Subgroups — #93751

Order 512.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12046	C₂	1	1	yes	maximal, characteristic
2	#7459	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆²	C₂	1	1	yes	maximal, characteristic, complemented
4	#2178	C₄	1	1	yes	characteristic
5	#2170	—	2	1	no	—
6	#1422	—	2	1	no	—
7	#1373	C₄	1	1	yes	characteristic
8	#1335	—	2	1	no	—
9	C₈×C₁₆	—	2	1	no	—
10	C₈×C₁₆	C₂²	1	1	yes	characteristic
11	#508	C₈	1	1	yes	characteristic
12	#504	—	4	1	no	—
13	#443	—	2	1	no	—
14	M₆₄	—	4	1	no	—
15	#368	C₈	1	1	yes	characteristic
16	#364	—	2	1	no	—
17	#359	—	2	1	no	—
18	#350	—	4	1	no	—
19	C₄×C₁₆	—	2	2	no	—
20	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₄×C₁₆	D₄	1	1	yes	characteristic
22	C₈²	D₄	1	1	yes	characteristic
23	#135	—	2	1	no	—
24	#131	—	4	1	no	—
25	Q₃₂	C₁₆	1	1	yes	characteristic, complemented
26	SD₃₂	—	2	1	no	—
27	D₁₆	C₁₆	1	1	yes	characteristic, complemented
28	M₃₂	—	4	1	no	—
29	M₃₂	—	8	1	no	—
30	C₂×C₁₆	—	2	4	no	—
31	C₂×C₁₆	—	8	1	no	—
32	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
33	C₂×C₁₆	M₁₆	1	1	yes	characteristic
34	C₂×C₁₆	D₈	1	1	yes	characteristic
35	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
36	#108	—	2	1	no	—
37	#104	—	4	1	no	—
38	C₄×C₈	—	2	1	no	—
39	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₃₂	—	8	1	no	—
41	#41	—	4	1	no	—
42	Q₁₆	—	2	1	no	—
43	SD₁₆	—	4	1	no	—
44	D₈	—	2	1	no	—
45	M₁₆	—	4	1	no	—
46	M₁₆	—	8	1	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	8	1	no	—
49	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₈	#102	1	1	yes	characteristic
51	C₄²	#104	1	1	yes	characteristic
52	C₁₆	—	2	8	no	—
53	C₁₆	—	2	1	no	—
54	C₁₆	—	2	1	no	—
55	C₁₆	—	8	1	no	—
56	C₁₆	C₂×C₁₆	1	1	yes	characteristic
57	C₁₆	M₃₂	1	1	yes	characteristic
58	C₁₆	D₁₆	1	1	yes	characteristic
59	C₁₆	SD₃₂	1	1	yes	characteristic
60	Q₈	—	4	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	8	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
66	C₈	—	2	4	no	—
67	C₈	—	8	1	no	—
68	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
69	C₈	#350	1	1	yes	characteristic
70	C₈	#357	1	1	yes	characteristic
71	C₈	#359	1	1	yes	characteristic
72	C₂²	—	8	1	no	—
73	C₂²	#1335	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	8	1	no	—
76	C₄	D₄⋊C₁₆	1	1	yes	characteristic
77	C₄	#1331	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	16	1	no	—
80	C₂	#7068	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₁₆
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂	#7068
prime radical	#508	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7459	C₂	semidirect
3. C₁₆²	C₂	semidirect
25. Q₃₂	C₁₆	semidirect
27. D₁₆	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#12046	2
#7459	2
C₁₆²	2