Subgroups — #135642

Order 512.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12014	C₂	1	2	yes	maximal
2	#8142	C₂	1	1	yes	maximal, characteristic
3	#7198	C₂	1	1	yes	maximal, characteristic
4	#7197	C₂	1	1	yes	maximal, characteristic
5	#7144	C₂	1	2	yes	maximal
6	#2163	C₂²	1	2	yes	—
7	#2162	C₂²	1	2	yes	—
8	#1746	C₂²	1	1	yes	characteristic
9	#1452	C₂²	1	1	yes	characteristic
10	#1450	C₂²	1	1	yes	characteristic
11	#1370	—	2	2	no	—
12	#1367	—	2	2	no	—
13	#1275	C₄	1	2	yes	—
14	#1274	C₄	1	2	yes	—
15	#431	C₂×C₄	1	2	yes	—
16	#431	C₂×C₄	1	2	yes	—
17	C₂×#97	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
19	#376	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₁₆	—	2	2	no	—
21	C₄⋊C₁₆	—	2	2	no	—
22	C₂²⋊C₁₆	—	2	2	no	—
23	C₂²⋊C₁₆	—	2	2	no	—
24	#346	—	2	2	no	—
25	C₄×C₁₆	—	2	2	no	—
26	#336	—	2	2	no	—
27	#322	—	2	2	no	—
28	#322	Q₈	1	2	yes	—
29	C₈²	D₄	1	2	yes	—
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	#41	1	2	yes	—
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₁₆	—	4	2	no	—
35	C₂×C₁₆	—	4	2	no	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₂²⋊C₈	—	2	2	no	—
39	C₂²⋊C₈	—	2	2	no	—
40	#97	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	C₄⋊C₄	1	4	yes	—
43	C₄×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₄×C₈	#41	1	2	yes	—
46	#95	—	2	2	no	—
47	C₂²×C₄	—	4	1	no	—
48	C₂²×C₄	#117	1	1	yes	characteristic
49	C₂²×C₄	#131	1	2	yes	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	—	4	2	no	—
53	C₂×C₈	—	4	2	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	#124	1	2	yes	—
56	C₂×C₈	#125	1	2	yes	—
57	C₄²	—	4	2	no	—
58	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄²	#117	1	1	yes	characteristic
60	C₄²	#131	1	2	yes	—
61	C₁₆	—	8	2	no	—
62	C₁₆	—	8	2	no	—
63	C₂³	#433	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	8	1	no	—
67	C₂×C₄	—	8	2	no	—
68	C₂×C₄	#382	1	1	yes	characteristic
69	C₂×C₄	#424	1	2	yes	—
70	C₂×C₄	#433	1	2	yes	—
71	C₂×C₄	#433	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	2	2	no	—
74	C₈	—	4	4	no	—
75	C₈	—	8	2	no	—
76	C₈	—	8	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	#1832	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	—	2	2	no	—
83	C₄	—	16	1	no	—
84	C₄	#1833	1	2	yes	—
85	C₂	—	4	1	no	—
86	C₂	#8783	1	1	yes	characteristic
87	C₂	#12885	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂³	#433
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12014	2
#8142	2
#7198	2
#7197	2
#7144	2