Finite Group Databasebeta

Subgroups — #8993

Order 256.

238 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 77 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	#3076	C₂	1	1	yes	maximal, characteristic
2	#2906	C₂	1	1	yes	maximal, characteristic
3	#1493	C₂	1	1	yes	maximal, characteristic, complemented
4	#1481	C₂	1	1	yes	maximal, characteristic, complemented
5	#1476	C₂	1	1	yes	maximal, characteristic
6	#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	—	2	1	no	—
9	#533	C₂²	1	1	yes	characteristic
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#479	—	2	1	no	—
12	#478	—	2	1	no	—
13	#477	C₄	1	1	yes	characteristic
14	#476	C₄	1	1	yes	characteristic
15	#475	C₄	1	1	yes	characteristic
16	#474	C₄	1	1	yes	characteristic
17	#431	C₂²	1	1	yes	characteristic
18	#427	—	2	1	no	—
19	#424	—	2	1	no	—
20	#424	—	2	2	no	—
21	#423	—	2	1	no	—
22	#423	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈	—	2	1	no	—
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	#416	—	2	1	no	—
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	Q₈⋊C₈	—	2	1	no	—
29	Q₈⋊C₈	—	2	1	no	—
30	Q₈⋊C₈	—	2	1	no	—
31	Q₈⋊C₈	—	2	2	no	—
32	D₄⋊C₈	—	2	1	no	—
33	C₂×#41	D₄	1	1	yes	characteristic
34	#131	—	4	2	no	—
35	#131	—	4	2	no	—
36	C₂×M₁₆	—	2	2	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
44	#125	—	2	1	no	—
45	C₂²⋊Q₈	—	2	1	no	—
46	C₂²⋊Q₈	—	2	1	no	—
47	C₄×Q₈	—	2	1	no	—
48	C₄×Q₈	—	2	1	no	—
49	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₈	D₄	1	1	yes	characteristic
51	C₄×D₄	—	2	1	no	—
52	C₄×D₄	—	2	1	no	—
53	C₄×D₄	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄	D₄	1	1	yes	characteristic
55	#117	—	2	1	no	—
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	#107	—	2	1	no	—
62	#106	—	2	1	no	—
63	#106	—	2	1	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₄⋊C₈	D₄	1	1	yes	characteristic
74	C₄⋊C₈	D₄	1	2	yes	—
75	#103	—	2	1	no	—
76	#103	—	2	1	no	—
77	#102	—	2	1	no	—
78	#102	—	2	1	no	—
79	C₂²⋊C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	#41	—	2	1	no	—
84	#41	—	2	1	no	—
85	#41	SD₁₆	1	2	yes	—
86	#41	Q₁₆	1	2	yes	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×Q₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	M₁₆	—	4	4	no	—
99	M₁₆	—	4	1	no	—
100	M₁₆	—	4	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₄⋊C₄	—	2	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	—	2	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
124	C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₂²⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	4	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	4	1	no	—
133	C₄²	C₂²×C₄	1	1	yes	characteristic
134	C₄²	C₂×D₄	1	1	yes	characteristic
135	C₄²	C₂×D₄	1	1	yes	characteristic
136	C₄²	C₂×D₄	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	C₂³	#123	1	1	yes	characteristic
139	Q₈	—	2	1	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	#103	1	2	yes	—
144	D₄	—	2	2	no	—
145	D₄	—	2	2	no	—
146	D₄	—	2	1	no	—
147	D₄	#103	1	2	yes	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄	#120	1	1	yes	characteristic
165	C₂×C₄	#121	1	1	yes	characteristic
166	C₂×C₄	#121	1	1	yes	characteristic
167	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
169	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄	#137	1	1	yes	characteristic
171	C₈	—	4	2	no	—
172	C₈	—	4	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	4	2	no	—
176	C₈	—	4	1	no	—
177	C₈	—	8	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	1	no	—
183	C₂²	#386	1	1	yes	characteristic
184	C₂²	#483	1	1	yes	characteristic
185	C₂²	#484	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	—	8	1	no	—
193	C₄	—	8	1	no	—
194	C₄	C₂×#103	1	1	yes	characteristic
195	C₄	#417	1	1	yes	characteristic
196	C₄	#443	1	1	yes	characteristic
197	C₄	#444	1	1	yes	characteristic
198	C₄	#450	1	1	yes	characteristic
199	C₄	#451	1	1	yes	characteristic
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	—	4	2	no	—
203	C₂	#1872	1	1	yes	characteristic
204	C₂	#1876	1	1	yes	characteristic
205	C₂	#1881	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#386
prime radical	C₂×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ 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
3. #1493	C₂	semidirect
4. #1481	C₂	semidirect
6. #1301	C₂	semidirect
7. #1277	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3076	2
#2906	2
#1493	2
#1481	2
#1476	2
#1301	2
#1277	2