Finite Group Databasebeta

← #8301

Subgroups — #8301

Order 256.

761 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2#1779C212yesmaximal, complemented
3C2×#324C212yesmaximal, complemented
4C2×#323C212yesmaximal, complemented
5C23×D4C411yescharacteristic
6C24×C4C411yescharacteristic
7C24×C4C2211yescharacteristic
8C2×C4×D4—24no—
9C2×C4×D4—24no—
10C2×C4×D4—24no—
11C22×C4⋊C4C2211yescharacteristic
12C22×C22⋊C4C411yescharacteristic
13C22×C22⋊C4C2211yescharacteristic
14C22×C42C411yescharacteristic
15C2×C22⋊C8—24no—
16C2×C22⋊C8C2214yes—
17#324—24no—
18#323—24no—
19C25C2×C411yescharacteristic
20C25C2×C411yescharacteristic
21C22×D4—22no—
22C22×D4—22no—
23C22×D4—22no—
24C22×D4—22no—
25C22×D4—22no—
26C22×D4—22no—
27C22×D4C814yescomplemented
28C23×C4—22no—
29C23×C4—22no—
30C23×C4—22no—
31C23×C4—22no—
32C23×C4—22no—
33C23×C4—44no—
34C23×C4—44no—
35C23×C4C2×C411yescharacteristic
36C23×C4C2×C411yescharacteristic
37C23×C4C2×C411yescharacteristic
38C23×C4C2×C411yescharacteristic
39C23×C4D414yes—
40C23×C4C2311yescharacteristic
41C22×C8—24no—
42C4×D4—232no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4D414yes—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4D414yes—
52C2×C42—22no—
53C2×C42—22no—
54C2×C42—22no—
55C22⋊C8—24no—
56C22⋊C8—24no—
57C22⋊C8—48no—
58C24—21no—
59C24—21no—
60C24—21no—
61C24—21no—
62C24—21no—
63C24—21no—
64C24—21no—
65C24—21no—
66C24—42no—
67C24—42no—
68C24—42no—
69C24—42no—
70C24C22⋊C412yes—
71C24C22⋊C412yes—
72C24C2×C812yes—
73C24C2×C812yes—
74C24M1612yes—
75C24M1612yes—
76C24C22×C411yescharacteristic
77C2×D4—28no—
78C2×D4—28no—
79C2×D4—28no—
80C2×D4—28no—
81C2×D4—28no—
82C2×D4—28no—
83C2×D4—28no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—24no—
91C22×C4—28no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—28no—
97C22×C4—28no—
98C22×C4—44no—
99C22×C4—42no—
100C22×C4—48no—
101C22×C4—48no—
102C22×C4—48no—
103C22×C4—48no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C22⋊C412yes—
109C22×C4C22⋊C412yes—
110C22×C4C22⋊C412yes—
111C22×C4C22⋊C412yes—
112C22×C4C2×C812yes—
113C22×C4M1612yes—
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C2×C8—48no—
118C2×C8—44no—
119C4⋊C4—28no—
120C22⋊C4—28no—
121C22⋊C4—28no—
122C42—28no—
123C23—22no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23—22no—
129C23—22no—
130C23—22no—
131C23—22no—
132C23—22no—
133C23—22no—
134C23—22no—
135C23—22no—
136C23—22no—
137C23—24no—
138C23—24no—
139C23—44no—
140C23—44no—
141C23—42no—
142C23—42no—
143C23—44no—
144C23—44no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—44no—
149C23—44no—
150C23—44no—
151C23—44no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23C22⋊C814yes—
158C23C22⋊C814yescomplemented
159C23C2×C22⋊C411yescharacteristic
160C23C2×C22⋊C411yescharacteristic
161C23C2×C22⋊C411yescharacteristic
162C23C22×C811yescharacteristic
163C23C2×M1611yescharacteristic
164C23C2×M1611yescharacteristic
165C23C2×M1611yescharacteristic
166D4—232no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—416no—
183C2×C4—416no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4C22⋊C814yescomplemented
191C2×C4#9914yes—
192C2×C4#12014yes—
193C8—84no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—22no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—28no—
203C22—24no—
204C22—24no—
205C22—44no—
206C22—44no—
207C22—44no—
208C22—44no—
209C22—44no—
210C22—44no—
211C22—44no—
212C22—44no—
213C22—44no—
214C22—44no—
215C22—44no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22#37911yescharacteristic
221C22C2×C22⋊C811yescharacteristic
222C22C2×C22⋊C811yescharacteristic
223C22C2×C22⋊C811yescharacteristic
224C22#40711yescharacteristic
225C22#40711yescharacteristic
226C22#40711yescharacteristic
227C22C2×#9912yes—
228C22#41312yes—
229C4—24no—
230C4—24no—
231C4—24no—
232C4—42no—
233C4—42no—
234C2—21no—
235C2—21no—
236C2—21no—
237C2—21no—
238C2—44no—
239C2—44no—
240C2#145912yes—
241C2#146512yes—
242C2#177911yescharacteristic
243C2#178111yescharacteristic
244C2#178511yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC23×D4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1779C2semidirect
3. C2×#324C2semidirect
4. C2×#323C2semidirect
27. C22×D4C8semidirect
158. C23C22⋊C8semidirect
190. C2×C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
#17792
C2×#3242
C2×#3232