Finite Group Databasebeta

Subgroups — #8149

Order 256.

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

normal subgroups only

This group has 277 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	C₂×#431	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#431	—	2	2	no	—
10	#431	—	2	2	no	—
11	C₂×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈	C₄	1	2	yes	—
14	C₂×C₂²⋊C₈	C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
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	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#336	C₄	1	2	yes	—
22	#336	C₄	1	2	yes	—
23	#322	—	2	2	no	—
24	#322	—	2	2	no	—
25	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄²	—	2	1	no	—
35	C₂×C₄²	—	2	1	no	—
36	C₂×C₄²	—	2	1	no	—
37	C₂×C₄²	—	2	1	no	—
38	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₄⋊C₈	—	2	4	no	—
43	C₂²⋊C₈	—	2	4	no	—
44	C₂²⋊C₈	—	2	4	no	—
45	C₂²⋊C₈	—	2	4	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	D₄	1	2	yes	—
57	C₄×C₈	D₄	1	2	yes	—
58	C₂⁴	C₄²	1	1	yes	characteristic
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	—	2	1	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	2	2	no	—
70	C₂²×C₄	C₄²	1	1	yes	characteristic
71	C₂²×C₄	C₄²	1	1	yes	characteristic
72	C₂²×C₄	C₄²	1	1	yes	characteristic
73	C₂²×C₄	M₁₆	1	4	yes	—
74	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	C₂²⋊C₄	1	4	yes	—
94	C₂×C₈	M₁₆	1	4	yes	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₄²	—	2	1	no	—
100	C₄²	—	2	1	no	—
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×D₄	1	1	yes	characteristic
103	C₄²	#41	1	1	yes	characteristic
104	C₄²	#41	1	1	yes	characteristic
105	C₂³	—	2	1	no	—
106	C₂³	—	2	1	no	—
107	C₂³	—	2	1	no	—
108	C₂³	—	2	1	no	—
109	C₂³	—	2	1	no	—
110	C₂³	—	2	1	no	—
111	C₂³	#97	1	2	yes	—
112	C₂³	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	#97	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄	#117	1	1	yes	characteristic
130	C₂×C₄	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄	C₂×M₁₆	1	2	yes	—
135	C₂×C₄	C₂×M₁₆	1	2	yes	—
136	C₂×C₄	#131	1	2	yes	—
137	C₂×C₄	#131	1	2	yes	—
138	C₂×C₄	#131	1	2	yes	—
139	C₂×C₄	#131	1	2	yes	—
140	C₈	—	2	4	no	—
141	C₈	—	2	4	no	—
142	C₈	—	4	2	no	—
143	C₈	—	4	2	no	—
144	C₈	—	4	2	no	—
145	C₈	—	4	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²	C₂×#97	1	1	yes	characteristic
155	C₂²	C₄×M₁₆	1	1	yes	characteristic
156	C₂²	#405	1	1	yes	characteristic
157	C₂²	#405	1	1	yes	characteristic
158	C₂²	#405	1	1	yes	characteristic
159	C₂²	#405	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	#408	1	2	yes	—
164	C₄	#433	1	2	yes	—
165	C₄	#435	1	2	yes	—
166	C₄	#435	1	2	yes	—
167	C₄	#435	1	2	yes	—
168	C₄	#436	1	2	yes	—
169	C₂	—	4	2	no	—
170	C₂	#1451	1	1	yes	characteristic
171	C₂	#1452	1	1	yes	characteristic
172	C₂	#1453	1	1	yes	characteristic
173	C₂	#1454	1	1	yes	characteristic
174	C₂	#1750	1	1	yes	characteristic
175	C₂	#1752	1	1	yes	characteristic
176	C₂	#1753	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂×#322	C₂	semidirect
5. C₂×#322	C₂	semidirect
6. #1273	C₂	semidirect
7. #1273	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431	2
C₄×C₂²⋊C₈	2
C₂×#322	2
#1273	2