Finite Group Databasebeta

Subgroups — #9054

Order 256.

235 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 71 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	#3167	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	#1980	C₂	1	1	yes	maximal, characteristic
4	#1503	C₂	1	1	yes	maximal, characteristic, complemented
5	#1480	C₂	1	1	yes	maximal, characteristic
6	#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	#498	C₄	1	1	yes	characteristic, complemented
11	#492	C₄	1	1	yes	characteristic, complemented
12	#487	C₄	1	1	yes	characteristic, complemented
13	#486	C₄	1	1	yes	characteristic, complemented
14	#465	—	2	2	no	—
15	#446	—	2	2	no	—
16	#446	—	2	1	no	—
17	C₈×Q₈	—	2	1	no	—
18	C₄×SD₁₆	—	2	2	no	—
19	#431	C₂²	1	1	yes	characteristic
20	#424	—	2	1	no	—
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×#104	—	2	1	no	—
23	C₂×#104	—	2	1	no	—
24	#405	—	2	1	no	—
25	C₄×M₁₆	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	#328	—	2	2	no	—
29	D₄⋊C₈	—	2	1	no	—
30	D₄⋊C₈	—	2	1	no	—
31	C₂×#41	—	2	1	no	—
32	C₂×SD₁₆	—	2	2	no	—
33	C₂×M₁₆	D₄	1	1	yes	characteristic
34	C₂×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
40	#127	C₂×C₄	1	1	yes	characteristic
41	#125	—	2	1	no	—
42	#124	—	2	1	no	—
43	C₂²⋊Q₈	—	2	1	no	—
44	C₂²⋊Q₈	D₄	1	2	yes	—
45	#121	—	2	1	no	—
46	#121	—	2	1	no	—
47	C₄×Q₈	—	2	1	no	—
48	C₄×Q₈	D₄	1	2	yes	—
49	C₄×D₄	—	2	1	no	—
50	C₄×D₄	—	2	1	no	—
51	#117	—	2	1	no	—
52	C₂×C₄⋊C₄	—	2	1	no	—
53	C₂×C₄²	—	2	1	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#106	—	2	2	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₄⋊C₈	D₄	1	1	yes	characteristic
62	#104	—	4	2	no	—
63	#104	—	4	2	no	—
64	#103	—	2	2	no	—
65	#102	—	2	2	no	—
66	C₂²⋊C₈	—	2	1	no	—
67	#97	—	2	2	no	—
68	#97	—	2	1	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	#95	—	2	1	no	—
76	#41	—	4	1	no	—
77	#41	—	4	1	no	—
78	#41	—	4	1	no	—
79	#41	—	4	1	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	C₂²⋊C₄	1	2	yes	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	4	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	#41	1	1	yes	characteristic
92	SD₁₆	—	4	4	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	SD₁₆	1	2	yes	complemented
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	4	1	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	4	1	no	—
122	C₄²	—	4	1	no	—
123	C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₄²	#41	1	1	yes	characteristic
127	C₂³	—	4	1	no	—
128	C₂³	#121	1	1	yes	characteristic
129	Q₈	—	4	2	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	4	1	no	—
139	D₄	—	8	1	no	—
140	D₄	—	8	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄	#120	1	1	yes	characteristic
156	C₂×C₄	#121	1	1	yes	characteristic
157	C₂×C₄	#123	1	1	yes	characteristic
158	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
159	C₂×C₄	#135	1	1	yes	characteristic
160	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₄	#137	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	—	4	1	no	—
165	C₈	—	4	1	no	—
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	8	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	8	1	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	1	no	—
174	C₂²	#386	1	1	yes	characteristic
175	C₂²	#460	1	1	yes	characteristic
176	C₂²	#464	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	8	2	no	—
183	C₄	—	8	1	no	—
184	C₄	C₄×SD₁₆	1	1	yes	characteristic
185	C₄	#440	1	1	yes	characteristic
186	C₄	#443	1	1	yes	characteristic
187	C₄	#444	1	1	yes	characteristic
188	C₄	#448	1	1	yes	characteristic
189	C₄	#449	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	—	16	1	no	—
193	C₂	#1876	1	1	yes	characteristic
194	C₂	#1880	1	1	yes	characteristic
195	C₂	#1884	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	#532	C₂²
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
2. #2962	C₂	semidirect
4. #1503	C₂	semidirect
6. #1279	C₂	semidirect
7. #1278	C₂	semidirect
10. #498	C₄	semidirect
11. #492	C₄	semidirect
12. #487	C₄	semidirect
13. #486	C₄	semidirect
96. M₁₆	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3167	2
#2962	2
#1980	2
#1503	2
#1480	2
#1279	2
#1278	2