Subgroups — #122949

Order 512.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12054	C₂	1	2	yes	maximal
2	C₈×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#7100	C₂	1	1	yes	maximal, characteristic
4	#7069	C₂	1	2	yes	maximal
5	C₁₆²	C₂	1	1	yes	maximal, characteristic
6	#2183	—	2	2	no	—
7	C₄×Q₃₂	C₄	1	1	yes	characteristic
8	C₈×Q₁₆	C₂²	1	2	yes	—
9	#1371	C₄	1	1	yes	characteristic
10	#1370	C₂²	1	2	yes	—
11	Q₈⋊C₁₆	—	2	2	no	—
12	#1332	C₄	1	2	yes	—
13	C₈×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₁₆	C₂²	1	1	yes	characteristic
15	C₈×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂	C₈	1	1	yes	characteristic
17	C₈×Q₈	—	2	2	no	—
18	C₄×Q₁₆	C₂×C₄	1	2	yes	—
19	#366	C₈	1	1	yes	characteristic
20	C₄⋊C₁₆	—	2	2	no	—
21	C₄⋊C₁₆	—	4	2	no	—
22	#358	C₈	1	2	yes	—
23	#346	—	4	2	no	—
24	C₄×C₁₆	—	2	1	no	—
25	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₄×C₁₆	D₄	1	1	yes	characteristic
28	C₄×C₁₆	D₄	1	1	yes	characteristic
29	#335	C₂×C₄	1	2	yes	—
30	Q₈⋊C₈	—	2	2	no	—
31	C₈²	C₂³	1	1	yes	characteristic
32	C₂×Q₁₆	C₂×C₈	1	2	yes	—
33	C₄×Q₈	—	2	2	no	—
34	Q₃₂	—	2	2	no	—
35	C₂×C₁₆	—	2	2	no	—
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	—	2	1	no	—
38	C₂×C₁₆	—	4	2	no	—
39	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
40	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
41	C₂×C₁₆	D₈	1	1	yes	characteristic
42	C₂×C₁₆	D₈	1	1	yes	characteristic
43	#107	C₂×C₈	1	2	yes	—
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	4	2	no	—
46	#103	—	2	2	no	—
47	C₄×C₈	—	4	2	no	—
48	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₄×C₈	C₂×D₄	1	1	yes	characteristic
50	C₄×C₈	#41	1	1	yes	characteristic
51	C₂×Q₈	—	2	2	no	—
52	Q₁₆	—	2	2	no	—
53	Q₁₆	—	2	2	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₈	#131	1	1	yes	characteristic
58	C₂×C₈	C₂×D₈	1	1	yes	characteristic
59	C₂×C₈	#135	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄²	—	4	2	no	—
63	C₄²	C₄×D₄	1	1	yes	characteristic
64	C₁₆	—	2	4	no	—
65	C₁₆	—	2	2	no	—
66	C₁₆	—	2	2	no	—
67	C₁₆	—	2	1	no	—
68	C₁₆	—	2	1	no	—
69	C₁₆	—	8	2	no	—
70	C₁₆	M₃₂	1	2	yes	—
71	C₁₆	Q₃₂	1	2	yes	complemented
72	Q₈	—	4	2	no	—
73	Q₈	—	4	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	C₈×D₄	1	1	yes	characteristic
77	C₂×C₄	C₄×D₈	1	1	yes	characteristic
78	C₂×C₄	#443	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	—	8	2	no	—
83	C₈	C₂×M₃₂	1	1	yes	characteristic
84	C₈	#504	1	1	yes	characteristic
85	C₈	C₂×Q₃₂	1	1	yes	characteristic
86	C₈	#508	1	1	yes	characteristic
87	C₂²	C₈×D₈	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	2	no	—
90	C₄	—	8	2	no	—
91	C₄	#2169	1	1	yes	characteristic
92	C₄	#2170	1	1	yes	characteristic
93	C₄	C₄×Q₃₂	1	1	yes	characteristic
94	C₄	#2178	1	1	yes	characteristic
95	C₂	#12039	1	1	yes	characteristic
96	C₂	C₈×Q₃₂	1	1	yes	characteristic
97	C₂	#12046	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle, prime radical	C₂²	C₈×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
71. C₁₆	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#12054	2
C₈×Q₃₂	2
#7100	2
#7069	2
C₁₆²	2