Finite Group Databasebeta

← #27888

Subgroups — #27888

Order 256.

314 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2629C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic
4#2077C211yesmaximal, characteristic
5#2076C211yesmaximal, characteristic
6#2075C211yesmaximal, characteristic
7#1992C211yesmaximal, characteristic, complemented
8#1991C211yesmaximal, characteristic, complemented
9#1921C211yesmaximal, characteristic, complemented
10#1920C211yesmaximal, characteristic, complemented
11#1919C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1847C211yesmaximal, characteristic, complemented
14#1834C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic
16C2×#125C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24C2×#97C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#398—21no—
27#398—21no—
28#396—21no—
29#396—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#388—21no—
33#388—21no—
34#387—21no—
35#387—21no—
36#384C2211yescharacteristic
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
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×C4C2311yescharacteristic
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64#125—22no—
65#125—22no—
66#117—22no—
67#117—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4D411yescharacteristic
91C2×C22⋊C4D411yescharacteristic
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—22no—
98#106—22no—
99#106—22no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102#97—22no—
103C4×C8—22no—
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#95—21no—
113#95—21no—
114C24C2×D411yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
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×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2411yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8#4112yes—
146C2×C8#4112yes—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
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—42no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—81no—
176C22⋊C4—81no—
177C22⋊C4—81no—
178C22⋊C4—81no—
179C42—21no—
180C42—21no—
181C42—81no—
182C42—81no—
183C42#4112yes—
184C42#4112yes—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23C22×D411yescharacteristic
193C2×C4—21no—
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—41no—
207C2×C4—41no—
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#12312yes—
235C2×C4#12312yes—
236C2×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4C2×#4111yescharacteristic
239C2×C4C2×#4111yescharacteristic
240C2×C42−1+411yescharacteristic
241C2×C42−1+411yescharacteristic
242C8—22no—
243C8—22no—
244C8—41no—
245C8—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22C2×#12311yescharacteristic
254C22#53211yescharacteristic
255C22#53611yescharacteristic
256C22#57611yescharacteristic
257C22#57711yescharacteristic
258C22#57711yescharacteristic
259C22#57811yescharacteristic
260C4—22no—
261C4—22no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4#55211yescharacteristic
274C4#55211yescharacteristic
275C4#55611yescharacteristic
276C4#55611yescharacteristic
277C2—81no—
278C2#249911yescharacteristic
279C2#309811yescharacteristic
280C2#309911yescharacteristic
281C2#317811yescharacteristic
282C2#317911yescharacteristic
283C2#324111yescharacteristic
284C2#324211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #1992C2semidirect
8. #1991C2semidirect
9. #1921C2semidirect
10. #1920C2semidirect
11. #1919C2semidirect
12. #1918C2semidirect
13. #1847C2semidirect
14. #1834C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#23142
#20772
#20762
#20752
#19922
#19912
#19212
#19202
#19192
#19182
#18472
#18342
#17502