Subgroups — #86280

Order 384.

1100 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#5730	C₂	1	2	yes	maximal, complemented
3	#5727	C₂	1	2	yes	maximal, complemented
4	#5690	C₂	1	4	yes	maximal, complemented
5	#5634	C₂	1	4	yes	maximal, complemented
6	D₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	#5579	C₂	1	2	yes	maximal, complemented
8	#5579	C₂	1	4	yes	maximal, complemented
9	Q₈×D₁₂	C₂	1	2	yes	maximal, complemented
10	#5521	C₂	1	2	yes	maximal
11	#5519	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
13	#5470	C₂	1	1	yes	maximal, characteristic, complemented
14	#3501	—	3	1	no	maximal, Sylow
15	C₂×D₃×Q₈	C₂²	1	4	yes	complemented
16	C₂²×D₁₂	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₃	C₂²	1	2	yes	—
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₃×#125	C₂²	1	2	yes	complemented
21	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
24	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
25	C₃×#117	C₂²	1	2	yes	—
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	#946	C₂²	1	8	yes	complemented
28	#946	C₂²	1	4	yes	complemented
29	C₂×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	#926	C₂²	1	2	yes	—
31	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
32	#924	C₂²	1	2	yes	—
33	#924	C₂²	1	2	yes	—
34	#897	C₂²	1	8	yes	complemented
35	#897	C₂²	1	4	yes	complemented
36	#896	C₂²	1	8	yes	complemented
37	#896	C₂²	1	8	yes	complemented
38	#896	C₂²	1	8	yes	complemented
39	#893	C₂²	1	4	yes	—
40	#893	C₂²	1	4	yes	—
41	D₃×C₄⋊C₄	C₂²	1	8	yes	complemented
42	D₃×C₄⋊C₄	C₂²	1	4	yes	complemented
43	#890	C₂²	1	4	yes	complemented
44	#888	C₂²	1	4	yes	complemented
45	D₃×C₂²⋊C₄	C₂²	1	8	yes	—
46	#878	C₂²	1	8	yes	—
47	C₄×D₁₂	C₂²	1	2	yes	—
48	C₄×D₁₂	C₂²	1	2	yes	—
49	C₄×D₁₂	C₂²	1	2	yes	—
50	C₄×D₁₂	C₂²	1	2	yes	—
51	C₄×D₁₂	C₂²	1	4	yes	—
52	C₄×D₁₂	C₂²	1	4	yes	—
53	#870	C₂²	1	2	yes	complemented
54	#870	C₂²	1	4	yes	complemented
55	#869	C₂²	1	1	yes	characteristic, complemented
56	#869	C₂²	1	1	yes	characteristic, complemented
57	#869	C₂²	1	4	yes	complemented
58	C₄×Dic₆	C₂²	1	2	yes	complemented
59	C₄×Dic₆	C₂²	1	2	yes	complemented
60	#554	—	3	2	no	—
61	#554	—	3	4	no	—
62	#554	—	3	2	no	—
63	#554	—	3	4	no	—
64	D₄×Q₈	—	3	2	no	—
65	D₄×Q₈	—	3	2	no	—
66	#544	—	3	2	no	—
67	#543	—	3	4	no	—
68	#533	—	3	1	no	—
69	#533	D₃	1	1	yes	characteristic, complemented
70	C₂×C₂²⋊Q₈	—	3	4	no	—
71	#518	—	3	2	no	—
72	C₂×C₄×D₄	—	3	1	no	—
73	C₂³×D₃	C₂³	1	2	yes	—
74	C₂×C₃×Q₈	C₂³	1	2	yes	—
75	C₂×C₃×Q₈	C₂³	1	2	yes	—
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₂²×Dic₃	C₂³	1	2	yes	—
80	D₃×Q₈	—	4	8	no	—
81	C₂×D₁₂	Q₈	1	8	yes	complemented
82	C₂×D₁₂	C₂³	1	2	yes	—
83	C₂×D₁₂	C₂³	1	2	yes	—
84	C₂×C₄×D₃	—	4	4	no	—
85	C₂×C₄×D₃	C₂³	1	4	yes	—
86	C₂×C₄×D₃	C₂³	1	4	yes	—
87	C₂×C₄×D₃	C₂³	1	8	yes	—
88	C₂×C₄×D₃	C₂³	1	8	yes	—
89	C₂×Dic₆	C₂³	1	2	yes	—
90	C₂×Dic₆	C₂³	1	2	yes	—
91	C₂×Dic₆	C₂³	1	8	yes	—
92	C₃×C₄⋊C₄	C₂³	1	2	yes	—
93	C₃×C₄⋊C₄	C₂³	1	2	yes	—
94	C₃×C₄⋊C₄	C₂³	1	4	yes	—
95	C₃×C₄⋊C₄	C₂³	1	2	yes	—
96	C₃×C₄⋊C₄	C₂³	1	2	yes	—
97	C₃×C₄⋊C₄	C₂³	1	4	yes	—
98	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
99	C₃×C₄²	C₂³	1	1	yes	characteristic
100	C₃×C₄²	C₂³	1	1	yes	characteristic
101	C₃×C₄²	C₂³	1	1	yes	characteristic
102	C₃×C₄²	C₂³	1	4	yes	—
103	C₃×C₄²	C₂³	1	1	yes	characteristic
104	#217	C₂³	1	4	yes	—
105	(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
106	(C₂×D₃)⋊C₄	C₂³	1	4	yes	—
107	(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
108	(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
109	C₄⋊Dic₃	C₂³	1	2	yes	—
110	C₄⋊Dic₃	C₂³	1	2	yes	—
111	C₄⋊Dic₃	C₂³	1	2	yes	—
112	C₄⋊Dic₃	C₂³	1	2	yes	—
113	C₄⋊Dic₃	C₂³	1	2	yes	—
114	C₄⋊Dic₃	C₂³	1	8	yes	—
115	C₄⋊Dic₃	C₂³	1	2	yes	—
116	Dic₃⋊C₄	C₂³	1	4	yes	—
117	Dic₃⋊C₄	C₂³	1	4	yes	—
118	Dic₃⋊C₄	C₂³	1	8	yes	—
119	Dic₃⋊C₄	C₂³	1	8	yes	—
120	C₄×Dic₃	C₂³	1	4	yes	—
121	C₂²×Q₈	—	3	4	no	—
122	C₂²×D₄	—	3	1	no	—
123	C₂³×C₄	—	3	2	no	—
124	C₄⋊Q₈	—	3	4	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	—	3	1	no	—
127	C₄⋊Q₈	—	3	4	no	—
128	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
129	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic, complemented
130	#125	—	3	4	no	—
131	#125	—	3	2	no	—
132	#125	—	3	4	no	—
133	#125	C₂×D₃	1	2	yes	complemented
134	C₂²⋊Q₈	—	3	4	no	—
135	C₂²⋊Q₈	—	3	8	no	—
136	C₂²⋊Q₈	—	3	2	no	—
137	C₂²⋊Q₈	—	3	2	no	—
138	C₂²⋊Q₈	—	3	4	no	—
139	C₂²⋊Q₈	—	3	8	no	—
140	C₂²⋊Q₈	—	3	8	no	—
141	C₂²⋊Q₈	—	3	8	no	—
142	C₂²⋊Q₈	—	3	8	no	—
143	C₂²⋊Q₈	—	3	8	no	—
144	C₂²⋊Q₈	C₂×D₃	1	2	yes	—
145	C₂²⋊Q₈	C₂×D₃	1	2	yes	—
146	C₄×Q₈	—	3	2	no	—
147	C₄×Q₈	—	3	2	no	—
148	C₄×Q₈	C₂×D₃	1	2	yes	complemented
149	C₄×Q₈	C₂×D₃	1	2	yes	complemented
150	C₄×D₄	—	3	4	no	—
151	C₄×D₄	—	3	4	no	—
152	C₄×D₄	—	3	2	no	—
153	C₄×D₄	—	3	2	no	—
154	C₄×D₄	—	3	2	no	—
155	C₄×D₄	—	3	2	no	—
156	C₄×D₄	—	3	4	no	—
157	C₄×D₄	—	3	4	no	—
158	#117	—	3	2	no	—
159	#117	C₂×D₃	1	2	yes	—
160	C₂×C₄⋊C₄	—	3	1	no	—
161	C₂×C₄⋊C₄	—	3	8	no	—
162	C₂×C₄⋊C₄	—	3	4	no	—
163	C₂×C₂²⋊C₄	—	3	2	no	—
164	C₂×C₂²⋊C₄	—	3	8	no	—
165	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
166	C₂³×C₃	C₂⁴	1	1	yes	characteristic
167	C₂²×D₃	—	4	2	no	—
168	C₂²×D₃	—	4	2	no	—
169	C₂²×D₃	C₂×Q₈	1	8	yes	complemented
170	C₂²×D₃	C₂⁴	1	4	yes	—
171	C₃×Q₈	—	2	4	no	—
172	C₃×Q₈	—	4	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	4	1	no	—
175	C₂×C₃×C₄	C₂×Q₈	1	4	yes	complemented
176	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂⁴	1	4	yes	—
179	C₂×C₃×C₄	C₂⁴	1	4	yes	—
180	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
184	C₂×Dic₃	—	4	2	no	—
185	C₂×Dic₃	C₂⁴	1	4	yes	—
186	C₂×Dic₃	C₂⁴	1	4	yes	—
187	C₂×Dic₃	C₂⁴	1	4	yes	—
188	D₁₂	C₂×Q₈	1	8	yes	complemented
189	D₁₂	C₂×Q₈	1	8	yes	complemented
190	C₄×D₃	—	2	8	no	—
191	C₄×D₃	—	4	4	no	—
192	C₄×D₃	—	4	8	no	—
193	C₄×D₃	—	4	8	no	—
194	Dic₆	—	2	4	no	—
195	Dic₆	—	4	2	no	—
196	Dic₆	—	4	8	no	—
197	C₂⁴	—	3	2	no	—
198	C₂×Q₈	—	3	2	no	—
199	C₂×Q₈	—	3	2	no	—
200	C₂×Q₈	—	3	8	no	—
201	C₂×Q₈	—	12	8	no	—
202	C₂×Q₈	C₂²×D₃	1	2	yes	—
203	C₂×Q₈	C₂²×D₃	1	2	yes	—
204	C₂×D₄	—	3	8	no	—
205	C₂×D₄	—	3	2	no	—
206	C₂×D₄	—	3	2	no	—
207	C₂²×C₄	—	3	2	no	—
208	C₂²×C₄	—	3	4	no	—
209	C₂²×C₄	—	3	8	no	—
210	C₂²×C₄	—	3	8	no	—
211	C₂²×C₄	—	3	4	no	—
212	C₂²×C₄	—	12	4	no	—
213	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
216	C₄⋊C₄	—	3	4	no	—
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	3	2	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	—	3	8	no	—
221	C₄⋊C₄	—	3	8	no	—
222	C₄⋊C₄	—	3	8	no	—
223	C₄⋊C₄	—	3	2	no	—
224	C₄⋊C₄	—	3	2	no	—
225	C₄⋊C₄	—	3	4	no	—
226	C₄⋊C₄	—	3	2	no	—
227	C₄⋊C₄	C₂²×D₃	1	4	yes	—
228	C₄⋊C₄	C₂²×D₃	1	2	yes	—
229	C₄⋊C₄	C₂²×D₃	1	2	yes	—
230	C₄⋊C₄	C₂²×D₃	1	2	yes	—
231	C₄⋊C₄	C₂²×D₃	1	4	yes	—
232	C₄⋊C₄	C₂²×D₃	1	2	yes	—
233	C₂²⋊C₄	—	3	4	no	—
234	C₂²⋊C₄	—	3	8	no	—
235	C₂²⋊C₄	—	3	8	no	—
236	C₂²⋊C₄	—	3	4	no	—
237	C₂²⋊C₄	—	3	4	no	—
238	C₂²⋊C₄	C₂²×D₃	1	4	yes	—
239	C₄²	—	3	4	no	—
240	C₄²	C₂²×D₃	1	1	yes	characteristic
241	C₄²	C₂²×D₃	1	1	yes	characteristic
242	C₄²	C₂²×D₃	1	1	yes	characteristic
243	C₄²	C₂²×D₃	1	1	yes	characteristic
244	C₄²	C₂²×D₃	1	4	yes	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	C₂²×Q₈	1	2	yes	—
248	C₂²×C₃	C₂⁵	1	1	yes	characteristic
249	C₂×D₃	—	2	4	no	—
250	C₂×D₃	—	2	4	no	—
251	C₂×D₃	—	4	8	no	—
252	C₂×D₃	C₂²×Q₈	1	8	yes	—
253	C₃×C₄	—	2	4	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	C₂²×Q₈	1	2	yes	—
259	C₃×C₄	C₂²×Q₈	1	2	yes	—
260	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
261	Dic₃	—	2	4	no	—
262	Dic₃	—	2	4	no	—
263	Dic₃	—	2	4	no	—
264	C₂³	—	3	4	no	—
265	C₂³	—	3	8	no	—
266	C₂³	—	12	2	no	—
267	C₂³	—	12	2	no	—
268	C₂³	C₂³×D₃	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	6	4	no	—
272	Q₈	—	12	8	no	—
273	Q₈	—	12	2	no	—
274	D₄	—	3	8	no	—
275	D₄	—	3	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	3	4	no	—
278	C₂×C₄	—	3	4	no	—
279	C₂×C₄	—	3	4	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	6	8	no	—
282	C₂×C₄	—	12	4	no	—
283	C₂×C₄	—	12	8	no	—
284	C₂×C₄	—	12	8	no	—
285	C₂×C₄	—	12	2	no	—
286	C₂×C₄	D₃×Q₈	1	4	yes	complemented
287	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₃	1	4	yes	—
291	C₂×C₄	C₂³×D₃	1	4	yes	—
292	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
297	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₃	#585	1	1	yes	characteristic
299	D₃	—	2	8	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	3	8	no	—
303	C₂²	—	6	4	no	—
304	C₂²	—	6	4	no	—
305	C₂²	—	12	8	no	—
306	C₂²	C₂×D₃×Q₈	1	2	yes	—
307	C₂²	C₂⁴×D₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	4	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	4	no	—
313	C₄	—	6	4	no	—
314	C₄	—	6	4	no	—
315	C₄	—	6	4	no	—
316	C₄	C₂×D₃×Q₈	1	2	yes	—
317	C₄	C₂×D₃×Q₈	1	2	yes	—
318	C₄	#1009	1	2	yes	—
319	C₃	#3501	1	1	yes	characteristic, complemented, Sylow
320	C₂	—	2	2	no	—
321	C₂	—	6	8	no	—
322	C₂	C₂²×D₃×Q₈	1	1	yes	characteristic
323	C₂	C₂×#1009	1	1	yes	characteristic
324	C₂	#5964	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×#533	C₂
prime radical	C₂²×D₁₂	C₂²
hypercenter	#533	D₃
hypocenter	C₃	#3501
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₃×#533	C₂	semidirect
2. #5730	C₂	semidirect
3. #5727	C₂	semidirect
4. #5690	C₂	semidirect
5. #5634	C₂	semidirect
6. D₃×C₂²⋊Q₈	C₂	semidirect
7. #5579	C₂	semidirect
8. #5579	C₂	semidirect
9. Q₈×D₁₂	C₂	semidirect
11. #5519	C₂	semidirect
13. #5470	C₂	semidirect
15. C₂×D₃×Q₈	C₂²	semidirect
18. C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×C₄⋊Q₈	C₂²	semidirect
20. C₃×#125	C₂²	semidirect
23. C₃×C₄×Q₈	C₂²	semidirect
24. C₃×C₄×Q₈	C₂²	semidirect
27. #946	C₂²	semidirect
28. #946	C₂²	semidirect
34. #897	C₂²	semidirect
35. #897	C₂²	semidirect
36. #896	C₂²	semidirect
37. #896	C₂²	semidirect
38. #896	C₂²	semidirect
41. D₃×C₄⋊C₄	C₂²	semidirect
42. D₃×C₄⋊C₄	C₂²	semidirect
43. #890	C₂²	semidirect
44. #888	C₂²	semidirect
53. #870	C₂²	semidirect
54. #870	C₂²	semidirect
55. #869	C₂²	semidirect
56. #869	C₂²	semidirect
57. #869	C₂²	semidirect
58. C₄×Dic₆	C₂²	semidirect
59. C₄×Dic₆	C₂²	semidirect
69. #533	D₃	semidirect
81. C₂×D₁₂	Q₈	semidirect
128. C₄⋊Q₈	C₂×D₃	semidirect
129. C₄⋊Q₈	C₂×D₃	semidirect
133. #125	C₂×D₃	semidirect
148. C₄×Q₈	C₂×D₃	semidirect
149. C₄×Q₈	C₂×D₃	semidirect
169. C₂²×D₃	C₂×Q₈	semidirect
175. C₂×C₃×C₄	C₂×Q₈	semidirect
188. D₁₂	C₂×Q₈	semidirect
189. D₁₂	C₂×Q₈	semidirect
286. C₂×C₄	D₃×Q₈	semidirect
319. C₃	#3501	semidirect

Maximal subgroups

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

subgroup	index
C₃×#533	2
#5730	2
#5727	2
#5690	2
#5634	2
D₃×C₂²⋊Q₈	2
#5579	2
Q₈×D₁₂	2
#5521	2
#5519	2
C₂×C₄×D₁₂	2
#5470	2
#3501	3