Subgroups — #19668

Order 256.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3231	C₂	1	4	yes	maximal
2	#3170	C₂	1	2	yes	maximal
3	#1718	C₂	1	4	yes	maximal, complemented
4	#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#1707	C₂	1	2	yes	maximal, complemented
7	#1450	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	4	yes	—
9	#500	C₂²	1	2	yes	—
10	#500	C₂²	1	4	yes	—
11	#498	C₂²	1	4	yes	—
12	#494	C₂²	1	2	yes	—
13	#487	C₂²	1	4	yes	—
14	#487	C₂²	1	4	yes	—
15	#487	C₂²	1	4	yes	—
16	#485	—	2	8	no	—
17	#479	—	2	8	no	—
18	#467	—	2	4	no	—
19	C₄×Q₁₆	—	2	4	no	—
20	#431	C₂²	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#322	C₂²	1	2	yes	—
23	C₈²	C₂²	1	1	yes	characteristic
24	C₈²	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆	—	2	4	no	—
26	C₂²×C₈	D₄	1	2	yes	—
27	C₄⋊Q₈	C₂³	1	4	yes	—
28	C₄⋊Q₈	C₂³	1	4	yes	—
29	#125	—	2	4	no	—
30	C₂²⋊Q₈	—	2	4	no	—
31	C₂²⋊Q₈	—	2	4	no	—
32	C₄×Q₈	—	2	4	no	—
33	C₄×Q₈	—	2	4	no	—
34	#117	—	2	4	no	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#107	—	2	4	no	—
37	#107	—	2	8	no	—
38	#106	—	2	8	no	—
39	C₄⋊C₈	D₄	1	4	yes	—
40	#103	—	2	8	no	—
41	#103	—	2	8	no	—
42	#103	—	2	8	no	—
43	C₂²⋊C₈	D₄	1	4	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₈	C₂³	1	2	yes	—
47	C₄×C₈	C₂³	1	2	yes	—
48	C₄×C₈	C₂³	1	1	yes	characteristic
49	C₂×Q₈	—	2	4	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂²×C₄	C₂×D₄	1	2	yes	—
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	Q₁₆	—	4	8	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₈	C₂×D₄	1	4	yes	—
57	C₂×C₈	C₂×D₄	1	4	yes	—
58	C₂×C₈	C₂×D₄	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄⋊C₄	—	2	4	no	—
61	C₄⋊C₄	—	2	4	no	—
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	—	4	4	no	—
65	C₂²⋊C₄	—	4	4	no	—
66	C₄²	—	4	4	no	—
67	C₄²	C₂×D₄	1	1	yes	characteristic
68	C₄²	C₂×D₄	1	2	yes	—
69	C₄²	C₂⁴	1	1	yes	characteristic
70	C₂³	#127	1	1	yes	characteristic
71	Q₈	—	4	8	no	—
72	Q₈	—	4	8	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	#127	1	2	yes	—
78	C₂×C₄	#127	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₄	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	2	4	no	—
83	C₈	—	2	4	no	—
84	C₈	#135	1	4	yes	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂×#127	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	8	4	no	—
91	C₄	—	8	4	no	—
92	C₄	C₂×#135	1	2	yes	—
93	C₄	#578	1	4	yes	—
94	C₂	—	4	1	no	—
95	C₂	#3147	1	1	yes	characteristic
96	C₂	#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1718	C₂	semidirect
4. #1713	C₂	semidirect
5. #1710	C₂	semidirect
6. #1707	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3231	2
#3170	2
#1718	2
#1713	2
#1710	2
#1707	2
#1450	2