Finite Group Databasebeta

← #23713

Subgroups — #23713

Order 256.

410 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#441C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic
3#2323C211yesmaximal, characteristic
4#2323C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#2056C211yesmaximal, characteristic, complemented
10#2036C211yesmaximal, characteristic
11#2035C211yesmaximal, characteristic
12#2001C211yesmaximal, characteristic
13#1776C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C4×#103C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#441—24no—
23#433—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#389—21no—
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#388—21no—
48#388—21no—
49#387—21no—
50#387—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C2×Q16—22no—
65C2×Q16—22no—
66C2×Q16—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79#117—22no—
80#117—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
99C2×C22⋊C4D411yescharacteristic
100C2×C22⋊C4D411yescharacteristic
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109#106—22no—
110C4⋊C8—22no—
111C4⋊C8—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120#97D414yes—
121C4×C8—22no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8#4112yes—
144C2×Q8#4112yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165Q16—28no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8C2×D412yes—
177C2×C8C2×D412yes—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4#4112yes—
197C4⋊C4#4112yes—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—42no—
208C42—81no—
209C42—81no—
210C42C2×D412yes—
211C42#4112yes—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
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—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4#12112yes—
270C2×C4#12112yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C8—22no—
278C8—22no—
279C8—41no—
280C8—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22C2×#12111yescharacteristic
289C22#52511yescharacteristic
290C22#52511yescharacteristic
291C22#57711yescharacteristic
292C22#57711yescharacteristic
293C22#57811yescharacteristic
294C22#57811yescharacteristic
295C4—22no—
296C4—22no—
297C4—42no—
298C4—42no—
299C4—41no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4#52511yescharacteristic
309C4#52911yescharacteristic
310C4#52911yescharacteristic
311C4#53211yescharacteristic
312C2—81no—
313C2#239211yescharacteristic
314C2#306311yescharacteristic
315C2#306311yescharacteristic
316C2#312011yescharacteristic
317C2#312111yescharacteristic
318C2#312511yescharacteristic
319C2#312711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#441C2semidirect
5. #2088C2semidirect
7. #2068C2semidirect
8. #2066C2semidirect
9. #2056C2semidirect
13. #1776C2semidirect
14. C4×#103C2semidirect
15. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4412
C2×#4332
#23232
#20882
#20772
#20682
#20662
#20562
#20362
#20352
#20012
#17762
C4×#1032