Finite Group Databasebeta

← #32822

Subgroups — #32822

Order 256.

313 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2864C211yesmaximal, characteristic, complemented
2#2861C211yesmaximal, characteristic
3#2097C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic
5#2074C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2067C211yesmaximal, characteristic
9#2066C211yesmaximal, characteristic, complemented
10#2065C211yesmaximal, characteristic
11#2059C211yesmaximal, characteristic, complemented
12#2059C211yesmaximal, characteristic, complemented
13#2042C211yesmaximal, characteristic
14#2042C211yesmaximal, characteristic
15#1831C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#388—21no—
43#388—21no—
44#387—21no—
45#387—21no—
46#387—21no—
47#387—21no—
48#382C2211yescharacteristic
49#382C2211yescharacteristic
50C43C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#125—41no—
66#125—41no—
67#124—41no—
68#124—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⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42D411yescharacteristic
88C2×C42D411yescharacteristic
89C2×C42D411yescharacteristic
90C2×C42D411yescharacteristic
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94#107—41no—
95#107—41no—
96#106—41no—
97#106—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100#103—41no—
101#103—41no—
102#103—41no—
103#103—41no—
104#102—41no—
105#102—41no—
106#102—41no—
107#102—41no—
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#95—21no—
119#95—21no—
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
147C2×C8—21no—
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—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
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—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C42—22no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—41no—
191C42—41no—
192C42—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23C22×D411yescharacteristic
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205D4—41no—
206D4—41no—
207D4—41no—
208D4—41no—
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—21no—
224C2×C4—21no—
225C2×C4—21no—
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—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4#13512yes—
248C2×C4H(5,𝔽2)11yescharacteristic
249C2×C4H(5,𝔽2)11yescharacteristic
250C2×C4H(5,𝔽2)11yescharacteristic
251C2×C42−1+411yescharacteristic
252C2×C42−1+411yescharacteristic
253C2×C42−1+411yescharacteristic
254C8—41no—
255C8—41no—
256C8—41no—
257C8—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#53511yescharacteristic
266C22#53611yescharacteristic
267C22#53711yescharacteristic
268C22C2×#13511yescharacteristic
269C22#57511yescharacteristic
270C22#57511yescharacteristic
271C22#57511yescharacteristic
272C4—22no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4#56211yescharacteristic
286C4#56211yescharacteristic
287C4#56311yescharacteristic
288C4#56311yescharacteristic
289C2—161no—
290C2#282011yescharacteristic
291C2#321211yescharacteristic
292C2#321911yescharacteristic
293C2#323311yescharacteristic
294C2#324011yescharacteristic
295C2#325011yescharacteristic
296C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2864C2semidirect
3. #2097C2semidirect
5. #2074C2semidirect
7. #2068C2semidirect
9. #2066C2semidirect
11. #2059C2semidirect
12. #2059C2semidirect
15. #1831C2semidirect

Maximal subgroups

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

subgroupindex
#28642
#28612
#20972
#20922
#20742
#20732
#20682
#20672
#20662
#20652
#20592
#20422
#18312