Finite Group Databasebeta

← #11222

Subgroups — #11222

Order 256.

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

normal subgroups only

This group has 595 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×#385C211yesmaximal, characteristic
3C2×#342C212yesmaximal
4C2×#342C211yesmaximal, characteristic
5C22⋊#95C212yesmaximal
6C24×C4C2211yescharacteristic
7C22×C4⋊C4—21no—
8C22×C22⋊C4C2212yes—
9C22×C22⋊C4C2212yes—
10C22×C22⋊C4C2211yescharacteristic
11C22×C22⋊C4C2211yescharacteristic
12#385—24no—
13#385—22no—
14#385C414yescomplemented
15#380—21no—
16#380—21no—
17C2×#95—21no—
18C2×#95—24no—
19#342—22no—
20#342—24no—
21C25C2311yescharacteristic
22C23×C4—21no—
23C23×C4—21no—
24C23×C4—22no—
25C23×C4—22no—
26C23×C4—21no—
27C23×C4—22no—
28C23×C4—44no—
29C23×C4C2×C412yes—
30C23×C4D414yes—
31C23×C4D411yescharacteristic
32C23×C4D411yescharacteristic
33C2×C4⋊C4—24no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—44no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
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—22no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4C2×C412yes—
54C2×C22⋊C4C2×C412yes—
55C2×C22⋊C4D412yes—
56C2×C22⋊C4D412yes—
57C2×C22⋊C4Q814yes—
58C2×C22⋊C4Q812yes—
59#95—22no—
60#95—22no—
61#95—24no—
62#95—48no—
63C24—42no—
64C24—42no—
65C24C22×C411yescharacteristic
66C24C2×D411yescharacteristic
67C24C2×D412yes—
68C24C2×D411yescharacteristic
69C24C2×D411yescharacteristic
70C24C2×Q811yescharacteristic
71C24C2×Q812yes—
72C24#4111yescharacteristic
73C24#4111yescharacteristic
74C24#4112yes—
75C24#4112yes—
76C22×C4—28no—
77C22×C4—24no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—24no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—44no—
92C22×C4—44no—
93C22×C4—48no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—44no—
98C22×C4—41no—
99C22×C4—42no—
100C22×C4—41no—
101C22×C4—44no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C22⋊C412yes—
110C22×C4C22⋊C412yes—
111C4⋊C4—48no—
112C4⋊C4—82no—
113C4⋊C4—82no—
114C22⋊C4—22no—
115C22⋊C4—24no—
116C22⋊C4—22no—
117C22⋊C4—24no—
118C22⋊C4—22no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4C4⋊C414yescomplemented
128C23—22no—
129C23—24no—
130C23—22no—
131C23—24no—
132C23—41no—
133C23—41no—
134C23—42no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23—44no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—44no—
144C23C2×C22⋊C411yescharacteristic
145C23C2×C4⋊C411yescharacteristic
146C23#11711yescharacteristic
147C23C4×D412yes—
148C23C4×Q812yes—
149C23#12011yescharacteristic
150C23#12111yescharacteristic
151C23#12112yes—
152C23#12112yes—
153C23#12111yescharacteristic
154C23C22⋊Q811yescharacteristic
155C23C22⋊Q812yes—
156C23C22⋊Q811yescharacteristic
157C23C22⋊Q812yes—
158C23C22⋊Q812yes—
159C23#12311yescharacteristic
160C23#12312yes—
161C23#12411yescharacteristic
162C23#12512yes—
163C23#12512yes—
164C23#12612yes—
165C23C4⋊Q812yes—
166C23C4⋊Q811yescharacteristic
167C2×C4—24no—
168C2×C4—28no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—48no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—48no—
187C2×C4—48no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—84no—
197C2×C4—84no—
198C22—22no—
199C22—24no—
200C22—24no—
201C22—22no—
202C22—21no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—22no—
208C22—21no—
209C22—22no—
210C22—22no—
211C22—24no—
212C22—24no—
213C22—22no—
214C22—41no—
215C22—41no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22#38011yescharacteristic
225C22#38611yescharacteristic
226C22#38712yes—
227C22#38912yes—
228C22#39111yescharacteristic
229C22#39411yescharacteristic
230C22#39511yescharacteristic
231C22#39612yes—
232C22#39811yescharacteristic
233C22#39812yes—
234C22#40011yescharacteristic
235C22B(4,𝔽2)12yes—
236C22#45812yes—
237C4—44no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—84no—
242C4—84no—
243C2—21no—
244C2—22no—
245C2—22no—
246C2—22no—
247C2—21no—
248C2—22no—
249C2—21no—
250C2—21no—
251C2#144111yescharacteristic
252C2#188911yescharacteristic
253C2#189011yescharacteristic
254C2#204412yes—
255C2#206212yes—
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
14. #385C4semidirect
127. 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×#3852
C2×#3422
C22⋊#952