Subgroups — #483725

Order 512.

419 conjugacy classes of subgroups.

The subgroup lattice has 281 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33499	C₂	1	1	yes	maximal, characteristic, complemented
2	#33489	C₂	1	1	yes	maximal, characteristic
3	#17902	C₂	1	1	yes	maximal, characteristic
4	#12058	C₂	1	2	yes	maximal, complemented
5	#12056	C₂	1	2	yes	maximal, complemented
6	#12051	C₂	1	2	yes	maximal, complemented
7	#12050	C₂	1	2	yes	maximal, complemented
8	#12014	C₂	1	1	yes	maximal, characteristic, complemented
9	#11638	C₂	1	1	yes	maximal, characteristic, complemented
10	#11631	C₂	1	1	yes	maximal, characteristic
11	#11412	C₂	1	1	yes	maximal, characteristic, complemented
12	#3101	C₄	1	1	yes	characteristic
13	#2960	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₄	C₂²	1	1	yes	characteristic
15	#2183	—	2	2	no	—
16	#2183	C₂²	1	2	yes	complemented
17	Q₈×C₁₆	C₂²	1	2	yes	complemented
18	#2169	—	2	2	no	—
19	#2169	C₂²	1	2	yes	—
20	#2169	C₂²	1	2	yes	—
21	#2164	—	2	2	no	—
22	#2163	C₂²	1	1	yes	characteristic
23	#2162	C₂²	1	1	yes	characteristic
24	#2162	C₂²	1	1	yes	characteristic
25	#2162	C₂²	1	1	yes	characteristic
26	#2112	—	2	2	no	—
27	#1582	C₄	1	2	yes	—
28	#1582	C₄	1	2	yes	—
29	C₈×SD₁₆	C₂²	1	2	yes	—
30	C₈×SD₁₆	C₂²	1	2	yes	—
31	#1570	C₄	1	1	yes	characteristic
32	#1486	C₄	1	1	yes	characteristic
33	#1481	C₄	1	1	yes	characteristic
34	#1450	C₂²	1	1	yes	characteristic
35	#1370	C₂²	1	2	yes	complemented
36	#1367	C₂²	1	2	yes	complemented
37	Q₈⋊C₁₆	C₂²	1	2	yes	complemented
38	Q₈⋊C₁₆	C₂²	1	2	yes	complemented
39	D₄⋊C₁₆	C₂²	1	2	yes	—
40	D₄⋊C₁₆	C₂²	1	2	yes	—
41	#1312	C₂²	1	1	yes	characteristic, complemented
42	#1312	C₂²	1	1	yes	characteristic, complemented
43	#1311	C₂²	1	1	yes	characteristic, complemented
44	C₈×C₁₆	C₂²	1	1	yes	characteristic, complemented
45	C₂³×C₈	—	2	1	no	—
46	#533	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₄	C₂×C₄	1	1	yes	characteristic
48	C₂×M₃₂	—	2	2	no	—
49	C₂×M₃₂	—	2	2	no	—
50	#483	C₈	1	2	yes	—
51	#476	C₈	1	2	yes	—
52	#461	C₈	1	2	yes	—
53	#450	C₈	1	2	yes	—
54	#446	—	2	1	no	—
55	#446	C₂×C₄	1	2	yes	—
56	C₈×Q₈	—	2	1	no	—
57	C₈×Q₈	C₂³	1	2	yes	—
58	C₄×SD₁₆	C₂×C₄	1	2	yes	—
59	C₄×SD₁₆	C₂×C₄	1	2	yes	—
60	C₈×D₄	—	2	1	no	—
61	C₈×D₄	—	2	1	no	—
62	C₈×D₄	C₂×C₄	1	2	yes	—
63	C₈×D₄	C₂³	1	2	yes	—
64	#431	C₂×C₄	1	1	yes	characteristic
65	#431	C₂×C₄	1	1	yes	characteristic
66	#424	—	2	1	no	—
67	#423	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₈	—	2	1	no	—
70	#405	—	2	1	no	—
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	—	2	1	no	—
75	C₄⋊C₁₆	D₄	1	2	yes	—
76	C₄⋊C₁₆	C₂³	1	2	yes	—
77	C₄⋊C₁₆	C₂³	1	2	yes	—
78	C₂²⋊C₁₆	—	2	1	no	—
79	C₂²⋊C₁₆	—	2	1	no	—
80	C₂²⋊C₁₆	—	2	1	no	—
81	C₂²⋊C₁₆	—	2	2	no	—
82	C₂²⋊C₁₆	—	2	2	no	—
83	C₂²⋊C₁₆	D₄	1	2	yes	—
84	#346	—	2	2	no	—
85	#346	—	2	1	no	—
86	C₄×C₁₆	—	2	2	no	—
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂³	1	1	yes	characteristic
89	C₄×C₁₆	C₂³	1	1	yes	characteristic
90	C₄×C₁₆	C₂³	1	1	yes	characteristic
91	C₄×C₁₆	C₂³	1	1	yes	characteristic
92	#334	C₂×C₄	1	2	yes	—
93	#334	C₂×C₄	1	2	yes	—
94	Q₈⋊C₈	C₂×C₄	1	2	yes	—
95	Q₈⋊C₈	C₂×C₄	1	2	yes	—
96	D₄⋊C₈	C₂×C₄	1	2	yes	—
97	D₄⋊C₈	C₂×C₄	1	2	yes	—
98	#322	C₂×C₄	1	1	yes	characteristic
99	#322	C₂×C₄	1	1	yes	characteristic
100	C₈²	C₂³	1	1	yes	characteristic
101	C₈²	C₂³	1	1	yes	characteristic
102	C₂²×D₄	C₂×C₈	1	1	yes	characteristic
103	C₂³×C₄	—	2	1	no	—
104	C₂×SD₁₆	C₂×C₈	1	2	yes	—
105	C₂×SD₁₆	C₂×C₈	1	2	yes	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	8	2	no	—
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
114	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
115	#125	—	2	1	no	—
116	C₂²⋊Q₈	—	2	1	no	—
117	C₂²⋊Q₈	C₂×C₈	1	2	yes	—
118	C₄×Q₈	—	2	1	no	—
119	C₄×Q₈	C₂²×C₄	1	2	yes	—
120	C₄×D₄	—	2	1	no	—
121	C₄×D₄	—	2	1	no	—
122	C₄×D₄	C₂×C₈	1	2	yes	—
123	C₄×D₄	C₂²×C₄	1	2	yes	—
124	#117	—	2	1	no	—
125	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	M₃₂	—	8	2	no	—
129	M₃₂	—	8	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	C₂×D₄	1	2	yes	—
135	C₂×C₁₆	C₂×D₄	1	2	yes	—
136	#106	C₂×C₈	1	2	yes	—
137	#106	C₂×C₈	1	2	yes	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	2	1	no	—
143	C₄⋊C₈	C₂×C₈	1	2	yes	—
144	C₄⋊C₈	C₂²×C₄	1	2	yes	—
145	C₄⋊C₈	C₂²×C₄	1	2	yes	—
146	#103	C₂×C₈	1	2	yes	—
147	#103	C₂×C₈	1	2	yes	—
148	#102	C₂×C₈	1	2	yes	—
149	#102	C₂×C₈	1	2	yes	—
150	C₂²⋊C₈	—	2	1	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	C₂×C₈	1	2	yes	—
154	#97	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₄×C₈	C₂×D₄	1	1	yes	characteristic
162	C₄×C₈	#41	1	1	yes	characteristic
163	C₄×C₈	C₂⁴	1	1	yes	characteristic
164	C₂⁴	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	C₂²×C₈	1	2	yes	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	C₂²×C₈	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	8	1	no	—
174	C₂²×C₄	—	8	1	no	—
175	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
177	C₂²×C₄	#131	1	1	yes	characteristic
178	SD₁₆	—	4	2	no	—
179	SD₁₆	—	4	2	no	—
180	M₁₆	—	8	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	8	2	no	—
190	C₂×C₈	—	8	1	no	—
191	C₂×C₈	C₄×D₄	1	1	yes	characteristic
192	C₂×C₈	C₄×D₄	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₈	1	2	yes	—
194	C₂×C₈	C₂²×C₈	1	2	yes	—
195	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₈	C₂×#41	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₂²×C₈	1	2	yes	—
204	C₄⋊C₄	C₂²×C₈	1	2	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	C₄×D₄	1	1	yes	characteristic
210	C₄²	C₂²×C₈	1	1	yes	characteristic
211	C₄²	#131	1	1	yes	characteristic
212	C₄²	C₂³×C₄	1	1	yes	characteristic
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	—	4	1	no	—
216	C₁₆	—	4	1	no	—
217	C₁₆	—	4	2	no	—
218	C₁₆	—	4	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	C₈×D₄	1	1	yes	characteristic
224	Q₈	—	2	2	no	—
225	Q₈	—	2	2	no	—
226	Q₈	—	8	1	no	—
227	D₄	—	2	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	4	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	C₈×D₄	1	1	yes	characteristic
242	C₂×C₄	C₈×D₄	1	1	yes	characteristic
243	C₂×C₄	C₈×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
246	C₂×C₄	C₂×#131	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	4	1	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	#577	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	4	no	—
264	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	2	no	—
272	C₄	#2949	1	1	yes	characteristic
273	C₄	#2957	1	1	yes	characteristic
274	C₄	#3407	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	—	8	2	no	—
277	C₂	#17855	1	1	yes	characteristic
278	C₂	#33433	1	1	yes	characteristic
279	C₂	#33442	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ 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. #33499	C₂	semidirect
4. #12058	C₂	semidirect
5. #12056	C₂	semidirect
6. #12051	C₂	semidirect
7. #12050	C₂	semidirect
8. #12014	C₂	semidirect
9. #11638	C₂	semidirect
11. #11412	C₂	semidirect
16. #2183	C₂²	semidirect
17. Q₈×C₁₆	C₂²	semidirect
35. #1370	C₂²	semidirect
36. #1367	C₂²	semidirect
37. Q₈⋊C₁₆	C₂²	semidirect
38. Q₈⋊C₁₆	C₂²	semidirect
41. #1312	C₂²	semidirect
42. #1312	C₂²	semidirect
43. #1311	C₂²	semidirect
44. C₈×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33499	2
#33489	2
#17902	2
#12058	2
#12056	2
#12051	2
#12050	2
#12014	2
#11638	2
#11631	2
#11412	2