Subgroups — #79385

Order 384.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5523	C₂	1	1	yes	maximal, characteristic
2	#5518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
4	#4849	C₂	1	1	yes	maximal, characteristic, complemented
5	#4847	C₂	1	1	yes	maximal, characteristic, complemented
6	#4841	C₂	1	1	yes	maximal, characteristic, complemented
7	#4829	C₂	1	1	yes	maximal, characteristic, complemented
8	#4733	C₂	1	1	yes	maximal, characteristic
9	#4733	C₂	1	1	yes	maximal, characteristic
10	#4722	C₂	1	1	yes	maximal, characteristic, complemented
11	#4721	C₂	1	1	yes	maximal, characteristic, complemented
12	#4716	C₂	1	1	yes	maximal, characteristic, complemented
13	#4710	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#204	C₂	1	1	yes	maximal, characteristic, complemented
16	#3120	—	3	1	no	maximal, Sylow
17	C₂×#235	C₂²	1	1	yes	characteristic
18	C₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	#924	C₂²	1	1	yes	characteristic
21	#924	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
24	C₂×#204	C₂²	1	1	yes	characteristic
25	C₂×#204	C₂²	1	1	yes	characteristic
26	#892	—	2	1	no	—
27	D₃×C₄⋊C₄	—	2	1	no	—
28	#889	—	2	1	no	—
29	#889	—	2	1	no	—
30	#67⋊C₄	—	2	1	no	—
31	#67⋊C₄	—	2	1	no	—
32	#879	—	2	1	no	—
33	#879	—	2	1	no	—
34	C₄×D₁₂	C₂²	1	1	yes	characteristic
35	C₄×D₁₂	C₂²	1	1	yes	characteristic
36	#870	C₂²	1	1	yes	characteristic
37	#869	C₂²	1	1	yes	characteristic
38	C₄×Dic₆	C₂²	1	1	yes	characteristic
39	C₄×Dic₆	C₂²	1	1	yes	characteristic
40	C₄×Dic₆	C₂²	1	1	yes	characteristic
41	C₄×Dic₆	C₂²	1	1	yes	characteristic
42	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₃×#97	C₂²	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
48	#821	C₂²	1	1	yes	characteristic
49	#821	C₂²	1	1	yes	characteristic
50	#818	C₂²	1	1	yes	characteristic
51	#818	C₂²	1	1	yes	characteristic
52	#817	C₂²	1	1	yes	characteristic
53	#817	C₂²	1	1	yes	characteristic
54	#816	C₂²	1	1	yes	characteristic
55	#816	C₂²	1	1	yes	characteristic
56	#816	C₂²	1	1	yes	characteristic
57	#816	C₂²	1	1	yes	characteristic
58	#816	C₂²	1	1	yes	characteristic
59	#816	C₂²	1	1	yes	characteristic
60	#541	—	3	1	no	—
61	#520	—	3	1	no	—
62	#484	—	3	1	no	—
63	#483	—	3	1	no	—
64	#479	—	3	1	no	—
65	#478	—	3	1	no	—
66	#462	—	3	1	no	—
67	#461	—	3	1	no	—
68	#452	—	3	1	no	—
69	#452	—	3	1	no	—
70	#441	—	3	1	no	—
71	#440	—	3	1	no	—
72	C₄×Q₁₆	—	3	1	no	—
73	C₄×SD₁₆	—	3	1	no	—
74	#433	D₃	1	1	yes	characteristic, complemented
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₂×#67	—	2	1	no	—
77	C₂²×Dic₃	—	2	1	no	—
78	#235	—	2	2	no	—
79	#235	—	2	2	no	—
80	C₂×D₁₂	C₂³	1	1	yes	characteristic
81	C₂×C₄×D₃	—	2	1	no	—
82	C₂×C₄×D₃	—	2	1	no	—
83	C₂×C₄×D₃	—	2	1	no	—
84	C₂×Dic₆	C₂³	1	1	yes	characteristic
85	C₂×Dic₆	C₂³	1	1	yes	characteristic
86	C₂×Dic₆	C₂³	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
91	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
92	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
93	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
94	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
95	C₃×C₄²	C₂³	1	1	yes	characteristic
96	C₃×C₄²	C₂³	1	1	yes	characteristic
97	#217	—	2	1	no	—
98	#217	—	2	1	no	—
99	(C₂×D₃)⋊C₄	—	2	1	no	—
100	(C₂×D₃)⋊C₄	—	2	1	no	—
101	C₄⋊Dic₃	—	2	1	no	—
102	C₄⋊Dic₃	—	2	1	no	—
103	C₄⋊Dic₃	—	2	1	no	—
104	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
107	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
108	Dic₃⋊C₄	—	2	1	no	—
109	Dic₃⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	Dic₃⋊C₄	—	2	1	no	—
112	Dic₃⋊C₄	—	2	1	no	—
113	Dic₃⋊C₄	—	2	1	no	—
114	Dic₃⋊C₄	—	2	1	no	—
115	Dic₃⋊C₄	—	2	1	no	—
116	C₄×Dic₃	—	2	1	no	—
117	C₄×Dic₃	—	2	1	no	—
118	C₄×Dic₃	—	2	1	no	—
119	C₄×Dic₃	—	2	1	no	—
120	Dic₁₂	—	2	2	no	—
121	Dic₁₂	—	2	2	no	—
122	#204	—	2	2	no	—
123	#204	—	2	2	no	—
124	C₂×#41	—	3	1	no	—
125	C₂×Q₁₆	—	3	1	no	—
126	C₂×Q₁₆	—	3	1	no	—
127	C₂×SD₁₆	—	3	1	no	—
128	C₂×SD₁₆	—	3	1	no	—
129	C₄⋊Q₈	—	3	1	no	—
130	#126	—	6	1	no	—
131	#126	—	6	1	no	—
132	#125	—	3	1	no	—
133	#125	—	6	1	no	—
134	#125	—	6	1	no	—
135	C₂²⋊Q₈	—	3	1	no	—
136	C₂²⋊Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×Q₈	—	3	1	no	—
140	C₄×Q₈	—	3	1	no	—
141	C₄×D₄	—	3	1	no	—
142	C₄×D₄	—	3	1	no	—
143	C₄×D₄	—	6	1	no	—
144	C₄×D₄	—	6	1	no	—
145	#117	—	6	1	no	—
146	#117	C₂×D₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	3	1	no	—
148	C₂×C₄⋊C₄	—	6	1	no	—
149	#107	—	3	1	no	—
150	#107	—	3	1	no	—
151	#106	—	3	1	no	—
152	#106	—	3	1	no	—
153	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
155	#103	—	3	1	no	—
156	#103	—	3	1	no	—
157	#103	—	3	1	no	—
158	#103	—	3	1	no	—
159	#103	—	3	1	no	—
160	#103	—	3	1	no	—
161	#102	—	3	1	no	—
162	#102	—	3	1	no	—
163	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
164	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
165	#97	C₂×D₃	1	1	yes	characteristic
166	C₄×C₈	C₂×D₃	1	1	yes	characteristic
167	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×D₃	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
177	#67	—	2	4	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	—	2	1	no	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	—	2	1	no	—
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	8	1	no	—
186	D₁₂	—	2	1	no	—
187	D₁₂	#41	1	2	yes	—
188	C₄×D₃	—	2	4	no	—
189	C₄×D₃	—	8	1	no	—
190	C₄×D₃	—	8	1	no	—
191	Dic₆	—	2	1	no	—
192	Dic₆	—	2	1	no	—
193	Dic₆	—	2	1	no	—
194	Dic₆	—	2	1	no	—
195	Dic₆	—	2	1	no	—
196	Dic₆	#41	1	2	yes	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	—	2	1	no	—
201	#41	—	6	2	no	—
202	#41	—	6	2	no	—
203	C₂×Q₈	—	3	1	no	—
204	C₂×Q₈	—	3	1	no	—
205	C₂×Q₈	—	3	1	no	—
206	C₂×D₄	—	3	1	no	—
207	C₂×D₄	—	6	1	no	—
208	C₂²×C₄	—	6	1	no	—
209	C₂²×C₄	—	6	1	no	—
210	C₂²×C₄	—	6	1	no	—
211	C₂²×C₄	—	6	1	no	—
212	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
213	Q₁₆	—	6	2	no	—
214	Q₁₆	—	6	2	no	—
215	SD₁₆	—	6	2	no	—
216	SD₁₆	—	6	2	no	—
217	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
219	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
220	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
221	C₄⋊C₄	—	3	1	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	3	1	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	6	1	no	—
226	C₄⋊C₄	—	6	1	no	—
227	C₄⋊C₄	—	6	1	no	—
228	C₄⋊C₄	—	6	1	no	—
229	C₄⋊C₄	—	6	1	no	—
230	C₄⋊C₄	—	6	1	no	—
231	C₄⋊C₄	—	6	1	no	—
232	C₄⋊C₄	—	6	1	no	—
233	C₄⋊C₄	—	6	1	no	—
234	C₄⋊C₄	—	6	1	no	—
235	C₄⋊C₄	—	6	1	no	—
236	C₄⋊C₄	D₁₂	1	1	yes	characteristic
237	C₄⋊C₄	D₁₂	1	1	yes	characteristic
238	C₂²⋊C₄	—	6	1	no	—
239	C₂²⋊C₄	—	6	1	no	—
240	C₂²⋊C₄	—	6	1	no	—
241	C₂²⋊C₄	—	6	1	no	—
242	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
243	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
244	C₄²	—	6	1	no	—
245	C₄²	—	6	1	no	—
246	C₄²	—	6	1	no	—
247	C₄²	—	6	1	no	—
248	C₄²	C₂²×D₃	1	1	yes	characteristic
249	C₄²	C₂²×D₃	1	1	yes	characteristic
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×D₃	—	2	2	no	—
255	C₂×D₃	—	4	1	no	—
256	C₂×D₃	—	4	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	C₂×#41	1	1	yes	characteristic
263	C₃×C₄	C₂×#41	1	1	yes	characteristic
264	Dic₃	—	2	2	no	—
265	Dic₃	—	4	1	no	—
266	Dic₃	—	4	1	no	—
267	Dic₃	—	4	1	no	—
268	Dic₃	—	4	1	no	—
269	Dic₃	—	4	1	no	—
270	Dic₃	—	4	1	no	—
271	C₂³	—	6	1	no	—
272	C₂³	C₂×D₁₂	1	1	yes	characteristic
273	Q₈	—	3	2	no	—
274	Q₈	—	6	1	no	—
275	Q₈	—	6	1	no	—
276	Q₈	—	6	1	no	—
277	Q₈	—	6	1	no	—
278	Q₈	—	6	1	no	—
279	D₄	—	3	2	no	—
280	D₄	—	6	4	no	—
281	D₄	—	6	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	6	4	no	—
285	C₂×C₄	—	6	1	no	—
286	C₂×C₄	—	6	1	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	—	6	1	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	24	1	no	—
293	C₂×C₄	—	24	1	no	—
294	C₂×C₄	—	24	1	no	—
295	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
296	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
297	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
298	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
299	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₂×C₃	—	4	1	no	—
306	C₂×C₃	#525	1	1	yes	characteristic
307	C₂×C₃	#577	1	1	yes	characteristic
308	C₂×C₃	#578	1	1	yes	characteristic
309	D₃	—	4	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	6	2	no	—
314	C₂²	—	12	1	no	—
315	C₂²	—	12	1	no	—
316	C₂²	C₂²×D₁₂	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	6	2	no	—
323	C₄	—	12	1	no	—
324	C₄	—	12	1	no	—
325	C₄	—	12	1	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	—	12	1	no	—
329	C₄	D₃×#41	1	1	yes	characteristic
330	C₄	D₃×#41	1	1	yes	characteristic
331	C₃	#3120	1	1	yes	characteristic, complemented, Sylow
332	C₂	—	4	1	no	—
333	C₂	—	12	2	no	—
334	C₂	#5524	1	1	yes	characteristic
335	C₂	#5751	1	1	yes	characteristic
336	C₂	#5753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#433	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#235	C₂²
hypercenter	#433	D₃
hypocenter	C₃	#3120
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
2. #5518	C₂	semidirect
3. C₃×#433	C₂	semidirect
4. #4849	C₂	semidirect
5. #4847	C₂	semidirect
6. #4841	C₂	semidirect
7. #4829	C₂	semidirect
10. #4722	C₂	semidirect
11. #4721	C₂	semidirect
12. #4716	C₂	semidirect
13. #4710	C₂	semidirect
14. C₄×Dic₁₂	C₂	semidirect
15. C₄×#204	C₂	semidirect
74. #433	D₃	semidirect
331. C₃	#3120	semidirect

Maximal subgroups

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

subgroup	index
#5523	2
#5518	2
C₃×#433	2
#4849	2
#4847	2
#4841	2
#4829	2
#4733	2
#4722	2
#4721	2
#4716	2
#4710	2
C₄×Dic₁₂	2
C₄×#204	2
#3120	3