Subgroups — #33289

Order 256.

431 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2868	C₂	1	2	yes	maximal
2	#1990	C₂	1	8	yes	maximal
3	#1834	C₂	1	4	yes	maximal
4	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈	—	2	2	no	—
6	C₂×C₄⋊Q₈	—	2	2	no	—
7	C₂×C₄⋊Q₈	C₂²	1	4	yes	—
8	C₂×C₄×C₈	C₂²	1	4	yes	—
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	#384	—	2	2	no	—
11	#384	—	2	2	no	—
12	#384	C₂²	1	4	yes	—
13	#384	C₂²	1	4	yes	—
14	C₄³	C₂²	1	1	yes	characteristic
15	#340	C₂²	1	16	yes	—
16	C₂²×Q₈	—	2	4	no	—
17	C₂²×C₈	C₂³	1	4	yes	—
18	C₄⋊Q₈	—	2	16	no	—
19	C₄⋊Q₈	—	4	8	no	—
20	C₄⋊Q₈	—	4	8	no	—
21	C₂×C₄⋊C₄	—	2	4	no	—
22	C₂×C₄⋊C₄	—	2	4	no	—
23	C₂×C₄⋊C₄	—	2	4	no	—
24	C₂×C₄⋊C₄	—	2	4	no	—
25	C₂×C₄⋊C₄	—	2	4	no	—
26	C₂×C₄⋊C₄	C₂³	1	8	yes	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₄²	D₄	1	2	yes	—
29	C₂×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	C₂×C₄²	C₂³	1	2	yes	—
32	C₄×C₈	—	2	8	no	—
33	C₄×C₈	Q₈	1	8	yes	complemented
34	C₂×Q₈	—	4	8	no	—
35	C₂×Q₈	—	4	8	no	—
36	C₂×Q₈	—	4	8	no	—
37	C₂²×C₄	—	2	8	no	—
38	C₂²×C₄	C₂×D₄	1	2	yes	—
39	C₂²×C₄	C₂×D₄	1	2	yes	—
40	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄	C₂⁴	1	1	yes	characteristic
43	C₂×C₈	—	2	8	no	—
44	C₂×C₈	C₂×Q₈	1	8	yes	—
45	C₄⋊C₄	—	2	16	no	—
46	C₄⋊C₄	—	4	8	no	—
47	C₄⋊C₄	—	4	8	no	—
48	C₄⋊C₄	—	4	8	no	—
49	C₄⋊C₄	—	4	8	no	—
50	C₄⋊C₄	—	4	8	no	—
51	C₄²	D₈	1	4	yes	—
52	C₄²	SD₁₆	1	8	yes	—
53	C₄²	Q₁₆	1	4	yes	—
54	C₄²	C₂×Q₈	1	4	yes	—
55	C₄²	#41	1	8	yes	—
56	C₂³	C₂²×D₄	1	1	yes	characteristic
57	Q₈	—	8	16	no	—
58	C₂×C₄	—	4	8	no	—
59	C₂×C₄	—	4	8	no	—
60	C₂×C₄	—	4	8	no	—
61	C₂×C₄	#124	1	4	yes	—
62	C₂×C₄	#124	1	4	yes	—
63	C₂×C₄	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄	C₂×D₈	1	4	yes	—
66	C₂×C₄	C₂×D₈	1	2	yes	—
67	C₂×C₄	C₂×SD₁₆	1	4	yes	—
68	C₂×C₄	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄	C₂×SD₁₆	1	4	yes	—
70	C₂×C₄	C₂×Q₁₆	1	4	yes	—
71	C₂×C₄	C₂×Q₁₆	1	2	yes	—
72	C₂×C₄	C₂²×Q₈	1	2	yes	—
73	C₂×C₄	C₂×#41	1	4	yes	—
74	C₈	—	2	8	no	—
75	C₂²	C₂×#124	1	2	yes	—
76	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂²	C₂²×D₈	1	1	yes	characteristic
78	C₂²	C₂²×SD₁₆	1	2	yes	—
79	C₂²	C₂²×Q₁₆	1	1	yes	characteristic
80	C₄	—	8	8	no	—
81	C₄	#486	1	8	yes	—
82	C₄	#487	1	8	yes	—
83	C₄	#498	1	4	yes	—
84	C₄	#500	1	4	yes	—
85	C₄	#533	1	4	yes	—
86	C₂	#2862	1	1	yes	characteristic
87	C₂	C₂×#486	1	2	yes	—
88	C₂	C₂×#487	1	2	yes	—
89	C₂	C₂×#498	1	1	yes	characteristic
90	C₂	C₂×#500	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
33. C₄×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2868	2
#1990	2
#1834	2
C₄²×C₈	2