Finite Group Databasebeta

← #26076

Subgroups — #26076

Order 256.

391 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#428C211yesmaximal, characteristic
3#2442C211yesmaximal, characteristic
4#2439C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2085C211yesmaximal, characteristic, complemented
7#2059C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#1944C211yesmaximal, characteristic, complemented
10#1941C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1853C211yesmaximal, characteristic
13#1846C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic
15#1753C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#501—24no—
22#428—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#400—21no—
37#398—21no—
38#396—21no—
39#393—21no—
40#391—21no—
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#387—21no—
46#385—21no—
47#382C2211yescharacteristic
48#382C2211yescharacteristic
49#380—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16Q814yescomplemented
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73#125—22no—
74#125—41no—
75#117—22no—
76#117—41no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—22no—
104#107—22no—
105#107—22no—
106#107—22no—
107#106—22no—
108#106—22no—
109#106—22no—
110#106—41no—
111#106—41no—
112#106D414yes—
113C22⋊C8Q814yescomplemented
114#97—22no—
115C4×C8—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128C24C2×D411yescharacteristic
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—81no—
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—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×Q812yes—
153C22×C4C2×Q812yes—
154C22×C4C2411yescharacteristic
155M16—28no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8C2×D412yes—
163C2×C8C2×Q812yes—
164C2×C8C2×Q812yes—
165C2×C8C2×Q812yes—
166C2×C8C2×Q812yes—
167C2×C8#4112yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4C2×D412yes—
191C4⋊C4C2×D412yes—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—42no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23C4⋊Q812yes—
209C23C22×D411yescharacteristic
210Q8—84no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
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×C4—41no—
252C2×C4—41no—
253C2×C4—84no—
254C2×C4C4⋊Q812yes—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×Q811yescharacteristic
257C2×C4C22×Q811yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C2×C42−1+411yescharacteristic
261C8—22no—
262C8—22no—
263C8—22no—
264C8C22⋊Q814yes—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22C2×C4⋊Q811yescharacteristic
273C22D4211yescharacteristic
274C22#54711yescharacteristic
275C22#57511yescharacteristic
276C22#57511yescharacteristic
277C22#57711yescharacteristic
278C22#57711yescharacteristic
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—42no—
285C4—42no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4C2×C22⋊Q811yescharacteristic
293C4#54411yescharacteristic
294C4D4×Q811yescharacteristic
295C4#55411yescharacteristic
296C2—42no—
297C2#244011yescharacteristic
298C2#316311yescharacteristic
299C2#316611yescharacteristic
300C2#327511yescharacteristic
301C2#327511yescharacteristic
302C2#334811yescharacteristic
303C2#335011yescharacteristic
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
1. C2×#501C2semidirect
5. #2085C2semidirect
6. #2085C2semidirect
7. #2059C2semidirect
8. #2058C2semidirect
9. #1944C2semidirect
10. #1941C2semidirect
13. #1846C2semidirect
66. C2×M16Q8semidirect
113. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4282
#24422
#24392
#20852
#20592
#20582
#19442
#19412
#18542
#18532
#18462
#18092
#17532