Subgroups — #123413

Order 512.

237 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12521	C₂	1	2	yes	maximal, complemented
2	#12517	C₂	1	1	yes	maximal, characteristic
3	#12516	C₂	1	1	yes	maximal, characteristic, complemented
4	#12509	C₂	1	2	yes	maximal
5	#7052	C₂	1	1	yes	maximal, characteristic, complemented
6	#2252	—	2	4	no	—
7	#2248	—	2	2	no	—
8	#2247	—	2	2	no	—
9	#2240	—	2	4	no	—
10	#1713	C₂²	1	2	yes	—
11	#1713	C₂²	1	1	yes	characteristic
12	#1712	C₂²	1	1	yes	characteristic
13	C₈×C₁₆	C₂²	1	2	yes	—
14	C₈×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂	—	4	4	no	—
16	C₂×SD₃₂	—	4	4	no	—
17	#500	—	2	2	no	—
18	#500	—	2	2	no	—
19	#500	—	2	2	no	—
20	#494	—	2	2	no	—
21	#494	—	2	1	no	—
22	#493	—	2	2	no	—
23	#493	—	2	1	no	—
24	#366	—	4	8	no	—
25	#357	—	4	8	no	—
26	C₄×C₁₆	D₄	1	4	yes	—
27	C₄×C₁₆	D₄	1	2	yes	—
28	C₈²	C₂³	1	1	yes	characteristic
29	C₂×Q₁₆	—	4	2	no	—
30	C₂×Q₁₆	—	4	4	no	—
31	C₂×D₈	—	4	4	no	—
32	C₂×D₈	—	4	2	no	—
33	C₄⋊Q₈	—	4	2	no	—
34	C₄⋊Q₈	—	4	1	no	—
35	#127	—	4	1	no	—
36	Q₃₂	—	8	8	no	—
37	SD₃₂	—	8	8	no	—
38	C₂×C₁₆	D₈	1	8	yes	—
39	C₂×C₁₆	D₈	1	4	yes	—
40	#107	—	4	4	no	—
41	#107	—	4	4	no	—
42	#107	—	4	4	no	—
43	C₄×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄×C₈	C₂×D₄	1	2	yes	—
45	C₂×Q₈	—	8	1	no	—
46	C₂×Q₈	—	8	2	no	—
47	C₂×D₄	—	8	1	no	—
48	C₂×D₄	—	8	2	no	—
49	Q₁₆	—	8	4	no	—
50	Q₁₆	—	8	8	no	—
51	D₈	—	8	4	no	—
52	D₈	—	8	8	no	—
53	C₂×C₈	C₂×D₈	1	2	yes	—
54	C₂×C₈	C₂×D₈	1	4	yes	—
55	C₄⋊C₄	—	8	2	no	—
56	C₄⋊C₄	—	8	2	no	—
57	C₄⋊C₄	—	8	2	no	—
58	C₄²	#127	1	1	yes	characteristic
59	C₁₆	—	2	8	no	—
60	C₁₆	D₁₆	1	8	yes	—
61	C₂³	—	16	1	no	—
62	Q₈	—	16	2	no	—
63	Q₈	—	16	4	no	—
64	D₄	—	16	2	no	—
65	D₄	—	16	4	no	—
66	C₂×C₄	—	16	2	no	—
67	C₂×C₄	—	16	1	no	—
68	C₂×C₄	#493	1	1	yes	characteristic
69	C₂×C₄	#493	1	2	yes	—
70	C₈	C₂×D₁₆	1	4	yes	—
71	C₈	#510	1	8	yes	—
72	C₂²	—	32	1	no	—
73	C₂²	—	32	2	no	—
74	C₂²	#1712	1	1	yes	characteristic
75	C₄	—	32	1	no	—
76	C₄	—	32	2	no	—
77	C₄	#2246	1	2	yes	—
78	C₄	#2251	1	4	yes	—
79	C₂	—	64	1	no	—
80	C₂	#12515	1	1	yes	characteristic
81	C₂	#12520	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#1712	C₂²
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #12521	C₂	semidirect
3. #12516	C₂	semidirect
5. #7052	C₂	semidirect

Maximal subgroups

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

subgroup	index
#12521	2
#12517	2
#12516	2
#12509	2
#7052	2