Subgroups — #149498

Order 512.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#11399	C₂	1	1	yes	maximal, characteristic
4	#7331	C₂	1	4	yes	maximal, complemented
5	C₂×M₆₄	C₄	1	2	yes	—
6	C₂²×C₃₂	C₄	1	2	yes	—
7	C₄×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	#1450	C₄	1	1	yes	characteristic
10	#1397	C₄	1	2	yes	—
11	C₄×C₃₂	C₄	1	2	yes	—
12	C₄×C₃₂	C₂²	1	2	yes	—
13	C₄×C₃₂	C₂²	1	2	yes	—
14	#1311	C₄	1	2	yes	—
15	C₈×C₁₆	C₂²	1	2	yes	—
16	C₂×M₃₂	C₈	1	2	yes	—
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	#431	C₈	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	M₆₄	—	2	4	no	—
22	C₂×C₃₂	—	2	4	no	—
23	C₂×C₃₂	C₂×C₄	1	4	yes	—
24	C₂×C₃₂	C₂×C₄	1	4	yes	—
25	#346	C₈	1	2	yes	—
26	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₄×C₁₆	C₂×C₄	1	2	yes	—
30	C₄×C₁₆	C₂³	1	1	yes	characteristic
31	#322	C₈	1	2	yes	—
32	C₈²	C₂×C₄	1	2	yes	—
33	C₂²×C₈	C₄²	1	1	yes	characteristic
34	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
36	M₃₂	—	2	4	no	—
37	C₂×C₁₆	C₄²	1	1	yes	characteristic
38	C₂×C₁₆	C₄²	1	1	yes	characteristic
39	C₂×C₁₆	C₂×C₈	1	4	yes	—
40	C₂×C₁₆	M₁₆	1	4	yes	—
41	C₂×C₁₆	M₁₆	1	4	yes	—
42	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊C₈	C₁₆	1	4	yes	—
45	C₂²⋊C₈	C₁₆	1	4	yes	—
46	C₄×C₈	C₄²	1	1	yes	characteristic
47	C₄×C₈	C₂×C₈	1	1	yes	characteristic
48	C₄×C₈	C₂×C₈	1	1	yes	characteristic
49	C₄×C₈	C₂×C₈	1	4	yes	—
50	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₃₂	—	2	4	no	—
52	C₃₂	—	2	4	no	—
53	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
54	C₂²×C₄	C₂×C₁₆	1	2	yes	—
55	C₂×C₈	C₄×C₈	1	1	yes	characteristic
56	C₂×C₈	C₄×C₈	1	1	yes	characteristic
57	C₂×C₈	#97	1	2	yes	—
58	C₂×C₈	#97	1	2	yes	—
59	C₂×C₈	C₂×C₁₆	1	8	yes	—
60	C₂×C₈	M₃₂	1	4	yes	—
61	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
63	C₄²	C₄×C₈	1	1	yes	characteristic
64	C₄²	C₂×C₁₆	1	2	yes	—
65	C₄²	C₂²×C₈	1	1	yes	characteristic
66	C₁₆	—	2	4	no	—
67	C₁₆	#97	1	2	yes	—
68	C₁₆	#97	1	2	yes	—
69	C₁₆	C₂×M₁₆	1	2	yes	—
70	C₁₆	C₂×M₁₆	1	2	yes	—
71	C₂³	C₄×C₁₆	1	1	yes	characteristic
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	#322	1	2	yes	—
74	C₂×C₄	C₄×C₁₆	1	2	yes	—
75	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
76	C₂×C₄	#346	1	2	yes	—
77	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
78	C₂×C₄	C₂²×C₁₆	1	2	yes	—
79	C₈	—	2	8	no	—
80	C₈	#322	1	2	yes	—
81	C₈	#346	1	2	yes	—
82	C₈	C₂×#97	1	1	yes	characteristic
83	C₈	C₂×#97	1	1	yes	characteristic
84	C₈	C₂×M₃₂	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	#1312	1	2	yes	—
87	C₂²	C₂×C₄×C₁₆	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	#1312	1	2	yes	—
90	C₄	C₂×#322	1	1	yes	characteristic
91	C₄	C₂×#346	1	1	yes	characteristic
92	C₄	#2258	1	4	yes	—
93	C₂	—	2	2	no	—
94	C₂	C₂×#1312	1	1	yes	characteristic
95	C₂	#13549	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₄×C₁₆	C₂³
commutator, socle	C₂²	C₂×C₄×C₁₆
prime radical	C₂³	C₄×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×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
4. #7331	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₆₄	2
C₂×C₄×C₃₂	2
#11399	2
#7331	2