Subgroups — H(3,𝔽₃)×#421 #13145022

Order 1728.

847 conjugacy classes of subgroups.

The subgroup lattice has 285 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
2	C₂×M₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₃)×#104	C₂	1	4	yes	maximal, complemented
5	C₃²×#421	C₃	1	4	yes	maximal, complemented
6	#41×H(3,𝔽₃)	—	2	1	no	—
7	#41×H(3,𝔽₃)	—	2	1	no	—
8	#41×H(3,𝔽₃)	C₄	1	2	yes	complemented
9	#41×H(3,𝔽₃)	C₂²	1	2	yes	—
10	C₂×Q₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented
11	C₂×D₄×H(3,𝔽₃)	—	2	1	no	—
12	C₂×D₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₄×H(3,𝔽₃)	—	2	1	no	—
14	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
15	M₁₆×H(3,𝔽₃)	—	2	1	no	—
16	M₁₆×H(3,𝔽₃)	C₂²	1	2	yes	complemented
17	C₂×C₈×H(3,𝔽₃)	—	2	1	no	—
18	C₄⋊C₄×H(3,𝔽₃)	—	2	1	no	—
19	C₂²⋊C₄×H(3,𝔽₃)	—	2	1	no	—
20	C₄²×H(3,𝔽₃)	C₂²	1	2	yes	complemented
21	C₂×C₃²×#41	C₂×C₃	1	4	yes	—
22	C₂×C₃²×M₁₆	C₂×C₃	1	4	yes	complemented
23	C₃²×#117	C₂×C₃	1	4	yes	complemented
24	C₃²×#104	C₂×C₃	1	16	yes	complemented
25	C₂³×H(3,𝔽₃)	—	2	1	no	—
26	C₂³×H(3,𝔽₃)	D₄	1	1	yes	characteristic
27	Q₈×H(3,𝔽₃)	—	2	1	no	—
28	Q₈×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
29	D₄×H(3,𝔽₃)	—	2	2	no	—
30	D₄×H(3,𝔽₃)	—	2	2	no	—
31	D₄×H(3,𝔽₃)	—	2	1	no	—
32	D₄×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
33	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
34	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
35	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
36	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
37	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
40	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
41	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
42	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
43	C₈×H(3,𝔽₃)	—	2	2	no	—
44	C₃×#421	—	3	4	no	—
45	C₃×#421	C₃²	1	1	yes	characteristic
46	C₃²×#41	—	2	4	no	—
47	C₃²×#41	—	2	4	no	—
48	C₃²×#41	C₃×C₄	1	8	yes	complemented
49	C₃²×#41	C₂²×C₃	1	8	yes	—
50	C₂×C₃²×Q₈	C₃×C₄	1	4	yes	complemented
51	C₂×C₃²×D₄	—	2	4	no	—
52	C₂×C₃²×D₄	C₃×C₄	1	4	yes	complemented
53	C₂²×C₃²×C₄	—	2	4	no	—
54	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
55	C₃²×M₁₆	—	2	4	no	—
56	C₃²×M₁₆	C₂²×C₃	1	8	yes	complemented
57	C₂×C₃²×C₈	—	2	4	no	—
58	C₃²×C₄⋊C₄	—	2	4	no	—
59	C₃²×C₂²⋊C₄	—	2	4	no	—
60	C₃²×C₄²	C₂²×C₃	1	8	yes	complemented
61	C₂²×H(3,𝔽₃)	—	2	2	no	—
62	C₂²×H(3,𝔽₃)	—	4	1	no	—
63	C₂²×H(3,𝔽₃)	—	4	2	no	—
64	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
67	C₄×H(3,𝔽₃)	—	2	2	no	—
68	C₄×H(3,𝔽₃)	—	4	2	no	—
69	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
70	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
71	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
72	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×#41	—	3	4	no	—
74	C₂×C₃×#41	C₂×C₃²	1	1	yes	characteristic
75	C₂×C₃×M₁₆	—	3	4	no	—
76	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
77	C₃×#117	—	3	4	no	—
78	C₃×#117	C₂×C₃²	1	1	yes	characteristic
79	C₃×#104	—	3	16	no	—
80	C₃×#104	C₂×C₃²	1	4	yes	—
81	C₂³×C₃²	—	2	4	no	—
82	C₂³×C₃²	C₃×D₄	1	4	yes	—
83	C₃²×Q₈	—	2	4	no	—
84	C₃²×Q₈	C₂×C₃×C₄	1	8	yes	—
85	C₃²×D₄	—	2	8	no	—
86	C₃²×D₄	—	2	8	no	—
87	C₃²×D₄	—	2	4	no	—
88	C₃²×D₄	C₂×C₃×C₄	1	8	yes	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	8	no	—
91	C₂×C₃²×C₄	—	2	8	no	—
92	C₂×C₃²×C₄	—	2	8	no	—
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
95	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
96	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
99	C₃²×C₈	—	2	8	no	—
100	#421	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×H(3,𝔽₃)	—	2	1	no	—
102	C₂×H(3,𝔽₃)	—	2	1	no	—
103	C₂×H(3,𝔽₃)	—	2	1	no	—
104	C₂×H(3,𝔽₃)	—	4	2	no	—
105	C₂×H(3,𝔽₃)	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₃×#41	—	2	1	no	—
107	C₃×#41	—	2	1	no	—
108	C₃×#41	—	3	8	no	—
109	C₃×#41	—	3	8	no	—
110	C₃×#41	—	6	4	no	—
111	C₃×#41	—	6	4	no	—
112	C₃×#41	C₃²×C₄	1	2	yes	—
113	C₃×#41	C₂²×C₃²	1	2	yes	—
114	C₂×C₃×Q₈	—	3	4	no	—
115	C₂×C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
116	C₂×C₃×D₄	—	2	1	no	—
117	C₂×C₃×D₄	—	3	4	no	—
118	C₂×C₃×D₄	—	6	4	no	—
119	C₂×C₃×D₄	C₃²×C₄	1	1	yes	characteristic
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	3	4	no	—
122	C₂²×C₃×C₄	—	6	4	no	—
123	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
124	C₃×M₁₆	—	2	1	no	—
125	C₃×M₁₆	—	3	8	no	—
126	C₃×M₁₆	—	6	4	no	—
127	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	6	4	no	—
130	C₃×C₄⋊C₄	—	2	1	no	—
131	C₃×C₄⋊C₄	—	6	4	no	—
132	C₃×C₂²⋊C₄	—	2	1	no	—
133	C₃×C₂²⋊C₄	—	6	4	no	—
134	C₃×C₄²	—	3	8	no	—
135	C₃×C₄²	C₂²×C₃²	1	2	yes	—
136	C₂²×C₃²	—	2	8	no	—
137	C₂²×C₃²	—	4	4	no	—
138	C₂²×C₃²	—	4	8	no	—
139	C₂²×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
140	C₂²×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
141	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
142	C₃²×C₄	—	2	8	no	—
143	C₃²×C₄	—	4	8	no	—
144	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
145	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
146	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
147	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
148	C₂×#41	C₂×H(3,𝔽₃)	1	1	yes	characteristic
149	C₂×M₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
150	#117	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
151	#104	C₂×H(3,𝔽₃)	1	4	yes	complemented
152	H(3,𝔽₃)	#421	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	—	3	4	no	—
155	C₂³×C₃	—	6	4	no	—
156	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
157	C₃×Q₈	—	2	1	no	—
158	C₃×Q₈	—	3	8	no	—
159	C₃×Q₈	—	6	4	no	—
160	C₃×Q₈	C₂×C₃²×C₄	1	2	yes	—
161	C₃×D₄	—	2	2	no	—
162	C₃×D₄	—	2	2	no	—
163	C₃×D₄	—	2	1	no	—
164	C₃×D₄	—	3	8	no	—
165	C₃×D₄	—	6	4	no	—
166	C₃×D₄	—	6	8	no	—
167	C₃×D₄	—	6	8	no	—
168	C₃×D₄	C₂×C₃²×C₄	1	2	yes	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	3	4	no	—
174	C₂×C₃×C₄	—	3	4	no	—
175	C₂×C₃×C₄	—	3	4	no	—
176	C₂×C₃×C₄	—	3	4	no	—
177	C₂×C₃×C₄	—	3	4	no	—
178	C₂×C₃×C₄	—	3	4	no	—
179	C₂×C₃×C₄	—	6	8	no	—
180	C₂×C₃×C₄	—	6	8	no	—
181	C₂×C₃×C₄	—	6	4	no	—
182	C₂×C₃×C₄	—	6	8	no	—
183	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	6	8	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	2	4	no	—
193	C₂×C₃²	—	2	4	no	—
194	C₂×C₃²	—	4	8	no	—
195	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
196	#41	—	2	1	no	—
197	#41	—	2	1	no	—
198	#41	C₄×H(3,𝔽₃)	1	2	yes	complemented
199	#41	C₂²×H(3,𝔽₃)	1	2	yes	—
200	C₂×Q₈	C₄×H(3,𝔽₃)	1	1	yes	characteristic, complemented
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₄×H(3,𝔽₃)	1	1	yes	characteristic, complemented
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
205	M₁₆	—	2	1	no	—
206	M₁₆	C₂²×H(3,𝔽₃)	1	2	yes	complemented
207	C₂×C₈	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₄²	C₂²×H(3,𝔽₃)	1	2	yes	complemented
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	3	4	no	—
213	C₂²×C₃	—	3	4	no	—
214	C₂²×C₃	—	3	4	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	2	no	—
217	C₂²×C₃	—	6	8	no	—
218	C₂²×C₃	—	12	4	no	—
219	C₂²×C₃	—	12	8	no	—
220	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	3	4	no	—
225	C₃×C₄	—	3	4	no	—
226	C₃×C₄	—	3	4	no	—
227	C₃×C₄	—	3	4	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	—	6	8	no	—
230	C₃×C₄	—	12	8	no	—
231	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
232	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
233	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
235	C₃²	C₃×#421	1	4	yes	complemented
236	C₂³	—	2	1	no	—
237	C₂³	D₄×H(3,𝔽₃)	1	1	yes	characteristic
238	Q₈	—	2	1	no	—
239	Q₈	C₂×C₄×H(3,𝔽₃)	1	2	yes	—
240	D₄	—	2	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	2	1	no	—
243	D₄	C₂×C₄×H(3,𝔽₃)	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
250	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
251	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
252	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
253	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	3	4	no	—
259	C₂×C₃	—	4	2	no	—
260	C₂×C₃	—	6	4	no	—
261	C₂×C₃	—	6	4	no	—
262	C₂×C₃	—	6	4	no	—
263	C₂×C₃	—	12	8	no	—
264	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
269	C₂²	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
270	C₂²	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
274	C₄	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
275	C₄	C₂²×C₄×H(3,𝔽₃)	1	1	yes	characteristic
276	C₄	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
277	C₃	—	3	4	no	—
278	C₃	C₃²×#421	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	C₂×C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×C₃²×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂×C₃	C₂×C₃²×C₂²⋊C₄
prime radical	C₂×#41×H(3,𝔽₃)	C₂
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₄×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×H(3,𝔽₃)	C₂	semidirect
3. H(3,𝔽₃)×#117	C₂	semidirect
4. H(3,𝔽₃)×#104	C₂	semidirect
5. C₃²×#421	C₃	semidirect
8. #41×H(3,𝔽₃)	C₄	semidirect
10. C₂×Q₈×H(3,𝔽₃)	C₄	semidirect
12. C₂×D₄×H(3,𝔽₃)	C₄	semidirect
16. M₁₆×H(3,𝔽₃)	C₂²	semidirect
20. C₄²×H(3,𝔽₃)	C₂²	semidirect
22. C₂×C₃²×M₁₆	C₂×C₃	semidirect
23. C₃²×#117	C₂×C₃	semidirect
24. C₃²×#104	C₂×C₃	semidirect
48. C₃²×#41	C₃×C₄	semidirect
50. C₂×C₃²×Q₈	C₃×C₄	semidirect
52. C₂×C₃²×D₄	C₃×C₄	semidirect
56. C₃²×M₁₆	C₂²×C₃	semidirect
60. C₃²×C₄²	C₂²×C₃	semidirect
100. #421	H(3,𝔽₃)	direct (N × H)
149. C₂×M₁₆	C₂×H(3,𝔽₃)	semidirect
150. #117	C₂×H(3,𝔽₃)	semidirect
151. #104	C₂×H(3,𝔽₃)	semidirect
152. H(3,𝔽₃)	#421	direct (N × H)
198. #41	C₄×H(3,𝔽₃)	semidirect
200. C₂×Q₈	C₄×H(3,𝔽₃)	semidirect
202. C₂×D₄	C₄×H(3,𝔽₃)	semidirect
206. M₁₆	C₂²×H(3,𝔽₃)	semidirect
210. C₄²	C₂²×H(3,𝔽₃)	semidirect
235. C₃²	C₃×#421	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
C₂×M₁₆×H(3,𝔽₃)	2
H(3,𝔽₃)×#117	2
H(3,𝔽₃)×#104	2
C₃²×#421	3

Subgroups — H(3,𝔽3)×#421 #13145022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#421 #13145022