Subgroups — #490652

Order 512.

396 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33734	C₂	1	1	yes	maximal, characteristic, complemented
2	#33480	C₂	1	1	yes	maximal, characteristic
3	#19220	C₂	1	2	yes	maximal
4	#12481	C₂	1	2	yes	maximal, complemented
5	#12473	C₂	1	2	yes	maximal, complemented
6	#12458	C₂	1	1	yes	maximal, characteristic, complemented
7	#12452	C₂	1	1	yes	maximal, characteristic, complemented
8	#12440	C₂	1	1	yes	maximal, characteristic, complemented
9	#12431	C₂	1	1	yes	maximal, characteristic, complemented
10	#12008	C₂	1	1	yes	maximal, characteristic, complemented
11	#11953	C₂	1	2	yes	maximal, complemented
12	#3233	—	2	2	no	—
13	#3170	C₂²	1	2	yes	—
14	#2966	C₂²	1	1	yes	characteristic
15	#2252	C₂²	1	1	yes	characteristic
16	#2247	C₂²	1	1	yes	characteristic
17	#2241	C₂²	1	2	yes	—
18	#2216	—	2	1	no	—
19	#2216	—	2	1	no	—
20	#2183	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×Q₃₂	—	2	1	no	—
23	C₄×Q₃₂	—	2	1	no	—
24	#2169	C₂²	1	1	yes	characteristic
25	#2168	—	2	1	no	—
26	D₄×C₁₆	—	2	1	no	—
27	D₄×C₁₆	C₂²	1	1	yes	characteristic
28	#2164	—	2	1	no	—
29	#2163	C₂²	1	1	yes	characteristic
30	#2108	—	2	1	no	—
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#1704	—	2	2	no	—
33	#1703	—	2	2	no	—
34	#1676	C₂²	1	2	yes	—
35	#1675	C₂²	1	2	yes	—
36	#1670	C₂²	1	2	yes	—
37	#1669	C₂²	1	2	yes	—
38	#1564	C₂²	1	2	yes	—
39	#1554	—	2	2	no	—
40	#1372	C₂²	1	2	yes	—
41	#1371	C₂²	1	1	yes	characteristic
42	#1371	C₂²	1	1	yes	characteristic
43	#1365	C₂²	1	2	yes	—
44	#1365	C₂²	1	2	yes	—
45	#1362	C₂²	1	2	yes	—
46	#1362	C₂²	1	2	yes	—
47	#533	—	2	2	no	—
48	#529	D₄	1	1	yes	characteristic
49	C₂×Q₃₂	—	2	1	no	—
50	C₂×Q₃₂	—	2	1	no	—
51	C₂×M₃₂	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	D₄	1	1	yes	characteristic
54	C₂²×C₁₆	D₄	1	1	yes	characteristic
55	#500	C₂³	1	2	yes	—
56	#499	—	4	2	no	—
57	#494	C₂³	1	2	yes	—
58	#488	—	4	2	no	—
59	#487	—	2	2	no	—
60	#485	—	4	2	no	—
61	#482	—	4	2	no	—
62	#477	—	2	2	no	—
63	#476	—	2	2	no	—
64	#475	—	2	2	no	—
65	#474	—	2	2	no	—
66	#467	—	2	2	no	—
67	#467	—	2	2	no	—
68	#446	D₄	1	1	yes	characteristic
69	C₈×Q₈	C₂³	1	1	yes	characteristic
70	C₄×Q₁₆	—	2	2	no	—
71	C₄×Q₁₆	—	2	2	no	—
72	#436	D₄	1	1	yes	characteristic
73	#435	—	2	1	no	—
74	C₈×D₄	—	2	1	no	—
75	C₈×D₄	C₂³	1	1	yes	characteristic
76	#433	—	2	1	no	—
77	#432	D₄	1	1	yes	characteristic
78	#431	—	2	2	no	—
79	#405	—	2	1	no	—
80	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	C₄⋊C₁₆	—	2	1	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
86	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
87	#358	—	2	2	no	—
88	#358	—	2	2	no	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₂²⋊C₁₆	—	2	1	no	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	#346	—	2	1	no	—
93	#346	—	2	1	no	—
94	C₄×C₁₆	—	2	1	no	—
95	C₄×C₁₆	D₄	1	1	yes	characteristic
96	C₄×C₁₆	D₄	1	1	yes	characteristic
97	C₄×C₁₆	C₂³	1	1	yes	characteristic
98	C₄×C₁₆	C₂³	1	1	yes	characteristic
99	#335	C₂³	1	2	yes	—
100	#335	C₂³	1	2	yes	—
101	#334	—	2	2	no	—
102	#333	—	2	2	no	—
103	#333	—	2	2	no	—
104	#332	—	2	2	no	—
105	#332	—	2	2	no	—
106	C₂×Q₁₆	—	2	2	no	—
107	C₂×Q₁₆	—	2	2	no	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	#41	1	1	yes	characteristic
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	#126	—	2	1	no	—
115	#125	—	4	2	no	—
116	#125	D₈	1	1	yes	characteristic
117	#124	D₈	1	1	yes	characteristic
118	#123	—	2	1	no	—
119	C₂²⋊Q₈	—	4	2	no	—
120	C₂²⋊Q₈	—	4	2	no	—
121	C₂²⋊Q₈	D₈	1	1	yes	characteristic
122	#121	D₈	1	1	yes	characteristic
123	C₄×Q₈	—	4	2	no	—
124	C₄×Q₈	—	4	2	no	—
125	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
126	C₄×D₄	—	2	1	no	—
127	C₄×D₄	C₂×D₄	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	#117	—	4	2	no	—
130	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
131	Q₃₂	—	4	2	no	—
132	Q₃₂	—	4	2	no	—
133	M₃₂	—	4	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	4	1	no	—
138	C₂×C₁₆	—	4	2	no	—
139	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
140	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
141	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
142	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	—	4	2	no	—
146	#107	—	4	2	no	—
147	#106	—	4	2	no	—
148	#106	—	4	2	no	—
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
155	#103	—	4	2	no	—
156	#103	—	4	2	no	—
157	#103	—	4	2	no	—
158	#103	—	4	2	no	—
159	#102	—	4	2	no	—
160	#102	—	4	2	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₂²⋊C₈	—	2	1	no	—
163	C₂²⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	4	2	no	—
165	#97	—	2	1	no	—
166	#97	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	4	2	no	—
169	C₄×C₈	C₂×D₄	1	1	yes	characteristic
170	C₄×C₈	C₂×D₄	1	1	yes	characteristic
171	C₄×C₈	#41	1	1	yes	characteristic
172	C₄×C₈	C₂⁴	1	1	yes	characteristic
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₂×D₈	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	#121	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
181	C₂²×C₄	#135	1	1	yes	characteristic
182	Q₁₆	—	4	2	no	—
183	Q₁₆	—	4	2	no	—
184	Q₁₆	—	8	2	no	—
185	M₁₆	—	4	2	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	#121	1	1	yes	characteristic
196	C₂×C₈	#121	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₈	C₂×#41	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
210	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	8	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	8	2	no	—
219	C₄²	#121	1	1	yes	characteristic
220	C₄²	C₂×D₈	1	1	yes	characteristic
221	C₄²	#135	1	1	yes	characteristic
222	C₄²	C₂²×D₄	1	1	yes	characteristic
223	C₁₆	—	2	1	no	—
224	C₁₆	—	2	1	no	—
225	C₁₆	—	2	1	no	—
226	C₁₆	—	4	1	no	—
227	C₁₆	—	4	1	no	—
228	C₁₆	#135	1	2	yes	—
229	C₂³	—	2	1	no	—
230	C₂³	#466	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	2	1	no	—
233	Q₈	—	8	2	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	16	2	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	1	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	#466	1	1	yes	characteristic
252	C₂×C₄	#466	1	1	yes	characteristic
253	C₂×C₄	#466	1	1	yes	characteristic
254	C₂×C₄	C₂×#121	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
256	C₂×C₄	C₂×#135	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	2	no	—
264	C₈	C₂×#135	1	1	yes	characteristic
265	C₈	#575	1	1	yes	characteristic
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	C₂×#466	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	16	2	no	—
281	C₄	—	16	2	no	—
282	C₄	#3035	1	1	yes	characteristic
283	C₄	#3067	1	1	yes	characteristic
284	C₄	#3417	1	1	yes	characteristic
285	C₄	#3417	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	8	1	no	—
288	C₂	#19125	1	1	yes	characteristic
289	C₂	#33608	1	1	yes	characteristic
290	C₂	#33617	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33734	C₂	semidirect
4. #12481	C₂	semidirect
5. #12473	C₂	semidirect
6. #12458	C₂	semidirect
7. #12452	C₂	semidirect
8. #12440	C₂	semidirect
9. #12431	C₂	semidirect
10. #12008	C₂	semidirect
11. #11953	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33734	2
#33480	2
#19220	2
#12481	2
#12473	2
#12458	2
#12452	2
#12440	2
#12431	2
#12008	2
#11953	2