Finite Group Databasebeta

← #26376324

Subgroups — #26376324

Order 12032.

4110 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246237C2116yesmaximal, complemented
2#20246002C212yesmaximal, complemented
3#20245986C214yesmaximal, complemented
4#20245771C214yesmaximal, complemented
5C2×#15688979C212yesmaximal
6#20245741C212yesmaximal, complemented
7C23×C22⋊C4×C47C211yesmaximal, characteristic
8C2×#13093983C2218yescomplemented
9C22×C22⋊Dic47C2214yes—
10C22×C22⋊Dic47C2218yes—
11#15689016C22132yescomplemented
12#15688998C22116yescomplemented
13#15688986C2218yescomplemented
14#15688983C2218yescomplemented
15#15688983C2218yescomplemented
16#15688979C2214yes—
17#15688979C2218yes—
18#15688979C2214yes—
19C22⋊C4×Dic47C22116yescomplemented
20C4×C22⋊Dic47C2218yescomplemented
21C4×C22⋊Dic47C2218yescomplemented
22C26×C47C2211yescharacteristic
23C24×C4×C47C2212yes—
24C22×C22⋊C4×C47C4116yescomplemented
25C22×C22⋊C4×C47C2211yescharacteristic
26C22×C22⋊C4×C47C2211yescharacteristic
27C22×C22⋊C4×C47C2218yes—
28C22×C22⋊C4×C47C2212yes—
29C23×Dic47C2318yes—
30#13093983—416no—
31C2×C4⋊Dic47C23116yescomplemented
32C2×C22⋊Dic47—216no—
33C2×C22⋊Dic47—416no—
34C2×C22⋊Dic47C2318yes—
35C2×C22⋊Dic47C23116yes—
36C2×C22⋊Dic47C2318yes—
37C2×C22⋊Dic47C23116yes—
38C2×C4×Dic47C23116yescomplemented
39C47⋊#95C23116yescomplemented
40C47⋊#95C23116yescomplemented
41C25×C47—28no—
42C25×C47—48no—
43C25×C47C2×C418yes—
44C25×C47C2311yescharacteristic
45C25×C47C2314yes—
46C25×C47C2311yescharacteristic
47C25×C47C2311yescharacteristic
48C23×C4×C47—48no—
49C23×C4×C47C2×C4116yescomplemented
50C23×C4×C47C2314yes—
51C23×C4×C47C2312yes—
52C23×C4×C47C2314yes—
53C23×C4×C47C2312yes—
54C2×C22⋊C4×C47C2×C418yes—
55C2×C22⋊C4×C47C2×C418yes—
56C2×C22⋊C4×C47C2×C4164yescomplemented
57C2×C22⋊C4×C47C2×C4116yescomplemented
58C2×C22⋊C4×C47C2314yes—
59C2×C22⋊C4×C47C2314yes—
60C2×C22⋊C4×C47C2318yes—
61C24×C47—24no—
62C24×C47—28no—
63C24×C47—232no—
64C24×C47—216no—
65C24×C47—24no—
66C24×C47—216no—
67C24×C47—432no—
68C24×C47—44no—
69C24×C47—48no—
70C24×C47—44no—
71C24×C47—464no—
72C24×C47C22×C414yes—
73C24×C47C22×C4116yes—
74C24×C47C22×C414yes—
75C24×C47C22×C414yes—
76C24×C47#4118yes—
77C24×C47C2412yes—
78C24×C47C2412yes—
79C24×C47C2411yescharacteristic
80C24×C47C2412yes—
81C22×C4×C47—216no—
82C22×C4×C47—432no—
83C22×C4×C47—44no—
84C22×C4×C47—48no—
85C22×C4×C47—44no—
86C22×C4×C47C22×C4116yes—
87C22×C4×C47C22×C418yes—
88C22×C4×C47C22×C418yes—
89C22×C4×C47C22×C4116yes—
90C22×C4×C47C2414yes—
91C22×C4×C47C2414yes—
92C22×Dic47—216no—
93C22×Dic47—416no—
94C22×Dic47C24116yes—
95C22⋊C4×C47C22×C4116yes—
96C22⋊C4×C47C22×C4116yes—
97C22⋊C4×C47C22×C4132yes—
98C22⋊Dic47—216no—
99C22⋊Dic47—216no—
100C22⋊Dic47—232no—
101C22⋊Dic47—416no—
102C4⋊Dic47—416no—
103C4×Dic47—416no—
104C23×C47—232no—
105C23×C47—28no—
106C23×C47—216no—
107C23×C47—24no—
108C23×C47—28no—
109C23×C47—24no—
110C23×C47—24no—
111C23×C47—232no—
112C23×C47—28no—
113C23×C47—232no—
114C23×C47—24no—
115C23×C47—28no—
116C23×C47—48no—
117C23×C47—416no—
118C23×C47—48no—
119C23×C47—416no—
120C23×C47—464no—
121C23×C47—4128no—
122C23×C47—416no—
123C23×C47#117116yescomplemented
124C23×C47C23×C414yes—
125C23×C47C23×C414yes—
126C23×C47C23×C412yes—
127C23×C47C23×C414yes—
128C23×C47C2×#4114yes—
129C23×C47C2×#4118yes—
130C23×C47H(5,𝔽2)18yes—
131C23×C47C2511yescharacteristic
132C2×C4×C47—232no—
133C2×C4×C47—28no—
134C2×C4×C47—28no—
135C2×C4×C47—48no—
136C2×C4×C47—48no—
137C2×C4×C47—416no—
138C2×C4×C47C23×C418yes—
139C2×C4×C47C23×C418yes—
140C2×Dic47—216no—
141C2×Dic47—232no—
142C2×Dic47—816no—
143#34176—471nomaximal, Sylow
144C22×C47—232no—
145C22×C47—24no—
146C22×C47—24no—
147C22×C47—24no—
148C22×C47—28no—
149C22×C47—216no—
150C22×C47—28no—
151C22×C47—216no—
152C22×C47—24no—
153C22×C47—28no—
154C22×C47—416no—
155C22×C47—464no—
156C22×C47—416no—
157C22×C47—48no—
158C22×C47C2×#11714yes—
159C22×C47C2×#11718yes—
160C22×C47#51818yes—
161C22×C47C24×C411yescharacteristic
162C22×C47C22×#4112yes—
163C22×C47C2×H(5,𝔽2)14yes—
164C4×C47—28no—
165C4×C47—28no—
166Dic47—416no—
167C23×C22⋊C4D4711yescharacteristic
168#2342—4716no—
169#2315—472no—
170#2312—474no—
171#2280—474no—
172C2×#379—472no—
173#2273—472no—
174C2×C47—24no—
175C2×C47—28no—
176C2×C47—28no—
177C2×C47—44no—
178C2×C47C22×#11711yescharacteristic
179C2×C47C2×#51812yes—
180C2×C47#349014yes—
181C26C2×D4711yescharacteristic
182C24×C4C2×D4712yes—
183C2×#117—478no—
184C22×C22⋊C4—474no—
185C22×C22⋊C4—478no—
186C22×C22⋊C4Dic47116yescomplemented
187C22×C22⋊C4C2×D4711yescharacteristic
188C22×C22⋊C4C2×D4711yescharacteristic
189C22×C22⋊C4C2×D4712yes—
190C22×C22⋊C4C2×D4718yes—
191#388—4732no—
192#385—4716no—
193#381—478no—
194#380—478no—
195#380—478no—
196#379—474no—
197#379—478no—
198#379—474no—
199C4×C22⋊C4—478no—
200C4×C22⋊C4—478no—
201C4×C22⋊C4—4716no—
202C47#3417611yescharacteristic, complemented, Sylow
203C25—28no—
204C25—48no—
205C25C2×Dic4718yes—
206C25C22×D4711yescharacteristic
207C25C22×D4714yes—
208C25C22×D4711yescharacteristic
209C25C22×D4711yescharacteristic
210C23×C4—48no—
211C23×C4—478no—
212C23×C4C2×Dic47116yescomplemented
213C23×C4C22×D4714yes—
214C23×C4C22×D4712yes—
215C23×C4C22×D4714yes—
216C23×C4C22×D4712yes—
217#117—18816no—
218C2×C4⋊C4—4716no—
219C2×C22⋊C4—4716no—
220C2×C22⋊C4—478no—
221C2×C22⋊C4—478no—
222C2×C22⋊C4—4716no—
223C2×C22⋊C4—9416no—
224C2×C22⋊C4—18816no—
225C2×C22⋊C4C2×Dic4718yes—
226C2×C22⋊C4C2×Dic4718yes—
227C2×C22⋊C4C2×Dic47164yescomplemented
228C2×C22⋊C4C2×Dic47116yescomplemented
229C2×C22⋊C4C22×D4714yes—
230C2×C22⋊C4C22×D4714yes—
231C2×C22⋊C4C22×D4718yes—
232C2×C42—4716no—
233#95—4716no—
234#95—4716no—
235C24—24no—
236C24—28no—
237C24—232no—
238C24—216no—
239C24—24no—
240C24—216no—
241C24—432no—
242C24—44no—
243C24—48no—
244C24—44no—
245C24—464no—
246C24#1062535018yes—
247C24C22×Dic4714yes—
248C24C22×Dic47116yes—
249C24C22×Dic4714yes—
250C24C22×Dic4714yes—
251C24C23×D4712yes—
252C24C23×D4711yescharacteristic
253C24C23×D4712yes—
254C24C23×D4712yes—
255C22×C4—216no—
256C22×C4—432no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4—44no—
260C22×C4—4716no—
261C22×C4—9416no—
262C22×C4—18816no—
263C22×C4C22×Dic47116yes—
264C22×C4C22×Dic4718yes—
265C22×C4C22×Dic4718yes—
266C22×C4C22×Dic47116yes—
267C22×C4C23×D4714yes—
268C22×C4C23×D4714yes—
269C4⋊C4—18816no—
270C22⋊C4—9416no—
271C22⋊C4—9416no—
272C22⋊C4—9432no—
273C22⋊C4—18816no—
274C22⋊C4C22×Dic47116yes—
275C22⋊C4C22×Dic47116yes—
276C22⋊C4C22×Dic47132yes—
277C42—18816no—
278C23—232no—
279C23—28no—
280C23—216no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—28no—
285C23—232no—
286C23—28no—
287C23—232no—
288C23—24no—
289C23—28no—
290C23—416no—
291C23—48no—
292C23—416no—
293C23—48no—
294C23—464no—
295C23—4128no—
296C23—416no—
297C23#13093983116yescomplemented
298C23C23×Dic4714yes—
299C23C23×Dic4714yes—
300C23C23×Dic4712yes—
301C23C23×Dic4714yes—
302C23C2×#1062535014yes—
303C23C2×#1062535018yes—
304C23#1309406918yes—
305C23C24×D4711yescharacteristic
306C2×C4—232no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—416no—
312C2×C4—9416no—
313C2×C4—9432no—
314C2×C4—37616no—
315C2×C4C23×Dic4718yes—
316C2×C4C23×Dic4718yes—
317C22—232no—
318C22—24no—
319C22—216no—
320C22—28no—
321C22—216no—
322C22—24no—
323C22—24no—
324C22—28no—
325C22—24no—
326C22—28no—
327C22—416no—
328C22—464no—
329C22—416no—
330C22—48no—
331C22C2×#1309398314yes—
332C22C2×#1309398318yes—
333C22#1568959818yes—
334C22C24×Dic4711yescharacteristic
335C22C22×#1062535012yes—
336C22C2×#1309406914yes—
337C4—28no—
338C4—28no—
339C4—18816no—
340C2—24no—
341C2—28no—
342C2—28no—
343C2—44no—
344C2C22×#1309398311yescharacteristic
345C2C2×#1568959812yes—
346C2#2025707714yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC23×C22⋊C4×C47C2
socle, cosocleC23×C47C25
prime radicalC26×C47C22
hypercenterC23×C22⋊C4D47
hypocenterC47#34176
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20246237C2semidirect
2. #20246002C2semidirect
3. #20245986C2semidirect
4. #20245771C2semidirect
6. #20245741C2semidirect
8. C2×#13093983C22semidirect
11. #15689016C22semidirect
12. #15688998C22semidirect
13. #15688986C22semidirect
14. #15688983C22semidirect
15. #15688983C22semidirect
19. C22⋊C4×Dic47C22semidirect
20. C4×C22⋊Dic47C22semidirect
21. C4×C22⋊Dic47C22semidirect
24. C22×C22⋊C4×C47C4semidirect
31. C2×C4⋊Dic47C23semidirect
38. C2×C4×Dic47C23semidirect
39. C47⋊#95C23semidirect
40. C47⋊#95C23semidirect
49. C23×C4×C47C2×C4semidirect
56. C2×C22⋊C4×C47C2×C4semidirect
57. C2×C22⋊C4×C47C2×C4semidirect
123. C23×C47#117semidirect
186. C22×C22⋊C4Dic47semidirect
202. C47#34176semidirect
212. C23×C4C2×Dic47semidirect
227. C2×C22⋊C4C2×Dic47semidirect
228. C2×C22⋊C4C2×Dic47semidirect
297. C23#13093983semidirect

Maximal subgroups

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

subgroupindex
#202462372
#202460022
#202459862
#202457712
C2×#156889792
#202457412
C23×C22⋊C4×C472
#3417647