Finite Group Databasebeta

Subgroups — #9228

Order 256.

243 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 85 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×#475	C₂	1	1	yes	maximal, characteristic
2	C₂×#446	C₂	1	1	yes	maximal, characteristic
3	#1956	C₂	1	1	yes	maximal, characteristic
4	#1803	C₂	1	1	yes	maximal, characteristic
5	C₂×#332	C₂	1	1	yes	maximal, characteristic
6	#1296	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
10	#475	—	2	2	no	—
11	#475	C₄	1	4	yes	complemented
12	#446	—	2	2	no	—
13	#446	—	2	2	no	—
14	C₂×#106	—	2	1	no	—
15	C₂×C₄⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#103	—	2	1	no	—
20	C₂×#103	—	2	1	no	—
21	C₂×#103	—	2	1	no	—
22	C₂×#103	—	2	1	no	—
23	C₂×#97	—	2	1	no	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#384	C₂²	1	1	yes	characteristic
26	#340	—	2	1	no	—
27	#332	—	2	2	no	—
28	C₂²×Q₈	—	2	1	no	—
29	C₂²×Q₈	D₄	1	1	yes	characteristic
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₄×Q₈	—	2	1	no	—
40	C₄×Q₈	—	2	1	no	—
41	C₄×Q₈	C₂×C₄	1	2	yes	—
42	C₄×Q₈	D₄	1	2	yes	—
43	C₂×C₄⋊C₄	—	2	1	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄⋊C₄	—	2	1	no	—
48	C₂×C₄⋊C₄	—	2	1	no	—
49	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₄²	—	2	1	no	—
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	#106	—	2	4	no	—
53	C₄⋊C₈	—	2	4	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	2	1	no	—
58	C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₄⋊C₈	D₄	1	2	yes	—
60	#103	—	2	4	no	—
61	#103	—	4	2	no	—
62	#103	—	4	2	no	—
63	#103	—	4	2	no	—
64	#97	—	2	4	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	1	no	—
70	C₂×Q₈	—	2	1	no	—
71	C₂×Q₈	—	2	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	4	1	no	—
76	C₂×Q₈	—	4	1	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×Q₈	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	#41	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	SD₁₆	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	8	1	no	—
113	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	C₂×D₄	1	1	yes	characteristic
119	C₄²	#41	1	1	yes	characteristic
120	C₂³	#121	1	1	yes	characteristic
121	Q₈	—	2	2	no	—
122	Q₈	—	2	2	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	8	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄	#120	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	#123	1	1	yes	characteristic
149	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄	#135	1	1	yes	characteristic
151	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄	#137	1	1	yes	characteristic
154	C₂×C₄	#137	1	1	yes	characteristic
155	C₂×C₄	#137	1	1	yes	characteristic
156	C₂×C₄	#137	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	4	2	no	—
159	C₈	—	4	1	no	—
160	C₈	—	4	2	no	—
161	C₈	—	8	1	no	—
162	C₂²	#386	1	1	yes	characteristic
163	C₂²	#460	1	1	yes	characteristic
164	C₂²	#464	1	1	yes	characteristic
165	C₂²	#468	1	1	yes	characteristic
166	C₂²	#470	1	1	yes	characteristic
167	C₂²	#471	1	1	yes	characteristic
168	C₂²	#473	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	8	2	no	—
174	C₄	—	16	1	no	—
175	C₄	#417	1	1	yes	characteristic
176	C₄	#419	1	1	yes	characteristic
177	C₄	C₄×SD₁₆	1	1	yes	characteristic
178	C₄	#440	1	1	yes	characteristic
179	C₄	#448	1	1	yes	characteristic
180	C₄	#449	1	1	yes	characteristic
181	C₄	#468	1	1	yes	characteristic
182	C₄	#470	1	1	yes	characteristic
183	C₂	#1686	1	1	yes	characteristic
184	C₂	#1690	1	1	yes	characteristic
185	C₂	#1700	1	1	yes	characteristic
186	C₂	#1704	1	1	yes	characteristic
187	C₂	#1880	1	1	yes	characteristic
188	C₂	#1894	1	1	yes	characteristic
189	C₂	#1908	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
11. #475	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#475	2
C₂×#446	2
#1956	2
#1803	2
C₂×#332	2
#1296	2
#1277	2