Finite Group Databasebeta

Subgroups — #8206

Order 256.

277 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×#436	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆	—	2	1	no	—
9	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
10	#436	—	2	2	no	—
11	#436	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#102	C₄	1	2	yes	—
15	C₂×C₂²⋊C₈	—	2	1	no	—
16	C₂×#97	—	2	1	no	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#340	C₄	1	2	yes	—
22	#336	—	2	1	no	—
23	D₄⋊C₈	C₄	1	4	yes	complemented
24	C₈²	—	2	2	no	—
25	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄	—	2	1	no	—
27	C₂×M₁₆	—	2	4	no	—
28	C₂×M₁₆	—	4	2	no	—
29	C₂×M₁₆	—	4	2	no	—
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₄×D₄	—	2	1	no	—
37	C₄×D₄	—	2	1	no	—
38	C₄×D₄	C₂×C₄	1	2	yes	—
39	C₄×D₄	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄	—	2	1	no	—
43	C₂×C₄²	—	2	1	no	—
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₄⋊C₈	C₂×C₄	1	2	yes	—
49	#102	—	2	4	no	—
50	C₂²⋊C₈	—	2	4	no	—
51	#97	—	4	2	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂×C₄	1	2	yes	—
56	C₄×C₈	C₂×C₄	1	2	yes	—
57	C₄×C₈	D₄	1	2	yes	—
58	#95	—	2	1	no	—
59	C₂⁴	—	2	1	no	—
60	C₂×D₄	—	2	1	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	2	2	no	—
63	C₂×D₄	C₄²	1	2	yes	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	4	1	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	—	4	1	no	—
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄	#41	1	1	yes	characteristic
75	M₁₆	—	4	8	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈	M₁₆	1	4	yes	complemented
91	C₂×C₈	D₈	1	2	yes	—
92	C₂×C₈	SD₁₆	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	C₄²	1	2	yes	—
97	C₂²⋊C₄	—	2	2	no	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₄²	—	4	1	no	—
100	C₄²	—	4	1	no	—
101	C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₄²	C₂×D₄	1	1	yes	characteristic
104	C₄²	#41	1	1	yes	characteristic
105	C₂³	—	2	2	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	2	no	—
110	C₂³	C₄×D₄	1	1	yes	characteristic
111	D₄	—	2	2	no	—
112	D₄	—	2	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄	#117	1	1	yes	characteristic
128	C₂×C₄	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂×M₁₆	1	2	yes	—
132	C₂×C₄	#131	1	2	yes	—
133	C₂×C₄	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄	#135	1	1	yes	characteristic
136	C₂×C₄	#135	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	2	2	no	—
139	C₈	—	2	4	no	—
140	C₈	—	4	2	no	—
141	C₈	—	4	2	no	—
142	C₈	#102	1	4	yes	complemented
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	—	8	2	no	—
147	C₂²	—	8	2	no	—
148	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²	#436	1	2	yes	—
150	C₂²	C₄×D₈	1	1	yes	characteristic
151	C₂²	C₄×SD₁₆	1	1	yes	characteristic
152	C₂²	#443	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	C₄×M₁₆	1	1	yes	characteristic
158	C₄	#405	1	1	yes	characteristic
159	C₄	C₂×#102	1	1	yes	characteristic
160	C₄	#416	1	1	yes	characteristic
161	C₄	#435	1	2	yes	—
162	C₄	C₄×D₈	1	1	yes	characteristic
163	C₄	C₄×SD₁₆	1	1	yes	characteristic
164	C₂	—	8	2	no	—
165	C₂	#1589	1	2	yes	—
166	C₂	#1590	1	2	yes	—
167	C₂	#1753	1	1	yes	characteristic
168	C₂	C₄×#102	1	1	yes	characteristic
169	C₂	#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂³	C₄×D₄
prime radical	C₂²×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1770	C₂	semidirect
5. C₂×C₈²	C₂	semidirect
6. #1279	C₂	semidirect
7. #1277	C₂	semidirect
11. #436	C₄	semidirect
23. D₄⋊C₈	C₄	semidirect
90. C₂×C₈	M₁₆	semidirect
142. C₈	#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436	2
#1770	2
C₄×#102	2
C₂×D₄⋊C₈	2
C₂×C₈²	2
#1279	2
#1277	2