Subgroups — #135809

Order 512.

173 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12024	C₂	1	2	yes	maximal
2	#12021	C₂	1	2	yes	maximal
3	#11401	C₂	1	1	yes	maximal, characteristic
4	#8796	C₂	1	2	yes	maximal
5	C₄×M₃₂	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#1568	C₂²	1	4	yes	—
8	#1450	C₂²	1	1	yes	characteristic
9	#1372	—	2	2	no	—
10	#1371	—	2	2	no	—
11	#1368	—	2	4	no	—
12	#1312	C₄	1	4	yes	—
13	#1287	—	2	4	no	—
14	C₂×M₃₂	Q₈	1	2	yes	—
15	C₂²×C₁₆	Q₈	1	2	yes	—
16	#432	—	2	4	no	—
17	#431	D₄	1	2	yes	—
18	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
19	#346	D₄	1	2	yes	—
20	C₄×C₁₆	C₂×C₄	1	2	yes	—
21	C₄×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₁₆	D₄	1	1	yes	characteristic
23	C₄×C₁₆	D₄	1	1	yes	characteristic
24	#335	—	2	4	no	—
25	#334	—	2	4	no	—
26	#322	Q₈	1	2	yes	—
27	C₈²	C₂×C₄	1	2	yes	—
28	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
31	M₃₂	—	4	2	no	—
32	C₂×C₁₆	—	4	2	no	—
33	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
34	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
35	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
36	C₄⋊C₈	—	2	2	no	—
37	C₄⋊C₈	—	4	4	no	—
38	C₂²⋊C₈	—	2	2	no	—
39	C₂²⋊C₈	—	4	4	no	—
40	#97	—	4	4	no	—
41	C₄×C₈	C₄⋊C₄	1	4	yes	—
42	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄×C₈	C₂×D₄	1	1	yes	characteristic
45	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
47	C₂²×C₄	#135	1	2	yes	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	—	8	4	no	—
52	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
56	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄²	#127	1	1	yes	characteristic
58	C₄²	#135	1	2	yes	—
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	2	no	—
61	C₁₆	—	2	4	no	—
62	C₂³	#499	1	1	yes	characteristic
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	#384	1	1	yes	characteristic
67	C₂×C₄	#427	1	2	yes	—
68	C₂×C₄	#495	1	2	yes	—
69	C₂×C₄	#499	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	4	4	no	—
73	C₈	—	16	4	no	—
74	C₈	#430	1	2	yes	—
75	C₈	#430	1	2	yes	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	1	no	—
78	C₂²	#1846	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	2	1	no	—
82	C₄	#1849	1	1	yes	characteristic
83	C₄	#1849	1	1	yes	characteristic
84	C₂	—	4	1	no	—
85	C₂	#8827	1	1	yes	characteristic
86	C₂	#12911	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#499
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#12024	2
#12021	2
#11401	2
#8796	2