Subgroups — #12658

Order 256.

175 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150	C₂	1	2	yes	maximal
2	C₂×#347	C₂	1	2	yes	maximal
3	C₄×#97	C₂	1	1	yes	maximal, characteristic
4	#1375	C₂	1	2	yes	maximal
5	C₂×M₃₂	C₂²	1	4	yes	—
6	C₂×#97	C₄	1	1	yes	characteristic
7	C₂×#97	C₄	1	1	yes	characteristic
8	C₂×#97	C₂²	1	1	yes	characteristic
9	C₂×#97	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄³	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆	C₄	1	8	yes	complemented
14	#347	—	2	4	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²	—	2	2	no	—
20	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	M₃₂	—	2	4	no	—
24	M₃₂	—	2	4	no	—
25	C₂×C₁₆	C₂×C₄	1	8	yes	—
26	#97	—	2	2	no	—
27	#97	—	2	2	no	—
28	#97	—	2	2	no	—
29	#97	D₄	1	2	yes	—
30	#97	Q₈	1	2	yes	—
31	C₄×C₈	—	2	1	no	—
32	C₄×C₈	—	2	1	no	—
33	C₄×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₄	—	2	1	no	—
35	C₂²×C₄	—	2	1	no	—
36	C₂²×C₄	M₁₆	1	2	yes	—
37	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	C₄²	1	2	yes	—
45	C₂×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₈	M₁₆	1	2	yes	—
47	C₂×C₈	C₂²×C₄	1	2	yes	—
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈	#41	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₄²	—	2	1	no	—
53	C₄²	—	4	4	no	—
54	C₄²	C₄²	1	2	yes	—
55	C₄²	C₄⋊C₄	1	2	yes	—
56	C₄²	M₁₆	1	2	yes	—
57	C₄²	M₁₆	1	1	yes	characteristic
58	C₄²	M₁₆	1	1	yes	characteristic
59	C₁₆	—	2	8	no	—
60	C₂³	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	#97	1	2	yes	—
67	C₂×C₄	#97	1	2	yes	—
68	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄	#117	1	1	yes	characteristic
71	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	2	2	no	—
76	C₈	—	2	2	no	—
77	C₈	C₄×D₄	1	2	yes	—
78	C₈	C₄×Q₈	1	2	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×#97	1	1	yes	characteristic
81	C₂²	#423	1	1	yes	characteristic
82	C₂²	#432	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	4	2	no	—
85	C₄	—	4	2	no	—
86	C₄	#347	1	4	yes	complemented
87	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄	C₄×M₁₆	1	1	yes	characteristic
89	C₄	#423	1	1	yes	characteristic
90	C₄	#432	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#1767	1	1	yes	characteristic
94	C₂	C₂×#347	1	2	yes	—

	subgroup	G/H
center, commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#97
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. C₄⋊C₁₆	C₄	semidirect
86. C₄	#347	semidirect

Maximal subgroups

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

subgroup	index
#2150	2
C₂×#347	2
C₄×#97	2
#1375	2