Subgroups — #135804

Order 512.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12022	C₂	1	4	yes	maximal
2	C₂×#1312	C₂	1	1	yes	maximal, characteristic
3	#8823	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆	C₂²	1	2	yes	—
5	#1848	—	2	2	no	—
6	#1567	—	2	2	no	—
7	#1567	C₂²	1	4	yes	—
8	C₂×C₈²	C₂²	1	1	yes	characteristic
9	#1371	—	2	8	no	—
10	#1312	C₄	1	4	yes	—
11	C₂²×C₁₆	Q₈	1	4	yes	—
12	C₂×#108	—	4	4	no	—
13	#423	—	2	4	no	—
14	C₂×C₄×C₈	D₄	1	1	yes	characteristic
15	C₂×C₄×C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
17	C₄×C₁₆	C₂×C₄	1	4	yes	—
18	C₄×C₁₆	D₄	1	4	yes	—
19	#335	—	2	8	no	—
20	#335	—	4	4	no	—
21	C₈²	C₂×C₄	1	2	yes	—
22	C₈²	Q₈	1	2	yes	—
23	C₂×M₁₆	—	4	4	no	—
24	C₂²×C₈	D₈	1	2	yes	—
25	C₂²×C₈	Q₁₆	1	2	yes	—
26	C₂²×C₈	C₂×Q₈	1	2	yes	—
27	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
28	C₂×C₁₆	—	4	4	no	—
29	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
30	#108	—	8	8	no	—
31	C₄⋊C₈	—	4	4	no	—
32	C₄⋊C₈	—	4	4	no	—
33	C₄×C₈	C₄⋊C₄	1	2	yes	—
34	C₄×C₈	C₄⋊C₄	1	2	yes	—
35	C₄×C₈	D₈	1	2	yes	—
36	C₄×C₈	Q₁₆	1	2	yes	—
37	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₈	C₂×D₄	1	2	yes	—
39	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
43	M₁₆	—	8	8	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	8	4	no	—
47	C₂×C₈	#107	1	4	yes	—
48	C₂×C₈	#107	1	4	yes	—
49	C₂×C₈	D₁₆	1	4	yes	—
50	C₂×C₈	Q₃₂	1	4	yes	—
51	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
52	C₂×C₈	C₄⋊Q₈	1	2	yes	—
53	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄²	#127	1	1	yes	characteristic
55	C₄²	C₂×D₈	1	1	yes	characteristic
56	C₄²	C₂×Q₁₆	1	1	yes	characteristic
57	C₁₆	—	2	8	no	—
58	C₂³	#500	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	#384	1	1	yes	characteristic
62	C₂×C₄	C₂×#107	1	1	yes	characteristic
63	C₂×C₄	C₂×#107	1	1	yes	characteristic
64	C₂×C₄	#493	1	1	yes	characteristic
65	C₂×C₄	#494	1	1	yes	characteristic
66	C₂×C₄	#500	1	1	yes	characteristic
67	C₂×C₄	C₂×D₁₆	1	2	yes	—
68	C₂×C₄	C₂×Q₃₂	1	2	yes	—
69	C₈	—	2	2	no	—
70	C₈	—	2	4	no	—
71	C₈	—	16	4	no	—
72	C₈	#366	1	8	yes	—
73	C₈	#430	1	4	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#1845	1	1	yes	characteristic
77	C₂²	#2252	1	2	yes	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	#1849	1	2	yes	—
81	C₄	C₂×#366	1	2	yes	—
82	C₄	#2161	1	2	yes	—
83	C₄	#2246	1	1	yes	characteristic
84	C₄	#2247	1	1	yes	characteristic
85	C₂	—	2	2	no	—
86	C₂	#8826	1	1	yes	characteristic
87	C₂	#12905	1	1	yes	characteristic
88	C₂	#12910	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1845
prime radical	C₂³	#500
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#12022	2
C₂×#1312	2
#8823	2