Finite Group Databasebeta

← #32527

Subgroups — #32527

Order 256.

315 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2514C211yesmaximal, characteristic, complemented
2#2513C211yesmaximal, characteristic
3#2097C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2055C211yesmaximal, characteristic
8#2055C211yesmaximal, characteristic
9#1968C211yesmaximal, characteristic
10#1949C211yesmaximal, characteristic, complemented
11#1949C211yesmaximal, characteristic, complemented
12#1941C211yesmaximal, characteristic, complemented
13#1939C211yesmaximal, characteristic, complemented
14#1927C211yesmaximal, characteristic, complemented
15#1926C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#401—21no—
33#400—21no—
34#397—21no—
35#397—21no—
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#383—21no—
48#382C2211yescharacteristic
49#382C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C2×SD16—22no—
62C2×SD16—41no—
63C2×SD16—41no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89#107—41no—
90#107—41no—
91#106—22no—
92#106—41no—
93#106—41no—
94C4⋊C8—22no—
95C4⋊C8—22no—
96C4⋊C8—41no—
97#103—22no—
98#103—41no—
99#103—41no—
100#102—22no—
101#102—41no—
102#102—41no—
103C4×C8—41no—
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—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147SD16—44no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8#4112yes—
159C2×C8#4112yes—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—81no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—42no—
189C42—81no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23C22×D411yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—41no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
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—21no—
221C2×C4—21no—
222C2×C4—21no—
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×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4#13512yes—
245C2×C4C2×#4111yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)11yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C2×C42−1+411yescharacteristic
250C2×C42−1+411yescharacteristic
251C8—22no—
252C8—22no—
253C8—41no—
254C8—41no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22#53711yescharacteristic
263C22#54711yescharacteristic
264C22#55011yescharacteristic
265C22C2×#13511yescharacteristic
266C22#57511yescharacteristic
267C22#57711yescharacteristic
268C22#57711yescharacteristic
269C4—22no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4#53811yescharacteristic
283C4#53911yescharacteristic
284C4#55611yescharacteristic
285C4#56311yescharacteristic
286C2—161no—
287C2#272711yescharacteristic
288C2#325011yescharacteristic
289C2#326011yescharacteristic
290C2#334011yescharacteristic
291C2#335111yescharacteristic
292C2#339411yescharacteristic
293C2#340111yescharacteristic
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. #2514C2semidirect
3. #2097C2semidirect
5. #2068C2semidirect
10. #1949C2semidirect
11. #1949C2semidirect
12. #1941C2semidirect
13. #1939C2semidirect
14. #1927C2semidirect

Maximal subgroups

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

subgroupindex
#25142
#25132
#20972
#20922
#20682
#20672
#20552
#19682
#19492
#19412
#19392
#19272
#19262