Finite Group Databasebeta

← C22×#15688882 #26213659

Subgroups — C22×#15688882 #26213659

Order 12032.

2830 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C211yesmaximal, characteristic
2C23×C4×Dic47C211yesmaximal, characteristic
3C2×#15688882C2112yesmaximal, direct product factor
4C22×#117×C47C211yesmaximal, characteristic
5C24×Dic47—21no—
6C22×#10625329—24no—
7C22×#10625329—23no—
8C22×#10625329C418yescomplemented
9C22×#10625329C2216yes—
10C23×C47⋊C8—21no—
11C22×C4×Dic47—23no—
12C22×C4×Dic47—24no—
13C22×C4×Dic47C418yescomplemented
14C22×C4×Dic47C2216yes—
15#15688882C22116yesdirect product factor
16C24×C4×C47C2211yescharacteristic
17C2×#117×C47—23no—
18C2×#117×C47—23no—
19C2×#117×C47C416yescomplemented
20C2×#117×C47C2216yes—
21C22×C4⋊C4×C47C412yescomplemented
22C22×C22⋊C4×C47—21no—
23C22×C42×C47—21no—
24C23×Dic47—216no—
25C23×Dic47—28no—
26C23×Dic47—26no—
27C2×#10625329—224no—
28C2×#10625329—212no—
29C2×#10625329—26no—
30C2×#10625329C2×C4124yescomplemented
31C2×#10625329C2314yes—
32C22×C47⋊C8—28no—
33C22×C47⋊C8—26no—
34C2×C4×Dic47—224no—
35C2×C4×Dic47—26no—
36C2×C4×Dic47—212no—
37C2×C4×Dic47C2×C4124yescomplemented
38C2×C4×Dic47C2314yes—
39C25×C47D411yescharacteristic
40C23×C4×C47—22no—
41C23×C4×C47C2×C414yes—
42C23×C4×C47C2×C414yes—
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2×C413yes—
45C23×C4×C47D413yes—
46C23×C4×C47D414yes—
47C23×C4×C47D413yes—
48C23×C4×C47D411yescharacteristic
49C23×C4×C47Q814yes—
50C23×C4×C47C2313yes—
51#117×C47—212no—
52#117×C47—26no—
53#117×C47—26no—
54#117×C47C2×C4112yescomplemented
55#117×C47C2314yes—
56C2×C4⋊C4×C47—26no—
57C2×C4⋊C4×C47C2×C4112yescomplemented
58C2×C22⋊C4×C47—26no—
59C2×C22⋊C4×C47—26no—
60C2×C42×C47—26no—
61C2×C42×C47—26no—
62C24×C47—28no—
63C24×C47C22⋊C414yes—
64C24×C47C22⋊C413yes—
65C24×C47C22⋊C411yescharacteristic
66C24×C47C4⋊C414yes—
67C24×C47C2×D413yes—
68C22×C4×C47—212no—
69C22×C4×C47—48no—
70C22×C4×C47C42112yes—
71C22×C4×C47C4214yes—
72C22×C4×C47C22⋊C4112yes—
73C22×C4×C47C22⋊C413yes—
74C22×C4×C47C22⋊C413yes—
75C22×C4×C47C22⋊C416yes—
76C22×C4×C47C22⋊C414yes—
77C22×C4×C47C22⋊C4112yes—
78C22×C4×C47C4⋊C4112yes—
79C22×C4×C47C4⋊C4112yes—
80C22×C4×C47C4⋊C414yes—
81C22×C4×C47C4⋊C4112yes—
82C22×C4×C47C4⋊C414yes—
83C22×C4×C47C22×C416yes—
84C22×C4×C47C22×C416yes—
85C22×C4×C47C22×C413yes—
86C22×C4×C47C22×C413yes—
87C22×C4×C47C2×D413yes—
88C22×C4×C47C2×D416yes—
89C22×C4×C47C2×D413yes—
90C22×C4×C47C2×D413yes—
91C22×C4×C47C2×Q816yes—
92C22×C4×C47C2411yescharacteristic
93C22×Dic47—264no—
94C22×Dic47—248no—
95C22×Dic47—224no—
96C22×Dic47—24no—
97C4⋊C4×C47—212no—
98C4⋊C4×C47C22×C418yescomplemented
99C22⋊C4×C47—212no—
100C22⋊C4×C47—24no—
101C42×C47—212no—
102C42×C47—24no—
103C4×Dic47—224no—
104C4×Dic47—24no—
105C4×Dic47C22×C418yescomplemented
106#10625329—224no—
107#10625329—24no—
108#10625329C22×C418yescomplemented
109C2×C47⋊C8—224no—
110C2×C47⋊C8—24no—
111C23×C47—216no—
112C23×C47—216no—
113C23×C47—212no—
114C23×C47—212no—
115C23×C47#95112yes—
116C23×C47#9514yes—
117C23×C47#10418yes—
118C23×C47C2×C22⋊C416yes—
119C23×C47C2×C22⋊C413yes—
120C23×C47C2×C22⋊C413yes—
121C23×C47C2×C4⋊C416yes—
122C23×C47C22×D411yescharacteristic
123C2×C4×C47—28no—
124C2×C4×C47—424no—
125C2×C4×C47#95124yes—
126C2×C4×C47#95112yes—
127C2×C4×C47#95112yes—
128C2×C4×C47C2×C4216yes—
129C2×C4×C47C2×C4216yes—
130C2×C4×C47C2×C22⋊C416yes—
131C2×C4×C47C2×C22⋊C416yes—
132C2×C4×C47C2×C22⋊C416yes—
133C2×C4×C47C2×C22⋊C413yes—
134C2×C4×C47C2×C22⋊C413yes—
135C2×C4×C47C2×C4⋊C416yes—
136C2×C4×C47C2×C4⋊C416yes—
137C2×C4×C47C2×C4⋊C416yes—
138C2×C4×C47C2×C4⋊C416yes—
139C2×C4×C47C2×C4⋊C416yes—
140C2×C4×C47C23×C411yescharacteristic
141C2×C4×C47C23×C411yescharacteristic
142C2×C4×C47C23×C411yescharacteristic
143C2×C4×C47C22×D411yescharacteristic
144C2×C4×C47C22×D411yescharacteristic
145C2×C4×C47C22×Q811yescharacteristic
146C2×Dic47—216no—
147C2×Dic47—296no—
148C2×Dic47—28no—
149C47⋊C8—28no—
150C22×#339—471nomaximal, Sylow
151C22×C47—22no—
152C22×C47—26no—
153C22×C47—212no—
154C22×C47—224no—
155C22×C47—212no—
156C22×C47#339116yescomplemented
157C22×C47C2×#9516yes—
158C22×C47C2×#9516yes—
159C22×C47C2×#104112yes—
160C22×C47C22×C22⋊C411yescharacteristic
161C22×C47C22×C22⋊C411yescharacteristic
162C22×C47C22×C4⋊C411yescharacteristic
163C4×C47—48no—
164C4×C47C2×#9516yes—
165C4×C47C2×#9516yes—
166C4×C47C22×C4211yescharacteristic
167C4×C47C22×C22⋊C411yescharacteristic
168C4×C47C22×C4⋊C411yescharacteristic
169C4×C47C22×C4⋊C411yescharacteristic
170Dic47—216no—
171C23×M16—471no—
172C22×#117D4711yescharacteristic
173C23×C42—471no—
174C2×#339—4712no—
175C2×C47—21no—
176C2×C47—21no—
177C2×C47—26no—
178C2×C47C2×#339112yescomplemented
179C2×C47C22×#9511yescharacteristic
180C2×C47C22×#10412yes—
181C24×C4—941no—
182C24×C4C2×D4711yescharacteristic
183C22×M16—478no—
184C22×M16—476no—
185C22×M16—944no—
186C22×M16—943no—
187C23×C8—941no—
188C2×#117—23no—
189C2×#117—23no—
190C2×#117Dic4716yescomplemented
191C2×#117C2×D4716yes—
192C22×C4⋊C4Dic4712yescomplemented
193C22×C22⋊C4—21no—
194C22×C42—21no—
195C22×C42—478no—
196C22×C42—476no—
197C22×C42—944no—
198C22×C42—943no—
199#339—4716no—
200C47C22×#33911yescharacteristic, complemented, Sylow
201C25C22⋊D4711yescharacteristic
202C23×C4—22no—
203C23×C4—9416no—
204C23×C4—948no—
205C23×C4—946no—
206C23×C4Dic9414yes—
207C23×C4C4×D4714yes—
208C23×C4C4×D4714yes—
209C23×C4D18814yes—
210C23×C4C2×Dic4711yescharacteristic
211C23×C4C2×Dic4713yes—
212C23×C4C22⋊D4713yes—
213C23×C4C22⋊D4713yes—
214C23×C4C22⋊D4711yescharacteristic
215C23×C4C22×D4713yes—
216C2×M16—4724no—
217C2×M16—474no—
218C2×M16—9424no—
219C2×M16—9412no—
220C2×M16—946no—
221C22×C8—948no—
222C22×C8—946no—
223#117—212no—
224#117—26no—
225#117—26no—
226#117C2×Dic47112yescomplemented
227#117C22×D4714yes—
228C2×C4⋊C4—26no—
229C2×C4⋊C4C2×Dic47112yescomplemented
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C42—26no—
233C2×C42—26no—
234C2×C42—4724no—
235C2×C42—474no—
236C2×C42—9424no—
237C2×C42—9412no—
238C2×C42—946no—
239C24—28no—
240C24Dic47⋊C414yes—
241C24(C2×D47)⋊C414yes—
242C24C22⋊Dic4713yes—
243C24C22⋊Dic4711yescharacteristic
244C24C2×C22⋊D4713yes—
245C22×C4—212no—
246C22×C4—48no—
247C22×C4—9448no—
248C22×C4—9464no—
249C22×C4—9424no—
250C22×C4—944no—
251C22×C4C4×Dic47112yes—
252C22×C4C4×Dic4714yes—
253C22×C4Dic47⋊C4112yes—
254C22×C4Dic47⋊C4112yes—
255C22×C4Dic47⋊C414yes—
256C22×C4C4⋊Dic47112yes—
257C22×C4C4⋊Dic4714yes—
258C22×C4(C2×D47)⋊C4112yes—
259C22×C4(C2×D47)⋊C414yes—
260C22×C4(C2×D47)⋊C4112yes—
261C22×C4C22⋊Dic4713yes—
262C22×C4C22⋊Dic4713yes—
263C22×C4C22⋊Dic4716yes—
264C22×C4C2×Dic9416yes—
265C22×C4C2×C4×D4716yes—
266C22×C4C2×C4×D4716yes—
267C22×C4C2×D18816yes—
268C22×C4C22×Dic4713yes—
269C22×C4C22×Dic4713yes—
270C22×C4C2×C22⋊D4713yes—
271C22×C4C2×C22⋊D4713yes—
272C22×C4C2×C22⋊D4713yes—
273C22×C4C23×D4711yescharacteristic
274M16—478no—
275M16—9424no—
276M16—944no—
277C2×C8—9424no—
278C2×C8—944no—
279C4⋊C4—212no—
280C4⋊C4C22×Dic4718yescomplemented
281C22⋊C4—212no—
282C22⋊C4—24no—
283C42—212no—
284C42—24no—
285C42—478no—
286C42—9424no—
287C42—944no—
288C23—212no—
289C23—212no—
290C23—216no—
291C23—216no—
292C23C47⋊#95112yes—
293C23C47⋊#9514yes—
294C23#1309395118yes—
295C23C2×(C2×D47)⋊C416yes—
296C23C2×C22⋊Dic4713yes—
297C23C2×C22⋊Dic4713yes—
298C23C2×Dic47⋊C416yes—
299C23C22×C22⋊D4711yescharacteristic
300C2×C4—28no—
301C2×C4—424no—
302C2×C4—9416no—
303C2×C4—9496no—
304C2×C4—948no—
305C2×C4C47⋊#95124yes—
306C2×C4C47⋊#95112yes—
307C2×C4C47⋊#95112yes—
308C2×C4C2×C4×Dic4716yes—
309C2×C4C2×C4×Dic4716yes—
310C2×C4C2×(C2×D47)⋊C416yes—
311C2×C4C2×(C2×D47)⋊C416yes—
312C2×C4C2×(C2×D47)⋊C416yes—
313C2×C4C2×C22⋊Dic4713yes—
314C2×C4C2×C22⋊Dic4713yes—
315C2×C4C2×Dic47⋊C416yes—
316C2×C4C2×Dic47⋊C416yes—
317C2×C4C2×Dic47⋊C416yes—
318C2×C4C2×C4⋊Dic4716yes—
319C2×C4C2×C4⋊Dic4716yes—
320C2×C4C22×C4×D4711yescharacteristic
321C2×C4C22×C4×D4711yescharacteristic
322C2×C4C23×Dic4711yescharacteristic
323C2×C4C22×C22⋊D4711yescharacteristic
324C2×C4C22×D18811yescharacteristic
325C2×C4C22×Dic9411yescharacteristic
326C8—948no—
327C22—22no—
328C22—26no—
329C22—212no—
330C22—224no—
331C22—212no—
332C22#15688882116yesdirect product factor
333C22C2×C47⋊#9516yes—
334C22C2×C47⋊#9516yes—
335C22C2×#13093951112yes—
336C22C22×(C2×D47)⋊C411yescharacteristic
337C22C22×C22⋊Dic4711yescharacteristic
338C22C22×Dic47⋊C411yescharacteristic
339C4—48no—
340C4—9416no—
341C4C2×C47⋊#9516yes—
342C4C2×C47⋊#9516yes—
343C4C22×C4×Dic4711yescharacteristic
344C4C22×(C2×D47)⋊C411yescharacteristic
345C4C22×Dic47⋊C411yescharacteristic
346C4C22×C4⋊Dic4711yescharacteristic
347C2—21no—
348C2—26no—
349C2—21no—
350C2C2×#15688882112yesdirect product factor
351C2C22×C47⋊#9511yescharacteristic
352C2C22×#1309395112yes—
subgroupG/H
centerC23×C4C22⋊D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC22×#117×C47C2
socleC24×C47C22⋊C4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC22×#117D47
hypocenterC47C22×#339
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#15688882C2direct (N × H)
8. C22×#10625329C4semidirect
13. C22×C4×Dic47C4semidirect
15. #15688882C22direct (N × H)
19. C2×#117×C47C4semidirect
21. C22×C4⋊C4×C47C4semidirect
30. C2×#10625329C2×C4semidirect
37. C2×C4×Dic47C2×C4semidirect
54. #117×C47C2×C4semidirect
57. C2×C4⋊C4×C47C2×C4semidirect
98. C4⋊C4×C47C22×C4semidirect
105. C4×Dic47C22×C4semidirect
108. #10625329C22×C4semidirect
156. C22×C47#339semidirect
178. C2×C47C2×#339semidirect
190. C2×#117Dic47semidirect
192. C22×C4⋊C4Dic47semidirect
200. C47C22×#339semidirect
226. #117C2×Dic47semidirect
229. C2×C4⋊C4C2×Dic47semidirect
280. C4⋊C4C22×Dic47semidirect
332. C22#15688882direct (N × H)
350. C2C2×#15688882direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253292
C23×C4×Dic472
C2×#156888822
C22×#117×C472
C22×#33947