Finite Group Databasebeta

Subgroups — #19203

Order 256.

387 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3208	C₂	1	2	yes	maximal, complemented
2	C₂×#493	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	#1688	C₂	1	4	yes	maximal, complemented
5	#1668	C₂	1	4	yes	maximal, complemented
6	#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈	—	2	1	no	—
9	C₂²×D₈	—	2	1	no	—
10	#533	C₂²	1	1	yes	characteristic
11	C₂×#127	C₂²	1	2	yes	—
12	#493	—	2	1	no	—
13	#493	—	2	1	no	—
14	#493	C₂²	1	2	yes	—
15	#493	C₂²	1	2	yes	—
16	#492	—	2	2	no	—
17	#486	—	2	2	no	—
18	#460	C₂²	1	8	yes	complemented
19	#450	—	2	4	no	—
20	#446	—	2	1	no	—
21	#446	C₂²	1	2	yes	complemented
22	C₈×Q₈	—	2	1	no	—
23	C₈×Q₈	C₂²	1	2	yes	complemented
24	#431	C₂²	1	2	yes	—
25	#424	—	2	1	no	—
26	#423	C₂²	1	1	yes	characteristic
27	#405	—	2	1	no	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#335	C₂²	1	2	yes	complemented
30	#334	C₂²	1	2	yes	complemented
31	#331	C₂²	1	8	yes	complemented
32	C₂²×D₄	—	2	2	no	—
33	C₂²×D₄	—	2	2	no	—
34	C₂²×D₄	—	2	2	no	—
35	C₂×SD₁₆	—	2	8	no	—
36	C₂×D₈	—	2	2	no	—
37	C₂×D₈	—	2	2	no	—
38	C₂×D₈	—	2	2	no	—
39	C₂×D₈	—	2	2	no	—
40	C₂×D₈	—	4	4	no	—
41	C₂×D₈	—	4	4	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	D₄	1	1	yes	characteristic
45	C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₄⋊Q₈	D₄	1	1	yes	characteristic
47	#127	—	2	2	no	—
48	#127	—	2	2	no	—
49	#127	—	2	2	no	—
50	#127	C₂³	1	4	yes	—
51	#125	—	2	1	no	—
52	C₂²⋊Q₈	—	2	1	no	—
53	C₂²⋊Q₈	D₄	1	2	yes	—
54	C₄×Q₈	—	2	1	no	—
55	C₄×Q₈	C₂³	1	2	yes	—
56	#117	—	2	1	no	—
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₄⋊C₈	—	2	2	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	C₂³	1	2	yes	—
62	C₄⋊C₈	C₂³	1	4	yes	—
63	#102	—	2	8	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	#97	—	2	2	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	—	2	2	no	—
68	C₄×C₈	D₄	1	1	yes	characteristic
69	C₄×C₈	D₄	1	1	yes	characteristic
70	C₄×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₈	C₂³	1	1	yes	characteristic
72	C₂⁴	—	4	2	no	—
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	C₂×D₄	1	2	yes	—
75	C₂×D₄	—	2	4	no	—
76	C₂×D₄	—	2	4	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	2	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	—	4	4	no	—
82	C₂×D₄	—	4	4	no	—
83	C₂×D₄	—	8	4	no	—
84	C₂×D₄	—	8	4	no	—
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	#41	1	1	yes	characteristic
88	SD₁₆	—	8	8	no	—
89	D₈	—	4	4	no	—
90	D₈	—	4	4	no	—
91	D₈	—	4	4	no	—
92	D₈	—	4	4	no	—
93	M₁₆	—	4	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	SD₁₆	1	4	yes	complemented
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	#41	1	1	yes	characteristic
113	C₄²	C₂⁴	1	1	yes	characteristic
114	C₂³	—	4	4	no	—
115	C₂³	—	8	4	no	—
116	C₂³	—	8	4	no	—
117	C₂³	—	8	4	no	—
118	C₂³	#121	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	D₄	—	4	4	no	—
122	D₄	—	4	4	no	—
123	D₄	—	4	8	no	—
124	D₄	—	8	4	no	—
125	D₄	—	8	4	no	—
126	D₄	—	8	4	no	—
127	D₄	—	8	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	#121	1	1	yes	characteristic
133	C₂×C₄	#121	1	1	yes	characteristic
134	C₂×C₄	#121	1	1	yes	characteristic
135	C₂×C₄	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂×#41	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	4	2	no	—
143	C₈	—	4	4	no	—
144	C₈	C₂×SD₁₆	1	2	yes	—
145	C₈	C₂×SD₁₆	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	4	no	—
150	C₂²	—	8	4	no	—
151	C₂²	—	16	8	no	—
152	C₂²	#460	1	2	yes	—
153	C₂²	C₂×#121	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	2	no	—
158	C₄	#460	1	2	yes	—
159	C₄	#472	1	2	yes	—
160	C₄	C₂²×SD₁₆	1	1	yes	characteristic
161	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₄	#576	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	16	4	no	—
165	C₂	C₂×#460	1	1	yes	characteristic
166	C₂	#3058	1	1	yes	characteristic
167	C₂	C₂×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#493	C₂
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
1. #3208	C₂	semidirect
3. #2960	C₂	semidirect
4. #1688	C₂	semidirect
5. #1668	C₂	semidirect
6. #1564	C₂	semidirect
7. #1545	C₂	semidirect
18. #460	C₂²	semidirect
21. #446	C₂²	semidirect
23. C₈×Q₈	C₂²	semidirect
29. #335	C₂²	semidirect
30. #334	C₂²	semidirect
31. #331	C₂²	semidirect
97. C₂×C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3208	2
C₂×#493	2
#2960	2
#1688	2
#1668	2
#1564	2
#1545	2