Subgroups — #151128

Order 512.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13653	C₂	1	1	yes	maximal, characteristic
2	#12085	C₂	1	2	yes	maximal
3	#7463	C₂	1	1	yes	maximal, characteristic
4	#7386	C₂	1	2	yes	maximal
5	#7330	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₆₄	C₄	1	1	yes	characteristic
7	C₄×Q₃₂	C₂²	1	2	yes	—
8	#1591	—	2	2	no	—
9	#1424	C₄	1	1	yes	characteristic
10	#1416	C₄	1	2	yes	—
11	#1397	C₂²	1	1	yes	characteristic
12	#1397	C₂²	1	1	yes	characteristic
13	#1372	C₂²	1	2	yes	—
14	#1332	—	2	2	no	—
15	#1311	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂	C₂×C₄	1	2	yes	—
17	#446	—	4	2	no	—
18	C₄×Q₁₆	—	2	2	no	—
19	Q₆₄	—	2	2	no	—
20	C₂×C₃₂	—	2	1	no	—
21	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
23	#366	C₂×C₄	1	2	yes	—
24	#358	—	2	2	no	—
25	C₄×C₁₆	D₄	1	1	yes	characteristic
26	C₄×C₁₆	C₂³	1	1	yes	characteristic
27	#335	—	2	2	no	—
28	Q₈⋊C₈	—	4	2	no	—
29	C₈²	D₄	1	1	yes	characteristic
30	C₂×Q₁₆	—	2	2	no	—
31	C₄×Q₈	—	4	2	no	—
32	Q₃₂	—	2	2	no	—
33	Q₃₂	—	2	2	no	—
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₁₆	#41	1	1	yes	characteristic
37	#107	—	2	2	no	—
38	C₄⋊C₈	—	4	2	no	—
39	C₄⋊C₈	—	8	2	no	—
40	#103	—	4	2	no	—
41	#97	—	8	2	no	—
42	C₄×C₈	D₈	1	2	yes	—
43	C₄×C₈	C₂×D₄	1	1	yes	characteristic
44	C₃₂	—	2	2	no	—
45	C₃₂	—	2	1	no	—
46	C₃₂	M₁₆	1	2	yes	—
47	C₂×Q₈	—	4	2	no	—
48	Q₁₆	—	4	2	no	—
49	Q₁₆	—	4	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	8	2	no	—
52	C₂×C₈	C₄×D₄	1	1	yes	characteristic
53	C₂×C₈	#135	1	1	yes	characteristic
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	8	2	no	—
56	C₄²	—	8	2	no	—
57	C₄²	C₂×D₈	1	1	yes	characteristic
58	C₁₆	—	2	1	no	—
59	C₁₆	—	4	1	no	—
60	C₁₆	C₂×M₁₆	1	1	yes	characteristic
61	C₁₆	#131	1	1	yes	characteristic
62	Q₈	—	8	2	no	—
63	Q₈	—	8	2	no	—
64	C₂×C₄	—	8	2	no	—
65	C₂×C₄	—	8	2	no	—
66	C₂×C₄	C₄×D₈	1	1	yes	characteristic
67	C₂×C₄	#509	1	2	yes	—
68	C₈	—	2	1	no	—
69	C₈	—	4	2	no	—
70	C₈	—	16	2	no	—
71	C₈	#436	1	1	yes	characteristic
72	C₈	#443	1	1	yes	characteristic
73	C₂²	#2177	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	16	2	no	—
76	C₄	—	16	2	no	—
77	C₄	#1589	1	1	yes	characteristic
78	C₄	#2179	1	1	yes	characteristic
79	C₂	#12086	1	1	yes	characteristic
80	C₂	#13657	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#13653	2
#12085	2
#7463	2
#7386	2
#7330	2