Subgroups — #99286

Order 512.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#8807	C₂	1	2	yes	maximal
2	#8133	C₂	1	1	yes	maximal, characteristic
3	#7018	C₂	1	4	yes	maximal
4	#1831	C₂²	1	4	yes	—
5	#1744	C₂²	1	2	yes	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×#335	—	2	2	no	—
8	C₂×#334	—	2	2	no	—
9	#1273	C₄	1	4	yes	—
10	C₂×C₄⋊C₈	—	2	4	no	—
11	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
12	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₈	D₄	1	1	yes	characteristic
14	C₂×C₄×C₈	D₄	1	1	yes	characteristic
15	C₂×C₄×C₈	Q₈	1	2	yes	—
16	C₄³	C₂³	1	1	yes	characteristic
17	#336	—	2	4	no	—
18	#336	—	2	8	no	—
19	#335	—	4	4	no	—
20	#334	—	4	4	no	—
21	C₂²×C₈	—	2	4	no	—
22	C₂²×C₈	—	4	4	no	—
23	C₂²×C₈	C₄⋊C₄	1	2	yes	—
24	C₂²×C₈	C₄⋊C₄	1	2	yes	—
25	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄²	#41	1	4	yes	—
29	C₄⋊C₈	—	4	4	no	—
30	C₄⋊C₈	—	4	4	no	—
31	C₄×C₈	—	2	4	no	—
32	C₄×C₈	—	2	8	no	—
33	C₄×C₈	—	4	2	no	—
34	C₄×C₈	D₈	1	2	yes	—
35	C₄×C₈	SD₁₆	1	4	yes	—
36	C₄×C₈	Q₁₆	1	2	yes	—
37	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄	#117	1	2	yes	—
39	C₂²×C₄	#124	1	2	yes	—
40	C₂²×C₄	#125	1	2	yes	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	4	8	no	—
46	C₂×C₈	—	4	4	no	—
47	C₂×C₈	—	8	8	no	—
48	C₂×C₈	—	8	4	no	—
49	C₂×C₈	#106	1	4	yes	—
50	C₂×C₈	#107	1	4	yes	—
51	C₄²	—	2	2	no	—
52	C₄²	—	4	4	no	—
53	C₄²	#131	1	4	yes	—
54	C₄²	C₂×D₈	1	1	yes	characteristic
55	C₄²	C₂×SD₁₆	1	2	yes	—
56	C₄²	C₂×Q₁₆	1	1	yes	characteristic
57	C₂³	#382	1	1	yes	characteristic
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	#424	1	2	yes	—
63	C₂×C₄	C₂×#106	1	1	yes	characteristic
64	C₂×C₄	C₂×#107	1	1	yes	characteristic
65	C₂×C₄	#430	1	2	yes	—
66	C₂×C₄	#433	1	4	yes	—
67	C₂×C₄	#486	1	2	yes	—
68	C₂×C₄	#487	1	2	yes	—
69	C₈	—	2	4	no	—
70	C₈	—	2	4	no	—
71	C₈	—	8	4	no	—
72	C₈	—	16	4	no	—
73	C₂²	#1832	1	1	yes	characteristic
74	C₂²	#1833	1	4	yes	—
75	C₂²	#1834	1	1	yes	characteristic
76	C₂²	#1838	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	4	2	no	—
81	C₄	#1570	1	2	yes	—
82	C₄	#1571	1	2	yes	—
83	C₂	#8782	1	2	yes	—
84	C₂	#8787	1	1	yes	characteristic
85	C₂	#8793	1	2	yes	—
86	C₂	#8798	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#8807	2
#8133	2
#7018	2