Subgroups — C₃²×#5727 #13179098

Order 1728.

1952 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#946	C₂	1	2	yes	maximal, complemented
4	C₃²×#896	C₂	1	4	yes	maximal, complemented
5	C₃²×#893	C₂	1	2	yes	maximal
6	C₃²×#888	C₂	1	2	yes	maximal, complemented
7	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
8	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×D₄×Q₈	—	3	1	no	maximal
10	C₃×#5727	C₃	1	4	yes	maximal, direct product factor
11	C₂×C₃³×Q₈	C₂²	1	2	yes	—
12	C₃²×D₃×Q₈	—	2	8	no	—
13	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
15	C₂×C₃²×C₄×D₃	C₂²	1	4	yes	—
16	C₂×C₃²×Dic₆	C₂²	1	4	yes	—
17	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
18	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃³×C₄²	C₂²	1	1	yes	characteristic
21	C₃²×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
22	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₃²×Dic₃⋊C₄	C₂²	1	4	yes	—
26	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
27	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
28	C₂²×C₃²×Q₈	—	3	2	no	—
29	C₂×C₃×D₃×Q₈	C₂×C₃	1	8	yes	complemented
30	C₃²×C₄⋊Q₈	—	2	4	no	—
31	C₃²×C₄⋊Q₈	—	3	2	no	—
32	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
33	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
34	C₃²×C₂²⋊Q₈	—	3	2	no	—
35	C₃²×C₂²⋊Q₈	—	3	4	no	—
36	C₃²×C₄×Q₈	—	3	1	no	—
37	C₃²×C₄×D₄	—	3	2	no	—
38	C₃²×C₄×D₄	—	3	1	no	—
39	C₃×#946	C₂×C₃	1	8	yes	complemented
40	C₃×#896	C₂×C₃	1	16	yes	complemented
41	C₃×#893	C₂×C₃	1	8	yes	—
42	C₃×#888	C₂×C₃	1	8	yes	complemented
43	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
44	C₃×C₄×Dic₆	C₂×C₃	1	4	yes	complemented
45	C₂²×C₃²×D₃	C₂³	1	2	yes	—
46	C₃³×Q₈	—	2	4	no	—
47	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃³×C₄	C₂³	1	4	yes	—
49	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
52	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
53	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
54	C₃²×D₁₂	Q₈	1	4	yes	complemented
55	C₃²×C₄×D₃	—	2	4	no	—
56	C₃²×C₄×D₃	—	2	8	no	—
57	C₃²×Dic₆	—	2	8	no	—
58	C₃²×Dic₆	D₄	1	4	yes	complemented
59	C₃×D₄×Q₈	—	3	4	no	—
60	#5727	C₃²	1	1	yes	characteristic, direct product factor
61	C₂×C₃²×Q₈	—	2	8	no	—
62	C₂×C₃²×Q₈	—	3	1	no	—
63	C₂×C₃²×Q₈	—	3	4	no	—
64	C₂×C₃²×Q₈	—	6	8	no	—
65	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
66	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
67	C₂×C₃²×D₄	—	3	1	no	—
68	C₂²×C₃²×C₄	—	3	4	no	—
69	C₂²×C₃²×C₄	—	3	2	no	—
70	C₃×D₃×Q₈	—	2	32	no	—
71	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
72	C₂×C₃×C₄×D₃	C₂²×C₃	1	16	yes	—
73	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
74	C₂×C₃×Dic₆	C₂²×C₃	1	16	yes	—
75	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	—
76	C₃²×C₄⋊C₄	—	2	8	no	—
77	C₃²×C₄⋊C₄	—	2	8	no	—
78	C₃²×C₄⋊C₄	—	3	1	no	—
79	C₃²×C₄⋊C₄	—	3	2	no	—
80	C₃²×C₄⋊C₄	—	3	1	no	—
81	C₃²×C₄⋊C₄	—	3	4	no	—
82	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
83	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
84	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
85	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
86	C₃²×C₂²⋊C₄	—	3	4	no	—
87	C₃²×C₂²⋊C₄	—	3	2	no	—
88	C₃²×C₄²	—	2	4	no	—
89	C₃²×C₄²	—	3	2	no	—
90	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
91	C₃²×C₄²	C₂²×C₃	1	4	yes	—
92	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	16	yes	—
93	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
94	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
95	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
96	C₃×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
97	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
98	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
99	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
100	C₂×C₃²×D₃	—	2	2	no	—
101	C₂×C₃²×D₃	—	2	2	no	—
102	C₂×C₃²×D₃	C₂×Q₈	1	4	yes	complemented
103	C₃³×C₄	—	2	4	no	—
104	C₃³×C₄	—	2	1	no	—
105	C₃³×C₄	C₂×D₄	1	2	yes	—
106	C₃³×C₄	C₂×Q₈	1	2	yes	complemented
107	C₃²×Dic₃	—	2	2	no	—
108	C₃²×Dic₃	—	2	2	no	—
109	C₃²×Dic₃	C₂×D₄	1	4	yes	—
110	C₂²×C₃×Q₈	—	3	8	no	—
111	C₂×D₃×Q₈	C₂×C₃²	1	2	yes	complemented
112	C₃×C₄⋊Q₈	—	2	4	no	—
113	C₃×C₄⋊Q₈	—	3	8	no	—
114	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
115	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
116	C₃×C₂²⋊Q₈	—	3	8	no	—
117	C₃×C₂²⋊Q₈	—	3	16	no	—
118	C₃×C₄×Q₈	—	3	4	no	—
119	C₃×C₄×D₄	—	3	8	no	—
120	C₃×C₄×D₄	—	3	4	no	—
121	#946	C₂×C₃²	1	2	yes	complemented
122	#896	C₂×C₃²	1	4	yes	complemented
123	#893	C₂×C₃²	1	2	yes	—
124	#888	C₂×C₃²	1	2	yes	complemented
125	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
126	C₄×Dic₆	C₂×C₃²	1	1	yes	characteristic, complemented
127	C₂³×C₃²	—	3	2	no	—
128	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
129	C₃²×Q₈	—	2	4	no	—
130	C₃²×Q₈	—	2	16	no	—
131	C₃²×Q₈	—	3	4	no	—
132	C₃²×Q₈	—	4	16	no	—
133	C₃²×Q₈	—	6	8	no	—
134	C₃²×D₄	—	3	4	no	—
135	C₂×C₃²×C₄	—	2	4	no	—
136	C₂×C₃²×C₄	—	2	4	no	—
137	C₂×C₃²×C₄	—	2	4	no	—
138	C₂×C₃²×C₄	—	2	16	no	—
139	C₂×C₃²×C₄	—	3	2	no	—
140	C₂×C₃²×C₄	—	3	2	no	—
141	C₂×C₃²×C₄	—	3	2	no	—
142	C₂×C₃²×C₄	—	6	8	no	—
143	C₂×C₃²×C₄	—	6	4	no	—
144	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
146	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
149	C₂×C₃²×C₄	C₂³×C₃	1	16	yes	—
150	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
151	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
152	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
153	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
154	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
155	C₃×D₁₂	C₃×Q₈	1	16	yes	complemented
156	C₃×C₄×D₃	—	2	16	no	—
157	C₃×C₄×D₃	—	2	32	no	—
158	C₃×Dic₆	—	2	32	no	—
159	C₃×Dic₆	C₃×D₄	1	16	yes	complemented
160	D₄×Q₈	—	3	1	no	Sylow
161	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
164	C₃²×D₃	—	2	4	no	—
165	C₂×C₃×Q₈	—	2	8	no	—
166	C₂×C₃×Q₈	—	3	4	no	—
167	C₂×C₃×Q₈	—	3	16	no	—
168	C₂×C₃×Q₈	—	6	32	no	—
169	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
170	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
171	C₂×C₃×D₄	—	3	4	no	—
172	C₂²×C₃×C₄	—	3	16	no	—
173	C₂²×C₃×C₄	—	3	8	no	—
174	D₃×Q₈	—	2	8	no	—
175	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
176	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
177	C₂×C₄×D₃	C₂²×C₃²	1	4	yes	—
178	C₂×Dic₆	C₂²×C₃²	1	4	yes	—
179	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic
180	C₃×C₄⋊C₄	—	2	8	no	—
181	C₃×C₄⋊C₄	—	2	8	no	—
182	C₃×C₄⋊C₄	—	3	8	no	—
183	C₃×C₄⋊C₄	—	3	16	no	—
184	C₃×C₄⋊C₄	—	3	4	no	—
185	C₃×C₄⋊C₄	—	3	4	no	—
186	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
187	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
188	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
189	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
190	C₃×C₂²⋊C₄	—	3	16	no	—
191	C₃×C₂²⋊C₄	—	3	8	no	—
192	C₃×C₄²	—	2	4	no	—
193	C₃×C₄²	—	3	8	no	—
194	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
195	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
196	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
197	(C₂×D₃)⋊C₄	C₂²×C₃²	1	4	yes	—
198	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
199	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
200	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
201	Dic₃⋊C₄	C₂²×C₃²	1	4	yes	—
202	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
203	C₂²×C₃²	—	2	4	no	—
204	C₂²×C₃²	—	3	4	no	—
205	C₂²×C₃²	—	6	2	no	—
206	C₂²×C₃²	—	6	2	no	—
207	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
208	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
209	C₂×C₃×D₃	—	2	8	no	—
210	C₂×C₃×D₃	—	2	8	no	—
211	C₂×C₃×D₃	C₂×C₃×Q₈	1	16	yes	complemented
212	C₃²×C₄	—	2	8	no	—
213	C₃²×C₄	—	2	4	no	—
214	C₃²×C₄	—	2	1	no	—
215	C₃²×C₄	—	2	4	no	—
216	C₃²×C₄	—	2	8	no	—
217	C₃²×C₄	—	2	8	no	—
218	C₃²×C₄	—	2	16	no	—
219	C₃²×C₄	—	3	4	no	—
220	C₃²×C₄	—	4	16	no	—
221	C₃²×C₄	—	6	2	no	—
222	C₃²×C₄	—	6	2	no	—
223	C₃²×C₄	D₃×D₄	1	2	yes	—
224	C₃²×C₄	D₃×Q₈	1	2	yes	complemented
225	C₃²×C₄	C₂×C₃×D₄	1	8	yes	—
226	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	complemented
227	C₃×Dic₃	—	2	8	no	—
228	C₃×Dic₃	—	2	8	no	—
229	C₃×Dic₃	C₂×C₃×D₄	1	16	yes	—
230	C₂²×Q₈	—	3	2	no	—
231	C₄⋊Q₈	—	3	2	no	—
232	C₄⋊Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
233	C₂²⋊Q₈	—	3	2	no	—
234	C₂²⋊Q₈	—	3	4	no	—
235	C₄×Q₈	—	3	1	no	—
236	C₄×D₄	—	3	2	no	—
237	C₄×D₄	—	3	1	no	—
238	C₃³	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
239	C₂³×C₃	—	3	8	no	—
240	C₂²×D₃	C₂³×C₃²	1	2	yes	—
241	C₃×Q₈	—	2	16	no	—
242	C₃×Q₈	—	2	4	no	—
243	C₃×Q₈	—	3	16	no	—
244	C₃×Q₈	—	4	16	no	—
245	C₃×Q₈	—	6	32	no	—
246	C₃×D₄	—	3	16	no	—
247	C₂×C₃×C₄	—	2	4	no	—
248	C₂×C₃×C₄	—	2	4	no	—
249	C₂×C₃×C₄	—	2	4	no	—
250	C₂×C₃×C₄	—	2	16	no	—
251	C₂×C₃×C₄	—	3	8	no	—
252	C₂×C₃×C₄	—	3	8	no	—
253	C₂×C₃×C₄	—	3	8	no	—
254	C₂×C₃×C₄	—	6	32	no	—
255	C₂×C₃×C₄	—	6	16	no	—
256	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
257	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
258	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
259	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
260	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
261	C₂×C₃×C₄	C₂³×C₃²	1	4	yes	—
262	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
264	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
265	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
266	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
267	D₁₂	C₃²×Q₈	1	4	yes	complemented
268	C₄×D₃	—	2	4	no	—
269	C₄×D₃	—	2	8	no	—
270	Dic₆	—	2	8	no	—
271	Dic₆	C₃²×D₄	1	4	yes	complemented
272	C₂×C₃²	—	2	4	no	—
273	C₂×C₃²	—	2	4	no	—
274	C₂×C₃²	—	2	4	no	—
275	C₂×C₃²	—	6	4	no	—
276	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
277	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
278	C₂×C₃²	#1007	1	1	yes	characteristic
279	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
280	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
281	C₂×C₃²	C₃×2₋¹⁺⁴	1	4	yes	—
282	C₃×D₃	—	2	16	no	—
283	C₂×Q₈	—	3	1	no	—
284	C₂×Q₈	—	3	4	no	—
285	C₂×Q₈	—	6	8	no	—
286	C₂×Q₈	C₂×C₃²×D₃	1	2	yes	—
287	C₂×D₄	—	3	1	no	—
288	C₂²×C₄	—	3	4	no	—
289	C₂²×C₄	—	3	2	no	—
290	C₄⋊C₄	—	3	1	no	—
291	C₄⋊C₄	—	3	2	no	—
292	C₄⋊C₄	—	3	1	no	—
293	C₄⋊C₄	—	3	4	no	—
294	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
295	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
296	C₂²⋊C₄	—	3	4	no	—
297	C₂²⋊C₄	—	3	2	no	—
298	C₄²	—	3	2	no	—
299	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
300	C₂²×C₃	—	2	4	no	—
301	C₂²×C₃	—	3	16	no	—
302	C₂²×C₃	—	6	8	no	—
303	C₂²×C₃	—	6	8	no	—
304	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
305	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
306	C₂×D₃	—	2	2	no	—
307	C₂×D₃	—	2	2	no	—
308	C₂×D₃	C₂×C₃²×Q₈	1	4	yes	complemented
309	C₃×C₄	—	2	8	no	—
310	C₃×C₄	—	2	16	no	—
311	C₃×C₄	—	2	4	no	—
312	C₃×C₄	—	2	4	no	—
313	C₃×C₄	—	2	1	no	—
314	C₃×C₄	—	2	8	no	—
315	C₃×C₄	—	2	8	no	—
316	C₃×C₄	—	3	16	no	—
317	C₃×C₄	—	4	16	no	—
318	C₃×C₄	—	6	8	no	—
319	C₃×C₄	—	6	8	no	—
320	C₃×C₄	C₃×D₃×D₄	1	8	yes	—
321	C₃×C₄	C₃×D₃×Q₈	1	8	yes	complemented
322	C₃×C₄	C₂×C₃²×D₄	1	2	yes	—
323	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	complemented
324	Dic₃	—	2	2	no	—
325	Dic₃	—	2	2	no	—
326	Dic₃	C₂×C₃²×D₄	1	4	yes	—
327	C₃²	—	2	4	no	—
328	C₃²	#5727	1	1	yes	characteristic, direct product factor
329	C₃²	C₃×D₄×Q₈	1	4	yes	complemented
330	C₂³	—	3	2	no	—
331	Q₈	—	2	4	no	—
332	Q₈	—	3	4	no	—
333	Q₈	—	6	8	no	—
334	D₄	—	3	4	no	—
335	C₂×C₄	—	3	2	no	—
336	C₂×C₄	—	3	2	no	—
337	C₂×C₄	—	3	2	no	—
338	C₂×C₄	—	6	8	no	—
339	C₂×C₄	—	6	4	no	—
340	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₃²×D₃	1	4	yes	—
342	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
344	C₂×C₃	—	2	4	no	—
345	C₂×C₃	—	2	4	no	—
346	C₂×C₃	—	2	4	no	—
347	C₂×C₃	—	6	16	no	—
348	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
349	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
350	C₂×C₃	C₃×#1007	1	4	yes	—
351	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
352	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
353	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
354	D₃	—	2	4	no	—
355	C₂²	—	3	4	no	—
356	C₂²	—	6	2	no	—
357	C₂²	—	6	2	no	—
358	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
359	C₄	—	2	4	no	—
360	C₄	—	2	1	no	—
361	C₄	—	3	4	no	—
362	C₄	—	6	2	no	—
363	C₄	—	6	2	no	—
364	C₄	C₃²×D₃×D₄	1	2	yes	—
365	C₄	C₃²×D₃×Q₈	1	2	yes	complemented
366	C₃	—	2	4	no	—
367	C₃	C₃×#5727	1	4	yes	direct product factor
368	C₃	C₃²×D₄×Q₈	1	1	yes	characteristic, complemented
369	C₂	—	6	4	no	—
370	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic
371	C₂	C₂×C₃²×D₃×Q₈	1	1	yes	characteristic
372	C₂	C₃²×#1007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄⋊Q₈	D₃
hypocenter	C₃	C₃²×D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₃×Q₈	C₂	semidirect
2. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃²×#946	C₂	semidirect
4. C₃²×#896	C₂	semidirect
6. C₃²×#888	C₂	semidirect
8. C₃²×C₄×Dic₆	C₂	semidirect
10. C₃×#5727	C₃	direct (N × H)
29. C₂×C₃×D₃×Q₈	C₂×C₃	semidirect
32. C₃²×C₄⋊Q₈	D₃	semidirect
33. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
39. C₃×#946	C₂×C₃	semidirect
40. C₃×#896	C₂×C₃	semidirect
42. C₃×#888	C₂×C₃	semidirect
44. C₃×C₄×Dic₆	C₂×C₃	semidirect
54. C₃²×D₁₂	Q₈	semidirect
58. C₃²×Dic₆	D₄	semidirect
60. #5727	C₃²	direct (N × H)
102. C₂×C₃²×D₃	C₂×Q₈	semidirect
106. C₃³×C₄	C₂×Q₈	semidirect
111. C₂×D₃×Q₈	C₂×C₃²	semidirect
114. C₃×C₄⋊Q₈	C₃×D₃	semidirect
115. C₃×C₄⋊Q₈	C₂×C₃²	semidirect
121. #946	C₂×C₃²	semidirect
122. #896	C₂×C₃²	semidirect
124. #888	C₂×C₃²	semidirect
126. C₄×Dic₆	C₂×C₃²	semidirect
155. C₃×D₁₂	C₃×Q₈	semidirect
159. C₃×Dic₆	C₃×D₄	semidirect
211. C₂×C₃×D₃	C₂×C₃×Q₈	semidirect
224. C₃²×C₄	D₃×Q₈	semidirect
226. C₃²×C₄	C₂×C₃×Q₈	semidirect
232. C₄⋊Q₈	C₃²×D₃	semidirect
238. C₃³	D₄×Q₈	semidirect
267. D₁₂	C₃²×Q₈	semidirect
271. Dic₆	C₃²×D₄	semidirect
308. C₂×D₃	C₂×C₃²×Q₈	semidirect
321. C₃×C₄	C₃×D₃×Q₈	semidirect
323. C₃×C₄	C₂×C₃²×Q₈	semidirect
328. C₃²	#5727	direct (N × H)
329. C₃²	C₃×D₄×Q₈	semidirect
365. C₄	C₃²×D₃×Q₈	semidirect
367. C₃	C₃×#5727	direct (N × H)
368. C₃	C₃²×D₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₃×Q₈	2
C₃³×C₄⋊Q₈	2
C₃²×#946	2
C₃²×#896	2
C₃²×#893	2
C₃²×#888	2
C₃²×C₄×D₁₂	2
C₃²×C₄×Dic₆	2
C₃²×D₄×Q₈	3
C₃×#5727	3

Subgroups — C32×#5727 #13179098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5727 #13179098