Subgroups — #93074

Order 512.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12013	C₂	1	2	yes	maximal
2	#8815	C₂	1	1	yes	maximal, characteristic
3	#2163	—	2	2	no	—
4	#1844	C₄	1	2	yes	—
5	#1834	—	2	2	no	—
6	C₂×C₈²	C₂²	1	1	yes	characteristic
7	#1370	—	2	2	no	—
8	#1367	C₄	1	4	yes	complemented
9	C₂×#106	—	4	2	no	—
10	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	D₄	1	2	yes	—
12	#384	—	2	2	no	—
13	C₄⋊C₁₆	—	2	4	no	—
14	C₂²⋊C₁₆	—	4	2	no	—
15	#346	—	4	2	no	—
16	#340	—	4	2	no	—
17	#340	—	4	2	no	—
18	C₈²	C₂×C₄	1	2	yes	—
19	C₈²	D₄	1	1	yes	characteristic
20	C₈²	Q₈	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	D₈	1	2	yes	—
23	C₂²×C₈	SD₁₆	1	2	yes	—
24	C₂×C₄⋊C₄	—	4	2	no	—
25	C₂×C₄⋊C₄	—	4	4	no	—
26	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₁₆	—	4	4	no	—
28	#106	—	8	4	no	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	C₄²	1	1	yes	characteristic
31	C₄×C₈	C₂²⋊C₄	1	2	yes	—
32	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₈	C₄⋊C₄	1	2	yes	—
34	C₄×C₈	SD₁₆	1	2	yes	—
35	C₄×C₈	Q₁₆	1	2	yes	—
36	C₂²×C₄	—	8	2	no	—
37	C₂²×C₄	#100	1	1	yes	characteristic
38	C₂²×C₄	#102	1	2	yes	—
39	C₂×C₈	—	2	4	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	#102	1	2	yes	—
45	C₂×C₈	#103	1	2	yes	—
46	C₂×C₈	#106	1	2	yes	—
47	C₂×C₈	#107	1	2	yes	—
48	C₄⋊C₄	—	8	4	no	—
49	C₄⋊C₄	—	8	4	no	—
50	C₄⋊C₄	—	8	4	no	—
51	C₄²	#95	1	1	yes	characteristic
52	C₄²	#101	1	1	yes	characteristic
53	C₄²	#103	1	2	yes	—
54	C₁₆	—	8	4	no	—
55	C₂³	#331	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	16	4	no	—
60	C₂×C₄	—	16	2	no	—
61	C₂×C₄	#332	1	2	yes	—
62	C₂×C₄	#333	1	1	yes	characteristic
63	C₂×C₄	#340	1	2	yes	—
64	C₂×C₄	#343	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₈	—	2	2	no	—
67	C₈	—	2	2	no	—
68	C₈	—	2	2	no	—
69	C₈	—	2	2	no	—
70	C₈	—	2	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	#1296	1	1	yes	characteristic
73	C₂²	#1359	1	2	yes	—
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	2	no	—
77	C₄	—	32	2	no	—
78	C₄	#1384	1	2	yes	—
79	C₄	#1389	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	#7268	1	1	yes	characteristic
82	C₂	#7279	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
8. #1367	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12013	2
#8815	2