Subgroups — #513292

Order 512.

378 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33958	C₂	1	2	yes	maximal, complemented
2	#13725	C₂	1	3	yes	maximal, complemented
3	#13685	C₂	1	6	yes	maximal, complemented
4	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₃₂	C₂	1	3	yes	maximal, complemented
6	#3391	—	2	2	no	—
7	C₂×D₃₂	C₂²	1	3	yes	—
8	C₂×D₃₂	C₂²	1	6	yes	—
9	#2246	C₂²	1	6	yes	—
10	#2206	—	2	6	no	—
11	Q₈×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×D₁₆	C₂²	1	6	yes	—
13	#1423	C₂²	1	1	yes	characteristic
14	C₄⋊C₃₂	C₂²	1	3	yes	—
15	#1415	C₂²	1	6	yes	—
16	C₄×C₃₂	C₂²	1	3	yes	—
17	#550	—	4	2	no	—
18	C₂×D₁₆	—	2	6	no	—
19	C₂×D₁₆	C₂³	1	6	yes	—
20	#493	—	2	6	no	—
21	#459	—	4	6	no	—
22	C₈×Q₈	D₄	1	1	yes	characteristic
23	C₄×D₈	—	2	6	no	—
24	D₃₂	—	2	6	no	—
25	D₃₂	—	4	6	no	—
26	C₂×C₃₂	C₂³	1	3	yes	—
27	C₂×C₃₂	C₂³	1	1	yes	characteristic
28	#366	C₂³	1	2	yes	—
29	C₄⋊C₁₆	D₄	1	3	yes	—
30	#357	—	2	6	no	—
31	C₄×C₁₆	C₂³	1	3	yes	—
32	C₂×#41	—	8	2	no	—
33	C₂×D₈	—	2	6	no	—
34	C₂×D₈	—	4	6	no	—
35	#127	—	4	6	no	—
36	#121	—	8	6	no	—
37	C₄×Q₈	D₈	1	1	yes	characteristic
38	C₄×D₄	—	4	6	no	—
39	D₁₆	—	2	6	no	—
40	D₁₆	—	2	6	no	—
41	D₁₆	—	8	6	no	—
42	C₂×C₁₆	C₂×D₄	1	3	yes	—
43	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
44	#107	—	2	2	no	—
45	C₄⋊C₈	D₈	1	3	yes	—
46	#102	—	4	6	no	—
47	C₄×C₈	C₂×D₄	1	3	yes	—
48	C₃₂	—	2	3	no	—
49	C₃₂	#41	1	2	yes	complemented
50	#41	—	16	8	no	—
51	C₂×Q₈	D₁₆	1	1	yes	characteristic
52	C₂×D₄	—	4	6	no	—
53	C₂×D₄	—	8	6	no	—
54	C₂×D₄	—	8	6	no	—
55	C₂²×C₄	—	8	6	no	—
56	D₈	—	4	6	no	—
57	D₈	—	4	6	no	—
58	D₈	—	16	6	no	—
59	C₂×C₈	C₂×D₈	1	3	yes	—
60	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	D₁₆	1	3	yes	—
63	C₂²⋊C₄	—	8	6	no	—
64	C₄²	C₂×D₈	1	3	yes	—
65	C₁₆	—	2	3	no	—
66	C₁₆	C₂×#41	1	1	yes	characteristic
67	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
68	C₂³	—	8	6	no	—
69	Q₈	D₃₂	1	4	yes	complemented
70	D₄	—	8	6	no	—
71	D₄	—	8	6	no	—
72	D₄	—	16	12	no	—
73	D₄	—	32	6	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	16	12	no	—
76	C₂×C₄	C₂×D₁₆	1	3	yes	—
77	C₂×C₄	C₂×D₁₆	1	3	yes	—
78	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
79	C₈	—	2	3	no	—
80	C₈	#550	1	1	yes	characteristic
81	C₈	#576	1	1	yes	characteristic
82	C₂²	—	16	6	no	—
83	C₂²	—	16	6	no	—
84	C₂²	—	16	6	no	—
85	C₂²	C₂²×D₁₆	1	1	yes	characteristic
86	C₄	—	2	3	no	—
87	C₄	—	16	2	no	—
88	C₄	C₂×D₃₂	1	6	yes	—
89	C₄	#3391	1	1	yes	characteristic
90	C₄	#3415	1	1	yes	characteristic
91	C₂	—	32	6	no	—
92	C₂	#33958	1	1	yes	characteristic
93	C₂	C₂²×D₃₂	1	1	yes	characteristic
94	C₂	#33982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33958	C₂	semidirect
2. #13725	C₂	semidirect
3. #13685	C₂	semidirect
4. Q₈×C₃₂	C₂	semidirect
5. C₄×D₃₂	C₂	semidirect
49. C₃₂	#41	semidirect
69. Q₈	D₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33958	2
#13725	2
#13685	2
Q₈×C₃₂	2
C₄×D₃₂	2