Subgroups — #79469

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5555	C₂	1	1	yes	maximal, characteristic
2	#5541	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
4	#5134	C₂	1	1	yes	maximal, characteristic, complemented
5	#5121	C₂	1	1	yes	maximal, characteristic, complemented
6	#5114	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#818	C₂	1	1	yes	maximal, characteristic, complemented
8	#4850	C₂	1	1	yes	maximal, characteristic, complemented
9	#4847	C₂	1	1	yes	maximal, characteristic, complemented
10	#4737	C₂	1	1	yes	maximal, characteristic, complemented
11	#4736	C₂	1	1	yes	maximal, characteristic, complemented
12	#4722	C₂	1	1	yes	maximal, characteristic, complemented
13	#4721	C₂	1	1	yes	maximal, characteristic, complemented
14	#4710	C₂	1	1	yes	maximal, characteristic, complemented
15	#4702	C₂	1	1	yes	maximal, characteristic, complemented
16	#3343	—	3	1	no	maximal, Sylow
17	C₂×#237	—	2	1	no	—
18	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	#935	—	2	1	no	—
22	#930	C₂²	1	1	yes	characteristic
23	#930	C₂²	1	1	yes	characteristic
24	#926	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	#924	C₂²	1	1	yes	characteristic
29	#924	C₂²	1	1	yes	characteristic
30	C₂×#204	C₂²	1	1	yes	characteristic
31	#897	—	2	1	no	—
32	#889	—	2	1	no	—
33	#886	—	2	1	no	—
34	#886	—	2	1	no	—
35	#879	—	2	1	no	—
36	#877	—	2	1	no	—
37	C₄×D₁₂	C₂²	1	1	yes	characteristic
38	#870	C₂²	1	1	yes	characteristic
39	#869	C₂²	1	1	yes	characteristic
40	C₄×Dic₆	C₂²	1	1	yes	characteristic
41	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₃×#97	C₂²	1	1	yes	characteristic
45	#821	C₂²	1	1	yes	characteristic
46	#821	C₂²	1	1	yes	characteristic
47	#821	C₂²	1	1	yes	characteristic
48	#818	C₂²	1	1	yes	characteristic
49	#818	C₂²	1	1	yes	characteristic
50	#818	C₂²	1	1	yes	characteristic
51	#818	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	#817	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	#552	—	3	1	no	—
61	#547	—	3	1	no	—
62	#501	—	3	1	no	—
63	#484	—	3	1	no	—
64	#483	—	3	1	no	—
65	#482	—	3	1	no	—
66	#480	—	3	1	no	—
67	#478	—	3	1	no	—
68	#477	—	3	1	no	—
69	#468	—	3	1	no	—
70	#465	—	3	1	no	—
71	#440	—	3	1	no	—
72	#435	D₃	1	1	yes	characteristic, complemented
73	#428	—	3	1	no	—
74	C₂×#107	—	3	1	no	—
75	C₂×C₃×D₄	D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	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	C₂²×Dic₃	—	2	1	no	—
81	C₂²×Dic₃	—	2	1	no	—
82	C₂²×Dic₃	—	2	1	no	—
83	#237	—	4	4	no	—
84	C₂×D₁₂	C₂³	1	1	yes	characteristic
85	C₂×C₄×D₃	—	2	1	no	—
86	C₂×Dic₆	—	2	1	no	—
87	C₂×Dic₆	C₂³	1	1	yes	characteristic
88	C₃×M₁₆	—	2	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
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×C₄⋊C₄	D₄	1	1	yes	characteristic
95	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
96	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
97	C₃×C₄²	C₂³	1	1	yes	characteristic
98	#217	—	2	1	no	—
99	#217	—	2	1	no	—
100	#217	—	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	C₄⋊Dic₃	—	2	1	no	—
105	C₄⋊Dic₃	—	2	1	no	—
106	C₄⋊Dic₃	—	2	1	no	—
107	C₄⋊Dic₃	—	2	1	no	—
108	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
109	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
110	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
111	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
112	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
113	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
114	Dic₃⋊C₄	—	2	1	no	—
115	Dic₃⋊C₄	—	2	1	no	—
116	Dic₃⋊C₄	—	2	1	no	—
117	Dic₃⋊C₄	—	2	1	no	—
118	Dic₃⋊C₄	—	2	1	no	—
119	C₄×Dic₃	—	2	1	no	—
120	C₄×Dic₃	—	2	1	no	—
121	#204	—	2	2	no	—
122	C₂×#41	—	6	1	no	—
123	C₂×SD₁₆	—	3	1	no	—
124	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
125	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
126	C₄⋊Q₈	—	3	1	no	—
127	#126	—	6	1	no	—
128	#125	—	3	1	no	—
129	#125	—	6	1	no	—
130	#123	—	6	1	no	—
131	#123	—	6	1	no	—
132	#123	—	6	1	no	—
133	C₂²⋊Q₈	—	3	1	no	—
134	C₂²⋊Q₈	—	3	1	no	—
135	C₂²⋊Q₈	—	6	1	no	—
136	#121	—	3	1	no	—
137	#121	—	3	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×D₄	—	3	1	no	—
140	C₄×D₄	C₂×D₃	1	1	yes	characteristic
141	#117	—	3	1	no	—
142	#117	—	6	1	no	—
143	C₂×C₄⋊C₄	—	3	1	no	—
144	C₂×C₄⋊C₄	—	3	1	no	—
145	C₂×C₄⋊C₄	—	3	1	no	—
146	#107	—	3	1	no	—
147	#107	—	3	1	no	—
148	#107	—	3	1	no	—
149	#107	—	3	1	no	—
150	#107	—	3	1	no	—
151	#107	—	3	1	no	—
152	#106	—	3	1	no	—
153	#106	—	3	1	no	—
154	#106	—	3	1	no	—
155	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
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	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
163	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
164	#97	C₂×D₃	1	1	yes	characteristic
165	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
166	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×D₃	—	2	1	no	—
168	C₃×D₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
178	#67	—	4	2	no	—
179	#67	—	4	2	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	—	2	1	no	—
186	C₂×Dic₃	—	2	1	no	—
187	C₂×Dic₃	—	4	2	no	—
188	C₂×Dic₃	—	4	2	no	—
189	C₂×Dic₃	—	4	2	no	—
190	D₁₂	—	2	1	no	—
191	D₁₂	—	2	1	no	—
192	C₄×D₃	—	4	2	no	—
193	Dic₆	—	2	1	no	—
194	Dic₆	—	2	1	no	—
195	Dic₆	—	4	2	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	—	2	1	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	#41	1	2	yes	complemented
200	#41	—	12	4	no	—
201	C₂×Q₈	—	3	1	no	—
202	C₂×Q₈	—	6	1	no	—
203	C₂×D₄	—	3	1	no	—
204	C₂×D₄	—	6	1	no	—
205	C₂×D₄	—	6	1	no	—
206	C₂×D₄	D₁₂	1	1	yes	characteristic
207	C₂²×C₄	—	6	1	no	—
208	C₂²×C₄	—	6	1	no	—
209	C₂²×C₄	—	6	1	no	—
210	C₂²×C₄	—	6	1	no	—
211	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
213	SD₁₆	—	6	2	no	—
214	M₁₆	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
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	C₄⋊C₄	—	3	1	no	—
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	—	3	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	D₁₂	1	1	yes	characteristic
236	C₂²⋊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	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	C₂²×D₃	1	1	yes	characteristic
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
253	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×D₃	—	4	1	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	C₂×#41	1	1	yes	characteristic
262	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
263	Dic₃	—	4	1	no	—
264	Dic₃	—	4	1	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	C₂³	—	6	1	no	—
271	C₂³	C₂×D₁₂	1	1	yes	characteristic
272	C₂³	C₂×D₁₂	1	1	yes	characteristic
273	Q₈	—	6	1	no	—
274	Q₈	—	6	1	no	—
275	Q₈	—	12	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	6	1	no	—
278	D₄	—	6	1	no	—
279	D₄	—	12	2	no	—
280	D₄	—	12	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	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	—	12	2	no	—
293	C₂×C₄	—	12	2	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	—	12	2	no	—
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	#237	1	2	yes	complemented
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	4	1	no	—
307	C₂×C₃	#547	1	1	yes	characteristic
308	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂×C₃	#578	1	1	yes	characteristic
310	D₃	—	8	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	12	1	no	—
317	C₂²	—	12	1	no	—
318	C₂²	—	12	1	no	—
319	C₂²	#908	1	2	yes	—
320	C₂²	C₂²×D₁₂	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	12	1	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	—	12	1	no	—
329	C₄	—	12	1	no	—
330	C₄	—	12	1	no	—
331	C₄	—	12	1	no	—
332	C₄	C₂×#237	1	1	yes	characteristic
333	C₄	#1010	1	1	yes	characteristic
334	C₃	#3343	1	1	yes	characteristic, complemented, Sylow
335	C₂	—	2	2	no	—
336	C₂	—	4	1	no	—
337	C₂	—	24	1	no	—
338	C₂	#5555	1	1	yes	characteristic
339	C₂	C₂×#908	1	1	yes	characteristic
340	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×#435	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5555	C₂
hypercenter	#435	D₃
hypocenter	C₃	#3343
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #5541	C₂	semidirect
3. C₃×#435	C₂	semidirect
4. #5134	C₂	semidirect
5. #5121	C₂	semidirect
6. #5114	C₂	semidirect
7. C₂×#818	C₂	semidirect
8. #4850	C₂	semidirect
9. #4847	C₂	semidirect
10. #4737	C₂	semidirect
11. #4736	C₂	semidirect
12. #4722	C₂	semidirect
13. #4721	C₂	semidirect
14. #4710	C₂	semidirect
15. #4702	C₂	semidirect
72. #435	D₃	semidirect
199. C₃×C₈	#41	semidirect
304. C₈	#237	semidirect
334. C₃	#3343	semidirect

Maximal subgroups

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

subgroup	index
#5555	2
#5541	2
C₃×#435	2
#5134	2
#5121	2
#5114	2
C₂×#818	2
#4850	2
#4847	2
#4737	2
#4736	2
#4722	2
#4721	2
#4710	2
#4702	2
#3343	3