Subgroups — #491189

Order 512.

396 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33726	C₂	1	1	yes	maximal, characteristic, complemented
2	#33480	C₂	1	1	yes	maximal, characteristic
3	#19361	C₂	1	2	yes	maximal
4	#12462	C₂	1	2	yes	maximal, complemented
5	#12458	C₂	1	1	yes	maximal, characteristic, complemented
6	#12457	C₂	1	1	yes	maximal, characteristic, complemented
7	#12456	C₂	1	2	yes	maximal, complemented
8	#12452	C₂	1	1	yes	maximal, characteristic, complemented
9	#12451	C₂	1	1	yes	maximal, characteristic, complemented
10	#12022	C₂	1	1	yes	maximal, characteristic, complemented
11	#11976	C₂	1	2	yes	maximal, complemented
12	#3252	—	2	2	no	—
13	#3160	C₂²	1	2	yes	—
14	#2966	C₂²	1	1	yes	characteristic
15	#2254	C₂²	1	2	yes	—
16	#2253	—	2	1	no	—
17	#2253	—	2	1	no	—
18	#2252	C₂²	1	1	yes	characteristic
19	#2252	C₂²	1	1	yes	characteristic
20	#2183	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	#2169	C₂²	1	1	yes	characteristic
23	#2168	—	2	1	no	—
24	D₄×C₁₆	—	2	1	no	—
25	D₄×C₁₆	C₂²	1	1	yes	characteristic
26	#2164	—	2	1	no	—
27	#2163	C₂²	1	1	yes	characteristic
28	#2158	—	2	1	no	—
29	#2158	—	2	1	no	—
30	#2108	—	2	1	no	—
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#1692	—	2	2	no	—
33	#1691	—	2	2	no	—
34	#1676	C₂²	1	2	yes	—
35	#1675	C₂²	1	2	yes	—
36	#1674	C₂²	1	2	yes	—
37	#1673	C₂²	1	2	yes	—
38	#1567	C₂²	1	2	yes	—
39	#1559	—	2	2	no	—
40	#1371	C₂²	1	1	yes	characteristic
41	#1371	C₂²	1	1	yes	characteristic
42	#1371	C₂²	1	2	yes	—
43	#1365	C₂²	1	2	yes	—
44	#1365	C₂²	1	2	yes	—
45	#1364	C₂²	1	2	yes	—
46	#1364	C₂²	1	2	yes	—
47	#533	—	2	2	no	—
48	#529	D₄	1	1	yes	characteristic
49	C₂×M₃₂	—	2	1	no	—
50	C₂²×C₁₆	—	2	1	no	—
51	C₂²×C₁₆	D₄	1	1	yes	characteristic
52	C₂²×C₁₆	D₄	1	1	yes	characteristic
53	#500	C₂³	1	2	yes	—
54	#500	C₂³	1	2	yes	—
55	#499	—	2	2	no	—
56	#499	—	2	2	no	—
57	#498	—	2	2	no	—
58	#479	—	4	2	no	—
59	#477	—	2	2	no	—
60	#476	—	2	2	no	—
61	#470	—	4	2	no	—
62	#468	—	4	2	no	—
63	#464	—	4	2	no	—
64	#462	—	2	2	no	—
65	#461	—	2	2	no	—
66	#446	D₄	1	1	yes	characteristic
67	C₈×Q₈	C₂³	1	1	yes	characteristic
68	#436	D₄	1	1	yes	characteristic
69	#435	—	2	1	no	—
70	C₈×D₄	—	2	1	no	—
71	C₈×D₄	C₂³	1	1	yes	characteristic
72	#433	—	2	1	no	—
73	#432	D₄	1	1	yes	characteristic
74	#427	—	2	2	no	—
75	#427	—	2	2	no	—
76	#423	—	2	2	no	—
77	#405	—	2	1	no	—
78	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
79	#367	—	2	2	no	—
80	#367	—	2	2	no	—
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	#366	—	2	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
88	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
100	#335	C₂³	1	2	yes	—
101	#335	C₂³	1	2	yes	—
102	#333	—	2	2	no	—
103	#333	—	2	2	no	—
104	#332	—	2	2	no	—
105	#332	—	2	2	no	—
106	C₂×Q₁₆	—	4	2	no	—
107	C₂×SD₁₆	—	4	2	no	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂×M₁₆	—	4	2	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	—	2	2	no	—
115	#126	—	2	1	no	—
116	#125	—	4	2	no	—
117	#125	D₈	1	1	yes	characteristic
118	#124	D₈	1	1	yes	characteristic
119	#123	—	2	1	no	—
120	C₂²⋊Q₈	—	4	2	no	—
121	C₂²⋊Q₈	—	4	2	no	—
122	C₂²⋊Q₈	D₈	1	1	yes	characteristic
123	#121	D₈	1	1	yes	characteristic
124	C₄×Q₈	—	4	2	no	—
125	C₄×Q₈	—	4	2	no	—
126	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
127	C₄×D₄	—	2	1	no	—
128	C₄×D₄	C₂×D₄	1	1	yes	characteristic
129	#117	—	2	1	no	—
130	#117	—	4	2	no	—
131	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
132	M₃₂	—	4	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	4	2	no	—
138	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
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	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	—	2	2	no	—
146	#107	—	4	2	no	—
147	#106	—	4	2	no	—
148	#106	—	4	2	no	—
149	#106	—	4	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	4	2	no	—
155	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
156	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
157	#103	—	4	2	no	—
158	#103	—	4	2	no	—
159	#103	—	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	#97	—	2	1	no	—
165	#97	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	C₂×D₄	1	1	yes	characteristic
168	C₄×C₈	C₂×D₄	1	1	yes	characteristic
169	C₄×C₈	#41	1	1	yes	characteristic
170	C₄×C₈	C₂⁴	1	1	yes	characteristic
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	C₂×D₈	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	#121	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
179	C₂²×C₄	#135	1	1	yes	characteristic
180	Q₁₆	—	16	2	no	—
181	SD₁₆	—	16	2	no	—
182	M₁₆	—	4	2	no	—
183	M₁₆	—	16	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	#121	1	1	yes	characteristic
194	C₂×C₈	#121	1	1	yes	characteristic
195	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₈	C₂×#41	1	1	yes	characteristic
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	8	2	no	—
206	C₄⋊C₄	—	8	2	no	—
207	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
208	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	8	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	8	2	no	—
217	C₄²	#121	1	1	yes	characteristic
218	C₄²	C₂×D₈	1	1	yes	characteristic
219	C₄²	#135	1	1	yes	characteristic
220	C₄²	C₂²×D₄	1	1	yes	characteristic
221	C₁₆	—	2	1	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	—	4	1	no	—
224	C₁₆	—	4	1	no	—
225	C₁₆	#137	1	2	yes	—
226	C₁₆	#137	1	2	yes	—
227	C₂³	—	2	1	no	—
228	C₂³	#466	1	1	yes	characteristic
229	Q₈	—	2	1	no	—
230	Q₈	—	2	1	no	—
231	Q₈	—	8	2	no	—
232	Q₈	—	8	2	no	—
233	Q₈	—	16	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	2	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	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	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	#466	1	1	yes	characteristic
250	C₂×C₄	#466	1	1	yes	characteristic
251	C₂×C₄	#466	1	1	yes	characteristic
252	C₂×C₄	C₂×#121	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
254	C₂×C₄	C₂×#135	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	8	2	no	—
260	C₈	—	8	2	no	—
261	C₈	#473	1	2	yes	—
262	C₈	C₂×#137	1	1	yes	characteristic
263	C₈	C₂×#137	1	1	yes	characteristic
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×#466	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	16	2	no	—
279	C₄	—	16	2	no	—
280	C₄	#3039	1	1	yes	characteristic
281	C₄	C₂×#473	1	1	yes	characteristic
282	C₄	#3415	1	1	yes	characteristic
283	C₄	#3417	1	1	yes	characteristic
284	C₂	—	4	1	no	—
285	C₂	—	8	1	no	—
286	C₂	#19271	1	1	yes	characteristic
287	C₂	#33610	1	1	yes	characteristic
288	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. #33726	C₂	semidirect
4. #12462	C₂	semidirect
5. #12458	C₂	semidirect
6. #12457	C₂	semidirect
7. #12456	C₂	semidirect
8. #12452	C₂	semidirect
9. #12451	C₂	semidirect
10. #12022	C₂	semidirect
11. #11976	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33726	2
#33480	2
#19361	2
#12462	2
#12458	2
#12457	2
#12456	2
#12452	2
#12451	2
#12022	2
#11976	2