Subgroups — #12313

Order 256.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3293	C₂	1	1	yes	maximal, characteristic
2	#2214	C₂	1	2	yes	maximal, complemented
3	#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	#1358	C₂	1	1	yes	maximal, characteristic, complemented
5	#1335	C₂	1	2	yes	maximal, complemented
6	#578	—	2	2	no	—
7	#510	—	2	2	no	—
8	#500	—	2	2	no	—
9	#494	C₂²	1	1	yes	characteristic
10	#494	C₂²	1	1	yes	characteristic
11	#473	—	2	2	no	—
12	#456	—	2	2	no	—
13	#443	C₂²	1	2	yes	—
14	#364	C₂²	1	2	yes	—
15	#362	—	2	2	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	2₋¹⁺⁴	—	2	2	no	—
18	#137	—	2	4	no	—
19	#137	—	4	4	no	—
20	#135	—	4	2	no	—
21	#135	D₄	1	2	yes	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	4	2	no	—
25	#131	—	2	2	no	—
26	C₄⋊Q₈	—	2	1	no	—
27	C₄⋊Q₈	D₄	1	2	yes	—
28	Q₃₂	—	4	2	no	—
29	SD₃₂	—	4	2	no	—
30	M₃₂	—	2	2	no	—
31	#108	D₄	1	2	yes	—
32	#107	—	4	4	no	—
33	#104	—	2	2	no	—
34	#101	—	2	2	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	C₂³	1	1	yes	characteristic
37	#41	—	2	2	no	—
38	#41	—	4	4	no	—
39	#41	—	8	2	no	—
40	C₂×Q₈	—	2	2	no	—
41	C₂×Q₈	—	4	4	no	—
42	C₂×Q₈	—	4	1	no	—
43	Q₁₆	—	4	4	no	—
44	Q₁₆	—	4	2	no	—
45	Q₁₆	—	4	2	no	—
46	Q₁₆	—	4	2	no	—
47	Q₁₆	—	4	2	no	—
48	Q₁₆	D₈	1	2	yes	—
49	SD₁₆	—	2	2	no	—
50	SD₁₆	—	4	4	no	—
51	D₈	D₈	1	2	yes	—
52	M₁₆	—	2	2	no	—
53	M₁₆	—	4	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	C₂×D₄	1	2	yes	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄²	C₂×D₄	1	1	yes	characteristic
60	C₁₆	—	4	2	no	—
61	Q₈	—	2	2	no	—
62	Q₈	—	4	4	no	—
63	Q₈	—	8	2	no	—
64	Q₈	—	8	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	2	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	8	4	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	#120	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	—	4	2	no	—
76	C₈	C₂×D₈	1	2	yes	—
77	C₈	AGL(1,ℤ/8)	1	2	yes	—
78	C₂²	—	4	2	no	—
79	C₂²	#453	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	2	no	—
82	C₄	—	8	2	no	—
83	C₄	—	16	1	no	—
84	C₄	#447	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	8	2	no	—
87	C₂	#1619	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ 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
2. #2214	C₂	semidirect
3. #1713	C₂	semidirect
4. #1358	C₂	semidirect
5. #1335	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3293	2
#2214	2
#1713	2
#1358	2
#1335	2