Subgroups — #135772

Order 512.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12888	C₂	1	4	yes	maximal
2	C₂×#1311	C₂	1	1	yes	maximal, characteristic
3	#8818	C₂	1	1	yes	maximal, characteristic
4	#8816	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₆	C₂²	1	2	yes	—
6	#1845	—	2	2	no	—
7	#1845	C₂²	1	2	yes	—
8	#1845	C₂²	1	2	yes	—
9	#1834	—	2	2	no	—
10	C₂×C₈²	C₂²	1	1	yes	characteristic
11	#1380	—	2	8	no	—
12	#1311	C₄	1	4	yes	complemented
13	C₂²×C₁₆	D₄	1	4	yes	—
14	C₂×#107	—	2	4	no	—
15	C₂×#107	—	2	4	no	—
16	C₂×#107	—	4	4	no	—
17	C₂×C₄×C₈	D₄	1	2	yes	—
18	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
19	#384	—	2	2	no	—
20	#384	—	2	2	no	—
21	C₄×C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₁₆	Q₈	1	4	yes	—
23	#340	—	4	4	no	—
24	C₈²	C₂×C₄	1	2	yes	—
25	C₈²	Q₈	1	2	yes	—
26	C₂²×C₈	D₈	1	4	yes	—
27	C₂²×C₈	C₂×D₄	1	2	yes	—
28	C₂×C₄⋊C₄	—	4	2	no	—
29	C₂×C₄⋊C₄	—	4	2	no	—
30	C₂×C₄⋊C₄	—	4	2	no	—
31	C₂×C₄⋊C₄	—	4	2	no	—
32	C₂×C₄⋊C₄	—	4	4	no	—
33	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
34	C₂×C₁₆	—	2	4	no	—
35	C₂×C₁₆	—	2	4	no	—
36	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
37	#107	—	4	8	no	—
38	#107	—	4	8	no	—
39	#107	—	8	8	no	—
40	C₄×C₈	C₄⋊C₄	1	4	yes	—
41	C₄×C₈	Q₁₆	1	4	yes	—
42	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
44	C₄×C₈	C₂×Q₈	1	2	yes	—
45	C₂²×C₄	—	8	2	no	—
46	C₂²×C₄	—	8	2	no	—
47	C₂²×C₄	#127	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₈	1	2	yes	—
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	#107	1	8	yes	—
52	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₈	C₄⋊Q₈	1	2	yes	—
54	C₂×C₈	#135	1	4	yes	—
55	C₂×C₈	#135	1	4	yes	—
56	C₄⋊C₄	—	8	4	no	—
57	C₄⋊C₄	—	8	4	no	—
58	C₄⋊C₄	—	8	8	no	—
59	C₄⋊C₄	—	8	4	no	—
60	C₄⋊C₄	—	8	4	no	—
61	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄²	C₄⋊Q₈	1	1	yes	characteristic
63	C₄²	C₂×Q₁₆	1	2	yes	—
64	C₁₆	—	2	8	no	—
65	C₂³	#493	1	1	yes	characteristic
66	C₂×C₄	—	16	2	no	—
67	C₂×C₄	—	16	2	no	—
68	C₂×C₄	—	16	2	no	—
69	C₂×C₄	—	16	4	no	—
70	C₂×C₄	—	16	2	no	—
71	C₂×C₄	#384	1	1	yes	characteristic
72	C₂×C₄	C₂×#107	1	2	yes	—
73	C₂×C₄	#494	1	1	yes	characteristic
74	C₂×C₄	#495	1	2	yes	—
75	C₂×C₄	#495	1	2	yes	—
76	C₂×C₄	#500	1	2	yes	—
77	C₂×C₄	#509	1	4	yes	—
78	C₂×C₄	#510	1	4	yes	—
79	C₈	—	2	8	no	—
80	C₈	#427	1	4	yes	—
81	C₈	#499	1	4	yes	—
82	C₂²	#1709	1	1	yes	characteristic
83	C₂²	#1709	1	1	yes	characteristic
84	C₂²	#1845	1	1	yes	characteristic
85	C₂²	#2250	1	2	yes	—
86	C₂²	#2251	1	2	yes	—
87	C₄	—	32	2	no	—
88	C₄	—	32	2	no	—
89	C₄	#1710	1	2	yes	—
90	C₄	#1846	1	2	yes	—
91	C₄	#2159	1	4	yes	—
92	C₄	#2255	1	4	yes	—
93	C₂	#8816	1	1	yes	characteristic
94	C₂	#12512	1	2	yes	—
95	C₂	#12513	1	1	yes	characteristic
96	C₂	#12514	1	1	yes	characteristic
97	C₂	#12908	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
12. #1311	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12888	2
C₂×#1311	2
#8818	2
#8816	2