Finite Group Databasebeta

Subgroups — #9311

Order 256.

257 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 63 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	#3115	C₂	1	1	yes	maximal, characteristic
2	#2875	C₂	1	1	yes	maximal, characteristic, complemented
3	#1530	C₂	1	1	yes	maximal, characteristic, complemented
4	#1488	C₂	1	1	yes	maximal, characteristic, complemented
5	#1484	C₂	1	1	yes	maximal, characteristic, complemented
6	#1302	C₂	1	1	yes	maximal, characteristic, complemented
7	#1291	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	—	2	1	no	—
9	#532	C₂²	1	1	yes	characteristic
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#482	—	2	1	no	—
12	#475	C₄	1	1	yes	characteristic
13	#474	C₄	1	1	yes	characteristic
14	#460	C₄	1	1	yes	characteristic
15	#459	C₄	1	1	yes	characteristic
16	#449	—	2	1	no	—
17	#442	—	2	1	no	—
18	#440	—	2	1	no	—
19	#432	C₂²	1	1	yes	characteristic
20	#432	C₂²	1	1	yes	characteristic
21	#431	C₂²	1	1	yes	characteristic
22	#405	—	2	1	no	—
23	#405	—	2	2	no	—
24	C₄×M₁₆	—	2	1	no	—
25	C₄×M₁₆	C₂²	1	1	yes	characteristic
26	C₂×#97	—	2	1	no	—
27	C₂×#97	C₂²	1	1	yes	characteristic
28	#329	—	2	1	no	—
29	#329	—	2	1	no	—
30	Q₈⋊C₈	—	2	1	no	—
31	D₄⋊C₈	—	2	1	no	—
32	D₄⋊C₈	—	2	2	no	—
33	C₂×#41	—	2	1	no	—
34	C₂×#41	D₄	1	1	yes	characteristic
35	C₂×SD₁₆	—	2	1	no	—
36	C₂×D₈	—	2	1	no	—
37	#131	—	2	4	no	—
38	#131	—	2	4	no	—
39	C₂×M₁₆	—	2	2	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	—	2	1	no	—
43	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
44	#127	C₂×C₄	1	1	yes	characteristic
45	#124	—	2	1	no	—
46	#121	—	2	1	no	—
47	#121	—	2	1	no	—
48	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
49	C₄×Q₈	D₄	1	1	yes	characteristic
50	C₄×D₄	—	2	1	no	—
51	C₄×D₄	—	2	1	no	—
52	C₄×D₄	—	2	1	no	—
53	C₄×D₄	—	2	1	no	—
54	C₄×D₄	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄	D₄	1	1	yes	characteristic
56	#117	—	2	1	no	—
57	#117	D₄	1	1	yes	characteristic
58	C₂×C₄²	—	2	1	no	—
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	#107	—	2	1	no	—
61	#106	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
66	#103	—	2	1	no	—
67	#102	—	2	1	no	—
68	#102	—	2	1	no	—
69	#102	—	2	1	no	—
70	C₂²⋊C₈	—	2	1	no	—
71	C₂²⋊C₈	—	2	1	no	—
72	C₂²⋊C₈	—	2	1	no	—
73	#97	—	2	2	no	—
74	#97	—	2	1	no	—
75	#97	—	2	1	no	—
76	#97	—	2	1	no	—
77	#97	—	2	1	no	—
78	#97	D₄	1	1	yes	characteristic
79	#97	D₄	1	1	yes	characteristic
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	D₄	1	2	yes	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	#41	—	8	2	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×D₄	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄	#41	1	1	yes	characteristic
100	C₂²×C₄	#41	1	1	yes	characteristic
101	SD₁₆	—	4	2	no	—
102	D₈	—	4	2	no	—
103	M₁₆	—	2	8	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	C₂²×C₄	1	1	yes	characteristic
131	C₄²	C₂×D₄	1	1	yes	characteristic
132	C₄²	C₂×D₄	1	1	yes	characteristic
133	C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂³	—	2	1	no	—
135	C₂³	—	4	1	no	—
136	C₂³	#123	1	1	yes	characteristic
137	Q₈	—	2	1	no	—
138	Q₈	—	8	1	no	—
139	Q₈	#104	1	2	yes	—
140	D₄	—	2	4	no	—
141	D₄	—	2	1	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	8	1	no	—
146	D₄	—	8	2	no	—
147	D₄	#104	1	2	yes	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄	#120	1	1	yes	characteristic
164	C₂×C₄	#121	1	1	yes	characteristic
165	C₂×C₄	#121	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	2	no	—
168	C₈	—	2	4	no	—
169	C₈	—	4	1	no	—
170	C₈	—	4	1	no	—
171	C₈	—	8	1	no	—
172	C₈	—	8	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	8	1	no	—
180	C₂²	—	8	1	no	—
181	C₂²	—	8	1	no	—
182	C₂²	#386	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	4	1	no	—
190	C₄	—	4	1	no	—
191	C₄	—	8	1	no	—
192	C₄	C₂×#104	1	1	yes	characteristic
193	C₄	C₂×#104	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	4	2	no	—
196	C₂	—	16	1	no	—
197	C₂	#1881	1	1	yes	characteristic
198	C₂	#1896	1	1	yes	characteristic
199	C₂	#1903	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#3115	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #2875	C₂	semidirect
3. #1530	C₂	semidirect
4. #1488	C₂	semidirect
5. #1484	C₂	semidirect
6. #1302	C₂	semidirect
7. #1291	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3115	2
#2875	2
#1530	2
#1488	2
#1484	2
#1302	2
#1291	2