Subgroups — #93862

Order 512.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12503	C₂	1	1	yes	maximal, characteristic
2	#7455	C₂	1	1	yes	maximal, characteristic, complemented
3	#7100	C₂	1	1	yes	maximal, characteristic, complemented
4	#2252	C₄	1	1	yes	characteristic
5	#2246	C₄	1	1	yes	characteristic
6	#2240	—	2	1	no	—
7	#1706	—	2	1	no	—
8	C₄⋊C₃₂	—	2	1	no	—
9	#1370	—	2	1	no	—
10	C₈×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×D₁₆	—	4	1	no	—
12	#500	—	2	1	no	—
13	#493	—	2	1	no	—
14	#492	—	4	1	no	—
15	#486	—	4	1	no	—
16	C₂×C₃₂	—	4	1	no	—
17	#366	—	4	1	no	—
18	C₄⋊C₁₆	—	4	1	no	—
19	#357	—	4	1	no	—
20	#357	—	4	1	no	—
21	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₄×C₁₆	D₄	1	1	yes	characteristic
23	C₈²	D₄	1	1	yes	characteristic
24	C₂×SD₁₆	—	8	1	no	—
25	C₂×SD₁₆	—	8	1	no	—
26	C₂×D₈	—	4	1	no	—
27	C₂×D₈	—	4	1	no	—
28	C₄⋊Q₈	—	4	1	no	—
29	#127	—	4	1	no	—
30	D₁₆	—	8	2	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	8	1	no	—
33	C₂×C₁₆	D₈	1	1	yes	characteristic
34	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
35	#107	—	4	1	no	—
36	#107	—	4	1	no	—
37	#102	—	8	2	no	—
38	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×C₈	D₈	1	1	yes	characteristic
40	C₄×C₈	SD₁₆	1	1	yes	characteristic
41	C₃₂	—	8	1	no	—
42	C₂×Q₈	—	8	1	no	—
43	C₂×D₄	—	8	1	no	—
44	C₂×D₄	—	8	1	no	—
45	C₂×D₄	—	8	1	no	—
46	SD₁₆	—	16	2	no	—
47	SD₁₆	—	16	1	no	—
48	SD₁₆	—	16	1	no	—
49	D₈	—	8	2	no	—
50	D₈	—	8	1	no	—
51	D₈	—	8	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	#100	1	1	yes	characteristic
54	C₂×C₈	#102	1	1	yes	characteristic
55	C₂×C₈	D₁₆	1	1	yes	characteristic
56	C₂×C₈	SD₃₂	1	1	yes	characteristic
57	C₄⋊C₄	—	8	1	no	—
58	C₄⋊C₄	—	8	1	no	—
59	C₄²	#102	1	1	yes	characteristic
60	C₁₆	—	2	2	no	—
61	C₁₆	—	2	1	no	—
62	C₁₆	—	2	1	no	—
63	C₁₆	—	16	1	no	—
64	C₂³	—	16	1	no	—
65	Q₈	—	16	1	no	—
66	Q₈	—	16	1	no	—
67	D₄	—	16	2	no	—
68	D₄	—	16	1	no	—
69	D₄	—	16	1	no	—
70	D₄	—	16	1	no	—
71	D₄	—	16	1	no	—
72	C₂×C₄	—	16	1	no	—
73	C₂×C₄	#331	1	1	yes	characteristic
74	C₂×C₄	#357	1	1	yes	characteristic
75	C₂×C₄	#361	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	4	1	no	—
79	C₈	#359	1	1	yes	characteristic
80	C₈	#360	1	1	yes	characteristic
81	C₈	D₃₂	1	1	yes	characteristic
82	C₈	SD₆₄	1	1	yes	characteristic
83	C₂²	—	32	1	no	—
84	C₂²	—	32	1	no	—
85	C₂²	—	32	1	no	—
86	C₂²	#1361	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	32	1	no	—
89	C₄	#1359	1	1	yes	characteristic
90	C₄	#1360	1	1	yes	characteristic
91	C₄	#1415	1	1	yes	characteristic
92	C₄	#1419	1	1	yes	characteristic
93	C₂	—	64	1	no	—
94	C₂	#7094	1	1	yes	characteristic
95	C₂	#7444	1	1	yes	characteristic
96	C₂	#7447	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1361
commutator	C₄×C₁₆	C₂×C₄
Frattini, cosocle	C₈×C₁₆	C₂²
prime radical	#2246	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7455	C₂	semidirect
3. #7100	C₂	semidirect

Maximal subgroups

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

subgroup	index
#12503	2
#7455	2
#7100	2