Subgroups — #79751

Order 384.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5568	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
4	#5165	C₂	1	1	yes	maximal, characteristic
5	#5162	C₂	1	1	yes	maximal, characteristic, complemented
6	#5021	C₂	1	1	yes	maximal, characteristic, complemented
7	#5016	C₂	1	1	yes	maximal, characteristic
8	#4890	C₂	1	1	yes	maximal, characteristic, complemented
9	#4875	C₂	1	1	yes	maximal, characteristic, complemented
10	#4794	C₂	1	1	yes	maximal, characteristic, complemented
11	#4792	C₂	1	1	yes	maximal, characteristic, complemented
12	#4757	C₂	1	1	yes	maximal, characteristic, complemented
13	#4756	C₂	1	1	yes	maximal, characteristic
14	#4714	C₂	1	1	yes	maximal, characteristic, complemented
15	#4701	C₂	1	1	yes	maximal, characteristic, complemented
16	#3367	—	3	1	no	maximal, Sylow
17	C₂²×Dic₆	—	2	1	no	—
18	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	#947	C₂²	1	1	yes	characteristic
22	#944	C₂²	1	1	yes	characteristic
23	D₄×Dic₃	C₂²	1	1	yes	characteristic
24	D₄×Dic₃	C₂²	1	1	yes	characteristic
25	C₂×#214	C₂²	1	1	yes	characteristic
26	C₂×#213	C₂²	1	1	yes	characteristic
27	#924	—	2	1	no	—
28	#898	—	2	1	no	—
29	#896	—	2	1	no	—
30	#893	C₂²	1	1	yes	characteristic
31	#888	C₂²	1	1	yes	characteristic
32	#888	C₂²	1	1	yes	characteristic
33	#887	C₂²	1	1	yes	characteristic
34	#878	—	2	1	no	—
35	#878	—	2	1	no	—
36	#876	—	2	1	no	—
37	#875	C₂²	1	1	yes	characteristic
38	#872	—	2	1	no	—
39	#869	C₂²	1	1	yes	characteristic
40	C₄×Dic₆	C₂²	1	1	yes	characteristic
41	C₃×#107	C₂²	1	1	yes	characteristic
42	C₃×#103	C₂²	1	1	yes	characteristic
43	C₃×#102	C₂²	1	1	yes	characteristic
44	C₃×#97	C₂²	1	1	yes	characteristic
45	#835	C₂²	1	1	yes	characteristic
46	#835	C₂²	1	1	yes	characteristic
47	#821	C₂²	1	1	yes	characteristic
48	#821	C₂²	1	1	yes	characteristic
49	#816	C₂²	1	1	yes	characteristic
50	#816	C₂²	1	1	yes	characteristic
51	#815	C₂²	1	1	yes	characteristic
52	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
54	C₈×Dic₃	C₂²	1	1	yes	characteristic
55	#810	C₂²	1	1	yes	characteristic
56	#809	C₂²	1	1	yes	characteristic
57	#807	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
60	#556	—	3	1	no	—
61	D₄×Q₈	—	3	1	no	—
62	#489	—	3	1	no	—
63	#489	—	3	1	no	—
64	#487	—	3	1	no	—
65	#477	—	3	1	no	—
66	#477	—	3	1	no	—
67	#474	—	3	1	no	—
68	#463	—	3	1	no	—
69	#463	—	3	1	no	—
70	#461	—	3	1	no	—
71	#446	—	3	1	no	—
72	#440	—	3	1	no	—
73	#440	D₃	1	1	yes	characteristic, complemented
74	C₄×D₈	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×Dic₃	—	2	1	no	—
79	C₂²×Dic₃	—	2	1	no	—
80	C₂×D₁₂	C₂³	1	1	yes	characteristic
81	C₂×C₄×D₃	—	2	1	no	—
82	C₂×Dic₆	—	2	1	no	—
83	C₂×Dic₆	—	2	1	no	—
84	C₂×Dic₆	—	2	1	no	—
85	C₂×Dic₆	—	4	4	no	—
86	C₂×Dic₆	D₄	1	1	yes	characteristic
87	C₂×Dic₆	C₂³	1	1	yes	characteristic
88	C₃×SD₁₆	—	2	2	no	—
89	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
91	C₃×C₄⋊C₄	—	2	1	no	—
92	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₃×C₂²⋊C₄	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	C₂³	1	1	yes	characteristic
97	#217	—	2	1	no	—
98	#217	—	2	1	no	—
99	#214	—	2	2	no	—
100	#213	—	2	2	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	—	2	1	no	—
107	C₄⋊Dic₃	D₄	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	D₄	1	1	yes	characteristic
116	Dic₃⋊C₄	D₄	1	1	yes	characteristic
117	C₄×Dic₃	—	2	1	no	—
118	C₄×Dic₃	C₂³	1	1	yes	characteristic
119	C₄×Dic₃	C₂³	1	1	yes	characteristic
120	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
122	C₂²×Q₈	—	6	1	no	—
123	C₂×SD₁₆	—	3	1	no	—
124	C₂×SD₁₆	C₂×D₃	1	1	yes	characteristic
125	C₂×D₈	—	3	1	no	—
126	C₄⋊Q₈	—	3	1	no	—
127	C₄⋊Q₈	—	3	1	no	—
128	C₄⋊Q₈	—	3	1	no	—
129	C₄⋊Q₈	—	3	1	no	—
130	#126	—	6	1	no	—
131	#126	—	6	1	no	—
132	#124	—	3	1	no	—
133	#124	—	3	1	no	—
134	C₂²⋊Q₈	—	6	1	no	—
135	C₂²⋊Q₈	—	6	1	no	—
136	C₂²⋊Q₈	—	6	1	no	—
137	C₂²⋊Q₈	—	6	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×Q₈	—	3	1	no	—
140	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
141	C₄×D₄	—	3	1	no	—
142	C₄×D₄	—	3	1	no	—
143	C₄×D₄	—	3	1	no	—
144	C₄×D₄	C₂×D₃	1	1	yes	characteristic
145	#117	—	6	1	no	—
146	#107	—	3	1	no	—
147	#107	—	3	1	no	—
148	#107	C₂×D₃	1	1	yes	characteristic
149	C₄⋊C₈	—	3	1	no	—
150	C₄⋊C₈	—	3	1	no	—
151	C₄⋊C₈	—	3	1	no	—
152	#103	—	3	1	no	—
153	#103	—	3	1	no	—
154	#103	—	3	1	no	—
155	#103	—	3	1	no	—
156	#103	—	3	1	no	—
157	#103	C₂×D₃	1	1	yes	characteristic
158	#102	—	3	1	no	—
159	#102	—	3	1	no	—
160	#102	—	3	1	no	—
161	#102	C₂×D₃	1	1	yes	characteristic
162	#97	—	3	1	no	—
163	#97	C₂×D₃	1	1	yes	characteristic
164	C₄×C₈	—	3	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂²×D₃	—	2	1	no	—
167	C₃×Q₈	—	2	1	no	—
168	C₃×Q₈	—	2	1	no	—
169	C₃×D₄	—	2	1	no	—
170	C₃×D₄	#41	1	2	yes	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	4	2	no	—
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	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	—	4	2	no	—
182	C₂×Dic₃	—	4	2	no	—
183	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
184	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
185	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
186	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
187	D₁₂	—	2	1	no	—
188	D₁₂	—	2	1	no	—
189	C₄×D₃	—	4	2	no	—
190	Dic₆	—	2	2	no	—
191	Dic₆	—	2	1	no	—
192	Dic₆	—	2	1	no	—
193	Dic₆	—	4	2	no	—
194	Dic₆	—	4	2	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	—	2	1	no	—
200	C₂×Q₈	—	3	1	no	—
201	C₂×Q₈	—	3	1	no	—
202	C₂×Q₈	—	6	1	no	—
203	C₂×Q₈	—	6	1	no	—
204	C₂×Q₈	—	6	1	no	—
205	C₂×Q₈	—	12	4	no	—
206	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
207	C₂×D₄	—	3	1	no	—
208	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	6	1	no	—
211	C₂²×C₄	—	6	1	no	—
212	C₂²×C₄	—	6	1	no	—
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	6	2	no	—
215	D₈	—	6	2	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	1	no	—
218	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
219	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
220	C₄⋊C₄	—	2	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	C₂²×D₃	1	1	yes	characteristic
233	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
234	C₂²⋊C₄	—	2	1	no	—
235	C₂²⋊C₄	—	6	1	no	—
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	C₄²	—	2	1	no	—
243	C₄²	—	3	1	no	—
244	C₄²	—	3	1	no	—
245	C₄²	—	6	1	no	—
246	C₄²	C₂²×D₃	1	1	yes	characteristic
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	1	no	—
250	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
251	C₂×D₃	—	4	1	no	—
252	C₂×D₃	—	4	1	no	—
253	C₂×D₃	—	4	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	4	1	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	C₂×#41	1	1	yes	characteristic
260	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
261	Dic₃	—	2	1	no	—
262	Dic₃	—	2	1	no	—
263	Dic₃	—	2	1	no	—
264	Dic₃	—	4	1	no	—
265	Dic₃	—	4	1	no	—
266	Dic₃	—	4	1	no	—
267	Dic₃	AGL(1,ℤ/8)	1	2	yes	—
268	C₂³	—	2	1	no	—
269	C₂³	—	6	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	6	2	no	—
273	Q₈	—	6	1	no	—
274	Q₈	—	6	1	no	—
275	Q₈	—	12	2	no	—
276	Q₈	—	12	2	no	—
277	Q₈	—	12	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	6	1	no	—
280	D₄	—	6	1	no	—
281	D₄	#235	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	3	1	no	—
286	C₂×C₄	—	3	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	—	12	2	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	—	12	2	no	—
296	C₂×C₄	D₃×D₄	1	1	yes	characteristic
297	C₂×C₄	D₃×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	6	1	no	—
302	C₈	—	6	1	no	—
303	C₂×C₃	—	4	2	no	—
304	C₂×C₃	#548	1	1	yes	characteristic
305	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂×C₃	#578	1	1	yes	characteristic
307	D₃	—	8	1	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	12	1	no	—
312	C₂²	—	12	1	no	—
313	C₂²	—	12	1	no	—
314	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	3	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	6	1	no	—
322	C₄	—	6	1	no	—
323	C₄	—	6	1	no	—
324	C₄	—	12	1	no	—
325	C₄	—	12	1	no	—
326	C₄	—	12	1	no	—
327	C₄	C₂×#235	1	1	yes	characteristic
328	C₄	#1010	1	1	yes	characteristic
329	C₃	#3367	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	4	2	no	—
331	C₂	—	24	1	no	—
332	C₂	#5552	1	1	yes	characteristic
333	C₂	D₃×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂	#5779	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#440	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#213	C₂²
hypercenter	#440	D₃
hypocenter	C₃	#3367
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #5568	C₂	semidirect
2. D₄×Dic₆	C₂	semidirect
3. C₃×#440	C₂	semidirect
5. #5162	C₂	semidirect
6. #5021	C₂	semidirect
8. #4890	C₂	semidirect
9. #4875	C₂	semidirect
10. #4794	C₂	semidirect
11. #4792	C₂	semidirect
12. #4757	C₂	semidirect
14. #4714	C₂	semidirect
15. #4701	C₂	semidirect
73. #440	D₃	semidirect
329. C₃	#3367	semidirect

Maximal subgroups

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

subgroup	index
#5568	2
D₄×Dic₆	2
C₃×#440	2
#5165	2
#5162	2
#5021	2
#5016	2
#4890	2
#4875	2
#4794	2
#4792	2
#4757	2
#4756	2
#4714	2
#4701	2
#3367	3