Finite Group Databasebeta

← #27342

Subgroups — #27342

Order 256.

359 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2489C211yesmaximal, characteristic
2#2344C211yesmaximal, characteristic
3#2075C211yesmaximal, characteristic
4#2068C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic
7#1989C211yesmaximal, characteristic, complemented
8#1935C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1928C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic
12#1921C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic
14#1852C211yesmaximal, characteristic
15#1776C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#117—21no—
19C22×C4⋊C4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#400—21no—
33#397—21no—
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#387—21no—
40#387—21no—
41#382—21no—
42#381—21no—
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66#125—22no—
67#125—41no—
68C22⋊Q8—22no—
69C22⋊Q8—41no—
70#117—44no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
90C2×C22⋊C4D411yescharacteristic
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—22no—
98#107—41no—
99#106—22no—
100C4⋊C8—41no—
101#103—22no—
102#103—22no—
103#103—22no—
104#103—41no—
105C22⋊C8—22no—
106C22⋊C8—41no—
107#97—22no—
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#95D411yescharacteristic
120#95D411yescharacteristic
121C24C2×D411yescharacteristic
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8#4112yes—
163C4⋊C4—24no—
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—21no—
174C4⋊C4—21no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4#4112yes—
181C4⋊C4#4112yes—
182C4⋊C4#4112yes—
183C22⋊C4—22no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—42no—
193C42—42no—
194C42—42no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23AGL(1,ℤ/8)12yes—
202C23C22×D411yescharacteristic
203Q8—41no—
204Q8—41no—
205Q8—41no—
206Q8—41no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
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—48no—
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—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4C2×#4111yescharacteristic
252C2×C4C2×#4111yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C2×C42−1+411yescharacteristic
257C8—22no—
258C8—41no—
259C8—41no—
260C8—41no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22#52511yescharacteristic
269C22#54611yescharacteristic
270C22#54711yescharacteristic
271C22C2×AGL(1,ℤ/8)11yescharacteristic
272C22#57511yescharacteristic
273C22#57811yescharacteristic
274C22#57811yescharacteristic
275C4—41no—
276C4—41no—
277C4—42no—
278C4—42no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—42no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4#52911yescharacteristic
289C4#54211yescharacteristic
290C4#55511yescharacteristic
291C4#55611yescharacteristic
292C2—42no—
293C2#247411yescharacteristic
294C2#311411yescharacteristic
295C2#312711yescharacteristic
296C2#330811yescharacteristic
297C2#331511yescharacteristic
298C2#334311yescharacteristic
299C2#335211yescharacteristic
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
4. #2068C2semidirect
5. #2059C2semidirect
7. #1989C2semidirect
8. #1935C2semidirect
9. #1928C2semidirect
10. #1928C2semidirect
12. #1921C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
#24892
#23442
#20752
#20682
#20592
#20352
#19892
#19352
#19282
#19242
#19212
#18722
#18522
#17762