Subgroups — #79855

Order 384.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5539	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#442	C₂	1	1	yes	maximal, characteristic
4	#5152	C₂	1	1	yes	maximal, characteristic
5	Dic₃×D₈	C₂	1	1	yes	maximal, characteristic
6	#5009	C₂	1	1	yes	maximal, characteristic, complemented
7	#5008	C₂	1	1	yes	maximal, characteristic, complemented
8	#4856	C₂	1	1	yes	maximal, characteristic, complemented
9	#4855	C₂	1	1	yes	maximal, characteristic, complemented
10	#4766	C₂	1	1	yes	maximal, characteristic, complemented
11	#4761	C₂	1	1	yes	maximal, characteristic, complemented
12	#4761	C₂	1	1	yes	maximal, characteristic, complemented
13	#4759	C₂	1	1	yes	maximal, characteristic, complemented
14	#4702	C₂	1	1	yes	maximal, characteristic, complemented
15	#4701	C₂	1	1	yes	maximal, characteristic, complemented
16	#3384	—	3	1	no	maximal, Sylow
17	C₂²×Dic₆	—	2	1	no	—
18	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	D₄×Dic₃	C₂²	1	1	yes	characteristic
22	D₄×Dic₃	C₂²	1	1	yes	characteristic
23	D₄×Dic₃	C₂²	1	1	yes	characteristic
24	D₄×Dic₃	C₂²	1	1	yes	characteristic
25	#924	—	2	1	no	—
26	#924	—	2	1	no	—
27	C₂×Dic₃⋊C₄	—	2	1	no	—
28	#890	C₂²	1	1	yes	characteristic
29	#888	C₂²	1	1	yes	characteristic
30	#888	C₂²	1	1	yes	characteristic
31	#888	C₂²	1	1	yes	characteristic
32	#878	—	2	1	no	—
33	#878	—	2	1	no	—
34	#878	—	2	1	no	—
35	#878	—	2	1	no	—
36	#870	C₂²	1	1	yes	characteristic
37	#869	C₂²	1	1	yes	characteristic
38	C₄×Dic₆	C₂²	1	1	yes	characteristic
39	C₃×#106	C₂²	1	1	yes	characteristic
40	C₃×#102	C₂²	1	1	yes	characteristic
41	C₃×#102	C₂²	1	1	yes	characteristic
42	C₃×#97	C₂²	1	1	yes	characteristic
43	#832	C₂²	1	1	yes	characteristic
44	#832	C₂²	1	1	yes	characteristic
45	#832	C₂²	1	1	yes	characteristic
46	#832	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#818	C₂²	1	1	yes	characteristic
49	#817	C₂²	1	1	yes	characteristic
50	#817	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	#808	C₂²	1	1	yes	characteristic
56	#808	C₂²	1	1	yes	characteristic
57	#808	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
60	#554	—	3	1	no	—
61	D₄×Q₈	—	3	1	no	—
62	#501	—	3	1	no	—
63	#501	—	3	1	no	—
64	#498	—	3	1	no	—
65	#478	—	3	1	no	—
66	#478	—	3	1	no	—
67	#476	—	3	1	no	—
68	#476	—	3	1	no	—
69	#476	—	3	1	no	—
70	#474	—	3	1	no	—
71	#446	—	3	1	no	—
72	#442	—	3	1	no	—
73	#442	D₃	1	1	yes	characteristic
74	C₄×D₈	—	3	1	no	—
75	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
76	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×Dic₃	—	2	1	no	—
80	C₂²×Dic₃	—	2	1	no	—
81	C₂²×Dic₃	—	2	1	no	—
82	C₂²×Dic₃	—	2	1	no	—
83	C₂×Dic₆	—	2	1	no	—
84	C₂×Dic₆	—	2	1	no	—
85	C₂×Dic₆	—	2	1	no	—
86	C₂×Dic₆	—	2	1	no	—
87	C₂×Dic₆	—	4	4	no	—
88	C₂×Dic₆	D₄	1	1	yes	characteristic
89	C₃×D₈	Q₈	1	4	yes	complemented
90	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄	—	2	1	no	—
96	C₃×C₄²	C₂³	1	1	yes	characteristic
97	#217	—	2	1	no	—
98	#217	—	2	1	no	—
99	#217	—	2	1	no	—
100	#217	—	2	1	no	—
101	C₄⋊Dic₃	—	2	1	no	—
102	C₄⋊Dic₃	—	2	1	no	—
103	C₄⋊Dic₃	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
115	Dic₃⋊C₄	D₄	1	1	yes	characteristic
116	C₄×Dic₃	C₂³	1	1	yes	characteristic
117	C₄×Dic₃	C₂³	1	1	yes	characteristic
118	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
119	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	C₂²×Q₈	—	6	1	no	—
121	C₂×D₈	C₂×D₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	3	1	no	—
123	C₄⋊Q₈	—	3	1	no	—
124	C₄⋊Q₈	—	3	1	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	#125	—	3	1	no	—
127	#125	—	3	1	no	—
128	C₂²⋊Q₈	—	6	1	no	—
129	C₂²⋊Q₈	—	6	1	no	—
130	C₂²⋊Q₈	—	6	1	no	—
131	C₂²⋊Q₈	—	6	1	no	—
132	C₂²⋊Q₈	—	6	1	no	—
133	C₂²⋊Q₈	—	6	1	no	—
134	C₄×Q₈	—	3	1	no	—
135	C₄×D₄	—	3	1	no	—
136	C₄×D₄	—	3	1	no	—
137	C₄×D₄	—	3	1	no	—
138	C₄×D₄	—	3	1	no	—
139	C₄×D₄	C₂×D₃	1	1	yes	characteristic
140	C₄×D₄	C₂×D₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	6	1	no	—
142	#107	—	3	1	no	—
143	#107	—	3	1	no	—
144	#107	—	3	1	no	—
145	#106	—	3	1	no	—
146	#106	—	3	1	no	—
147	#106	—	3	1	no	—
148	#106	—	3	1	no	—
149	#106	—	3	1	no	—
150	#106	C₂×D₃	1	1	yes	characteristic
151	C₄⋊C₈	—	3	1	no	—
152	C₄⋊C₈	—	3	1	no	—
153	C₄⋊C₈	—	3	1	no	—
154	#102	—	3	1	no	—
155	#102	—	3	1	no	—
156	#102	—	3	1	no	—
157	#102	—	3	1	no	—
158	#102	C₂×D₃	1	1	yes	characteristic
159	#102	C₂×D₃	1	1	yes	characteristic
160	#97	—	3	1	no	—
161	#97	C₂×D₃	1	1	yes	characteristic
162	C₄×C₈	—	3	1	no	—
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₃×D₄	—	2	1	no	—
166	C₃×D₄	—	2	1	no	—
167	C₃×D₄	C₂×Q₈	1	2	yes	—
168	C₃×D₄	C₂×Q₈	1	2	yes	complemented
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	8	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	C₂⁴	1	1	yes	characteristic
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	4	2	no	—
181	C₂×Dic₃	—	4	2	no	—
182	C₂×Dic₃	—	4	2	no	—
183	C₂×Dic₃	—	8	1	no	—
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	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
188	Dic₆	—	2	2	no	—
189	Dic₆	—	4	2	no	—
190	Dic₆	—	4	2	no	—
191	Dic₆	—	4	2	no	—
192	Dic₆	—	8	1	no	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	C₂×Q₈	1	2	yes	complemented
195	C₃⋊C₈	—	2	1	no	—
196	C₃⋊C₈	—	2	1	no	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	6	1	no	—
199	C₂×Q₈	—	6	1	no	—
200	C₂×Q₈	—	6	1	no	—
201	C₂×Q₈	—	6	1	no	—
202	C₂×Q₈	—	12	4	no	—
203	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
204	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
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	D₈	Dic₆	1	4	yes	complemented
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	—	3	1	no	—
214	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	—	3	1	no	—
221	C₄⋊C₄	—	3	1	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄⋊C₄	—	6	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	C₂²×D₃	1	1	yes	characteristic
232	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
233	C₂²⋊C₄	—	2	1	no	—
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	C₄²	—	3	1	no	—
240	C₄²	—	3	1	no	—
241	C₄²	C₂²×D₃	1	1	yes	characteristic
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	4	1	no	—
245	C₂²×C₃	—	4	1	no	—
246	C₂²×C₃	—	4	1	no	—
247	C₂²×C₃	—	4	1	no	—
248	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	4	1	no	—
252	C₃×C₄	—	4	1	no	—
253	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
254	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
255	Dic₃	—	2	1	no	—
256	Dic₃	—	2	1	no	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	4	1	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	4	1	no	—
261	Dic₃	—	4	1	no	—
262	Dic₃	AGL(1,ℤ/8)	1	2	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	Q₈	—	6	2	no	—
266	Q₈	—	12	2	no	—
267	Q₈	—	12	2	no	—
268	Q₈	—	12	2	no	—
269	Q₈	—	24	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	2	1	no	—
272	D₄	C₂×Dic₆	1	2	yes	—
273	D₄	C₂×Dic₆	1	2	yes	complemented
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	12	2	no	—
287	C₂×C₄	—	12	2	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	—	24	1	no	—
290	C₂×C₄	D₃×D₄	1	1	yes	characteristic
291	C₂×C₄	D₃×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	6	1	no	—
295	C₈	—	6	1	no	—
296	C₈	C₂×Dic₆	1	2	yes	complemented
297	C₂×C₃	—	4	2	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
300	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₂×C₃	#577	1	1	yes	characteristic
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	3	2	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	6	1	no	—
315	C₄	—	6	1	no	—
316	C₄	—	6	1	no	—
317	C₄	—	12	1	no	—
318	C₄	—	12	1	no	—
319	C₄	—	12	1	no	—
320	C₄	—	12	1	no	—
321	C₄	C₂²×Dic₆	1	1	yes	characteristic
322	C₄	#1010	1	1	yes	characteristic
323	C₃	#3384	1	1	yes	characteristic, complemented, Sylow
324	C₂	—	4	2	no	—
325	C₂	—	4	2	no	—
326	C₂	D₄×Dic₆	1	1	yes	characteristic
327	C₂	D₃×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂	#5775	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×#442	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₈	C₂²
hypercenter	#442	D₃
hypocenter	C₃	#3384
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #5539	C₂	semidirect
2. D₄×Dic₆	C₂	semidirect
6. #5009	C₂	semidirect
7. #5008	C₂	semidirect
8. #4856	C₂	semidirect
9. #4855	C₂	semidirect
10. #4766	C₂	semidirect
11. #4761	C₂	semidirect
12. #4761	C₂	semidirect
13. #4759	C₂	semidirect
14. #4702	C₂	semidirect
15. #4701	C₂	semidirect
89. C₃×D₈	Q₈	semidirect
168. C₃×D₄	C₂×Q₈	semidirect
194. C₃×C₈	C₂×Q₈	semidirect
211. D₈	Dic₆	semidirect
273. D₄	C₂×Dic₆	semidirect
296. C₈	C₂×Dic₆	semidirect
323. C₃	#3384	semidirect

Maximal subgroups

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

subgroup	index
#5539	2
D₄×Dic₆	2
C₃×#442	2
#5152	2
Dic₃×D₈	2
#5009	2
#5008	2
#4856	2
#4855	2
#4766	2
#4761	2
#4759	2
#4702	2
#4701	2
#3384	3