Subgroups — #8142

Order 256.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1746	C₂	1	1	yes	maximal, characteristic
2	#1452	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	#1275	C₂	1	2	yes	maximal
5	#1274	C₂	1	2	yes	maximal
6	#431	C₂²	1	2	yes	—
7	#431	C₂²	1	2	yes	—
8	C₂×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#376	C₂²	1	1	yes	characteristic
11	#336	C₄	1	4	yes	—
12	#322	C₄	1	2	yes	complemented
13	#322	C₄	1	4	yes	complemented
14	C₈²	C₄	1	2	yes	complemented
15	C₂²×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂³	1	1	yes	characteristic
19	C₄⋊C₈	—	2	2	no	—
20	C₄⋊C₈	—	2	2	no	—
21	C₂²⋊C₈	—	2	2	no	—
22	C₂²⋊C₈	—	2	2	no	—
23	#97	—	2	2	no	—
24	C₄×C₈	C₂×C₄	1	4	yes	—
25	C₄×C₈	C₂×C₄	1	4	yes	—
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂×C₄	1	2	yes	—
29	#95	—	2	2	no	—
30	C₂²×C₄	—	2	2	no	—
31	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄	#41	1	2	yes	—
33	C₂×C₈	—	2	4	no	—
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	4	no	—
38	C₂×C₈	C₄²	1	2	yes	—
39	C₂×C₈	C₄²	1	2	yes	—
40	C₄²	—	2	4	no	—
41	C₄²	C₂²×C₄	1	1	yes	characteristic
42	C₄²	C₂²×C₄	1	1	yes	characteristic
43	C₄²	#41	1	2	yes	—
44	C₂³	#117	1	1	yes	characteristic
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	—	4	4	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄	#117	1	2	yes	—
50	C₂×C₄	#117	1	2	yes	—
51	C₂×C₄	#117	1	1	yes	characteristic
52	C₂×C₄	#131	1	4	yes	—
53	C₈	—	2	2	no	—
54	C₈	—	2	2	no	—
55	C₈	—	4	4	no	—
56	C₈	—	4	4	no	—
57	C₈	—	4	2	no	—
58	C₂²	—	2	2	no	—
59	C₂²	#376	1	1	yes	characteristic
60	C₂²	#433	1	2	yes	—
61	C₄	—	2	2	no	—
62	C₄	—	2	2	no	—
63	C₄	—	8	2	no	—
64	C₄	#405	1	2	yes	—
65	C₄	#433	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#1746	1	1	yes	characteristic
68	C₂	#1747	1	2	yes	—

	subgroup	G/H
center, commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#376
prime radical	C₂³	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #322	C₄	semidirect
13. #322	C₄	semidirect
14. C₈²	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1746	2
#1452	2
#1450	2
#1275	2
#1274	2