Finite Group Databasebeta

← #27474

Subgroups — #27474

Order 256.

325 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2495C211yesmaximal, characteristic
2#2345C211yesmaximal, characteristic
3#2086C211yesmaximal, characteristic, complemented
4#2085C211yesmaximal, characteristic, complemented
5#2077C211yesmaximal, characteristic
6#2076C211yesmaximal, characteristic
7#1949C211yesmaximal, characteristic, complemented
8#1943C211yesmaximal, characteristic, complemented
9#1924C211yesmaximal, characteristic
10#1920C211yesmaximal, characteristic, complemented
11#1919C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic
14#1852C211yesmaximal, characteristic
15#1776C211yesmaximal, characteristic, complemented
16C22×C4⋊C4C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#400—21no—
28#397—21no—
29#389C2211yescharacteristic
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#388—21no—
34#387—21no—
35#387—21no—
36#383—21no—
37#382C2211yescharacteristic
38#380—21no—
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46C4×C22⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C23×C4—21no—
60C23×C4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4D411yescharacteristic
84C2×C22⋊C4D411yescharacteristic
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90#107—22no—
91#107—22no—
92#107—22no—
93#106—22no—
94#106—22no—
95#106—22no—
96C4⋊C8—22no—
97C22⋊C8—22no—
98C22⋊C8—22no—
99#97—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95D411yescharacteristic
113#95D411yescharacteristic
114C24C2×D411yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8#4112yes—
149C2×C8#4112yes—
150C4⋊C4—24no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4#4112yes—
170C4⋊C4#4112yes—
171C22⋊C4—22no—
172C22⋊C4—41no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—81no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—42no—
181C42—81no—
182C42—81no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23#13712yes—
190C23C22×D411yescharacteristic
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—84no—
235C2×C4C2×#4111yescharacteristic
236C2×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4C2×#4111yescharacteristic
239C2×C42−1+411yescharacteristic
240C2×C42−1+411yescharacteristic
241C8—22no—
242C8—22no—
243C8—41no—
244C8—41no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22#52511yescharacteristic
253C22#54711yescharacteristic
254C22#54711yescharacteristic
255C22C2×#13711yescharacteristic
256C22#57511yescharacteristic
257C22#57611yescharacteristic
258C22#57711yescharacteristic
259C4—41no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—42no—
264C4—42no—
265C4—41no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4#52911yescharacteristic
273C4#54111yescharacteristic
274C4#55211yescharacteristic
275C4#55611yescharacteristic
276C2—42no—
277C2#247811yescharacteristic
278C2#311711yescharacteristic
279C2#311811yescharacteristic
280C2#334411yescharacteristic
281C2#334611yescharacteristic
282C2#334711yescharacteristic
283C2#335111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2086C2semidirect
4. #2085C2semidirect
7. #1949C2semidirect
8. #1943C2semidirect
10. #1920C2semidirect
11. #1919C2semidirect
12. #1918C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
#24952
#23452
#20862
#20852
#20772
#20762
#19492
#19432
#19242
#19202
#19192
#19182
#18532
#18522
#17762