Subgroups — #81055

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	#5727	C₂	1	1	yes	maximal, characteristic, complemented
2	#5539	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
4	#5081	C₂	1	1	yes	maximal, characteristic, complemented
5	#5014	C₂	1	1	yes	maximal, characteristic
6	#4871	C₂	1	1	yes	maximal, characteristic, complemented
7	#4870	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	#4847	C₂	1	1	yes	maximal, characteristic, complemented
11	#4828	C₂	1	1	yes	maximal, characteristic, complemented
12	#4785	C₂	1	1	yes	maximal, characteristic
13	#4761	C₂	1	1	yes	maximal, characteristic, complemented
14	#4759	C₂	1	1	yes	maximal, characteristic, complemented
15	#4756	C₂	1	1	yes	maximal, characteristic
16	#3384	—	3	1	no	maximal, Sylow
17	C₂×D₃×Q₈	—	2	1	no	—
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
20	#946	—	2	1	no	—
21	D₄×Dic₃	C₂²	1	1	yes	characteristic
22	D₄×Dic₃	C₂²	1	1	yes	characteristic
23	C₂×#213	C₂²	1	1	yes	characteristic
24	#924	—	2	1	no	—
25	C₂×Dic₃⋊C₄	—	2	1	no	—
26	#896	—	2	1	no	—
27	#896	—	2	1	no	—
28	#893	C₂²	1	1	yes	characteristic
29	#893	C₂²	1	1	yes	characteristic
30	#890	C₂²	1	1	yes	characteristic
31	#888	C₂²	1	1	yes	characteristic
32	#888	C₂²	1	1	yes	characteristic
33	#888	C₂²	1	1	yes	characteristic
34	#878	—	2	1	no	—
35	#878	—	2	1	no	—
36	C₄×D₁₂	C₂²	1	1	yes	characteristic
37	#870	C₂²	1	1	yes	characteristic
38	C₄×Dic₆	C₂²	1	1	yes	characteristic
39	C₃×#106	C₂²	1	1	yes	characteristic
40	C₃×#106	C₂²	1	1	yes	characteristic
41	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×#102	C₂²	1	1	yes	characteristic
43	C₃×#102	C₂²	1	1	yes	characteristic
44	#832	C₂²	1	1	yes	characteristic
45	#821	C₂²	1	1	yes	characteristic
46	#821	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#817	C₂²	1	1	yes	characteristic
49	#815	C₂²	1	1	yes	characteristic
50	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₃	C₂²	1	1	yes	characteristic
53	#809	C₂²	1	1	yes	characteristic
54	#808	C₂²	1	1	yes	characteristic
55	#808	C₂²	1	1	yes	characteristic
56	#808	C₂²	1	1	yes	characteristic
57	#807	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	#803	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	D₃	1	1	yes	characteristic, complemented
70	#474	—	3	1	no	—
71	#446	—	3	1	no	—
72	#442	—	3	1	no	—
73	#442	—	3	1	no	—
74	C₄×D₈	—	3	1	no	—
75	C₂×C₃×Q₈	—	2	1	no	—
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	D₃×Q₈	—	4	4	no	—
81	C₂×D₁₂	C₂³	1	1	yes	characteristic
82	C₂×C₄×D₃	—	2	1	no	—
83	C₂×C₄×D₃	—	2	1	no	—
84	C₂×C₄×D₃	—	2	1	no	—
85	C₂×Dic₆	—	2	1	no	—
86	C₂×Dic₆	—	2	1	no	—
87	C₂×Dic₆	—	2	1	no	—
88	C₂×Dic₆	D₄	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	—	2	1	no	—
92	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
93	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
94	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
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	#213	Q₈	1	4	yes	complemented
100	(C₂×D₃)⋊C₄	—	2	1	no	—
101	(C₂×D₃)⋊C₄	—	2	1	no	—
102	(C₂×D₃)⋊C₄	—	2	1	no	—
103	C₄⋊Dic₃	—	2	1	no	—
104	C₄⋊Dic₃	D₄	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	—	3	1	no	—
122	C₄⋊Q₈	—	3	1	no	—
123	C₄⋊Q₈	—	3	1	no	—
124	C₄⋊Q₈	—	3	1	no	—
125	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
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	C₂×D₃	1	1	yes	characteristic
150	#106	C₂×D₃	1	1	yes	characteristic
151	C₄⋊C₈	—	3	1	no	—
152	C₄⋊C₈	—	3	1	no	—
153	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₂³×C₃	—	2	1	no	—
164	C₂²×D₃	—	2	1	no	—
165	C₃×Q₈	—	4	2	no	—
166	C₃×D₄	—	2	1	no	—
167	C₃×D₄	C₂×Q₈	1	2	yes	complemented
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	8	1	no	—
172	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
175	C₂×Dic₃	—	2	1	no	—
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	4	2	no	—
179	C₂×Dic₃	—	8	1	no	—
180	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
181	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
183	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
184	D₁₂	—	2	1	no	—
185	D₁₂	C₂×Q₈	1	2	yes	—
186	C₄×D₃	—	4	2	no	—
187	C₄×D₃	—	4	2	no	—
188	C₄×D₃	—	4	2	no	—
189	Dic₆	—	2	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	—	2	1	no	—
195	C₃⋊C₈	—	2	1	no	—
196	C₃⋊C₈	C₂×Q₈	1	2	yes	complemented
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	—	3	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	—	3	1	no	—
204	C₂×D₄	C₂²×D₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	6	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	—	3	4	no	—
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	—	2	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	C₂²×D₃	1	1	yes	characteristic
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	—	6	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	—	4	1	no	—
244	C₂²×C₃	—	4	1	no	—
245	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×D₃	—	2	2	no	—
247	C₂×D₃	—	4	1	no	—
248	C₂×D₃	—	4	1	no	—
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	—	4	1	no	—
254	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
255	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
256	Dic₃	—	2	1	no	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	2	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	—	6	1	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	6	2	no	—
267	Q₈	—	12	2	no	—
268	Q₈	—	12	2	no	—
269	Q₈	—	24	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	3	2	no	—
272	D₄	—	6	1	no	—
273	D₄	D₃×Q₈	1	2	yes	complemented
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	12	2	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	—	2	1	no	—
295	C₈	—	3	2	no	—
296	C₈	—	6	1	no	—
297	C₂×C₃	—	4	2	no	—
298	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
299	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂×C₃	#577	1	1	yes	characteristic
301	D₃	—	4	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	6	2	no	—
306	C₂²	—	12	1	no	—
307	C₂²	—	12	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	—	4	1	no	—
315	C₄	—	6	1	no	—
316	C₄	—	6	1	no	—
317	C₄	—	6	1	no	—
318	C₄	—	12	1	no	—
319	C₄	—	12	1	no	—
320	C₄	—	12	1	no	—
321	C₄	C₂×D₃×Q₈	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	—	12	2	no	—
326	C₂	#5632	1	1	yes	characteristic
327	C₂	#5762	1	1	yes	characteristic
328	C₂	D₃×AGL(1,ℤ/8)	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×#476	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#213	C₂²
hypercenter	#476	D₃
hypocenter	C₃	#3384
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #5727	C₂	semidirect
2. #5539	C₂	semidirect
3. C₃×#476	C₂	semidirect
4. #5081	C₂	semidirect
6. #4871	C₂	semidirect
7. #4870	C₂	semidirect
8. #4856	C₂	semidirect
9. #4855	C₂	semidirect
10. #4847	C₂	semidirect
11. #4828	C₂	semidirect
13. #4761	C₂	semidirect
14. #4759	C₂	semidirect
69. #476	D₃	semidirect
99. #213	Q₈	semidirect
167. C₃×D₄	C₂×Q₈	semidirect
196. C₃⋊C₈	C₂×Q₈	semidirect
273. D₄	D₃×Q₈	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
#5727	2
#5539	2
C₃×#476	2
#5081	2
#5014	2
#4871	2
#4870	2
#4856	2
#4855	2
#4847	2
#4828	2
#4785	2
#4761	2
#4759	2
#4756	2
#3384	3