Subgroups — #353045

Order 512.

762 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33312	C₂	1	2	yes	maximal
2	#33289	C₂	1	4	yes	maximal
3	#8816	C₂	1	8	yes	maximal
4	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	4	no	—
6	#2868	C₂²	1	4	yes	—
7	#1990	—	2	16	no	—
8	#1845	—	2	4	no	—
9	#1845	C₂²	1	8	yes	—
10	#1834	C₂²	1	16	yes	—
11	C₄²×C₈	C₂²	1	2	yes	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₄⋊Q₈	—	2	8	no	—
16	#500	—	4	16	no	—
17	C₂×#107	—	2	16	no	—
18	C₂×C₄×C₈	D₄	1	8	yes	—
19	C₂×C₄×C₈	D₄	1	2	yes	—
20	C₂×C₄×C₈	D₄	1	2	yes	—
21	C₂×C₄×C₈	C₂³	1	4	yes	—
22	C₂×C₄×C₈	C₂³	1	2	yes	—
23	#384	—	2	4	no	—
24	#384	—	2	8	no	—
25	#384	C₂³	1	8	yes	—
26	C₄³	C₂³	1	1	yes	characteristic
27	#340	—	2	32	no	—
28	C₈²	Q₈	1	16	yes	complemented
29	C₂²×Q₈	—	4	4	no	—
30	C₂²×C₈	C₂×D₄	1	8	yes	—
31	C₂²×C₈	C₂×D₄	1	4	yes	—
32	C₄⋊Q₈	—	4	16	no	—
33	C₄⋊Q₈	—	4	32	no	—
34	C₂×C₄⋊C₄	—	2	16	no	—
35	C₂×C₄⋊C₄	—	2	8	no	—
36	C₂×C₄⋊C₄	—	4	4	no	—
37	C₂×C₄⋊C₄	—	4	8	no	—
38	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×D₄	1	4	yes	—
41	C₂×C₄²	C₂⁴	1	1	yes	characteristic
42	#107	—	4	32	no	—
43	C₄×C₈	—	2	16	no	—
44	C₄×C₈	D₈	1	8	yes	—
45	C₄×C₈	Q₁₆	1	8	yes	—
46	C₄×C₈	C₂×Q₈	1	16	yes	—
47	C₄×C₈	C₂×Q₈	1	8	yes	—
48	C₂×Q₈	—	8	8	no	—
49	C₂×Q₈	—	8	16	no	—
50	C₂²×C₄	—	4	8	no	—
51	C₂²×C₄	#127	1	2	yes	—
52	C₂²×C₄	#127	1	2	yes	—
53	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₄	C₂²×D₄	1	2	yes	—
55	C₂×C₈	—	2	16	no	—
56	C₂×C₈	C₄⋊Q₈	1	16	yes	—
57	C₂×C₈	C₄⋊Q₈	1	8	yes	—
58	C₂×C₈	C₂×D₈	1	8	yes	—
59	C₂×C₈	C₂×Q₁₆	1	8	yes	—
60	C₄⋊C₄	—	4	16	no	—
61	C₄⋊C₄	—	4	32	no	—
62	C₄⋊C₄	—	8	8	no	—
63	C₄⋊C₄	—	8	16	no	—
64	C₄²	C₂×D₈	1	4	yes	—
65	C₄²	C₂×Q₁₆	1	4	yes	—
66	C₄²	#135	1	16	yes	—
67	C₄²	C₂²×Q₈	1	4	yes	—
68	C₂³	C₂×#127	1	1	yes	characteristic
69	Q₈	—	16	16	no	—
70	C₂×C₄	—	8	8	no	—
71	C₂×C₄	—	8	16	no	—
72	C₂×C₄	#493	1	4	yes	—
73	C₂×C₄	#494	1	4	yes	—
74	C₂×C₄	#495	1	8	yes	—
75	C₂×C₄	#495	1	8	yes	—
76	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
77	C₂×C₄	C₂×C₄⋊Q₈	1	4	yes	—
78	C₂×C₄	C₂²×D₈	1	2	yes	—
79	C₂×C₄	C₂²×Q₁₆	1	2	yes	—
80	C₂×C₄	C₂×#135	1	8	yes	—
81	C₈	—	2	16	no	—
82	C₈	#500	1	16	yes	—
83	C₂²	#2868	1	1	yes	characteristic
84	C₂²	C₂×#493	1	1	yes	characteristic
85	C₂²	C₂×#494	1	1	yes	characteristic
86	C₂²	C₂×#495	1	4	yes	—
87	C₄	—	16	8	no	—
88	C₄	#1709	1	8	yes	—
89	C₄	#1710	1	8	yes	—
90	C₄	C₂×#500	1	4	yes	—
91	C₄	#3160	1	8	yes	—
92	C₂	C₂×#1709	1	2	yes	—
93	C₂	C₂×#1710	1	2	yes	—
94	C₂	#33312	1	1	yes	characteristic
95	C₂	#33313	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
28. C₈²	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33312	2
#33289	2
#8816	2
C₄×C₈²	2