Subgroups — #135681

Order 512.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12901	C₂	1	1	yes	maximal, characteristic
2	#12880	C₂	1	1	yes	maximal, characteristic
3	#8138	C₂	1	1	yes	maximal, characteristic
4	#7153	C₂	1	4	yes	maximal
5	C₂×#364	—	2	2	no	—
6	#2150	C₂²	1	2	yes	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	#1726	C₂²	1	1	yes	characteristic
9	#1450	C₂²	1	2	yes	—
10	#1374	—	2	2	no	—
11	#1275	C₄	1	2	yes	—
12	#1274	C₄	1	2	yes	—
13	C₂×M₃₂	—	2	2	no	—
14	C₂²×C₁₆	—	2	2	no	—
15	#431	C₂×C₄	1	4	yes	—
16	C₂×#97	C₂×C₄	1	1	yes	characteristic
17	C₂×#97	Q₈	1	2	yes	—
18	C₂×C₄×C₈	D₄	1	2	yes	—
19	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
20	#376	C₂×C₄	1	1	yes	characteristic
21	#364	—	4	4	no	—
22	C₄⋊C₁₆	—	2	4	no	—
23	C₄⋊C₁₆	—	2	4	no	—
24	#322	—	2	2	no	—
25	C₈²	—	2	2	no	—
26	C₂²×C₈	C₄⋊C₄	1	2	yes	—
27	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄²	C₄⋊C₄	1	2	yes	—
30	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
31	M₃₂	—	4	4	no	—
32	C₂×C₁₆	—	4	2	no	—
33	C₂×C₁₆	—	4	2	no	—
34	C₂×C₁₆	—	4	2	no	—
35	C₂×C₁₆	—	4	2	no	—
36	C₄⋊C₈	—	2	4	no	—
37	C₂²⋊C₈	—	2	4	no	—
38	#97	—	2	4	no	—
39	#97	—	2	2	no	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	4	no	—
43	C₄×C₈	#41	1	2	yes	—
44	C₄×C₈	#41	1	2	yes	—
45	#95	—	2	2	no	—
46	C₂²×C₄	—	2	2	no	—
47	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄	#131	1	2	yes	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	4	4	no	—
53	C₂×C₈	#108	1	4	yes	—
54	C₂×C₈	#124	1	1	yes	characteristic
55	C₂×C₈	#124	1	1	yes	characteristic
56	C₂×C₈	#125	1	2	yes	—
57	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×C₈	#137	1	2	yes	—
59	C₄²	—	2	2	no	—
60	C₄²	—	2	2	no	—
61	C₄²	#117	1	2	yes	—
62	C₄²	#131	1	2	yes	—
63	C₁₆	—	8	2	no	—
64	C₁₆	—	8	2	no	—
65	C₂³	#424	1	1	yes	characteristic
66	C₂×C₄	—	4	2	no	—
67	C₂×C₄	—	4	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	#382	1	1	yes	characteristic
70	C₂×C₄	#424	1	1	yes	characteristic
71	C₂×C₄	#428	1	2	yes	—
72	C₂×C₄	C₂×#108	1	2	yes	—
73	C₂×C₄	#433	1	4	yes	—
74	C₈	—	2	4	no	—
75	C₈	—	2	2	no	—
76	C₈	—	8	4	no	—
77	C₈	#489	1	2	yes	—
78	C₈	#490	1	1	yes	characteristic
79	C₈	#491	1	1	yes	characteristic
80	C₂²	—	2	2	no	—
81	C₂²	#1574	1	2	yes	—
82	C₂²	#1832	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	8	2	no	—
85	C₄	#1573	1	2	yes	—
86	C₄	#1836	1	1	yes	characteristic
87	C₄	#1837	1	1	yes	characteristic
88	C₂	—	2	2	no	—
89	C₂	#8789	1	1	yes	characteristic
90	C₂	#12885	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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
#12901	2
#12880	2
#8138	2
#7153	2