Subgroups — #99295

Order 512.

412 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#8805	C₂	1	6	yes	maximal
2	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
3	#1842	C₂²	1	4	yes	—
4	C₄²×C₈	C₂²	1	3	yes	—
5	C₂×#334	—	2	12	no	—
6	C₂×C₈²	C₄	1	3	yes	—
7	C₂×C₈²	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₈	—	2	12	no	—
9	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
10	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
11	C₂×C₄×C₈	D₄	1	6	yes	—
12	C₂×C₄×C₈	Q₈	1	6	yes	—
13	C₄³	C₂³	1	1	yes	characteristic
14	#334	—	4	24	no	—
15	C₈²	—	2	6	no	—
16	C₈²	C₈	1	4	yes	complemented
17	C₂²×C₈	—	4	4	no	—
18	C₂²×C₈	C₄⋊C₄	1	6	yes	—
19	C₂²×C₈	C₄⋊C₄	1	6	yes	—
20	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄²	C₂×D₄	1	3	yes	—
22	C₂×C₄²	C₂×Q₈	1	3	yes	—
23	C₄⋊C₈	—	4	24	no	—
24	C₄×C₈	—	2	12	no	—
25	C₄×C₈	—	2	6	no	—
26	C₄×C₈	C₂×C₈	1	6	yes	—
27	C₄×C₈	M₁₆	1	6	yes	—
28	C₄×C₈	SD₁₆	1	24	yes	—
29	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
30	C₂²×C₄	#127	1	1	yes	characteristic
31	C₂²×C₄	C₄⋊Q₈	1	3	yes	—
32	C₂×C₈	—	2	6	no	—
33	C₂×C₈	—	2	6	no	—
34	C₂×C₈	—	8	12	no	—
35	C₂×C₈	C₄⋊C₈	1	12	yes	—
36	C₂×C₈	#106	1	12	yes	—
37	C₂×C₈	#106	1	12	yes	—
38	C₂×C₈	#108	1	12	yes	—
39	C₄²	—	2	6	no	—
40	C₄²	C₂²×C₈	1	1	yes	characteristic
41	C₄²	C₂×M₁₆	1	3	yes	—
42	C₄²	C₂×SD₁₆	1	12	yes	—
43	C₂³	#384	1	1	yes	characteristic
44	C₂×C₄	—	2	6	no	—
45	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
46	C₂×C₄	#423	1	3	yes	—
47	C₂×C₄	C₂×#106	1	6	yes	—
48	C₂×C₄	C₂×#108	1	6	yes	—
49	C₂×C₄	#492	1	6	yes	—
50	C₂×C₄	#498	1	6	yes	—
51	C₈	—	2	12	no	—
52	C₈	—	16	4	no	—
53	C₈	#334	1	24	yes	complemented
54	C₂²	#1842	1	1	yes	characteristic
55	C₂²	#1844	1	3	yes	—
56	C₂²	#1848	1	3	yes	—
57	C₄	—	2	6	no	—
58	C₄	C₂×#334	1	6	yes	—
59	C₄	#1566	1	6	yes	—
60	C₄	#1705	1	4	yes	—
61	C₂	#8805	1	3	yes	—
62	C₂	#8813	1	1	yes	characteristic
63	C₂	#8822	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. C₈²	C₈	semidirect
53. C₈	#334	semidirect

Maximal subgroups

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

subgroup	index
#8805	2
C₄×C₈²	2