Finite Group Databasebeta

Subgroups — #19205

Order 256.

339 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 121 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	#3246	C₂	1	2	yes	maximal, complemented
2	#3147	C₂	1	1	yes	maximal, characteristic
3	#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	#1690	C₂	1	2	yes	maximal, complemented
5	#1688	C₂	1	2	yes	maximal, complemented
6	#1666	C₂	1	4	yes	maximal, complemented
7	C₂×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	#1547	C₂	1	2	yes	maximal, complemented
9	C₂×#135	—	2	1	no	—
10	C₂×#135	—	2	1	no	—
11	#533	C₂²	1	1	yes	characteristic
12	#532	C₂²	1	2	yes	—
13	#495	—	2	1	no	—
14	#495	—	2	1	no	—
15	#494	C₂²	1	1	yes	characteristic
16	#493	C₂²	1	1	yes	characteristic
17	#492	C₂²	1	2	yes	—
18	#476	—	2	2	no	—
19	#475	C₂²	1	4	yes	—
20	#465	—	2	4	no	—
21	#461	—	2	2	no	—
22	#460	C₂²	1	4	yes	—
23	#446	—	2	1	no	—
24	#446	C₂²	1	2	yes	—
25	C₈×Q₈	—	2	1	no	—
26	C₈×Q₈	C₂²	1	2	yes	—
27	#424	—	2	1	no	—
28	#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	2	yes	—
30	#405	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#334	C₂²	1	2	yes	—
33	#334	C₂²	1	2	yes	—
34	#332	C₂²	1	4	yes	—
35	#331	C₂²	1	4	yes	—
36	C₂×#41	—	2	2	no	—
37	#135	—	4	4	no	—
38	#135	—	4	4	no	—
39	C₂×Q₁₆	—	2	1	no	—
40	C₂×Q₁₆	—	2	1	no	—
41	C₂×SD₁₆	—	2	4	no	—
42	C₂×SD₁₆	—	2	2	no	—
43	C₂×SD₁₆	—	2	2	no	—
44	C₂×D₈	—	2	1	no	—
45	C₂×D₈	—	2	1	no	—
46	C₂×M₁₆	—	2	1	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₈	D₄	1	1	yes	characteristic
50	C₄⋊Q₈	D₄	1	1	yes	characteristic
51	C₄⋊Q₈	D₄	1	1	yes	characteristic
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	#127	C₂³	1	2	yes	—
54	#125	—	2	1	no	—
55	#124	—	2	2	no	—
56	C₂²⋊Q₈	—	2	1	no	—
57	C₂²⋊Q₈	D₄	1	2	yes	—
58	#121	—	2	2	no	—
59	#121	—	2	2	no	—
60	C₄×Q₈	—	2	1	no	—
61	C₄×Q₈	C₂³	1	2	yes	—
62	C₄×D₄	—	2	2	no	—
63	C₄×D₄	—	2	2	no	—
64	#117	—	2	1	no	—
65	C₂×C₄²	C₂³	1	1	yes	characteristic
66	#106	—	2	4	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	C₂³	1	2	yes	—
72	C₄⋊C₈	C₂³	1	4	yes	—
73	#103	—	2	4	no	—
74	#102	—	2	4	no	—
75	#97	—	2	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	D₄	1	1	yes	characteristic
78	C₄×C₈	D₄	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	#41	—	4	4	no	—
82	#41	—	4	4	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	C₂×D₄	1	2	yes	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	#41	1	1	yes	characteristic
94	Q₁₆	—	4	2	no	—
95	Q₁₆	—	4	2	no	—
96	SD₁₆	—	4	4	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	8	4	no	—
99	D₈	—	4	2	no	—
100	D₈	—	4	2	no	—
101	M₁₆	—	4	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	SD₁₆	1	4	yes	complemented
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	4	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	#41	1	1	yes	characteristic
126	C₄²	C₂⁴	1	1	yes	characteristic
127	C₂³	—	4	2	no	—
128	C₂³	#121	1	1	yes	characteristic
129	Q₈	—	2	4	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	4	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	#121	1	1	yes	characteristic
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	C₂×SD₁₆	1	2	yes	—
149	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂×#41	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	4	no	—
157	C₈	C₂×SD₁₆	1	2	yes	—
158	C₈	C₂×SD₁₆	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	8	2	no	—
164	C₂²	#460	1	2	yes	—
165	C₂²	C₂×#121	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	#460	1	2	yes	—
173	C₄	C₂²×SD₁₆	1	1	yes	characteristic
174	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₄	#577	1	2	yes	—
176	C₂	—	2	2	no	—
177	C₂	—	16	2	no	—
178	C₂	C₂×#460	1	1	yes	characteristic
179	C₂	#3060	1	1	yes	characteristic
180	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3147	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. #3246	C₂	semidirect
3. #2960	C₂	semidirect
4. #1690	C₂	semidirect
5. #1688	C₂	semidirect
6. #1666	C₂	semidirect
7. C₂×#334	C₂	semidirect
8. #1547	C₂	semidirect
106. 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
#3246	2
#3147	2
#2960	2
#1690	2
#1688	2
#1666	2
C₂×#334	2
#1547	2