Finite Group Databasebeta

← #32253

Subgroups — #32253

Order 256.

321 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2552C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic
6#2065C211yesmaximal, characteristic
7#2025C211yesmaximal, characteristic
8#2025C211yesmaximal, characteristic
9#1992C211yesmaximal, characteristic, complemented
10#1989C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic
12#1926C211yesmaximal, characteristic
13#1919C211yesmaximal, characteristic, complemented
14#1919C211yesmaximal, characteristic, complemented
15#1917C211yesmaximal, characteristic, complemented
16C2×#126—21no—
17C2×#125C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#398—21no—
28#398—21no—
29#397—21no—
30#396—21no—
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#384—21no—
40#384C2211yescharacteristic
41#384C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
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#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64#126—44no—
65#125—22no—
66#125—22no—
67#125—41no—
68#125—41no—
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⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#106—22no—
101#106—22no—
102C4⋊C8—41no—
103#102—22no—
104#102—22no—
105#102—41no—
106#102—41no—
107#97—41no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95D411yescharacteristic
114#95D411yescharacteristic
115C24—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—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×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
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—21no—
146C2×C8—21no—
147C2×C8—21no—
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—41no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4#4112yes—
174C4⋊C4#4112yes—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—42no—
188C42—42no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23C22×D411yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
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×C4—41no—
238C2×C4—41no—
239C2×C4AGL(1,ℤ/8)12yes—
240C2×C4C2×#4111yescharacteristic
241C2×C4C2×#4111yescharacteristic
242C2×C4H(5,𝔽2)11yescharacteristic
243C2×C42−1+411yescharacteristic
244C2×C42−1+411yescharacteristic
245C2×C42−1+411yescharacteristic
246C8—41no—
247C8—41no—
248C8—41no—
249C8—41no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22#53611yescharacteristic
258C22#54611yescharacteristic
259C22#54811yescharacteristic
260C22C2×AGL(1,ℤ/8)11yescharacteristic
261C22#57511yescharacteristic
262C22#57711yescharacteristic
263C22#57711yescharacteristic
264C4—22no—
265C4—41no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—42no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4#54111yescharacteristic
278C4#54211yescharacteristic
279C4#55611yescharacteristic
280C4#56311yescharacteristic
281C2—161no—
282C2#271611yescharacteristic
283C2#323411yescharacteristic
284C2#324411yescharacteristic
285C2#330711yescharacteristic
286C2#333211yescharacteristic
287C2#336811yescharacteristic
288C2#337611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2552C2semidirect
3. #2099C2semidirect
4. #2084C2semidirect
9. #1992C2semidirect
10. #1989C2semidirect
13. #1919C2semidirect
14. #1919C2semidirect
15. #1917C2semidirect

Maximal subgroups

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

subgroupindex
#25522
#25122
#20992
#20842
#20712
#20652
#20252
#19922
#19892
#19362
#19262
#19192
#19172