Subgroups — #135767

Order 512.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12886	C₂	1	4	yes	maximal
2	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#8822	C₂	1	1	yes	maximal, characteristic
4	#8813	C₂	1	1	yes	maximal, characteristic
5	C₄×M₃₂	C₂²	1	2	yes	—
6	#1848	C₂²	1	2	yes	—
7	#1844	—	2	2	no	—
8	#1844	C₂²	1	2	yes	—
9	#1566	—	2	2	no	—
10	C₂×C₈²	C₂²	1	1	yes	characteristic
11	#1383	—	2	8	no	—
12	#1311	C₄	1	2	yes	complemented
13	C₈×C₁₆	C₄	1	2	yes	complemented
14	C₂×M₃₂	D₄	1	4	yes	—
15	C₂×#108	—	2	4	no	—
16	C₂×#106	—	2	4	no	—
17	C₂×#106	—	4	4	no	—
18	#423	—	2	2	no	—
19	C₂×C₄×C₈	D₄	1	2	yes	—
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	#384	—	2	2	no	—
22	#346	Q₈	1	4	yes	—
23	C₄×C₁₆	C₂×C₄	1	4	yes	—
24	#334	—	4	4	no	—
25	C₈²	C₂×C₄	1	1	yes	characteristic
26	C₈²	C₂×C₄	1	1	yes	characteristic
27	C₈²	Q₈	1	2	yes	—
28	C₂×M₁₆	—	4	2	no	—
29	C₂²×C₈	SD₁₆	1	4	yes	—
30	C₂²×C₈	C₂×D₄	1	2	yes	—
31	C₂×C₄⋊C₄	—	4	2	no	—
32	C₂×C₄⋊C₄	—	4	4	no	—
33	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
34	M₃₂	—	2	8	no	—
35	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
36	#108	—	4	8	no	—
37	#106	—	4	8	no	—
38	#106	—	8	8	no	—
39	C₄⋊C₈	—	4	4	no	—
40	C₄×C₈	C₄⋊C₄	1	2	yes	—
41	C₄×C₈	C₄⋊C₄	1	2	yes	—
42	C₄×C₈	SD₁₆	1	4	yes	—
43	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₄×C₈	C₂×Q₈	1	2	yes	—
46	C₂²×C₄	—	8	2	no	—
47	C₂²×C₄	#127	1	1	yes	characteristic
48	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
49	M₁₆	—	8	4	no	—
50	C₂×C₈	—	2	8	no	—
51	C₂×C₈	—	8	2	no	—
52	C₂×C₈	#106	1	4	yes	—
53	C₂×C₈	#106	1	4	yes	—
54	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
55	C₂×C₈	C₄⋊Q₈	1	2	yes	—
56	C₂×C₈	#135	1	8	yes	—
57	C₄⋊C₄	—	8	8	no	—
58	C₄⋊C₄	—	8	4	no	—
59	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄²	C₄⋊Q₈	1	1	yes	characteristic
61	C₄²	C₂×SD₁₆	1	2	yes	—
62	C₁₆	—	2	8	no	—
63	C₂³	#492	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	16	4	no	—
66	C₂×C₄	—	16	2	no	—
67	C₂×C₄	#384	1	1	yes	characteristic
68	C₂×C₄	C₂×#106	1	2	yes	—
69	C₂×C₄	#492	1	1	yes	characteristic
70	C₂×C₄	#495	1	4	yes	—
71	C₂×C₄	#498	1	2	yes	—
72	C₈	—	2	4	no	—
73	C₈	—	16	2	no	—
74	C₈	#365	1	8	yes	—
75	C₈	#427	1	4	yes	—
76	C₈	#499	1	4	yes	—
77	C₂²	—	2	2	no	—
78	C₂²	#1708	1	2	yes	—
79	C₂²	#1844	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	32	2	no	—
82	C₄	#1708	1	2	yes	—
83	C₄	#1846	1	2	yes	—
84	C₄	C₂×#365	1	4	yes	—
85	C₂	—	2	2	no	—
86	C₂	#8815	1	1	yes	characteristic
87	C₂	#12903	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1844
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #1311	C₄	semidirect
13. C₈×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12886	2
C₈×M₃₂	2
#8822	2
#8813	2