Finite Group Databasebeta

Subgroups — #9055

Order 256.

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

normal subgroups only

This group has 250 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	#2935	C₂	1	1	yes	maximal, characteristic
3	#1986	C₂	1	1	yes	maximal, characteristic, complemented
4	#1502	C₂	1	1	yes	maximal, characteristic
5	#1481	C₂	1	1	yes	maximal, characteristic, complemented
6	#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	—	2	1	no	—
9	#533	C₂²	1	1	yes	characteristic
10	#532	C₂²	1	1	yes	characteristic
11	#498	C₄	1	1	yes	characteristic, complemented
12	#492	C₄	1	1	yes	characteristic, complemented
13	#487	C₄	1	1	yes	characteristic, complemented
14	#486	C₄	1	1	yes	characteristic, complemented
15	#465	—	2	2	no	—
16	C₄×SD₁₆	—	2	2	no	—
17	#436	—	2	1	no	—
18	#435	—	2	2	no	—
19	C₈×D₄	—	2	1	no	—
20	#431	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	#408	—	2	1	no	—
23	C₄×M₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	#339	—	2	1	no	—
27	#339	—	2	1	no	—
28	#327	—	2	2	no	—
29	Q₈⋊C₈	—	2	1	no	—
30	Q₈⋊C₈	—	2	1	no	—
31	C₂×#41	—	2	1	no	—
32	C₂×SD₁₆	—	2	2	no	—
33	#131	—	4	2	no	—
34	#131	—	4	2	no	—
35	C₂×M₁₆	—	2	2	no	—
36	C₂×M₁₆	D₄	1	1	yes	characteristic
37	C₂×M₁₆	D₄	1	1	yes	characteristic
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	2	1	no	—
41	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
44	#127	C₂×C₄	1	1	yes	characteristic
45	#125	—	2	1	no	—
46	#124	—	2	1	no	—
47	C₂²⋊Q₈	—	2	1	no	—
48	C₂²⋊Q₈	—	2	1	no	—
49	#121	—	2	1	no	—
50	#121	D₄	1	2	yes	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×D₄	—	2	1	no	—
54	C₄×D₄	D₄	1	2	yes	—
55	#117	—	2	1	no	—
56	C₂×C₄⋊C₄	—	2	1	no	—
57	C₂×C₄²	—	2	1	no	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	#106	—	2	2	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₄⋊C₈	D₄	1	1	yes	characteristic
64	#103	—	2	2	no	—
65	#102	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₂²⋊C₈	—	2	1	no	—
69	#97	—	2	1	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	—	2	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₄	—	2	2	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	—	4	4	no	—
97	M₁₆	SD₁₆	1	2	yes	complemented
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	—	2	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	4	1	no	—
122	C₄²	—	4	1	no	—
123	C₄²	—	4	1	no	—
124	C₄²	—	4	1	no	—
125	C₄²	C₂²×C₄	1	1	yes	characteristic
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂×D₄	1	1	yes	characteristic
128	C₄²	#41	1	1	yes	characteristic
129	C₂³	—	2	2	no	—
130	C₂³	#121	1	1	yes	characteristic
131	Q₈	—	4	2	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄	#120	1	1	yes	characteristic
158	C₂×C₄	#121	1	1	yes	characteristic
159	C₂×C₄	#123	1	1	yes	characteristic
160	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₄	#135	1	1	yes	characteristic
162	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄	#137	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	4	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	8	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	#386	1	1	yes	characteristic
177	C₂²	#461	1	1	yes	characteristic
178	C₂²	#463	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	2	no	—
184	C₄	—	8	1	no	—
185	C₄	—	8	1	no	—
186	C₄	C₄×SD₁₆	1	1	yes	characteristic
187	C₄	#440	1	1	yes	characteristic
188	C₄	#443	1	1	yes	characteristic
189	C₄	#444	1	1	yes	characteristic
190	C₄	#450	1	1	yes	characteristic
191	C₄	#452	1	1	yes	characteristic
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	—	8	2	no	—
195	C₂	#1876	1	1	yes	characteristic
196	C₂	#1877	1	1	yes	characteristic
197	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
3. #1986	C₂	semidirect
5. #1481	C₂	semidirect
6. #1279	C₂	semidirect
7. #1278	C₂	semidirect
11. #498	C₄	semidirect
12. #492	C₄	semidirect
13. #487	C₄	semidirect
14. #486	C₄	semidirect
97. 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
#2935	2
#1986	2
#1502	2
#1481	2
#1279	2
#1278	2