Finite Group Databasebeta

← #27104

Subgroups — #27104

Order 256.

339 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2494C211yesmaximal, characteristic
2#2362C211yesmaximal, characteristic
3#2087C211yesmaximal, characteristic, complemented
4#2085C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic
6#2035C211yesmaximal, characteristic
7#1933C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1923C211yesmaximal, characteristic
10#1918C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1854C211yesmaximal, characteristic
13#1825C211yesmaximal, characteristic
14#1805C211yesmaximal, characteristic, complemented
15C4×#106C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#401—21no—
32#397—21no—
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#388—21no—
40#387—21no—
41#387—21no—
42#382C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4—21no—
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×C4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C4×Q8—22no—
68C4×Q8—41no—
69#117—22no—
70#117—41no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
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⋊C4—21no—
87C2×C22⋊C4D411yescharacteristic
88C2×C22⋊C4D411yescharacteristic
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#107—22no—
96#107—41no—
97#106—22no—
98#106—22no—
99C4⋊C8—41no—
100#103—22no—
101#103—22no—
102#103—22no—
103#103—41no—
104C22⋊C8—22no—
105C22⋊C8—41no—
106C4×C8—22no—
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
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8#4112yes—
127C22×C4—21no—
128C22×C4—21no—
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×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#4112yes—
159C4⋊C4—24no—
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—21no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—81no—
176C4⋊C4#4112yes—
177C4⋊C4#4112yes—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—81no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—42no—
189C42—42no—
190C42—42no—
191C42—81no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23C22×D411yescharacteristic
200Q8—22no—
201Q8—22no—
202Q8—41no—
203Q8—41no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
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—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×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4#13512yes—
247C2×C4C2×#4111yescharacteristic
248C2×C4C2×#4111yescharacteristic
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4H(5,𝔽2)11yescharacteristic
252C2×C42−1+411yescharacteristic
253C8—22no—
254C8—41no—
255C8—41no—
256C8—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22#52511yescharacteristic
265C22#54611yescharacteristic
266C22#54711yescharacteristic
267C22C2×#13511yescharacteristic
268C22#57511yescharacteristic
269C22#57711yescharacteristic
270C22#57811yescharacteristic
271C4—22no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#52911yescharacteristic
285C4#54211yescharacteristic
286C4#55211yescharacteristic
287C4#55311yescharacteristic
288C2—81no—
289C2#246811yescharacteristic
290C2#310211yescharacteristic
291C2#312111yescharacteristic
292C2#330211yescharacteristic
293C2#332611yescharacteristic
294C2#333711yescharacteristic
295C2#334911yescharacteristic
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. #2087C2semidirect
4. #2085C2semidirect
7. #1933C2semidirect
8. #1928C2semidirect
10. #1918C2semidirect
11. #1865C2semidirect
14. #1805C2semidirect
15. C4×#106C2semidirect

Maximal subgroups

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

subgroupindex
#24942
#23622
#20872
#20852
#20362
#20352
#19332
#19282
#19232
#19182
#18652
#18542
#18252
#18052
C4×#1062