Subgroups — #11404

Order 256.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#322	C₂	1	1	yes	maximal, characteristic
4	#1313	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂	C₄	1	2	yes	—
6	C₂×M₃₂	C₄	1	2	yes	—
7	C₂×C₄×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#346	C₂²	1	2	yes	—
10	#346	C₂²	1	2	yes	—
11	C₄×C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆	C₄	1	2	yes	—
13	#322	C₄	1	2	yes	—
14	#322	C₂²	1	2	yes	—
15	C₂²×C₈	—	2	2	no	—
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	M₃₂	—	2	4	no	—
20	M₃₂	C₈	1	8	yes	complemented
21	C₂×C₁₆	C₂×C₄	1	4	yes	—
22	C₂×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈	—	2	2	no	—
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	C₂×C₄	1	4	yes	—
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₄	C₄²	1	1	yes	characteristic
30	C₂²×C₄	M₁₆	1	2	yes	—
31	C₂×C₈	—	2	4	no	—
32	C₂×C₈	—	4	4	no	—
33	C₂×C₈	C₄²	1	1	yes	characteristic
34	C₂×C₈	C₄²	1	1	yes	characteristic
35	C₂×C₈	C₂×C₈	1	4	yes	—
36	C₂×C₈	M₁₆	1	4	yes	—
37	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₄²	C₄²	1	1	yes	characteristic
40	C₄²	M₁₆	1	2	yes	—
41	C₄²	C₂²×C₄	1	1	yes	characteristic
42	C₁₆	—	2	4	no	—
43	C₁₆	C₂×C₈	1	8	yes	complemented
44	C₂³	#97	1	1	yes	characteristic
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	C₄×C₈	1	2	yes	—
47	C₂×C₄	#97	1	1	yes	characteristic
48	C₂×C₄	#97	1	2	yes	—
49	C₂×C₄	#97	1	2	yes	—
50	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄	C₂×M₁₆	1	2	yes	—
52	C₈	—	4	4	no	—
53	C₈	C₄×C₈	1	2	yes	—
54	C₈	#97	1	2	yes	—
55	C₈	C₂²×C₈	1	2	yes	—
56	C₈	C₂×M₁₆	1	2	yes	—
57	C₂²	—	2	2	no	—
58	C₂²	#322	1	2	yes	—
59	C₂²	C₂×#97	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#322	1	2	yes	—
62	C₄	#347	1	4	yes	—
63	C₄	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄	C₂×#97	1	1	yes	characteristic
65	C₂	—	2	2	no	—
66	C₂	C₂×#322	1	1	yes	characteristic
67	C₂	C₂×#347	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#97
prime radical	C₂³	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
4. #1313	C₂	semidirect
20. M₃₂	C₈	semidirect
43. C₁₆	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
C₂×#322	2
#1313	2