Finite Group Databasebeta

Subgroups — #9672

Order 256.

233 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 73 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×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	#1479	C₂	1	1	yes	maximal, characteristic, complemented
4	#1478	C₂	1	1	yes	maximal, characteristic
5	#1461	C₂	1	1	yes	maximal, characteristic, complemented
6	#1460	C₂	1	1	yes	maximal, characteristic
7	#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄	C₂²	1	1	yes	characteristic
11	C₄×SD₁₆	—	2	1	no	—
12	C₄×SD₁₆	—	2	1	no	—
13	C₄×SD₁₆	—	2	1	no	—
14	C₄×SD₁₆	—	2	1	no	—
15	#431	C₂²	1	1	yes	characteristic
16	#431	C₂²	1	1	yes	characteristic
17	#431	C₂²	1	1	yes	characteristic
18	#431	C₂²	1	1	yes	characteristic
19	C₂×#106	C₄	1	1	yes	characteristic
20	C₂×#103	C₄	1	1	yes	characteristic
21	C₂×#102	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#335	—	2	1	no	—
24	#334	—	2	1	no	—
25	Q₈⋊C₈	—	2	2	no	—
26	D₄⋊C₈	—	2	2	no	—
27	#324	—	2	2	no	—
28	#323	—	2	2	no	—
29	#322	—	2	1	no	—
30	#322	—	2	1	no	—
31	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄	—	2	1	no	—
34	C₂×SD₁₆	—	2	2	no	—
35	C₂×SD₁₆	—	2	2	no	—
36	C₂×SD₁₆	—	2	1	no	—
37	C₂×SD₁₆	—	2	1	no	—
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈	—	2	1	no	—
41	C₄×Q₈	—	2	1	no	—
42	C₄×Q₈	—	2	1	no	—
43	C₄×Q₈	—	2	1	no	—
44	C₄×D₄	—	2	1	no	—
45	C₄×D₄	—	2	1	no	—
46	C₄×D₄	—	2	1	no	—
47	C₄×D₄	—	2	1	no	—
48	C₂×C₄⋊C₄	—	2	1	no	—
49	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊C₄	—	2	1	no	—
52	C₂×C₄²	—	2	1	no	—
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	#106	—	2	1	no	—
55	#106	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	2	1	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	D₄	1	2	yes	—
60	#103	—	2	1	no	—
61	#103	—	2	1	no	—
62	#102	—	2	1	no	—
63	#102	—	2	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	2	1	no	—
66	C₂²⋊C₈	—	2	1	no	—
67	C₂²⋊C₈	D₄	1	2	yes	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	D₄	1	1	yes	characteristic
73	C₄×C₈	D₄	1	1	yes	characteristic
74	C₄×C₈	D₄	1	1	yes	characteristic
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₂⁴	—	2	1	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	M₁₆	1	2	yes	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	2	1	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	M₁₆	1	2	yes	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	4	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	8	2	no	—
95	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	SD₁₆	—	2	4	no	—
99	SD₁₆	—	2	4	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₂²⋊C₄	—	4	1	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₄²	—	4	1	no	—
121	C₄²	—	4	1	no	—
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	#41	1	1	yes	characteristic
126	C₂³	—	2	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	4	1	no	—
129	C₂³	—	8	1	no	—
130	C₂³	—	8	1	no	—
131	C₂³	C₄×D₄	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	D₄	—	2	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄	#121	1	1	yes	characteristic
154	C₂×C₄	#121	1	1	yes	characteristic
155	C₂×C₄	#124	1	1	yes	characteristic
156	C₂×C₄	#127	1	1	yes	characteristic
157	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
159	C₂×C₄	#131	1	1	yes	characteristic
160	C₂×C₄	#131	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	4	2	no	—
166	C₈	—	4	2	no	—
167	C₈	—	8	1	no	—
168	C₈	—	8	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	8	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	8	1	no	—
175	C₂²	—	8	1	no	—
176	C₂²	#390	1	1	yes	characteristic
177	C₂²	#435	1	1	yes	characteristic
178	C₂²	#435	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	8	1	no	—
185	C₄	#407	1	1	yes	characteristic
186	C₄	#408	1	1	yes	characteristic
187	C₄	#435	1	1	yes	characteristic
188	C₄	#435	1	1	yes	characteristic
189	C₄	#471	1	2	yes	—
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	—	8	2	no	—
193	C₂	#1955	1	1	yes	characteristic
194	C₂	#1976	1	1	yes	characteristic
195	C₂	#1977	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	C₂²×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
2. #1564	C₂	semidirect
3. #1479	C₂	semidirect
5. #1461	C₂	semidirect
7. #1452	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆	2
#1564	2
#1479	2
#1478	2
#1461	2
#1460	2
#1452	2