Finite Group Databasebeta

← C22×#15688884 #26213661

Subgroups — C22×#15688884 #26213661

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
1C22×#13093983C211yesmaximal, characteristic
2C23×C4×Dic47C211yesmaximal, characteristic
3C2×#15688884C2112yesmaximal, direct product factor
4C23×M16×C47C211yesmaximal, characteristic
5C24×Dic47—21no—
6C2×#13093983—23no—
7C2×#13093983—23no—
8C2×#13093983C416yescomplemented
9C2×#13093983C2216yes—
10C22×C4⋊Dic47C412yescomplemented
11C22×C22⋊Dic47—21no—
12C22×C4×Dic47—23no—
13C22×C4×Dic47—24no—
14C22×C4×Dic47—21no—
15C22×C4×Dic47C418yescomplemented
16C22×C4×Dic47C2216yes—
17#15688884C22116yesdirect product factor
18C24×C4×C47C2211yescharacteristic
19C22×M16×C47—24no—
20C22×M16×C47—23no—
21C22×M16×C47C418yescomplemented
22C22×M16×C47C2216yes—
23C23×C8×C47—21no—
24C23×Dic47—216no—
25C23×Dic47—28no—
26C23×Dic47—26no—
27C23×Dic47—22no—
28#13093983—212no—
29#13093983—26no—
30#13093983—26no—
31#13093983C2×C4112yescomplemented
32#13093983C2314yes—
33C2×C4⋊Dic47—26no—
34C2×C4⋊Dic47C2×C4112yescomplemented
35C2×C22⋊Dic47—26no—
36C2×C22⋊Dic47—26no—
37C2×C4×Dic47—224no—
38C2×C4×Dic47—26no—
39C2×C4×Dic47—26no—
40C2×C4×Dic47—212no—
41C2×C4×Dic47—26no—
42C2×C4×Dic47C2×C4124yescomplemented
43C2×C4×Dic47C2314yes—
44C25×C47D411yescharacteristic
45C23×C4×C47C2×C414yes—
46C23×C4×C47C2×C414yes—
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47C2×C413yes—
49C23×C4×C47D413yes—
50C23×C4×C47D414yes—
51C23×C4×C47D413yes—
52C23×C4×C47D411yescharacteristic
53C23×C4×C47Q814yes—
54C23×C4×C47C2313yes—
55C2×M16×C47—224no—
56C2×M16×C47—212no—
57C2×M16×C47—26no—
58C2×M16×C47C2×C4124yescomplemented
59C2×M16×C47C2314yes—
60C22×C8×C47—28no—
61C22×C8×C47—26no—
62C24×C47—28no—
63C24×C47C22⋊C413yes—
64C24×C47C22⋊C414yes—
65C24×C47C22⋊C411yescharacteristic
66C24×C47C4⋊C414yes—
67C24×C47C2×D413yes—
68C22×C4×C47C42112yes—
69C22×C4×C47C4214yes—
70C22×C4×C47C22⋊C4112yes—
71C22×C4×C47C22⋊C413yes—
72C22×C4×C47C22⋊C413yes—
73C22×C4×C47C22⋊C416yes—
74C22×C4×C47C22⋊C414yes—
75C22×C4×C47C22⋊C4112yes—
76C22×C4×C47C4⋊C4112yes—
77C22×C4×C47C4⋊C4112yes—
78C22×C4×C47C4⋊C4112yes—
79C22×C4×C47C4⋊C414yes—
80C22×C4×C47C4⋊C414yes—
81C22×C4×C47C22×C416yes—
82C22×C4×C47C22×C416yes—
83C22×C4×C47C22×C413yes—
84C22×C4×C47C22×C413yes—
85C22×C4×C47C2×D413yes—
86C22×C4×C47C2×D413yes—
87C22×C4×C47C2×D416yes—
88C22×C4×C47C2×D413yes—
89C22×C4×C47C2×Q816yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47—264no—
92C22×Dic47—248no—
93C22×Dic47—224no—
94C22×Dic47—24no—
95C22×Dic47—212no—
96C22×Dic47—48no—
97M16×C47—224no—
98M16×C47—24no—
99M16×C47C22×C418yescomplemented
100C2×C8×C47—224no—
101C2×C8×C47—24no—
102C22⋊Dic47—212no—
103C22⋊Dic47—24no—
104C4⋊Dic47—212no—
105C4⋊Dic47C22×C418yescomplemented
106C4×Dic47—224no—
107C4×Dic47—212no—
108C4×Dic47—24no—
109C4×Dic47—24no—
110C4×Dic47C22×C418yescomplemented
111C23×C47—216no—
112C23×C47—216no—
113C23×C47—212no—
114C23×C47—212no—
115C23×C47#95112yes—
116C23×C47#9514yes—
117C23×C47#10418yes—
118C23×C47C2×C22⋊C413yes—
119C23×C47C2×C22⋊C416yes—
120C23×C47C2×C22⋊C413yes—
121C23×C47C2×C4⋊C416yes—
122C23×C47C22×D411yescharacteristic
123C2×C4×C47#95124yes—
124C2×C4×C47#95112yes—
125C2×C4×C47#95112yes—
126C2×C4×C47C2×C4216yes—
127C2×C4×C47C2×C4216yes—
128C2×C4×C47C2×C22⋊C416yes—
129C2×C4×C47C2×C22⋊C413yes—
130C2×C4×C47C2×C22⋊C413yes—
131C2×C4×C47C2×C22⋊C416yes—
132C2×C4×C47C2×C22⋊C416yes—
133C2×C4×C47C2×C4⋊C416yes—
134C2×C4×C47C2×C4⋊C416yes—
135C2×C4×C47C2×C4⋊C416yes—
136C2×C4×C47C2×C4⋊C416yes—
137C2×C4×C47C2×C4⋊C416yes—
138C2×C4×C47C23×C411yescharacteristic
139C2×C4×C47C23×C411yescharacteristic
140C2×C4×C47C23×C411yescharacteristic
141C2×C4×C47C22×D411yescharacteristic
142C2×C4×C47C22×D411yescharacteristic
143C2×C4×C47C22×Q811yescharacteristic
144C2×Dic47—216no—
145C2×Dic47—296no—
146C2×Dic47—28no—
147C2×Dic47—28no—
148C2×Dic47—424no—
149C8×C47—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×C47C2×#9516yes—
164C4×C47C2×#9516yes—
165C4×C47C22×C4211yescharacteristic
166C4×C47C22×C22⋊C411yescharacteristic
167C4×C47C22×C4⋊C411yescharacteristic
168C4×C47C22×C4⋊C411yescharacteristic
169Dic47—216no—
170Dic47—48no—
171C23×M16D4711yescharacteristic
172C22×#117—471no—
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—24no—
184C22×M16—23no—
185C22×M16Dic4718yescomplemented
186C22×M16C2×D4716yes—
187C23×C8—21no—
188C2×#117—476no—
189C2×#117—476no—
190C2×#117—943no—
191C2×#117—943no—
192C22×C4⋊C4—472no—
193C22×C22⋊C4—941no—
194C22×C42—478no—
195C22×C42—476no—
196C22×C42—943no—
197C22×C42—944no—
198C22×C42—941no—
199#339—4716no—
200C47C22×#33911yescharacteristic, complemented, Sylow
201C25D18811yescharacteristic
202C23×C4—9416no—
203C23×C4—948no—
204C23×C4—946no—
205C23×C4—942no—
206C23×C4Dic9414yes—
207C23×C4C4×D4711yescharacteristic
208C23×C4C4×D4713yes—
209C23×C4C4×D4714yes—
210C23×C4D18813yes—
211C23×C4C2×Dic4714yes—
212C23×C4C22⋊D4713yes—
213C23×C4C22⋊D4714yes—
214C23×C4C22⋊D4711yescharacteristic
215C23×C4C22×D4713yes—
216C2×M16—224no—
217C2×M16—212no—
218C2×M16—26no—
219C2×M16C2×Dic47124yescomplemented
220C2×M16C22×D4714yes—
221C22×C8—28no—
222C22×C8—26no—
223#117—4712no—
224#117—474no—
225#117—9412no—
226#117—946no—
227#117—946no—
228C2×C4⋊C4—4712no—
229C2×C4⋊C4—946no—
230C2×C22⋊C4—946no—
231C2×C22⋊C4—946no—
232C2×C42—4724no—
233C2×C42—474no—
234C2×C42—946no—
235C2×C42—9424no—
236C2×C42—946no—
237C2×C42—946no—
238C2×C42—9412no—
239C24—28no—
240C24C4⋊Dic4714yes—
241C24(C2×D47)⋊C413yes—
242C24(C2×D47)⋊C414yes—
243C24(C2×D47)⋊C411yescharacteristic
244C24C2×D18813yes—
245C22×C4—9448no—
246C22×C4—9464no—
247C22×C4—9424no—
248C22×C4—944no—
249C22×C4—9412no—
250C22×C4—1888no—
251C22×C4C4×Dic47112yes—
252C22×C4C4×Dic4714yes—
253C22×C4Dic47⋊C4112yes—
254C22×C4Dic47⋊C4112yes—
255C22×C4Dic47⋊C414yes—
256C22×C4Dic47⋊C414yes—
257C22×C4C4⋊Dic47112yes—
258C22×C4(C2×D47)⋊C4112yes—
259C22×C4(C2×D47)⋊C413yes—
260C22×C4(C2×D47)⋊C413yes—
261C22×C4(C2×D47)⋊C416yes—
262C22×C4C22⋊Dic4714yes—
263C22×C4C22⋊Dic47112yes—
264C22×C4C2×Dic9416yes—
265C22×C4C2×C4×D4716yes—
266C22×C4C2×C4×D4713yes—
267C22×C4C2×C4×D4713yes—
268C22×C4C2×D18813yes—
269C22×C4C22×Dic4716yes—
270C22×C4C2×C22⋊D4713yes—
271C22×C4C2×C22⋊D4716yes—
272C22×C4C2×C22⋊D4713yes—
273C22×C4C23×D4711yescharacteristic
274M16—224no—
275M16—24no—
276M16C22×Dic4718yescomplemented
277C2×C8—224no—
278C2×C8—24no—
279C4⋊C4—478no—
280C4⋊C4—9412no—
281C22⋊C4—9412no—
282C22⋊C4—944no—
283C42—478no—
284C42—9424no—
285C42—9412no—
286C42—944no—
287C42—944no—
288C23—212no—
289C23—212no—
290C23—216no—
291C23—216no—
292C23C47⋊#95112yes—
293C23C47⋊#9514yes—
294C23#1309395018yes—
295C23C2×(C2×D47)⋊C413yes—
296C23C2×(C2×D47)⋊C416yes—
297C23C2×(C2×D47)⋊C413yes—
298C23C2×C4⋊Dic4716yes—
299C23C22×D18811yescharacteristic
300C2×C4—9416no—
301C2×C4—9496no—
302C2×C4—948no—
303C2×C4—948no—
304C2×C4—18824no—
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)⋊C413yes—
312C2×C4C2×(C2×D47)⋊C413yes—
313C2×C4C2×C22⋊Dic4716yes—
314C2×C4C2×C22⋊Dic4716yes—
315C2×C4C2×Dic47⋊C416yes—
316C2×C4C2×Dic47⋊C416yes—
317C2×C4C2×Dic47⋊C416yes—
318C2×C4C2×Dic47⋊C416yes—
319C2×C4C2×C4⋊Dic4716yes—
320C2×C4C22×C4×D4711yescharacteristic
321C2×C4C22×C4×D4711yescharacteristic
322C2×C4C23×Dic4711yescharacteristic
323C2×C4C22×C22⋊D4711yescharacteristic
324C2×C4C22×C22⋊D4711yescharacteristic
325C2×C4C22×Dic9411yescharacteristic
326C8—28no—
327C22—22no—
328C22—26no—
329C22—212no—
330C22—224no—
331C22—212no—
332C22#15688884116yesdirect product factor
333C22C2×C47⋊#9516yes—
334C22C2×C47⋊#9516yes—
335C22C2×#13093950112yes—
336C22C22×(C2×D47)⋊C411yescharacteristic
337C22C22×(C2×D47)⋊C411yescharacteristic
338C22C22×C4⋊Dic4711yescharacteristic
339C4—9416no—
340C4—1888no—
341C4C2×C47⋊#9516yes—
342C4C2×C47⋊#9516yes—
343C4C22×C4×Dic4711yescharacteristic
344C4C22×C22⋊Dic4711yescharacteristic
345C4C22×Dic47⋊C411yescharacteristic
346C4C22×Dic47⋊C411yescharacteristic
347C2—21no—
348C2—26no—
349C2—21no—
350C2C2×#15688884112yesdirect product factor
351C2C22×C47⋊#9511yescharacteristic
352C2C22×#1309395012yes—
subgroupG/H
centerC23×C4C22⋊D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC23×M16×C47C2
socleC24×C47C22⋊C4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC23×M16D47
hypocenterC47C22×#339
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#15688884C2direct (N × H)
8. C2×#13093983C4semidirect
10. C22×C4⋊Dic47C4semidirect
15. C22×C4×Dic47C4semidirect
17. #15688884C22direct (N × H)
21. C22×M16×C47C4semidirect
31. #13093983C2×C4semidirect
34. C2×C4⋊Dic47C2×C4semidirect
42. C2×C4×Dic47C2×C4semidirect
58. C2×M16×C47C2×C4semidirect
99. M16×C47C22×C4semidirect
105. C4⋊Dic47C22×C4semidirect
110. C4×Dic47C22×C4semidirect
156. C22×C47#339semidirect
178. C2×C47C2×#339semidirect
185. C22×M16Dic47semidirect
200. C47C22×#339semidirect
219. C2×M16C2×Dic47semidirect
276. M16C22×Dic47semidirect
332. C22#15688884direct (N × H)
350. C2C2×#15688884direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939832
C23×C4×Dic472
C2×#156888842
C23×M16×C472
C22×#33947