Subgroups — #93271

Order 512.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#364	C₂	1	2	yes	maximal
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#364	C₄	1	4	yes	—
4	C₄×M₃₂	—	2	2	no	—
5	C₂×C₄×C₁₆	—	2	2	no	—
6	C₂×C₄×C₁₆	C₄	1	2	yes	—
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	#1376	—	2	2	no	—
9	C₂×M₃₂	—	2	4	no	—
10	C₂²×C₁₆	—	2	2	no	—
11	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₈	D₄	1	2	yes	—
14	C₂×C₄×C₈	D₄	1	1	yes	characteristic
15	C₄³	Q₈	1	1	yes	characteristic
16	#364	—	2	4	no	—
17	#364	C₈	1	8	yes	—
18	#346	—	4	2	no	—
19	C₄×C₁₆	—	2	8	no	—
20	C₄×C₁₆	—	2	2	no	—
21	C₄×C₁₆	—	4	2	no	—
22	C₄×C₁₆	C₈	1	4	yes	—
23	C₂²×C₈	C₄²	1	1	yes	characteristic
24	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄²	—	2	2	no	—
27	C₂×C₄²	C₄⋊C₄	1	2	yes	—
28	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
29	M₃₂	—	2	8	no	—
30	M₃₂	—	4	4	no	—
31	C₂×C₁₆	—	2	8	no	—
32	C₂×C₁₆	—	2	4	no	—
33	C₂×C₁₆	—	4	4	no	—
34	C₄×C₈	—	2	4	no	—
35	C₄×C₈	C₂×C₈	1	2	yes	—
36	C₄×C₈	C₂×C₈	1	4	yes	—
37	C₄×C₈	M₁₆	1	2	yes	—
38	C₄×C₈	M₁₆	1	4	yes	—
39	C₄×C₈	D₈	1	1	yes	characteristic
40	C₄×C₈	SD₁₆	1	2	yes	—
41	C₄×C₈	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄	—	2	2	no	—
43	C₂²×C₄	#95	1	1	yes	characteristic
44	C₂²×C₄	#108	1	2	yes	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	C₄×C₈	1	1	yes	characteristic
47	C₂×C₈	C₄×C₈	1	2	yes	—
48	C₂×C₈	#97	1	1	yes	characteristic
49	C₂×C₈	#97	1	2	yes	—
50	C₂×C₈	C₂²⋊C₈	1	2	yes	—
51	C₂×C₈	C₂²⋊C₈	1	4	yes	—
52	C₂×C₈	#102	1	2	yes	—
53	C₂×C₈	#103	1	2	yes	—
54	C₂×C₈	#104	1	4	yes	—
55	C₄²	—	2	8	no	—
56	C₄²	—	2	2	no	—
57	C₄²	C₄⋊C₈	1	4	yes	—
58	C₄²	C₄⋊C₈	1	2	yes	—
59	C₄²	#106	1	1	yes	characteristic
60	C₄²	#107	1	1	yes	characteristic
61	C₁₆	—	2	8	no	—
62	C₁₆	—	4	8	no	—
63	C₂³	#341	1	1	yes	characteristic
64	C₂×C₄	—	2	8	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	#334	1	2	yes	—
68	C₂×C₄	#335	1	2	yes	—
69	C₂×C₄	#336	1	1	yes	characteristic
70	C₂×C₄	#336	1	2	yes	—
71	C₂×C₄	#339	1	2	yes	—
72	C₂×C₄	#340	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₈²	1	1	yes	characteristic
75	C₈	#322	1	1	yes	characteristic
76	C₈	#322	1	2	yes	—
77	C₈	D₄⋊C₈	1	4	yes	—
78	C₈	Q₈⋊C₈	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	#1277	1	1	yes	characteristic
81	C₂²	#1278	1	2	yes	—
82	C₄	—	2	8	no	—
83	C₄	—	2	2	no	—
84	C₄	#1273	1	1	yes	characteristic
85	C₄	#1276	1	1	yes	characteristic
86	C₄	#1279	1	2	yes	—
87	C₄	#1373	1	4	yes	—
88	C₂	—	2	2	no	—
89	C₂	#7015	1	1	yes	characteristic
90	C₂	#7160	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1277
prime radical	C₂³	#341
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×#364	2
C₄²×C₁₆	2