Subgroups — #151058

Order 512.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13549	C₂	1	1	yes	maximal, characteristic
2	#12027	C₂	1	2	yes	maximal
3	#7464	C₂	1	2	yes	maximal, complemented
4	#7462	C₂	1	2	yes	maximal, complemented
5	C₂×M₆₄	C₄	1	1	yes	characteristic
6	C₂²×C₃₂	C₄	1	1	yes	characteristic
7	#2166	—	2	2	no	—
8	#2108	C₂²	1	1	yes	characteristic
9	#1397	C₄	1	1	yes	characteristic
10	#1397	C₂²	1	1	yes	characteristic
11	C₄×C₃₂	C₄	1	1	yes	characteristic
12	C₄×C₃₂	C₂²	1	1	yes	characteristic
13	#1373	C₂²	1	4	yes	—
14	#1369	—	2	2	no	—
15	C₂×M₃₂	Q₈	1	1	yes	characteristic
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	#405	D₄	1	1	yes	characteristic
18	M₆₄	—	2	2	no	—
19	C₂×C₃₂	—	2	2	no	—
20	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
24	#364	—	2	4	no	—
25	#349	—	4	2	no	—
26	#347	—	4	2	no	—
27	#346	D₄	1	1	yes	characteristic
28	#346	Q₈	1	1	yes	characteristic
29	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₄×C₁₆	C₂³	1	1	yes	characteristic
31	C₂×M₁₆	D₈	1	1	yes	characteristic
32	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
33	#117	Q₁₆	1	1	yes	characteristic
34	M₃₂	—	2	2	no	—
35	M₃₂	—	4	4	no	—
36	C₂×C₁₆	—	2	1	no	—
37	C₂×C₁₆	—	2	1	no	—
38	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
42	#97	D₈	1	1	yes	characteristic
43	#97	Q₁₆	1	1	yes	characteristic
44	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₈	C₂×D₄	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	2	1	no	—
48	C₃₂	M₁₆	1	2	yes	—
49	C₃₂	M₁₆	1	2	yes	—
50	C₂²×C₄	#107	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	#107	1	1	yes	characteristic
55	C₂×C₈	#107	1	1	yes	characteristic
56	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈	C₂×D₈	1	1	yes	characteristic
58	C₄⋊C₄	—	2	1	no	—
59	C₂²⋊C₄	—	2	1	no	—
60	C₄²	#107	1	1	yes	characteristic
61	C₄²	C₂×Q₁₆	1	1	yes	characteristic
62	C₁₆	—	2	1	no	—
63	C₁₆	—	2	1	no	—
64	C₁₆	—	8	4	no	—
65	C₁₆	#108	1	2	yes	—
66	C₁₆	C₂×M₁₆	1	1	yes	characteristic
67	C₁₆	C₂×M₁₆	1	1	yes	characteristic
68	C₂³	#368	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	#368	1	1	yes	characteristic
74	C₂×C₄	C₂×#107	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	2	1	no	—
77	C₈	#423	1	1	yes	characteristic
78	C₈	C₂×#108	1	1	yes	characteristic
79	C₈	#508	1	2	yes	—
80	C₂²	—	2	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	C₂×#368	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	#1567	1	1	yes	characteristic
87	C₄	#2158	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	4	1	no	—
90	C₂	#12024	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2158
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	#12024
prime radical	C₂³	#368
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #97 ◁ #405 ◁ #2108 ◁ 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
3. #7464	C₂	semidirect
4. #7462	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13549	2
#12027	2
#7464	2
#7462	2