yes	—
179	C₃×#41	—	2	4	no	—
180	C₃×#41	—	6	16	no	—
181	C₂×C₃×Q₈	—	3	4	no	—
182	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
183	C₂×C₃×D₄	—	2	4	no	—
184	C₂×C₃×D₄	—	3	4	no	—
185	C₂×C₃×D₄	—	3	4	no	—
186	C₂×C₃×D₄	—	3	8	no	—
187	C₂×C₃×D₄	—	3	4	no	—
188	C₂×C₃×D₄	—	3	8	no	—
189	C₂×C₃×D₄	—	3	8	no	—
190	C₂×C₃×D₄	—	6	16	no	—
191	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
192	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
193	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
194	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
195	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
196	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
197	C₂²×C₃×C₄	—	3	4	no	—
198	C₂²×C₃×C₄	—	3	8	no	—
199	C₂²×C₃×C₄	—	3	4	no	—
200	C₂²×C₃×C₄	—	3	8	no	—
201	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
202	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
203	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
204	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
205	C₃×C₄⋊C₄	—	3	4	no	—
206	C₃×C₄⋊C₄	—	3	4	no	—
207	C₃×C₄⋊C₄	—	3	8	no	—
208	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
209	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
210	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
211	C₃×C₂²⋊C₄	—	3	4	no	—
212	C₃×C₂²⋊C₄	—	3	8	no	—
213	C₃×C₂²⋊C₄	—	3	8	no	—
214	C₃×C₂²⋊C₄	—	3	8	no	—
215	C₃×C₂²⋊C₄	—	3	8	no	—
216	C₃×C₂²⋊C₄	—	3	4	no	—
217	C₃×C₂²⋊C₄	—	3	8	no	—
218	C₃×C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic
219	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
220	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
221	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
222	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
223	C₃×C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic
224	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
225	C₃×C₄²	—	3	4	no	—
226	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
227	C₂²×C₃²	—	2	4	no	—
228	C₂²×C₃²	—	2	8	no	—
229	C₂²×C₃²	—	2	8	no	—
230	C₂²×C₃²	—	2	4	no	—
231	C₂²×C₃²	—	2	4	no	—
232	C₂²×C₃²	—	2	8	no	—
233	C₂²×C₃²	—	2	4	no	—
234	C₂²×C₃²	—	2	8	no	—
235	C₂²×C₃²	—	2	4	no	—
236	C₂²×C₃²	—	2	8	no	—
237	C₂²×C₃²	—	4	32	no	—
238	C₂²×C₃²	C₂×C₃×D₄	1	16	yes	—
239	C₂²×C₃²	C₃×#41	1	8	yes	complemented
240	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
241	C₃²×C₄	—	2	4	no	—
242	C₃²×C₄	—	2	4	no	—
243	C₃²×C₄	—	2	8	no	—
244	C₃²×C₄	—	2	4	no	—
245	C₃²×C₄	—	2	4	no	—
246	C₃²×C₄	—	2	8	no	—
247	C₃²×C₄	C₂×C₃×D₄	1	8	yes	—
248	C₂×#41	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
249	C₂²×D₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
250	#124	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
251	#123	C₂×H(3,𝔽₃)	1	2	yes	complemented
252	C₂²⋊Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
253	#121	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
254	#121	C₂×H(3,𝔽₃)	1	2	yes	complemented
255	#120	C₂×H(3,𝔽₃)	1	2	yes	complemented
256	C₄×D₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
257	C₄×D₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
258	C₂×C₂²⋊C₄	C₂×H(3,𝔽₃)	1	2	yes	complemented
259	H(3,𝔽₃)	#546	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³×C₃	—	2	4	no	—
261	C₂³×C₃	—	2	4	no	—
262	C₂³×C₃	—	3	4	no	—
263	C₂³×C₃	—	3	8	no	—
264	C₂³×C₃	—	3	8	no	—
265	C₂³×C₃	—	3	4	no	—
266	C₂³×C₃	—	4	2	no	—
267	C₂³×C₃	—	6	16	no	—
268	C₂³×C₃	—	6	16	no	—
269	C₂³×C₃	—	12	8	no	—
270	C₂³×C₃	C₂³×C₃²	1	2	yes	—
271	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
272	C₂³×C₃	C₂³×C₃²	1	2	yes	—
273	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
274	C₃×Q₈	—	2	2	no	—
275	C₃×Q₈	—	6	8	no	—
276	C₃×D₄	—	2	4	no	—
277	C₃×D₄	—	2	4	no	—
278	C₃×D₄	—	2	2	no	—
279	C₃×D₄	—	2	2	no	—
280	C₃×D₄	—	3	16	no	—
281	C₃×D₄	—	4	2	no	—
282	C₃×D₄	—	6	8	no	—
283	C₃×D₄	—	6	8	no	—
284	C₃×D₄	—	6	16	no	—
285	C₃×D₄	—	6	16	no	—
286	C₃×D₄	—	12	8	no	—
287	C₃×D₄	C₃²×D₄	1	4	yes	—
288	C₂×C₃×C₄	—	2	4	no	—
289	C₂×C₃×C₄	—	2	2	no	—
290	C₂×C₃×C₄	—	2	2	no	—
291	C₂×C₃×C₄	—	3	4	no	—
292	C₂×C₃×C₄	—	3	4	no	—
293	C₂×C₃×C₄	—	3	4	no	—
294	C₂×C₃×C₄	—	3	8	no	—
295	C₂×C₃×C₄	—	3	4	no	—
296	C₂×C₃×C₄	—	3	8	no	—
297	C₂×C₃×C₄	—	3	4	no	—
298	C₂×C₃×C₄	—	4	2	no	—
299	C₂×C₃×C₄	—	6	8	no	—
300	C₂×C₃×C₄	—	6	8	no	—
301	C₂×C₃×C₄	—	6	16	no	—
302	C₂×C₃×C₄	—	12	8	no	—
303	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
304	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
305	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
306	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
307	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
308	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
309	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
310	C₂×C₃²	—	2	8	no	—
311	C₂×C₃²	—	2	16	no	—
312	C₂×C₃²	—	4	4	no	—
313	C₂×C₃²	—	4	8	no	—
314	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
315	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
316	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
317	C₂⁴	C₂²×H(3,𝔽₃)	1	2	yes	—
318	#41	—	2	4	no	—
319	C₂×Q₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic, complemented
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
322	C₂×D₄	C₂²×H(3,𝔽₃)	1	2	yes	—
323	C₂×D₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
324	C₂×D₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
325	C₂×D₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
326	C₂×D₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
327	C₂²×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
328	C₂²×C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
329	C₂²×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic, complemented
330	C₂²×C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
331	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic, complemented
332	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic, complemented
333	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
334	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
335	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	—
336	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
337	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
338	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
339	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic, complemented
340	C₂²⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	complemented
341	C₄²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic, complemented
342	C₂²×C₃	—	2	1	no	—
343	C₂²×C₃	—	2	2	no	—
344	C₂²×C₃	—	2	2	no	—
345	C₂²×C₃	—	2	1	no	—
346	C₂²×C₃	—	2	1	no	—
347	C₂²×C₃	—	2	2	no	—
348	C₂²×C₃	—	2	1	no	—
349	C₂²×C₃	—	2	2	no	—
350	C₂²×C₃	—	2	1	no	—
351	C₂²×C₃	—	2	2	no	—
352	C₂²×C₃	—	3	4	no	—
353	C₂²×C₃	—	3	16	no	—
354	C₂²×C₃	—	3	8	no	—
355	C₂²×C₃	—	4	8	no	—
356	C₂²×C₃	—	6	4	no	—
357	C₂²×C₃	—	6	8	no	—
358	C₂²×C₃	—	6	8	no	—
359	C₂²×C₃	—	6	4	no	—
360	C₂²×C₃	—	6	4	no	—
361	C₂²×C₃	—	6	8	no	—
362	C₂²×C₃	—	6	4	no	—
363	C₂²×C₃	—	6	4	no	—
364	C₂²×C₃	—	6	8	no	—
365	C₂²×C₃	—	6	8	no	—
366	C₂²×C₃	—	12	32	no	—
367	C₂²×C₃	C₂×C₃²×D₄	1	4	yes	—
368	C₂²×C₃	C₃²×#41	1	2	yes	—
369	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
370	C₃×C₄	—	2	1	no	—
371	C₃×C₄	—	2	1	no	—
372	C₃×C₄	—	2	2	no	—
373	C₃×C₄	—	2	1	no	—
374	C₃×C₄	—	2	1	no	—
375	C₃×C₄	—	2	2	no	—
376	C₃×C₄	—	3	8	no	—
377	C₃×C₄	—	6	4	no	—
378	C₃×C₄	—	6	4	no	—
379	C₃×C₄	—	6	8	no	—
380	C₃×C₄	—	6	4	no	—
381	C₃×C₄	—	6	8	no	—
382	C₃×C₄	—	6	4	no	—
383	C₃×C₄	C₂×C₃²×D₄	1	2	yes	—
384	C₃²	C₃×#546	1	4	yes	complemented
385	C₂³	—	2	4	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	4	2	no	—
388	C₂³	C₂³×H(3,𝔽₃)	1	2	yes	—
389	C₂³	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
390	C₂³	C₂³×H(3,𝔽₃)	1	2	yes	—
391	C₂³	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
392	Q₈	—	2	2	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	4	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	2	no	—
397	D₄	—	4	2	no	—
398	D₄	D₄×H(3,𝔽₃)	1	4	yes	complemented
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
404	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
405	C₂×C₄	C₂³×H(3,𝔽₃)	1	2	yes	—
406	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
407	C₂×C₄	C₂³×H(3,𝔽₃)	1	2	yes	—
408	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
409	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
410	C₂×C₃	—	2	2	no	—
411	C₂×C₃	—	2	4	no	—
412	C₂×C₃	—	3	4	no	—
413	C₂×C₃	—	3	4	no	—
414	C₂×C₃	—	3	4	no	—
415	C₂×C₃	—	4	2	no	—
416	C₂×C₃	—	4	1	no	—
417	C₂×C₃	—	6	8	no	—
418	C₂×C₃	—	6	16	no	—
419	C₂×C₃	—	12	8	no	—
420	C₂×C₃	—	12	4	no	—
421	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
422	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
423	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
424	C₂²	—	2	1	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	4	8	no	—
435	C₂²	C₂×D₄×H(3,𝔽₃)	1	4	yes	—
436	C₂²	#41×H(3,𝔽₃)	1	2	yes	complemented
437	C₂²	C₂⁴×H(3,𝔽₃)	1	1	yes	characteristic
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	1	no	—
443	C₄	—	2	2	no	—
444	C₄	C₂×D₄×H(3,𝔽₃)	1	2	yes	—
445	C₃	—	3	4	no	—
446	C₃	C₃²×#546	1	1	yes	characteristic
447	C₂	—	2	2	no	—
448	C₂	—	2	4	no	—
449	C₂	—	4	2	no	—
450	C₂	—	4	1	no	—
451	C₂	C₂²×D₄×H(3,𝔽₃)	1	1	yes	characteristic
452	C₂	C₂×#41×H(3,𝔽₃)	1	1	yes	characteristic
453	C₂	H(3,𝔽₃)×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃²
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#41×H(3,𝔽₃)	C₂	semidirect
2. C₂²×D₄×H(3,𝔽₃)	C₂	semidirect
3. H(3,𝔽₃)×#124	C₂	semidirect
4. H(3,𝔽₃)×#123	C₂	semidirect
5. H(3,𝔽₃)×C₂²⋊Q₈	C₂	semidirect
6. H(3,𝔽₃)×#121	C₂	semidirect
7. H(3,𝔽₃)×#121	C₂	semidirect
8. H(3,𝔽₃)×#120	C₂	semidirect
9. C₄×D₄×H(3,𝔽₃)	C₂	semidirect
10. C₄×D₄×H(3,𝔽₃)	C₂	semidirect
11. C₂×C₂²⋊C₄×H(3,𝔽₃)	C₂	semidirect
12. C₃²×#546	C₃	semidirect
15. C₂×Q₈×H(3,𝔽₃)	C₂²	semidirect
21. C₂×D₄×H(3,𝔽₃)	C₂²	semidirect
22. C₂×D₄×H(3,𝔽₃)	C₂²	semidirect
24. C₂²×C₄×H(3,𝔽₃)	C₂²	semidirect
25. C₂²×C₄×H(3,𝔽₃)	C₂²	semidirect
26. C₂²×C₄×H(3,𝔽₃)	C₂²	semidirect
27. C₄⋊C₄×H(3,𝔽₃)	C₂²	semidirect
28. C₄⋊C₄×H(3,𝔽₃)	C₂²	semidirect
29. C₄⋊C₄×H(3,𝔽₃)	C₂²	semidirect
32. C₂²⋊C₄×H(3,𝔽₃)	C₂²	semidirect
33. C₂²⋊C₄×H(3,𝔽₃)	C₂²	semidirect
34. C₂²⋊C₄×H(3,𝔽₃)	C₂²	semidirect
35. C₂²⋊C₄×H(3,𝔽₃)	C₂²	semidirect
36. C₂²⋊C₄×H(3,𝔽₃)	C₂²	semidirect
37. C₄²×H(3,𝔽₃)	C₂²	semidirect
38. C₂×C₃²×#41	C₂×C₃	semidirect
39. C₂²×C₃²×D₄	C₂×C₃	semidirect
40. C₃²×#124	C₂×C₃	semidirect
41. C₃²×#123	C₂×C₃	semidirect
42. C₃²×C₂²⋊Q₈	C₂×C₃	semidirect
43. C₃²×#121	C₂×C₃	semidirect
44. C₃²×#121	C₂×C₃	semidirect
45. C₃²×#120	C₂×C₃	semidirect
46. C₃²×C₄×D₄	C₂×C₃	semidirect
47. C₃²×C₄×D₄	C₂×C₃	semidirect
48. C₂×C₃²×C₂²⋊C₄	C₂×C₃	semidirect
62. D₄×H(3,𝔽₃)	D₄	semidirect
78. C₂×C₃²×Q₈	C₂²×C₃	semidirect
84. C₂×C₃²×D₄	C₂²×C₃	semidirect
85. C₂×C₃²×D₄	C₂²×C₃	semidirect
87. C₂²×C₃²×C₄	C₂²×C₃	semidirect
88. C₂²×C₃²×C₄	C₂²×C₃	semidirect
89. C₂²×C₃²×C₄	C₂²×C₃	semidirect
90. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
91. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
92. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
95. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
96. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
97. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
98. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
99. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
100. C₃²×C₄²	C₂²×C₃	semidirect
113. C₂²×H(3,𝔽₃)	#41	semidirect
157. C₃²×D₄	C₃×D₄	semidirect
169. #546	H(3,𝔽₃)	direct (N × H)
239. C₂²×C₃²	C₃×#41	semidirect
248. C₂×#41	C₂×H(3,𝔽₃)	semidirect
249. C₂²×D₄	C₂×H(3,𝔽₃)	semidirect
250. #124	C₂×H(3,𝔽₃)	semidirect
251. #123	C₂×H(3,𝔽₃)	semidirect
252. C₂²⋊Q₈	C₂×H(3,𝔽₃)	semidirect
253. #121	C₂×H(3,𝔽₃)	semidirect
254. #121	C₂×H(3,𝔽₃)	semidirect
255. #120	C₂×H(3,𝔽₃)	semidirect
256. C₄×D₄	C₂×H(3,𝔽₃)	semidirect
257. C₄×D₄	C₂×H(3,𝔽₃)	semidirect
258. C₂×C₂²⋊C₄	C₂×H(3,𝔽₃)	semidirect
259. H(3,𝔽₃)	#546	direct (N × H)
319. C₂×Q₈	C₂²×H(3,𝔽₃)	semidirect
325. C₂×D₄	C₂²×H(3,𝔽₃)	semidirect
326. C₂×D₄	C₂²×H(3,𝔽₃)	semidirect
328. C₂²×C₄	C₂²×H(3,𝔽₃)	semidirect
329. C₂²×C₄	C₂²×H(3,𝔽₃)	semidirect
330. C₂²×C₄	C₂²×H(3,𝔽₃)	semidirect
331. C₄⋊C₄	C₂²×H(3,𝔽₃)	semidirect
332. C₄⋊C₄	C₂²×H(3,𝔽₃)	semidirect
333. C₄⋊C₄	C₂²×H(3,𝔽₃)	semidirect
336. C₂²⋊C₄	C₂²×H(3,𝔽₃)	semidirect
337. C₂²⋊C₄	C₂²×H(3,𝔽₃)	semidirect
338. C₂²⋊C₄	C₂²×H(3,𝔽₃)	semidirect
339. C₂²⋊C₄	C₂²×H(3,𝔽₃)	semidirect
340. C₂²⋊C₄	C₂²×H(3,𝔽₃)	semidirect
341. C₄²	C₂²×H(3,𝔽₃)	semidirect
384. C₃²	C₃×#546	semidirect
398. D₄	D₄×H(3,𝔽₃)	semidirect
436. C₂²	#41×H(3,𝔽₃)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#41×H(3,𝔽₃)	2
C₂²×D₄×H(3,𝔽₃)	2
H(3,𝔽₃)×#124	2
H(3,𝔽₃)×#123	2
H(3,𝔽₃)×C₂²⋊Q₈	2
H(3,𝔽₃)×#121	2
H(3,𝔽₃)×#120	2
C₄×D₄×H(3,𝔽₃)	2
C₂×C₂²⋊C₄×H(3,𝔽₃)	2
C₃²×#546	3

Subgroups — H(3,𝔽3)×#546 #13154621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#546 #13154621