Subgroups — #490651

Order 512.

429 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33733	C₂	1	1	yes	maximal, characteristic, complemented
2	#33480	C₂	1	1	yes	maximal, characteristic, complemented
3	#19222	C₂	1	2	yes	maximal, complemented
4	#12480	C₂	1	2	yes	maximal, complemented
5	#12472	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	#12439	C₂	1	1	yes	maximal, characteristic, complemented
9	#12430	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
11	#11952	C₂	1	2	yes	maximal, complemented
12	#3247	—	2	2	no	—
13	#3169	C₂²	1	2	yes	—
14	#2966	C₂²	1	1	yes	characteristic
15	#2252	C₂²	1	1	yes	characteristic
16	#2246	C₂²	1	1	yes	characteristic
17	#2240	C₂²	1	2	yes	—
18	#2215	—	2	1	no	—
19	#2215	—	2	1	no	—
20	#2183	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×D₁₆	—	2	1	no	—
23	C₄×D₁₆	—	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	#1698	—	2	2	no	—
33	#1697	—	2	2	no	—
34	#1676	C₂²	1	2	yes	—
35	#1675	C₂²	1	2	yes	—
36	#1668	C₂²	1	2	yes	—
37	#1667	C₂²	1	2	yes	—
38	C₂×#335	C₂²	1	2	yes	—
39	#1551	—	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	#1361	C₂²	1	2	yes	—
46	#1361	C₂²	1	2	yes	—
47	#532	—	2	2	no	—
48	#529	D₄	1	1	yes	characteristic
49	C₂×D₁₆	—	2	1	no	—
50	C₂×D₁₆	—	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	#493	C₂³	1	2	yes	—
57	#486	—	2	2	no	—
58	#478	—	4	2	no	—
59	#477	—	2	2	no	—
60	#476	—	2	2	no	—
61	#466	—	2	2	no	—
62	#466	—	2	2	no	—
63	#466	—	4	2	no	—
64	#465	—	4	2	no	—
65	#463	—	4	2	no	—
66	#460	—	2	2	no	—
67	#459	—	2	2	no	—
68	#446	D₄	1	1	yes	characteristic
69	C₈×Q₈	C₂³	1	1	yes	characteristic
70	C₄×D₈	—	2	2	no	—
71	C₄×D₈	—	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	C₂×C₄⋊C₈	—	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	#357	—	2	2	no	—
88	#357	—	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	—	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	#331	—	2	2	no	—
105	#331	—	2	2	no	—
106	C₂×#41	—	4	2	no	—
107	C₂×SD₁₆	—	4	2	no	—
108	C₂×D₈	—	2	2	no	—
109	C₂×D₈	—	2	2	no	—
110	C₂×D₈	—	4	2	no	—
111	C₂×M₁₆	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	4	2	no	—
114	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	2	no	—
117	#127	—	2	2	no	—
118	#126	—	2	1	no	—
119	#125	D₈	1	1	yes	characteristic
120	#124	—	4	2	no	—
121	#124	D₈	1	1	yes	characteristic
122	#123	—	2	1	no	—
123	C₂²⋊Q₈	D₈	1	1	yes	characteristic
124	#121	—	4	2	no	—
125	#121	—	4	2	no	—
126	#121	D₈	1	1	yes	characteristic
127	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
128	C₄×D₄	—	2	1	no	—
129	C₄×D₄	—	4	2	no	—
130	C₄×D₄	—	4	2	no	—
131	C₄×D₄	C₂×D₄	1	1	yes	characteristic
132	#117	—	2	1	no	—
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	D₁₆	—	4	2	no	—
135	D₁₆	—	4	2	no	—
136	M₃₂	—	4	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	4	1	no	—
141	C₂×C₁₆	—	4	2	no	—
142	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
143	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
144	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
145	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
146	#107	—	2	2	no	—
147	#107	—	2	2	no	—
148	#107	—	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	#102	—	4	2	no	—
159	#102	—	4	2	no	—
160	#102	—	4	2	no	—
161	#102	—	4	2	no	—
162	#102	—	4	2	no	—
163	C₂²⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	#97	—	2	1	no	—
167	#97	—	2	1	no	—
168	C₄×C₈	—	2	1	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	#41	—	16	4	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	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	8	2	no	—
181	C₂×D₄	C₂×D₈	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	8	2	no	—
184	C₂²×C₄	#121	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
186	C₂²×C₄	#135	1	1	yes	characteristic
187	SD₁₆	—	16	2	no	—
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	D₈	—	8	2	no	—
191	D₈	—	16	2	no	—
192	M₁₆	—	4	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	16	2	no	—
203	C₂×C₈	#121	1	1	yes	characteristic
204	C₂×C₈	#121	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₈	C₂×#41	1	1	yes	characteristic
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
214	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	8	2	no	—
221	C₂²⋊C₄	—	8	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	#121	1	1	yes	characteristic
224	C₄²	C₂×D₈	1	1	yes	characteristic
225	C₄²	#135	1	1	yes	characteristic
226	C₄²	C₂²×D₄	1	1	yes	characteristic
227	C₁₆	—	2	1	no	—
228	C₁₆	—	2	1	no	—
229	C₁₆	—	2	1	no	—
230	C₁₆	—	4	1	no	—
231	C₁₆	—	4	1	no	—
232	C₁₆	#135	1	2	yes	—
233	C₂³	—	2	1	no	—
234	C₂³	—	8	2	no	—
235	C₂³	#466	1	1	yes	characteristic
236	Q₈	—	2	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	16	2	no	—
239	D₄	—	2	1	no	—
240	D₄	—	2	1	no	—
241	D₄	—	4	2	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	D₄	—	16	2	no	—
247	D₄	—	16	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	16	4	no	—
260	C₂×C₄	#466	1	1	yes	characteristic
261	C₂×C₄	#466	1	1	yes	characteristic
262	C₂×C₄	#466	1	1	yes	characteristic
263	C₂×C₄	C₂×#121	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂×#135	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	8	2	no	—
272	C₈	—	8	2	no	—
273	C₈	C₂×#135	1	1	yes	characteristic
274	C₈	#575	1	1	yes	characteristic
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	16	2	no	—
281	C₂²	—	16	2	no	—
282	C₂²	—	16	2	no	—
283	C₂²	#2221	1	2	yes	—
284	C₂²	C₂×#466	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	16	2	no	—
293	C₄	#3035	1	1	yes	characteristic
294	C₄	#3067	1	1	yes	characteristic
295	C₄	#3415	1	1	yes	characteristic
296	C₄	#3417	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	1	no	—
299	C₂	—	32	2	no	—
300	C₂	#19125	1	1	yes	characteristic
301	C₂	#33610	1	1	yes	characteristic
302	C₂	C₂×#2221	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
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. #33733	C₂	semidirect
2. #33480	C₂	semidirect
3. #19222	C₂	semidirect
4. #12480	C₂	semidirect
5. #12472	C₂	semidirect
6. #12458	C₂	semidirect
7. #12452	C₂	semidirect
8. #12439	C₂	semidirect
9. #12430	C₂	semidirect
10. C₂×#1371	C₂	semidirect
11. #11952	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33733	2
#33480	2
#19222	2
#12480	2
#12472	2
#12458	2
#12452	2
#12439	2
#12430	2
C₂×#1371	2
#11952	2