Finite Group Databasebeta

← #27477

Subgroups — #27477

Order 256.

343 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2497C211yesmaximal, characteristic
2#2344C211yesmaximal, characteristic
3#2086C211yesmaximal, characteristic, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic
6#2076C211yesmaximal, characteristic
7#1948C211yesmaximal, characteristic, complemented
8#1944C211yesmaximal, characteristic, complemented
9#1924C211yesmaximal, characteristic
10#1920C211yesmaximal, characteristic, complemented
11#1919C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic
14#1853C211yesmaximal, characteristic
15#1776C211yesmaximal, characteristic, complemented
16C2×#117—21no—
17C22×C4⋊C4C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#400—21no—
29#396—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#387—21no—
36#387—21no—
37#387—21no—
38#384C2211yescharacteristic
39#381—21no—
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
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
65#117—44no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
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⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4D411yescharacteristic
87C2×C22⋊C4D411yescharacteristic
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93#107—22no—
94#107—22no—
95#106—22no—
96#106—22no—
97#106—22no—
98#106—22no—
99C4⋊C8—22no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102#97—22no—
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#95—21no—
113#95D411yescharacteristic
114#95D411yescharacteristic
115C24C2×D411yescharacteristic
116C22×C4—24no—
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×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
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—21no—
149C2×C8#4112yes—
150C2×C8#4112yes—
151C4⋊C4—24no—
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—21no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4#4112yes—
173C4⋊C4#4112yes—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—42no—
177C22⋊C4—81no—
178C22⋊C4—81no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—42no—
184C42—42no—
185C42—81no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23#13712yes—
193C23C22×D411yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
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—21no—
209C2×C4—21no—
210C2×C4—48no—
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—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4AGL(1,ℤ/8)12yes—
238C2×C4C2×#4111yescharacteristic
239C2×C4C2×#4111yescharacteristic
240C2×C4C2×#4111yescharacteristic
241C2×C4C2×#4111yescharacteristic
242C2×C42−1+411yescharacteristic
243C2×C42−1+411yescharacteristic
244C8—22no—
245C8—22no—
246C8—41no—
247C8—41no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22#52511yescharacteristic
256C22#54711yescharacteristic
257C22#54711yescharacteristic
258C22C2×AGL(1,ℤ/8)11yescharacteristic
259C22C2×#13711yescharacteristic
260C22#57711yescharacteristic
261C22#57711yescharacteristic
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—42no—
267C4—42no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4#52911yescharacteristic
276C4#54111yescharacteristic
277C4#55211yescharacteristic
278C4#55611yescharacteristic
279C2—42no—
280C2#247811yescharacteristic
281C2#311011yescharacteristic
282C2#312611yescharacteristic
283C2#334411yescharacteristic
284C2#334411yescharacteristic
285C2#334511yescharacteristic
286C2#335411yescharacteristic
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. #2084C2semidirect
7. #1948C2semidirect
8. #1944C2semidirect
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
#24972
#23442
#20862
#20842
#20762
#19482
#19442
#19242
#19202
#19192
#19182
#18532
#17762