Subgroups — #92899

Order 512.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11411	C₂	1	1	yes	maximal, characteristic
2	#11402	C₂	1	1	yes	maximal, characteristic
3	C₂×#1311	C₂	1	1	yes	maximal, characteristic
4	#2162	C₄	1	2	yes	—
5	C₄×M₃₂	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	C₄	1	2	yes	—
7	C₂×C₈²	C₂²	1	1	yes	characteristic
8	#1312	—	2	1	no	—
9	#1312	—	2	1	no	—
10	#1312	—	2	1	no	—
11	#1311	—	2	1	no	—
12	#1311	—	2	1	no	—
13	#1311	—	2	1	no	—
14	C₂×M₃₂	—	2	2	no	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₁₆	—	2	2	no	—
20	C₂²⋊C₁₆	C₈	1	4	yes	—
21	#346	C₈	1	4	yes	—
22	C₄×C₁₆	—	2	2	no	—
23	C₄×C₁₆	—	2	2	no	—
24	C₄×C₁₆	—	2	2	no	—
25	C₄×C₁₆	C₈	1	4	yes	—
26	C₈²	D₄	1	1	yes	characteristic
27	C₈²	D₄	1	1	yes	characteristic
28	C₈²	D₄	1	1	yes	characteristic
29	C₈²	Q₈	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₈	1	2	yes	—
31	C₂²×C₈	M₁₆	1	2	yes	—
32	C₂²×C₈	M₁₆	1	2	yes	—
33	C₂×C₄²	C₄²	1	1	yes	characteristic
34	M₃₂	—	4	4	no	—
35	C₂×C₁₆	—	2	8	no	—
36	C₂×C₁₆	—	2	4	no	—
37	C₂×C₁₆	—	2	4	no	—
38	C₂×C₁₆	—	2	4	no	—
39	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₈	C₂×C₈	1	2	yes	—
46	C₄×C₈	C₂×C₈	1	2	yes	—
47	C₄×C₈	M₁₆	1	2	yes	—
48	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
49	C₂²×C₄	#97	1	1	yes	characteristic
50	C₂²×C₄	#97	1	1	yes	characteristic
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	C₂²⋊C₈	1	2	yes	—
55	C₂×C₈	C₂²⋊C₈	1	2	yes	—
56	C₂×C₈	C₂²⋊C₈	1	2	yes	—
57	C₂×C₈	C₄⋊C₈	1	2	yes	—
58	C₂×C₈	C₄⋊C₈	1	2	yes	—
59	C₂×C₈	C₄⋊C₈	1	2	yes	—
60	C₄²	#95	1	1	yes	characteristic
61	C₄²	C₄×C₈	1	1	yes	characteristic
62	C₄²	C₄×C₈	1	1	yes	characteristic
63	C₄²	#97	1	1	yes	characteristic
64	C₁₆	—	2	8	no	—
65	C₁₆	—	4	4	no	—
66	C₁₆	—	4	4	no	—
67	C₂³	#322	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	C₈²	1	1	yes	characteristic
71	C₂×C₄	#322	1	1	yes	characteristic
72	C₂×C₄	#322	1	1	yes	characteristic
73	C₂×C₄	#336	1	1	yes	characteristic
74	C₂×C₄	#336	1	1	yes	characteristic
75	C₂×C₄	#336	1	1	yes	characteristic
76	C₂×C₄	#347	1	4	yes	—
77	C₈	—	2	2	no	—
78	C₈	—	2	2	no	—
79	C₈	—	2	2	no	—
80	C₈	—	2	2	no	—
81	C₈	—	2	2	no	—
82	C₈	#364	1	4	yes	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	1	no	—
85	C₂²	#1273	1	1	yes	characteristic
86	C₂²	#1313	1	2	yes	—
87	C₄	—	2	1	no	—
88	C₄	—	2	1	no	—
89	C₄	#1313	1	2	yes	—
90	C₄	#1375	1	2	yes	—
91	C₄	#1376	1	2	yes	—
92	C₄	#1378	1	2	yes	—
93	C₂	—	2	2	no	—
94	C₂	#7137	1	1	yes	characteristic
95	C₂	#7138	1	1	yes	characteristic
96	C₂	#7140	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#95
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#11411	2
#11402	2
C₂×#1311	2