Finite Group Databasebeta

← #11203

Subgroups — #11203

Order 256.

597 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 131 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×#385C211yesmaximal, characteristic
2C2×#380C211yesmaximal, characteristic
3C2×#342C212yesmaximal
4C2×#342C212yesmaximal
5C22⋊#95C211yesmaximal, characteristic
6C24×C4C2211yescharacteristic
7C22×C4⋊C4—21no—
8C22×C4⋊C4C412yescomplemented
9C22×C22⋊C4C2211yescharacteristic
10C22×C22⋊C4C2212yes—
11C22×C22⋊C4C2212yes—
12C22×C22⋊C4C2211yescharacteristic
13#385—21no—
14#385—22no—
15#385—21no—
16#380—24no—
17C2×#95—21no—
18C2×#95—22no—
19C2×#95C412yescomplemented
20#342—24no—
21#342—24no—
22C25C2311yescharacteristic
23C23×C4—22no—
24C23×C4—22no—
25C23×C4—22no—
26C23×C4—22no—
27C23×C4—44no—
28C23×C4C2×C412yes—
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2×C412yes—
32C23×C4D411yescharacteristic
33C23×C4D412yes—
34C23×C4D411yescharacteristic
35C2×C4⋊C4—24no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—44no—
39C2×C4⋊C4—44no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
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⋊C4—22no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4D414yes—
59C2×C22⋊C4D412yes—
60C2×C22⋊C4Q814yes—
61C2×C22⋊C4Q812yes—
62#95—24no—
63#95—22no—
64#95—22no—
65#95—44no—
66C24—42no—
67C24—42no—
68C24C22×C411yescharacteristic
69C24C2×D411yescharacteristic
70C24C2×D411yescharacteristic
71C24C2×D411yescharacteristic
72C24C2×D412yes—
73C24C2×Q811yescharacteristic
74C24C2×Q812yes—
75C24#4111yescharacteristic
76C24#4111yescharacteristic
77C24#4112yes—
78C24#4112yes—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—24no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—24no—
85C22×C4—21no—
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—24no—
89C22×C4—24no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—48no—
100C22×C4—48no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—44no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C22⋊C412yes—
114C22×C4C22⋊C412yes—
115C22×C4C4⋊C414yescomplemented
116C4⋊C4—48no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—41no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4—41no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C23—22no—
135C23—22no—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—44no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—44no—
150C23C2×C22⋊C411yescharacteristic
151C23C2×C4⋊C411yescharacteristic
152C23#11711yescharacteristic
153C23C4×D412yes—
154C23C4×Q812yes—
155C23#12011yescharacteristic
156C23#12111yescharacteristic
157C23#12111yescharacteristic
158C23#12112yes—
159C23#12112yes—
160C23C22⋊Q811yescharacteristic
161C23C22⋊Q811yescharacteristic
162C23C22⋊Q812yes—
163C23C22⋊Q812yes—
164C23C22⋊Q812yes—
165C23#12311yescharacteristic
166C23#12312yes—
167C23#12411yescharacteristic
168C23#12512yes—
169C23#12512yes—
170C23#12612yes—
171C23C4⋊Q811yescharacteristic
172C23C4⋊Q812yes—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—48no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—48no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—84no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#9914yes—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—22no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—24no—
221C22—22no—
222C22—24no—
223C22—22no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22#38011yescharacteristic
233C22#38611yescharacteristic
234C22#38712yes—
235C22#38912yes—
236C22#39111yescharacteristic
237C22#39411yescharacteristic
238C22#39511yescharacteristic
239C22#39612yes—
240C22#39811yescharacteristic
241C22#39812yes—
242C22#40011yescharacteristic
243C22C2×#9912yes—
244C22#41012yes—
245C4—24no—
246C4—44no—
247C4—42no—
248C4—84no—
249C4—84no—
250C4—81no—
251C4—81no—
252C2—21no—
253C2—22no—
254C2—22no—
255C2—22no—
256C2—21no—
257C2—21no—
258C2—22no—
259C2—21no—
260C2#144111yescharacteristic
261C2#179912yes—
262C2#189811yescharacteristic
263C2#189911yescharacteristic
264C2#199612yes—
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC24C22×C4
Frattini, prime radical, cosocleC25C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C22×C4⋊C4C4semidirect
19. C2×#95C4semidirect
115. C22×C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3852
C2×#3802
C2×#3422
C22⋊#952