Finite Group Databasebeta

← #8712

Subgroups — #8712

Order 256.

470 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#380C211yesmaximal, characteristic
2#1779C211yesmaximal, characteristic
3C22⋊#95C211yesmaximal, characteristic
4#1294C214yesmaximal, complemented
5C24×C4C2211yescharacteristic
6C22×C4⋊C4C411yescharacteristic, complemented
7C22×C4⋊C4C411yescharacteristic, complemented
8C22×C22⋊C4—21no—
9C22×C22⋊C4—21no—
10C2×C22⋊C8—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8C2212yes—
13#380—21no—
14#380—21no—
15#380C412yescomplemented
16#380C2212yes—
17C2×#95—21no—
18C2×#95—21no—
19C2×#95C2212yes—
20C25D411yescharacteristic
21C23×C4—24no—
22C23×C4—21no—
23C23×C4—21no—
24C23×C4—22no—
25C23×C4—42no—
26C23×C4—42no—
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4C2311yescharacteristic
33C22×C8—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—42no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4C2×C412yes—
39C2×C4⋊C4C2×C412yes—
40C2×C22⋊C4—24no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—42no—
44C2×C22⋊C4—42no—
45C2×C22⋊C4—42no—
46C2×C22⋊C4—42no—
47C22⋊C8—24no—
48C22⋊C8—42no—
49C22⋊C8—42no—
50#95—22no—
51#95—22no—
52#95—24no—
53#95—42no—
54C24—22no—
55C24—22no—
56C24—22no—
57C24—22no—
58C24—41no—
59C24—41no—
60C24C22⋊C411yescharacteristic
61C24C22⋊C411yescharacteristic
62C24D812yes—
63C24SD1612yes—
64C24C2×D411yescharacteristic
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—24no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—416no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22×C411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4114yes—
103C2×C8—42no—
104C2×C8—42no—
105C2×C8—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C22⋊C4—48no—
111C22⋊C4—44no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—24no—
123C23—24no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—21no—
130C23—24no—
131C23—24no—
132C23—42no—
133C23—42no—
134C23—44no—
135C23—44no—
136C23—44no—
137C23—44no—
138C23#9912yes—
139C23#10212yes—
140C23#10212yes—
141C23#10412yes—
142C23C2×C22⋊C411yescharacteristic
143C23#12312yes—
144C23C2×D811yescharacteristic
145C23C2×SD1611yescharacteristic
146C23#13712yes—
147C2×C4—28no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—44no—
161C2×C4—44no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—84no—
175C2×C4—84no—
176C2×C4—82no—
177C2×C4#9912yes—
178C2×C4#11712yes—
179C2×C4#12312yes—
180C8—82no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—21no—
187C22—21no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—21no—
194C22—21no—
195C22—22no—
196C22—22no—
197C22—22no—
198C22—48no—
199C22—48no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22#32714yes—
205C22#38111yescharacteristic
206C22C2×#9911yescharacteristic
207C22C2×#9911yescharacteristic
208C22C2×#10211yescharacteristic
209C22#41911yescharacteristic
210C22C2×#10411yescharacteristic
211C22#42111yescharacteristic
212C22#48012yes—
213C22#48312yes—
214C4—22no—
215C4—22no—
216C4—22no—
217C4—41no—
218C4—84no—
219C4—82no—
220C4—82no—
221C2—22no—
222C2—24no—
223C2—22no—
224C2—42no—
225C2C2×#32711yescharacteristic
226C2#150811yescharacteristic
227C2#151111yescharacteristic
228C2#151911yescharacteristic
229C2#181911yescharacteristic
230C2#182311yescharacteristic
231C2#182611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
4. #1294C2semidirect
6. C22×C4⋊C4C4semidirect
7. C22×C4⋊C4C4semidirect
15. #380C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3802
#17792
C22⋊#952
#12942