Subgroups — #7266

Order 256.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#347	C₂	1	2	yes	maximal
2	#1836	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂	—	2	2	no	—
4	C₂×#97	C₂²	1	1	yes	characteristic
5	#384	C₄	1	1	yes	characteristic
6	#382	C₄	1	1	yes	characteristic
7	#347	—	2	2	no	—
8	#347	C₄	1	4	yes	complemented
9	#340	—	2	2	no	—
10	C₂²×C₈	D₄	1	2	yes	—
11	C₂×C₄⋊C₄	—	2	1	no	—
12	C₂×C₄⋊C₄	—	2	1	no	—
13	C₂×C₄⋊C₄	—	4	1	no	—
14	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
15	M₃₂	—	2	4	no	—
16	M₃₂	—	2	4	no	—
17	C₂×C₁₆	—	2	4	no	—
18	#97	C₂×C₄	1	2	yes	—
19	#97	D₄	1	1	yes	characteristic
20	#97	Q₈	1	1	yes	characteristic
21	#95	—	4	1	no	—
22	C₂²×C₄	—	2	1	no	—
23	C₂²×C₄	—	4	1	no	—
24	C₂²×C₄	—	4	1	no	—
25	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	C₂²⋊C₄	1	2	yes	—
29	C₂×C₈	C₄⋊C₄	1	2	yes	—
30	C₄⋊C₄	—	4	2	no	—
31	C₄⋊C₄	—	4	2	no	—
32	C₄⋊C₄	—	8	2	no	—
33	C₄²	—	2	1	no	—
34	C₄²	C₄²	1	1	yes	characteristic
35	C₄²	C₄⋊C₄	1	1	yes	characteristic
36	C₁₆	—	2	8	no	—
37	C₂³	#100	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	—	8	2	no	—
43	C₂×C₄	—	8	1	no	—
44	C₂×C₄	—	8	1	no	—
45	C₂×C₄	—	8	1	no	—
46	C₂×C₄	—	8	1	no	—
47	C₂×C₄	#95	1	1	yes	characteristic
48	C₂×C₄	#101	1	1	yes	characteristic
49	C₂×C₄	#104	1	2	yes	—
50	C₂×C₄	#104	1	2	yes	—
51	C₈	—	2	2	no	—
52	C₈	—	2	2	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	#329	1	1	yes	characteristic
56	C₂²	#329	1	1	yes	characteristic
57	C₂²	#343	1	1	yes	characteristic
58	C₄	—	4	1	no	—
59	C₄	—	4	1	no	—
60	C₄	—	16	1	no	—
61	C₄	—	16	1	no	—
62	C₄	#330	1	2	yes	—
63	C₄	#339	1	2	yes	—
64	C₂	—	2	1	no	—
65	C₂	—	2	1	no	—
66	C₂	#1295	1	1	yes	characteristic
67	C₂	#1355	1	1	yes	characteristic
68	C₂	#1356	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#329
commutator	C₄²	C₄²
Frattini, cosocle	C₂×#97	C₂²
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. #347	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#347	2
#1836	2