Subgroups — #93758

Order 512.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12044	C₂	1	1	yes	maximal, characteristic
2	#7332	C₂	1	1	yes	maximal, characteristic
3	#7099	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₃₂	C₄	1	1	yes	characteristic
5	C₈×Q₁₆	—	2	1	no	—
6	#1397	—	2	1	no	—
7	#1372	C₄	1	1	yes	characteristic
8	#1367	—	2	1	no	—
9	#1332	—	2	1	no	—
10	#1311	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂	C₈	1	1	yes	characteristic
12	C₈×Q₈	—	4	1	no	—
13	C₄×Q₁₆	—	2	1	no	—
14	C₂×C₃₂	—	2	2	no	—
15	#366	C₈	1	1	yes	characteristic
16	C₄⋊C₁₆	—	4	1	no	—
17	#358	—	2	1	no	—
18	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₄×C₁₆	D₄	1	1	yes	characteristic
20	#335	—	2	1	no	—
21	Q₈⋊C₈	—	4	1	no	—
22	C₈²	D₄	1	1	yes	characteristic
23	C₂×Q₁₆	—	2	1	no	—
24	C₄×Q₈	—	4	1	no	—
25	Q₃₂	—	2	1	no	—
26	Q₃₂	C₁₆	1	2	yes	complemented
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	—	8	1	no	—
29	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
30	C₂×C₁₆	M₁₆	1	1	yes	characteristic
31	#107	—	2	1	no	—
32	C₄⋊C₈	—	4	1	no	—
33	C₄⋊C₈	—	8	1	no	—
34	#103	—	4	1	no	—
35	C₄×C₈	—	8	1	no	—
36	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₈	D₈	1	1	yes	characteristic
38	C₄×C₈	SD₁₆	1	1	yes	characteristic
39	C₃₂	—	4	2	no	—
40	C₂×Q₈	—	4	1	no	—
41	Q₁₆	—	2	2	no	—
42	Q₁₆	—	4	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	8	1	no	—
46	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₈	#104	1	1	yes	characteristic
48	C₄⋊C₄	—	4	1	no	—
49	C₄⋊C₄	—	8	1	no	—
50	C₄²	—	8	1	no	—
51	C₄²	#102	1	1	yes	characteristic
52	C₁₆	—	2	2	no	—
53	C₁₆	—	2	1	no	—
54	C₁₆	—	16	1	no	—
55	C₁₆	C₂×C₁₆	1	1	yes	characteristic
56	C₁₆	M₃₂	1	1	yes	characteristic
57	Q₈	—	4	2	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄	#360	1	1	yes	characteristic
63	C₂×C₄	#361	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	4	1	no	—
66	C₈	—	4	1	no	—
67	C₈	—	16	1	no	—
68	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
69	C₈	#350	1	1	yes	characteristic
70	C₂²	#1333	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	1	no	—
73	C₄	—	8	2	no	—
74	C₄	—	16	1	no	—
75	C₄	D₄⋊C₁₆	1	1	yes	characteristic
76	C₄	#1336	1	1	yes	characteristic
77	C₂	#7070	1	1	yes	characteristic
78	C₂	#7389	1	1	yes	characteristic
79	C₂	#7391	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#1333
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	#1311	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
26. Q₃₂	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#12044	2
#7332	2
#7099	2