Finite Group Databasebeta

← #26374878

Subgroups — #26374878

Order 12032.

2390 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093981C211yesmaximal, characteristic
2#20245993C218yesmaximal, complemented
3#20245988C214yesmaximal, complemented
4#20245983C214yesmaximal
5#20245885C214yesmaximal, complemented
6#20245883C214yesmaximal, complemented
7#20245791C212yesmaximal, complemented
8#20245767C212yesmaximal
9C4×#13093983C211yesmaximal, characteristic, complemented
10C4×#117×C47C211yesmaximal, characteristic, complemented
11C23×C4×D47C2211yescharacteristic
12C2×#13093983C2212yes—
13C2×#13093981C4116yescomplemented
14C2×#13093981C2212yes—
15C2×#13093981C2212yes—
16C2×#13093981C2214yes—
17C22×Dic47⋊C4C2212yes—
18C22×(C2×D47)⋊C4C2212yes—
19C22×C4×Dic47C2211yescharacteristic
20#15689014C22116yes—
21#15689012C22116yes—
22#15689007C22116yescomplemented
23#15689005C22116yescomplemented
24#15688992C2214yescomplemented
25#15688988C2214yescomplemented
26#15688984C2214yes—
27#15688981C2218yes—
28#15688980C2218yes—
29#15688978C2214yes—
30C4×Dic47⋊C4C2218yescomplemented
31C4×C4⋊Dic47C2214yescomplemented
32C4×(C2×D47)⋊C4C2218yes—
33C4×C22⋊Dic47C2214yes—
34#15688969C2212yescomplemented
35C42×Dic47C2212yescomplemented
36C2×#117×C47C2212yes—
37C22×C42×C47C2211yescharacteristic
38C4×C4⋊C4×C47C2214yescomplemented
39C4×C22⋊C4×C47C2214yes—
40C47×#376C2212yescomplemented
41C43×C47C2212yescomplemented
42C24×D47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2312yes—
45C22×C4×D47—28no—
46C22×C4×D47C2×C418yes—
47C22×C4×D47C2312yes—
48C22×C4×D47C2312yes—
49#13093983—28no—
50#13093981C2×C4116yes—
51#13093981C2×C4116yes—
52#13093981C2×C4132yes—
53C2×C4⋊Dic47C2314yes—
54C2×Dic47⋊C4C2×C4116yes—
55C2×Dic47⋊C4C23116yes—
56C2×Dic47⋊C4C2314yes—
57C2×Dic47⋊C4C2314yes—
58C2×C22⋊Dic47C2314yes—
59C2×(C2×D47)⋊C4C2×C4116yes—
60C2×(C2×D47)⋊C4C23116yes—
61C2×(C2×D47)⋊C4C2314yes—
62C2×(C2×D47)⋊C4C2314yes—
63C2×C4×Dic47C2×C418yes—
64C2×C4×Dic47C2314yes—
65C2×C4×Dic47C2318yes—
66C2×C4×Dic47C2312yes—
67C2×C4×Dic47C2312yes—
68C47⋊#95C2318yes—
69C47⋊#95C23116yes—
70C47⋊#95C23116yes—
71C23×C4×C47C2312yes—
72C23×C4×C47C2311yescharacteristic
73#117×C47—28no—
74C2×C4⋊C4×C47C2314yes—
75C2×C22⋊C4×C47C2314yes—
76C2×C42×C47C2×C418yes—
77C2×C42×C47C2314yes—
78C2×C42×C47C2318yes—
79C2×C42×C47C2311yescharacteristic
80C2×C42×C47C2311yescharacteristic
81C2×C42×C47C2312yes—
82#95×C47C2318yes—
83C24×C47C2411yescharacteristic
84C23×D47—24no—
85C23×D47—44no—
86C23×D47C22×C414yes—
87C23×D47C2412yes—
88C22×C4×C47—44no—
89C22×C4×C47C22×C418yes—
90C22×C4×C47C22×C414yes—
91C22×C4×C47#4118yes—
92C22×C4×C47C2418yes—
93C22×C4×C47C2412yes—
94C22×C4×C47C2412yes—
95C22×C4×C47C2411yescharacteristic
96C22×C4×C47C2411yescharacteristic
97C22×Dic47—24no—
98C22×Dic47—44no—
99C22×Dic47C22×C414yes—
100C22×Dic47C22×C418yes—
101C22×Dic47C2418yes—
102C22×Dic47C2412yes—
103C22×Dic47C2414yes—
104C2×C4×D47—232no—
105C2×C4×D47—28no—
106C2×C4×D47—28no—
107C2×C4×D47C22×C418yes—
108C2×C4×D47C22×C418yes—
109C4⋊C4×C47—28no—
110C22⋊C4×C47—28no—
111C42×C47—28no—
112C42×C47—48no—
113C42×C47C22×C414yes—
114C42×C47C22×C414yes—
115C42×C47C22×C418yes—
116C22⋊Dic47—28no—
117(C2×D47)⋊C4—416no—
118(C2×D47)⋊C4C22×C4116yes—
119(C2×D47)⋊C4C22×C4116yes—
120C4⋊Dic47—28no—
121Dic47⋊C4—416no—
122Dic47⋊C4C22×C4116yes—
123Dic47⋊C4C22×C4116yes—
124C4×Dic47—28no—
125C4×Dic47—48no—
126C4×Dic47C22×C418yes—
127C4×Dic47C22×C418yes—
128C23×C47—24no—
129C23×C47C23×C412yes—
130C23×C47C2×#4114yes—
131C23×C47C2511yescharacteristic
132C22×D47—28no—
133C22×D47—24no—
134C22×D47—28no—
135C22×D47—48no—
136C22×D47—416no—
137C22×D47C23×C414yes—
138C2×C4×C47—24no—
139C2×C4×C47—24no—
140C2×C4×C47—28no—
141C2×C4×C47—216no—
142C2×C4×C47—48no—
143C2×C4×C47#117116yes—
144C2×C4×C47C23×C414yes—
145C2×C4×C47C23×C414yes—
146C2×C4×C47C23×C412yes—
147C2×C4×C47C23×C412yes—
148C2×C4×C47C2×#4114yes—
149C2×C4×C47C2×#4114yes—
150C2×Dic47—28no—
151C2×Dic47—24no—
152C2×Dic47—28no—
153C2×Dic47—28no—
154C2×Dic47—216no—
155C2×Dic47—48no—
156C2×Dic47C23×C418yes—
157C2×Dic47C23×C414yes—
158C4×D47—216no—
159C4×D47—216no—
160#34048—471nomaximal, Sylow
161C22×C47—24no—
162C22×C47—28no—
163C22×C47C2×#11714yes—
164C22×C47C24×C411yescharacteristic
165C22×C47C22×#4112yes—
166C22×C47#58514yes—
167C2×D47—24no—
168C2×D47—28no—
169C2×D47—416no—
170C2×D47—48no—
171C4×C47—24no—
172C4×C47—24no—
173C4×C47—48no—
174C4×C47C2×#11714yes—
175C4×C47C2×#11714yes—
176Dic47—24no—
177Dic47—28no—
178Dic47—48no—
179C22×#117—471no—
180#2314—478no—
181#2313—474no—
182#2312—474no—
183#2298—474no—
184#2298—474no—
185#2285—472no—
186#2280—472no—
187C4×#117—471no—
188C4×#117D4711yescharacteristic, complemented
189C2×C47—24no—
190C2×C47C22×#11711yescharacteristic
191C2×C47#342812yes—
192C2×C47#345514yes—
193D47—44no—
194C24×C4—471no—
195C2×#117—472no—
196C2×#117—4716no—
197C2×#117—474no—
198C2×#117—472no—
199C2×#117—472no—
200C2×#117C2×D4712yes—
201C22×C4⋊C4—472no—
202C22×C22⋊C4—472no—
203C22×C42—471no—
204C22×C42C2×D4711yescharacteristic
205#388—4716no—
206#388—4716no—
207#387—4716no—
208#387—4716no—
209#384—474no—
210#382—474no—
211#381—474no—
212#380—478no—
213#380—478no—
214#379—474no—
215C4×C4⋊C4—474no—
216C4×C4⋊C4—478no—
217C4×C4⋊C4C2×D4714yescomplemented
218C4×C22⋊C4—474no—
219C4×C22⋊C4—478no—
220C4×C22⋊C4C2×D4714yes—
221#376—472no—
222#376C2×D4712yescomplemented
223C43—472no—
224C43C2×D4712yescomplemented
225C47#3404811yescharacteristic, complemented, Sylow
226C25—471no—
227C23×C4—471no—
228C23×C4—472no—
229C23×C4—478no—
230C23×C4—472no—
231C23×C4—472no—
232C23×C4—948no—
233C23×C4C22×D4712yes—
234C23×C4C22×D4711yescharacteristic
235#117—28no—
236#117—4732no—
237#117—4716no—
238#117—4716no—
239#117—948no—
240C2×C4⋊C4—4716no—
241C2×C4⋊C4—474no—
242C2×C4⋊C4—474no—
243C2×C4⋊C4—474no—
244C2×C4⋊C4—4716no—
245C2×C4⋊C4C22×D4714yes—
246C2×C22⋊C4—4716no—
247C2×C22⋊C4—474no—
248C2×C22⋊C4—474no—
249C2×C22⋊C4—474no—
250C2×C22⋊C4—4716no—
251C2×C22⋊C4C22×D4714yes—
252C2×C42—478no—
253C2×C42—472no—
254C2×C42—472no—
255C2×C42—474no—
256C2×C42—478no—
257C2×C42C4×D4718yes—
258C2×C42C22×D4714yes—
259C2×C42C22×D4718yes—
260C2×C42C22×D4712yes—
261C2×C42C22×D4711yescharacteristic
262C2×C42C22×D4711yescharacteristic
263#95—478no—
264#95—4716no—
265#95—4716no—
266#95C22×D4718yes—
267C24—472no—
268C24—474no—
269C24—944no—
270C24—1884no—
271C24C23×D4711yescharacteristic
272C22×C4—44no—
273C22×C4—474no—
274C22×C4—472no—
275C22×C4—474no—
276C22×C4—478no—
277C22×C4—478no—
278C22×C4—478no—
279C22×C4—478no—
280C22×C4—944no—
281C22×C4—9432no—
282C22×C4—948no—
283C22×C4—948no—
284C22×C4—1884no—
285C22×C4C2×C4×D4718yes—
286C22×C4C2×C4×D4714yes—
287C22×C4#1062535018yes—
288C22×C4C23×D4718yes—
289C22×C4C23×D4712yes—
290C22×C4C23×D4712yes—
291C22×C4C23×D4711yescharacteristic
292C22×C4C23×D4711yescharacteristic
293C4⋊C4—28no—
294C4⋊C4—4716no—
295C4⋊C4—4716no—
296C4⋊C4—948no—
297C4⋊C4—18816no—
298C22⋊C4—28no—
299C22⋊C4—4716no—
300C22⋊C4—4716no—
301C22⋊C4—948no—
302C22⋊C4—18816no—
303C42—28no—
304C42—48no—
305C42—478no—
306C42—478no—
307C42—948no—
308C42—1888no—
309C42C2×C4×D4718yes—
310C42C2×C4×D4714yes—
311C42C2×C4×D4714yes—
312C23—24no—
313C23—474no—
314C23—948no—
315C23—944no—
316C23—948no—
317C23—1888no—
318C23—18816no—
319C23C22×C4×D4712yes—
320C23C2×#1062535014yes—
321C23C24×D4711yescharacteristic
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—216no—
326C2×C4—48no—
327C2×C4—478no—
328C2×C4—474no—
329C2×C4—948no—
330C2×C4—944no—
331C2×C4—948no—
332C2×C4—9416no—
333C2×C4—9416no—
334C2×C4—948no—
335C2×C4—9416no—
336C2×C4—1888no—
337C2×C4#13093981116yes—
338C2×C4C22×C4×D4714yes—
339C2×C4C22×C4×D4714yes—
340C2×C4C22×C4×D4712yes—
341C2×C4C22×C4×D4712yes—
342C2×C4C2×#1062535014yes—
343C2×C4C2×#1062535014yes—
344C22—28no—
345C22—24no—
346C22—944no—
347C22—948no—
348C22—18816no—
349C22—1888no—
350C22C2×#1309398114yes—
351C22C23×C4×D4711yescharacteristic
352C22C22×#1062535012yes—
353C22#1568995914yes—
354C4—24no—
355C4—24no—
356C4—48no—
357C4—944no—
358C4—948no—
359C4—1888no—
360C4C2×#1309398114yes—
361C4C2×#1309398114yes—
362C2—24no—
363C2—1884no—
364C2C22×#1309398111yescharacteristic
365C2#2025658912yes—
366C2#2025677314yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC4×#117×C47C2
socle, cosocleC23×C47C25
prime radicalC24×D47C23
hypercenterC4×#117D47
hypocenterC47#34048
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #20245993C2semidirect
3. #20245988C2semidirect
5. #20245885C2semidirect
6. #20245883C2semidirect
7. #20245791C2semidirect
9. C4×#13093983C2semidirect
10. C4×#117×C47C2semidirect
13. C2×#13093981C4semidirect
22. #15689007C22semidirect
23. #15689005C22semidirect
24. #15688992C22semidirect
25. #15688988C22semidirect
30. C4×Dic47⋊C4C22semidirect
31. C4×C4⋊Dic47C22semidirect
34. #15688969C22semidirect
35. C42×Dic47C22semidirect
38. C4×C4⋊C4×C47C22semidirect
40. C47×#376C22semidirect
41. C43×C47C22semidirect
188. C4×#117D47semidirect
217. C4×C4⋊C4C2×D47semidirect
222. #376C2×D47semidirect
224. C43C2×D47semidirect
225. C47#34048semidirect

Maximal subgroups

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

subgroupindex
C22×#130939812
#202459932
#202459882
#202459832
#202458852
#202458832
#202457912
#202457672
C4×#130939832
C4×#117×C472
#3404847