Finite Group Databasebeta

Subgroups — #9224

Order 256.

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

normal subgroups only

This group has 235 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×#477	C₂	1	1	yes	maximal, characteristic
2	C₂×#446	C₂	1	1	yes	maximal, characteristic
3	#1990	C₂	1	1	yes	maximal, characteristic
4	#1803	C₂	1	1	yes	maximal, characteristic
5	C₂×#333	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	#477	—	2	2	no	—
11	#477	C₄	1	4	yes	complemented
12	#446	—	2	2	no	—
13	#446	—	2	2	no	—
14	C₂×#107	—	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×#97	—	2	1	no	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#384	C₂²	1	1	yes	characteristic
24	#340	—	2	1	no	—
25	#340	—	2	1	no	—
26	#340	—	2	1	no	—
27	#333	—	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	#107	—	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	#97	—	2	4	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₂×Q₈	—	2	1	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	2	1	no	—
69	C₂×Q₈	—	2	1	no	—
70	C₂×Q₈	—	2	1	no	—
71	C₂×Q₈	—	4	1	no	—
72	C₂×Q₈	—	8	1	no	—
73	C₂×Q₈	—	8	1	no	—
74	C₂×Q₈	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	4	1	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄	#41	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	Q₁₆	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	8	1	no	—
109	C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	#41	1	1	yes	characteristic
116	C₂³	#121	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	8	2	no	—
122	Q₈	—	8	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	8	1	no	—
137	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄	#120	1	1	yes	characteristic
141	C₂×C₄	#121	1	1	yes	characteristic
142	C₂×C₄	#123	1	1	yes	characteristic
143	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄	#135	1	1	yes	characteristic
145	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄	#137	1	1	yes	characteristic
147	C₂×C₄	#137	1	1	yes	characteristic
148	C₂×C₄	#137	1	1	yes	characteristic
149	C₂×C₄	#137	1	1	yes	characteristic
150	C₂×C₄	#137	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	4	2	no	—
153	C₈	—	4	1	no	—
154	C₈	—	4	2	no	—
155	C₈	—	8	1	no	—
156	C₂²	#386	1	1	yes	characteristic
157	C₂²	#462	1	1	yes	characteristic
158	C₂²	#464	1	1	yes	characteristic
159	C₂²	#468	1	1	yes	characteristic
160	C₂²	#470	1	1	yes	characteristic
161	C₂²	#471	1	1	yes	characteristic
162	C₂²	#472	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	8	2	no	—
168	C₄	—	16	1	no	—
169	C₄	#417	1	1	yes	characteristic
170	C₄	#419	1	1	yes	characteristic
171	C₄	C₄×Q₁₆	1	1	yes	characteristic
172	C₄	#441	1	1	yes	characteristic
173	C₄	#451	1	1	yes	characteristic
174	C₄	#452	1	1	yes	characteristic
175	C₄	#468	1	1	yes	characteristic
176	C₄	#470	1	1	yes	characteristic
177	C₂	#1692	1	1	yes	characteristic
178	C₂	#1696	1	1	yes	characteristic
179	C₂	#1698	1	1	yes	characteristic
180	C₂	#1704	1	1	yes	characteristic
181	C₂	#1878	1	1	yes	characteristic
182	C₂	#1894	1	1	yes	characteristic
183	C₂	#1907	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. #477	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#477	2
C₂×#446	2
#1990	2
#1803	2
C₂×#333	2
#1296	2
#1277	2