Subgroups — #151060

Order 512.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13642	C₂	1	1	yes	maximal, characteristic
2	#12021	C₂	1	1	yes	maximal, characteristic
3	#12002	C₂	1	1	yes	maximal, characteristic
4	#7457	C₂	1	4	yes	maximal, complemented
5	C₂×#364	—	2	1	no	—
6	C₄×M₃₂	C₂²	1	1	yes	characteristic
7	#1568	—	2	1	no	—
8	#1422	C₄	1	2	yes	—
9	#1400	C₄	1	2	yes	—
10	#1398	C₂²	1	2	yes	—
11	#1369	—	2	1	no	—
12	#1369	C₂²	1	2	yes	—
13	#1368	—	2	1	no	—
14	#1368	C₂²	1	2	yes	—
15	C₂×M₃₂	—	4	1	no	—
16	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
17	C₂×M₃₂	Q₈	1	1	yes	characteristic
18	#432	—	4	1	no	—
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	M₆₄	C₂×C₄	1	4	yes	—
21	#364	—	2	2	no	—
22	#364	—	4	1	no	—
23	#346	D₄	1	1	yes	characteristic
24	#346	C₂³	1	1	yes	characteristic
25	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₄×C₁₆	Q₈	1	1	yes	characteristic
27	#335	—	4	1	no	—
28	#334	—	2	2	no	—
29	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₈	SD₁₆	1	1	yes	characteristic
31	C₂×C₄²	SD₁₆	1	1	yes	characteristic
32	M₃₂	—	2	2	no	—
33	M₃₂	—	4	1	no	—
34	M₃₂	—	4	2	no	—
35	M₃₂	—	8	1	no	—
36	C₂×C₁₆	—	8	1	no	—
37	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
41	C₄⋊C₈	—	4	2	no	—
42	C₂²⋊C₈	—	8	1	no	—
43	#97	—	8	1	no	—
44	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₈	SD₁₆	1	1	yes	characteristic
46	C₄×C₈	SD₁₆	1	1	yes	characteristic
47	C₄×C₈	C₂×D₄	1	1	yes	characteristic
48	C₃₂	—	2	4	no	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	#106	1	1	yes	characteristic
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	8	2	no	—
56	C₂×C₈	#106	1	1	yes	characteristic
57	C₂×C₈	#106	1	1	yes	characteristic
58	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	#106	1	1	yes	characteristic
62	C₄²	C₂×SD₁₆	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	2	1	no	—
66	C₁₆	—	8	2	no	—
67	C₁₆	#131	1	2	yes	—
68	C₂³	#365	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#365	1	1	yes	characteristic
75	C₂×C₄	C₂×#106	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	16	2	no	—
79	C₈	#365	1	1	yes	characteristic
80	C₈	#365	1	1	yes	characteristic
81	C₈	#424	1	1	yes	characteristic
82	C₈	#430	1	1	yes	characteristic
83	C₂²	—	2	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	C₂×#365	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	4	1	no	—
89	C₄	#1570	1	1	yes	characteristic
90	C₄	C₂×#365	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	4	1	no	—
93	C₂	#12030	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#12030
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	#346	C₂³
prime radical	C₂³	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7457	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13642	2
#12021	2
#12002	2
#7457	2