Subgroups — #151733

Order 512.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13733	C₂	1	2	yes	maximal
2	#13731	C₂	1	2	yes	maximal
3	#12522	C₂	1	1	yes	maximal, characteristic
4	#12521	C₂	1	1	yes	maximal, characteristic
5	#7329	C₂	1	1	yes	maximal, characteristic
6	#2254	—	2	1	no	—
7	#2252	—	2	1	no	—
8	#2252	C₂²	1	1	yes	characteristic
9	#2252	C₂²	1	2	yes	—
10	#2252	C₂²	1	1	yes	characteristic
11	#1713	—	2	1	no	—
12	#1710	—	2	1	no	—
13	#1424	—	2	4	no	—
14	#1423	—	2	4	no	—
15	C₄×C₃₂	C₂²	1	2	yes	—
16	C₈×C₁₆	C₂²	1	1	yes	characteristic
17	#500	—	2	1	no	—
18	#500	—	2	1	no	—
19	#500	—	2	2	no	—
20	#500	—	4	1	no	—
21	#494	—	4	1	no	—
22	#487	—	4	1	no	—
23	#487	—	4	1	no	—
24	C₂×C₃₂	D₄	1	4	yes	—
25	#367	—	4	2	no	—
26	#366	—	2	2	no	—
27	#366	—	2	2	no	—
28	#366	—	2	2	no	—
29	#366	—	2	2	no	—
30	#366	—	4	2	no	—
31	C₄×C₁₆	D₄	1	1	yes	characteristic
32	C₄×C₁₆	C₂³	1	1	yes	characteristic
33	C₈²	D₄	1	1	yes	characteristic
34	C₂×Q₁₆	—	8	2	no	—
35	C₄⋊Q₈	—	4	1	no	—
36	C₄⋊Q₈	—	4	1	no	—
37	C₄⋊Q₈	—	4	2	no	—
38	C₂×C₁₆	D₈	1	2	yes	—
39	C₂×C₁₆	C₂×D₄	1	2	yes	—
40	#107	—	4	1	no	—
41	#107	—	4	2	no	—
42	#107	—	4	1	no	—
43	#107	—	4	2	no	—
44	#107	—	4	1	no	—
45	#107	—	4	1	no	—
46	#107	—	8	2	no	—
47	#103	—	8	2	no	—
48	#103	—	8	2	no	—
49	C₄×C₈	D₈	1	1	yes	characteristic
50	C₄×C₈	D₈	1	1	yes	characteristic
51	C₄×C₈	C₂×D₄	1	1	yes	characteristic
52	C₃₂	Q₁₆	1	8	yes	complemented
53	C₂×Q₈	—	8	1	no	—
54	C₂×Q₈	—	8	1	no	—
55	C₂×Q₈	—	8	2	no	—
56	Q₁₆	—	16	4	no	—
57	C₂×C₈	D₁₆	1	2	yes	—
58	C₂×C₈	D₁₆	1	2	yes	—
59	C₂×C₈	#127	1	1	yes	characteristic
60	C₂×C₈	C₂×D₈	1	1	yes	characteristic
61	C₄⋊C₄	—	8	2	no	—
62	C₄⋊C₄	—	8	1	no	—
63	C₄⋊C₄	—	8	1	no	—
64	C₄⋊C₄	—	8	1	no	—
65	C₄⋊C₄	—	8	1	no	—
66	C₄⋊C₄	—	8	2	no	—
67	C₄²	C₂×D₈	1	1	yes	characteristic
68	C₁₆	Q₃₂	1	4	yes	—
69	C₁₆	C₂×Q₁₆	1	4	yes	—
70	Q₈	—	16	4	no	—
71	Q₈	—	16	2	no	—
72	Q₈	—	16	2	no	—
73	C₂×C₄	—	16	1	no	—
74	C₂×C₄	—	16	2	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	#493	1	1	yes	characteristic
77	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
78	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	2	2	no	—
81	C₈	#494	1	2	yes	—
82	C₈	C₂×Q₃₂	1	2	yes	—
83	C₂²	#2246	1	1	yes	characteristic
84	C₄	—	32	2	no	—
85	C₄	—	32	1	no	—
86	C₄	—	32	1	no	—
87	C₄	#1713	1	1	yes	characteristic
88	C₄	#2247	1	1	yes	characteristic
89	C₄	#2263	1	2	yes	—
90	C₄	#2264	1	2	yes	—
91	C₂	#12521	1	1	yes	characteristic
92	C₂	#13729	1	1	yes	characteristic
93	C₂	#13730	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
52. C₃₂	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#13733	2
#13731	2
#12522	2
#12521	2
#7329	2