Finite Group Databasebeta

← #11333

Subgroups — #11333

Order 256.

363 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2911C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic
3#2317C211yesmaximal, characteristic
4C2×#344C211yesmaximal, characteristic
5#1740C211yesmaximal, characteristic
6#1736C211yesmaximal, characteristic
7#1733C211yesmaximal, characteristic
8C22×M16C2211yescharacteristic
9C22×M16C2211yescharacteristic
10C2×#117C2211yescharacteristic
11C2×#117C2211yescharacteristic
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15#428—22no—
16#428—22no—
17#428—21no—
18#428—21no—
19#428—21no—
20#428—21no—
21#427—22no—
22C2×#107—21no—
23C2×#106—21no—
24#389—22no—
25#385—22no—
26#384—21no—
27#382—21no—
28#344C414yescomplemented
29#342—22no—
30#340—22no—
31#339—22no—
32C23×C4—21no—
33C23×C4C2311yescharacteristic
34C2×M16—21no—
35C2×M16—21no—
36C2×M16—21no—
37C2×M16—21no—
38C2×M16—21no—
39C2×M16—21no—
40C2×M16—21no—
41C2×M16C2×C412yes—
42C2×M16C2×C412yes—
43C2×M16D412yes—
44C2×M16Q812yes—
45C2×M16Q812yes—
46C22×C8—21no—
47C22×C8D412yes—
48#117—21no—
49#117—21no—
50#117—21no—
51#117—21no—
52#117—21no—
53#117—21no—
54#117—21no—
55#117—21no—
56#117—21no—
57#117—21no—
58#117—21no—
59#117—21no—
60#117—21no—
61#117C2×C412yes—
62#117Q812yes—
63#117Q812yes—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4D411yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4D412yes—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85#107—24no—
86#107—22no—
87#107—22no—
88#106—24no—
89#106—22no—
90#106—22no—
91#95—24no—
92C24C2×D411yescharacteristic
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—82no—
103C22×C4C22×C411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×Q811yescharacteristic
110C22×C4C2×Q811yescharacteristic
111C22×C4C2×Q811yescharacteristic
112C22×C4C2×Q811yescharacteristic
113C22×C4#4111yescharacteristic
114C22×C4#4111yescharacteristic
115C22×C4#4111yescharacteristic
116C22×C4#4111yescharacteristic
117M16—22no—
118M16—22no—
119M16—22no—
120M16—22no—
121M16—22no—
122M16—22no—
123M16—22no—
124M16C4⋊C414yescomplemented
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C4⋊C414yescomplemented
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—84no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—41no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4—41no—
154C22⋊C4—41no—
155C22⋊C4C4⋊C414yes—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—41no—
160C42—41no—
161C42—41no—
162C42—41no—
163C42—41no—
164C23—41no—
165C23—41no—
166C23C2×C4⋊C411yescharacteristic
167C23#12011yescharacteristic
168C23#12111yescharacteristic
169C23C22⋊Q811yescharacteristic
170C23#12311yescharacteristic
171C23#12711yescharacteristic
172C23C4⋊Q811yescharacteristic
173C2×C4—24no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4C2×C4⋊C411yescharacteristic
192C2×C4C2×C4⋊C411yescharacteristic
193C2×C4C4×D411yescharacteristic
194C2×C4C4×D411yescharacteristic
195C2×C4C4×D411yescharacteristic
196C2×C4C4×Q811yescharacteristic
197C2×C4#12011yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12111yescharacteristic
201C2×C4C22⋊Q811yescharacteristic
202C2×C4C22⋊Q811yescharacteristic
203C2×C4C22⋊Q811yescharacteristic
204C2×C4C22⋊Q811yescharacteristic
205C2×C4C22⋊Q811yescharacteristic
206C2×C4C22⋊Q811yescharacteristic
207C2×C4C22⋊Q811yescharacteristic
208C2×C4C22⋊Q811yescharacteristic
209C2×C4#12311yescharacteristic
210C2×C4#12311yescharacteristic
211C2×C4#12511yescharacteristic
212C2×C4#12511yescharacteristic
213C2×C4#12511yescharacteristic
214C2×C4C4⋊Q811yescharacteristic
215C2×C4C4⋊Q811yescharacteristic
216C2×C4C4⋊Q811yescharacteristic
217C2×C4C4⋊Q811yescharacteristic
218C2×C4C4⋊Q811yescharacteristic
219C8—24no—
220C8—24no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22#38411yescharacteristic
232C22#38511yescharacteristic
233C22#38911yescharacteristic
234C22#39211yescharacteristic
235C22#39511yescharacteristic
236C22#39811yescharacteristic
237C22#39911yescharacteristic
238C4—44no—
239C4—82no—
240C4—81no—
241C4—82no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4#38511yescharacteristic
246C4#38511yescharacteristic
247C4#38911yescharacteristic
248C4#38911yescharacteristic
249C4#39511yescharacteristic
250C4#39511yescharacteristic
251C4#39811yescharacteristic
252C4#39811yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—21no—
257C2—21no—
258C2—21no—
259C2#144411yescharacteristic
260C2#200811yescharacteristic
261C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #344C4semidirect
124. M16C4⋊C4semidirect
130. C2×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#29112
C2×#4282
#23172
C2×#3442
#17402
#17362
#17332