Subgroups — #491480

Order 512.

508 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33726	C₂	1	4	yes	maximal
2	#19604	C₂	1	2	yes	maximal
3	#12524	C₂	1	8	yes	maximal, complemented
4	C₂×#1311	C₂	1	1	yes	maximal, characteristic
5	#3170	—	2	4	no	—
6	#3160	C₂²	1	4	yes	—
7	#2254	C₂²	1	8	yes	—
8	#2253	—	2	8	no	—
9	#2252	C₂²	1	8	yes	—
10	#2158	—	2	8	no	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1713	C₂²	1	4	yes	—
13	#1710	C₂²	1	4	yes	—
14	#1707	—	2	4	no	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1311	C₂²	1	4	yes	—
17	#533	—	2	4	no	—
18	C₂²×C₁₆	D₄	1	4	yes	—
19	#500	—	2	4	no	—
20	#500	C₂³	1	8	yes	—
21	#499	—	2	8	no	—
22	#498	—	2	4	no	—
23	#494	—	2	4	no	—
24	#487	—	2	8	no	—
25	#467	—	4	8	no	—
26	C₄×Q₁₆	—	4	8	no	—
27	#427	—	2	8	no	—
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	#367	—	2	16	no	—
31	#366	—	2	16	no	—
32	C₄×C₁₆	D₄	1	4	yes	—
33	C₄×C₁₆	C₂³	1	4	yes	—
34	C₈²	D₄	1	2	yes	—
35	C₈²	C₂³	1	2	yes	—
36	C₂×Q₁₆	—	4	8	no	—
37	C₂²×C₈	D₈	1	4	yes	—
38	C₂²×C₈	C₂×D₄	1	2	yes	—
39	C₄⋊Q₈	—	2	8	no	—
40	#125	—	4	4	no	—
41	C₂²⋊Q₈	—	4	8	no	—
42	C₄×Q₈	—	4	8	no	—
43	#117	—	4	4	no	—
44	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
45	C₂×C₁₆	Q₁₆	1	16	yes	complemented
46	C₂×C₁₆	C₂×D₄	1	8	yes	—
47	#107	—	2	16	no	—
48	#107	—	4	8	no	—
49	#106	—	4	8	no	—
50	#103	—	4	16	no	—
51	C₄×C₈	D₈	1	4	yes	—
52	C₄×C₈	C₂×D₄	1	2	yes	—
53	C₄×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄×C₈	C₂×D₄	1	4	yes	—
55	C₄×C₈	C₂⁴	1	1	yes	characteristic
56	C₂×Q₈	—	4	8	no	—
57	C₂²×C₄	#127	1	1	yes	characteristic
58	C₂²×C₄	C₂×D₈	1	2	yes	—
59	Q₁₆	—	8	16	no	—
60	C₂×C₈	—	4	4	no	—
61	C₂×C₈	#127	1	2	yes	—
62	C₂×C₈	C₂×D₈	1	8	yes	—
63	C₂×C₈	C₂×Q₁₆	1	8	yes	—
64	C₂×C₈	C₂²×D₄	1	2	yes	—
65	C₄⋊C₄	—	4	8	no	—
66	C₄⋊C₄	—	4	8	no	—
67	C₄⋊C₄	—	8	8	no	—
68	C₂²⋊C₄	—	8	4	no	—
69	C₄²	—	8	4	no	—
70	C₄²	#127	1	1	yes	characteristic
71	C₄²	C₂×D₈	1	2	yes	—
72	C₄²	C₂²×D₄	1	1	yes	characteristic
73	C₁₆	C₂×Q₁₆	1	16	yes	—
74	C₂³	#493	1	1	yes	characteristic
75	Q₈	—	8	16	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	8	8	no	—
78	C₂×C₄	#493	1	2	yes	—
79	C₂×C₄	#493	1	1	yes	characteristic
80	C₂×C₄	#494	1	4	yes	—
81	C₂×C₄	C₂×#127	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₈	1	2	yes	—
83	C₈	—	2	8	no	—
84	C₈	#494	1	4	yes	—
85	C₈	C₂²×Q₁₆	1	4	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	#1713	1	2	yes	—
88	C₂²	C₂×#493	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	16	8	no	—
91	C₄	#1713	1	2	yes	—
92	C₄	C₂×#494	1	2	yes	—
93	C₄	#3414	1	4	yes	—
94	C₂	—	2	2	no	—
95	C₂	C₂×#1713	1	1	yes	characteristic
96	C₂	#33716	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #12524	C₂	semidirect
45. C₂×C₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#33726	2
#19604	2
#12524	2
C₂×#1311	2