Subgroups — #79299

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5479	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
4	#4847	C₂	1	1	yes	maximal, characteristic, complemented
5	#4845	C₂	1	1	yes	maximal, characteristic, complemented
6	#4844	C₂	1	1	yes	maximal, characteristic, complemented
7	#4829	C₂	1	1	yes	maximal, characteristic, complemented
8	#4737	C₂	1	1	yes	maximal, characteristic, complemented
9	#4734	C₂	1	1	yes	maximal, characteristic
10	#4733	C₂	1	1	yes	maximal, characteristic, complemented
11	#4723	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#204	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#204	C₂	1	1	yes	maximal, characteristic, complemented
16	#3102	—	3	1	no	maximal, Sylow
17	C₂×#235	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	#930	C₂²	1	1	yes	characteristic
20	#929	—	2	1	no	—
21	C₄×#67	—	2	1	no	—
22	C₄×#67	—	2	1	no	—
23	C₄×#67	—	2	1	no	—
24	#926	C₂²	1	1	yes	characteristic
25	#924	C₂²	1	1	yes	characteristic
26	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
27	C₂×D₂₄	C₂²	1	1	yes	characteristic
28	C₂×#204	C₂²	1	1	yes	characteristic
29	C₂×#204	C₂²	1	1	yes	characteristic
30	#876	—	2	1	no	—
31	#875	C₂²	1	1	yes	characteristic
32	C₄×D₁₂	C₂²	1	1	yes	characteristic
33	C₄×D₁₂	C₂²	1	1	yes	characteristic
34	C₄×D₁₂	C₂²	1	1	yes	characteristic
35	#872	—	2	1	no	—
36	#872	—	2	1	no	—
37	C₄²×D₃	—	2	1	no	—
38	#870	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×C₄×C₈	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	#821	C₂²	1	1	yes	characteristic
51	#821	C₂²	1	1	yes	characteristic
52	#818	C₂²	1	1	yes	characteristic
53	#818	C₂²	1	1	yes	characteristic
54	#817	C₂²	1	1	yes	characteristic
55	#817	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	#529	—	3	1	no	—
61	C₄×#41	—	3	1	no	—
62	#485	—	3	1	no	—
63	#482	—	3	1	no	—
64	#479	—	3	1	no	—
65	#478	—	3	1	no	—
66	#464	—	3	1	no	—
67	#463	—	3	1	no	—
68	#452	—	3	1	no	—
69	#449	—	3	1	no	—
70	C₄×Q₁₆	—	3	1	no	—
71	C₄×SD₁₆	—	3	1	no	—
72	C₄×SD₁₆	—	3	1	no	—
73	C₄×D₈	—	3	1	no	—
74	#431	D₃	1	1	yes	characteristic, complemented
75	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂×#67	—	2	1	no	—
79	C₂×#67	—	2	1	no	—
80	#235	—	2	2	no	—
81	#235	—	2	2	no	—
82	C₂×D₁₂	C₂³	1	1	yes	characteristic
83	C₂×D₁₂	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₃	—	2	1	no	—
85	C₂×C₄×D₃	—	2	1	no	—
86	C₂×C₄×D₃	—	2	1	no	—
87	C₂×C₄×D₃	—	2	1	no	—
88	C₂×Dic₆	C₂³	1	1	yes	characteristic
89	C₂×Dic₆	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
94	C₃×C₄²	D₄	1	1	yes	characteristic
95	C₃×C₄²	D₄	1	1	yes	characteristic
96	C₃×C₄²	C₂³	1	1	yes	characteristic
97	C₃×C₄²	C₂³	1	1	yes	characteristic
98	#217	—	2	1	no	—
99	#217	—	2	1	no	—
100	(C₂×D₃)⋊C₄	—	2	1	no	—
101	(C₂×D₃)⋊C₄	—	2	1	no	—
102	(C₂×D₃)⋊C₄	—	2	1	no	—
103	(C₂×D₃)⋊C₄	—	2	1	no	—
104	(C₂×D₃)⋊C₄	—	2	1	no	—
105	(C₂×D₃)⋊C₄	—	2	1	no	—
106	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
107	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
108	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
109	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	D₂₄	—	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×SD₁₆	—	3	1	no	—
127	C₂×SD₁₆	—	3	1	no	—
128	C₂×D₈	—	3	1	no	—
129	#126	—	6	1	no	—
130	#125	—	3	1	no	—
131	#124	—	3	1	no	—
132	#123	—	6	1	no	—
133	C₂²⋊Q₈	—	3	1	no	—
134	#121	—	3	1	no	—
135	C₄×Q₈	—	3	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×D₄	—	3	1	no	—
139	C₄×D₄	—	3	1	no	—
140	C₄×D₄	—	3	1	no	—
141	C₄×D₄	—	6	1	no	—
142	C₄×D₄	—	6	1	no	—
143	C₄×D₄	—	6	1	no	—
144	#117	—	3	1	no	—
145	#117	—	6	1	no	—
146	#117	—	6	1	no	—
147	C₂×C₄²	—	6	1	no	—
148	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
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	#102	—	3	1	no	—
160	#102	—	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	C₄×C₈	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	C₂²×D₃	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	4	1	no	—
174	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
180	#67	—	2	4	no	—
181	#67	—	4	2	no	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	—	2	1	no	—
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	2	1	no	—
186	C₂×Dic₃	—	2	1	no	—
187	C₂×Dic₃	—	2	1	no	—
188	D₁₂	—	2	1	no	—
189	D₁₂	—	2	1	no	—
190	D₁₂	—	2	1	no	—
191	D₁₂	#41	1	2	yes	—
192	C₄×D₃	—	2	4	no	—
193	C₄×D₃	—	4	2	no	—
194	C₄×D₃	—	4	2	no	—
195	C₄×D₃	—	4	2	no	—
196	Dic₆	—	2	1	no	—
197	Dic₆	—	2	1	no	—
198	Dic₆	—	2	1	no	—
199	Dic₆	#41	1	2	yes	—
200	C₃×C₈	—	2	1	no	—
201	C₃×C₈	—	2	1	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	2	1	no	—
204	#41	—	6	2	no	—
205	#41	—	6	2	no	—
206	C₂×Q₈	—	3	1	no	—
207	C₂×Q₈	—	3	1	no	—
208	C₂×D₄	—	3	1	no	—
209	C₂×D₄	—	3	1	no	—
210	C₂×D₄	—	6	1	no	—
211	C₂×D₄	—	6	1	no	—
212	C₂²×C₄	—	6	1	no	—
213	C₂²×C₄	—	6	1	no	—
214	C₂²×C₄	—	6	1	no	—
215	C₂²×C₄	—	6	1	no	—
216	C₂²×C₄	D₁₂	1	1	yes	characteristic
217	C₂²×C₄	D₁₂	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
219	Q₁₆	—	6	2	no	—
220	SD₁₆	—	6	2	no	—
221	SD₁₆	—	6	2	no	—
222	D₈	—	6	2	no	—
223	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	3	1	no	—
230	C₄⋊C₄	—	3	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	—	6	1	no	—
237	C₂²⋊C₄	—	6	1	no	—
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	—	6	1	no	—
243	C₂²⋊C₄	—	6	1	no	—
244	C₂²⋊C₄	—	6	1	no	—
245	C₄²	—	6	1	no	—
246	C₄²	—	6	1	no	—
247	C₄²	—	6	1	no	—
248	C₄²	—	6	1	no	—
249	C₄²	D₁₂	1	1	yes	characteristic
250	C₄²	D₁₂	1	1	yes	characteristic
251	C₄²	C₂²×D₃	1	1	yes	characteristic
252	C₄²	C₂²×D₃	1	1	yes	characteristic
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
257	C₂×D₃	—	2	2	no	—
258	C₂×D₃	—	4	1	no	—
259	C₂×D₃	—	4	1	no	—
260	C₂×D₃	—	4	1	no	—
261	C₂×D₃	—	4	1	no	—
262	C₂×D₃	—	4	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	#135	1	2	yes	—
268	C₃×C₄	C₂×#41	1	1	yes	characteristic
269	C₃×C₄	C₂×#41	1	1	yes	characteristic
270	Dic₃	—	2	2	no	—
271	Dic₃	—	4	1	no	—
272	Dic₃	—	4	1	no	—
273	Dic₃	—	4	1	no	—
274	Dic₃	—	4	1	no	—
275	Dic₃	—	4	1	no	—
276	C₂³	—	6	1	no	—
277	C₂³	—	6	1	no	—
278	C₂³	C₂×D₁₂	1	1	yes	characteristic
279	Q₈	—	3	2	no	—
280	Q₈	—	6	1	no	—
281	Q₈	—	6	1	no	—
282	Q₈	—	6	1	no	—
283	D₄	—	3	2	no	—
284	D₄	—	6	4	no	—
285	D₄	—	6	1	no	—
286	D₄	—	6	1	no	—
287	D₄	—	6	1	no	—
288	D₄	—	12	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	6	1	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	—	12	2	no	—
302	C₂×C₄	—	12	2	no	—
303	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
304	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
305	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
306	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₂×C₃	—	4	1	no	—
314	C₂×C₃	#525	1	1	yes	characteristic
315	C₂×C₃	C₂×#135	1	1	yes	characteristic
316	C₂×C₃	#575	1	1	yes	characteristic
317	D₃	—	4	2	no	—
318	D₃	—	8	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	6	2	no	—
323	C₂²	—	12	1	no	—
324	C₂²	—	12	1	no	—
325	C₂²	—	12	1	no	—
326	C₂²	—	12	1	no	—
327	C₂²	—	12	1	no	—
328	C₂²	C₂²×D₁₂	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	6	2	no	—
334	C₄	—	12	1	no	—
335	C₄	—	12	1	no	—
336	C₄	—	12	1	no	—
337	C₄	—	12	1	no	—
338	C₄	—	12	1	no	—
339	C₄	#904	1	2	yes	—
340	C₄	D₃×#41	1	1	yes	characteristic
341	C₄	D₃×#41	1	1	yes	characteristic
342	C₃	#3102	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	4	1	no	—
344	C₂	—	12	2	no	—
345	C₂	—	24	1	no	—
346	C₂	#5524	1	1	yes	characteristic
347	C₂	C₂×#904	1	1	yes	characteristic
348	C₂	#5748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#431	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#4734	C₂
hypercenter	#431	D₃
hypocenter	C₃	#3102
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
1. #5479	C₂	semidirect
2. C₄×#235	C₂	semidirect
3. C₃×#431	C₂	semidirect
4. #4847	C₂	semidirect
5. #4845	C₂	semidirect
6. #4844	C₂	semidirect
7. #4829	C₂	semidirect
8. #4737	C₂	semidirect
10. #4733	C₂	semidirect
11. #4723	C₂	semidirect
12. C₄×Dic₁₂	C₂	semidirect
13. C₄×D₂₄	C₂	semidirect
14. C₄×#204	C₂	semidirect
15. C₄×#204	C₂	semidirect
74. #431	D₃	semidirect
342. C₃	#3102	semidirect

Maximal subgroups

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

subgroup	index
#5479	2
C₄×#235	2
C₃×#431	2
#4847	2
#4845	2
#4844	2
#4829	2
#4737	2
#4734	2
#4733	2
#4723	2
C₄×Dic₁₂	2
C₄×D₂₄	2
C₄×#204	2
#3102	3