Finite Group Databasebeta

← #11070

Subgroups — #11070

Order 256.

601 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#2315C211yesmaximal, characteristic
2C2×#381C212yesmaximal
3C2×#381C211yesmaximal, characteristic
4C2×#342C212yesmaximal
5C2×#342C211yesmaximal, characteristic
6C24×C4C2211yescharacteristic
7C22×C22⋊C4C2212yes—
8C22×C22⋊C4C2211yescharacteristic
9C22×C22⋊C4C2212yes—
10C22×C22⋊C4C2211yescharacteristic
11#385—24no—
12#381—24no—
13#381—24no—
14#381—22no—
15#381C414yescomplemented
16#380—22no—
17C2×#95—24no—
18C2×#95—22no—
19#342—22no—
20#342—24no—
21C25C2311yescharacteristic
22C23×C4—21no—
23C23×C4—22no—
24C23×C4—21no—
25C23×C4—22no—
26C23×C4—21no—
27C23×C4—21no—
28C23×C4—22no—
29C23×C4—22no—
30C23×C4—21no—
31C23×C4—22no—
32C23×C4C2×C412yes—
33C23×C4D412yes—
34C23×C4D414yes—
35C23×C4D412yes—
36C23×C4Q812yes—
37C2×C4⋊C4—24no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4C2×C412yes—
55C2×C22⋊C4C2×C412yes—
56C2×C22⋊C4D414yes—
57C2×C22⋊C4D412yes—
58#95—216no—
59#95—24no—
60#95—28no—
61C24—44no—
62C24C22×C411yescharacteristic
63C24C2×D411yescharacteristic
64C24C2×D412yes—
65C24C2×D411yescharacteristic
66C24C2×D411yescharacteristic
67C24C2×D412yes—
68C24C2×Q811yescharacteristic
69C24#4111yescharacteristic
70C24#4111yescharacteristic
71C24#4112yes—
72C24#4112yes—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—24no—
76C22×C4—24no—
77C22×C4—24no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—24no—
84C22×C4—24no—
85C22×C4—24no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—41no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—41no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—44no—
103C22×C4—48no—
104C22×C4—44no—
105C22×C4—48no—
106C22×C4—42no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4C22⋊C414yes—
111C22×C4C4⋊C414yescomplemented
112C4⋊C4—84no—
113C22⋊C4—24no—
114C22⋊C4—24no—
115C22⋊C4—24no—
116C22⋊C4—24no—
117C22⋊C4—24no—
118C22⋊C4—42no—
119C22⋊C4—42no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C23—24no—
126C23—28no—
127C23—42no—
128C23—42no—
129C23—42no—
130C23—44no—
131C23—42no—
132C23—44no—
133C23—42no—
134C23—42no—
135C23—44no—
136C23C2×C22⋊C411yescharacteristic
137C23C2×C4⋊C411yescharacteristic
138C23#11711yescharacteristic
139C23C4×D412yes—
140C23C4×D412yes—
141C23#12012yes—
142C23#12011yescharacteristic
143C23#12111yescharacteristic
144C23#12112yes—
145C23#12112yes—
146C23#12111yescharacteristic
147C23#12112yes—
148C23C22⋊Q811yescharacteristic
149C23C22⋊Q812yes—
150C23C22⋊Q811yescharacteristic
151C23C22⋊Q812yes—
152C23#12311yescharacteristic
153C23#12312yes—
154C23#12312yes—
155C23#12412yes—
156C23#12411yescharacteristic
157C23#12612yes—
158C23#12711yescharacteristic
159C2×C4—24no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—42no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—48no—
175C2×C4—48no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—48no—
180C2×C4—82no—
181C2×C4—84no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—84no—
185C2×C4—84no—
186C22—22no—
187C22—24no—
188C22—22no—
189C22—24no—
190C22—24no—
191C22—24no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—24no—
196C22—22no—
197C22—24no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22#38011yescharacteristic
209C22#38512yes—
210C22#38611yescharacteristic
211C22#38812yes—
212C22#39011yescharacteristic
213C22#39211yescharacteristic
214C22#39311yescharacteristic
215C22#39412yes—
216C22#39612yes—
217C22#39711yescharacteristic
218C22#39911yescharacteristic
219C22#45814yes—
220C4—44no—
221C4—44no—
222C4—82no—
223C4—84no—
224C4—84no—
225C2—21no—
226C2—22no—
227C2—21no—
228C2—22no—
229C2—22no—
230C2—22no—
231C2—21no—
232C2—21no—
233C2#143911yescharacteristic
234C2#189012yes—
235C2#202212yes—
236C2#203012yes—
subgroupG/H
center, socleC23#120
commutatorC24C22×C4
Frattini, prime radical, cosocleC25C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #381C4semidirect
111. C22×C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23152
C2×#3812
C2×#3422