Finite Group Databasebeta

← #27083895

Subgroups — #27083895

Order 12032.

2178 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256793C211yesmaximal, characteristic, complemented
2C2×#15689597C211yesmaximal, characteristic
3#20253220C2116yesmaximal, complemented
4#20253187C218yesmaximal, complemented
5#20251978C212yesmaximal, complemented
6C2×#15689129C212yesmaximal
7C47×#2905C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9#15689754—22no—
10#15689751C2218yescomplemented
11#15689749—22no—
12#15689680C2214yescomplemented
13#15689676—24no—
14#15689597—24no—
15#15689597C2218yes—
16C2×C4×D188C2214yes—
17C2×#13093986—22no—
18C2×#13093983C2212yes—
19C2×#13093983C2211yescharacteristic
20C2×C22⋊C4×D47—22no—
21#15689424C22132yescomplemented
22#15689414C22116yescomplemented
23#15689404C22116yescomplemented
24#15689165C2218yescomplemented
25#15689165C2218yescomplemented
26#15689161C2218yescomplemented
27#15689129C2218yes—
28#15689129C2218yes—
29C2×#13093946C2218yes—
30C22×M16×C47C2212yes—
31C2×#117×C47C2211yescharacteristic
32C47×#424C2218yescomplemented
33C47×#423C2214yescomplemented
34C24×D47—21no—
35C22×D188—21no—
36C22×D188—22no—
37C22×D188—24no—
38C22×D188Q818yescomplemented
39C22×D188C2312yes—
40C22×D188C2314yes—
41C22×C4×D47—24no—
42#13094024C2318yescomplemented
43#13094023—28no—
44#13094016C2318yescomplemented
45#13094015—28no—
46#13094003—28no—
47#13094001—216no—
48C4×Dic94—28no—
49C4×D188—28no—
50C4×D188C23116yes—
51#13093986—216no—
52#13093983—22no—
53#13093983—22no—
54#13093983—22no—
55#13093983C2314yes—
56#13093983C2314yes—
57#13093983C2314yes—
58C2×C4⋊Dic47C2314yes—
59C2×C4⋊Dic47C2312yes—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—21no—
62C2×(C2×D47)⋊C4—24no—
63C22⋊C4×D47—216no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—22no—
66#13093958C23116yescomplemented
67#13093956C23116yescomplemented
68#13093946C23132yes—
69C23×C4×C47C2311yescharacteristic
70C2×M16×C47—28no—
71C2×M16×C47C2314yes—
72C2×M16×C47C2314yes—
73C22×C8×C47C2314yes—
74#117×C47D414yes—
75#117×C47C2314yes—
76C2×C4⋊C4×C47D412yes—
77C2×C22⋊C4×C47D412yes—
78C2×C42×C47C2312yes—
79C4⋊C8×C47C23116yescomplemented
80C24×C47C2×D411yescharacteristic
81C23×D47—22no—
82C23×D47—28no—
83C23×D47—24no—
84C23×D47—84no—
85C22×C4×C47C2×D414yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×Q814yes—
89C22×C4×C47#4114yes—
90C22×C4×C47C2411yescharacteristic
91C22×C4×C47C2412yes—
92C22×Dic47—22no—
93C22×Dic47—24no—
94C2×D188—28no—
95C2×D188—216no—
96C2×D188—24no—
97C2×D188—22no—
98C2×D188—28no—
99C2×D188—24no—
100C2×D188C2×Q818yes—
101C2×D188C2×Q8116yes—
102C2×D188C2414yes—
103C2×C4×D47—216no—
104C2×C4×D47—88no—
105C2×Dic94—28no—
106M16×C47—28no—
107M16×C47—28no—
108C2×C8×C47—28no—
109C2×C8×C47C2418yes—
110C4⋊C4×C47C2×D418yes—
111C22⋊C4×C47C2×D418yes—
112C42×C47C2×D414yes—
113C42×C47C2414yes—
114C22⋊Dic47—24no—
115C22⋊Dic47—24no—
116C22⋊Dic47—24no—
117(C2×D47)⋊C4—216no—
118C4⋊Dic47—22no—
119C4⋊Dic47—24no—
120C4⋊Dic47—28no—
121C4⋊Dic47C2418yes—
122C4⋊Dic47C2414yes—
123Dic47⋊C4—216no—
124C4×Dic47—24no—
125C4×Dic47—24no—
126C4×Dic47—24no—
127C23×C47—42no—
128C23×C47C22⋊Q814yes—
129C23×C47C22×D412yes—
130C23×C47C22×D411yescharacteristic
131C22×D47—28no—
132C22×D47—24no—
133C22×D47—216no—
134C22×D47—48no—
135C22×D47—88no—
136C22×D47—816no—
137C2×C4×C47—44no—
138C2×C4×C47C22⋊Q814yes—
139C2×C4×C47C22⋊Q818yes—
140C2×C4×C47C22×D418yes—
141C2×C4×C47C22×D411yescharacteristic
142C2×C4×C47C22×D412yes—
143C2×C4×C47C22×Q812yes—
144C2×C4×C47C22×Q814yes—
145C2×C4×C47C2×#4114yes—
146C2×C4×C47C2×#4112yes—
147C2×C4×C47C2511yescharacteristic
148C2×Dic47—28no—
149C2×Dic47—24no—
150C2×Dic47—44no—
151C2×Dic47—84no—
152D188—216no—
153D188—28no—
154D188—24no—
155D188C22×Q818yes—
156C4×D47—816no—
157Dic94—88no—
158C8×C47—28no—
159#60490—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—24no—
162C22×C47—44no—
163C22×C47C2×C22⋊Q812yes—
164C22×C47C2×C22⋊Q814yes—
165C22×C47C23×D411yescharacteristic
166C2×D47—28no—
167C2×D47—416no—
168C2×D47—48no—
169C2×D47—816no—
170C4×C47—28no—
171C4×C47C2×C22⋊Q812yes—
172C4×C47C2×C22⋊Q814yes—
173C4×C47C23×Q811yescharacteristic
174C4×C47C22×#4111yescharacteristic
175Dic47—48no—
176Dic47—44no—
177#3458—471no—
178C2×#518—471no—
179#3081—4716no—
180#3077—478no—
181#2911—472no—
182#2905D4711yescharacteristic, complemented
183C2×#418—472no—
184C2×C47—22no—
185C2×C47—24no—
186C2×C47C22×C22⋊Q811yescharacteristic
187C2×C47#358512yes—
188D47—48no—
189C23×D4—471no—
190C22×M16C2×D4712yes—
191#544—478no—
192#544—942no—
193#543—942no—
194#533—474no—
195#533—944no—
196#518—478no—
197#518—944no—
198C2×C4×D4—474no—
199C2×C4×D4—942no—
200C2×#117—472no—
201C2×#117—471no—
202C2×#117C2×D4711yescharacteristic
203C22×C22⋊C4—942no—
204#478—4732no—
205#476—4716no—
206#474—4716no—
207#428—478no—
208#428—478no—
209#427—478no—
210#424C2×D4718yescomplemented
211#423C2×D4714yescomplemented
212#418—478no—
213#418—478no—
214C2×#102—478no—
215C47#6049011yescharacteristic, complemented, Sylow
216C25—941no—
217C22×D4—478no—
218C22×D4—472no—
219C22×D4—474no—
220C22×D4—941no—
221C22×D4—942no—
222C22×D4—944no—
223C23×C4—944no—
224C23×C4C22×D4711yescharacteristic
225C2×M16—28no—
226C2×M16C22×D4714yes—
227C2×M16C22×D4714yes—
228C22×C8C22×D4714yes—
229C4⋊Q8—478no—
230C4⋊Q8—948no—
231#125—478no—
232#125—948no—
233C22⋊Q8—948no—
234C22⋊Q8—9416no—
235C4×Q8—948no—
236C4×D4—4716no—
237C4×D4—9416no—
238C4×D4—948no—
239#117—474no—
240#117—474no—
241#117—474no—
242#117—942no—
243#117—942no—
244#117—942no—
245#117D18814yes—
246#117C22×D4714yes—
247C2×C4⋊C4—472no—
248C2×C4⋊C4—474no—
249C2×C4⋊C4D18812yes—
250C2×C22⋊C4—941no—
251C2×C22⋊C4—942no—
252C2×C22⋊C4—944no—
253C2×C22⋊C4—9416no—
254C2×C22⋊C4D18812yes—
255C2×C42—941no—
256C2×C42—942no—
257C2×C42C22×D4712yes—
258#107—4716no—
259#106—4716no—
260C4⋊C8C22×D47116yescomplemented
261#102—4732no—
262C24—948no—
263C24—944no—
264C24—942no—
265C24—3764no—
266C24C2×D18811yescharacteristic
267C2×Q8—948no—
268C2×D4—4716no—
269C2×D4—478no—
270C2×D4—474no—
271C2×D4—9416no—
272C2×D4—948no—
273C2×D4—942no—
274C2×D4—944no—
275C2×D4—948no—
276C2×D4—944no—
277C22×C4—944no—
278C22×C4—942no—
279C22×C4—9416no—
280C22×C4—3768no—
281C22×C4C2×D18814yes—
282C22×C4C2×D18811yescharacteristic
283C22×C4C2×D18812yes—
284C22×C4#1062535214yes—
285C22×C4Q8×D4714yes—
286C22×C4C23×D4711yescharacteristic
287C22×C4C23×D4712yes—
288M16—28no—
289M16—28no—
290C2×C8—28no—
291C2×C8C23×D4718yes—
292C4⋊C4—478no—
293C4⋊C4—474no—
294C4⋊C4—9416no—
295C4⋊C4—942no—
296C4⋊C4—944no—
297C4⋊C4—948no—
298C4⋊C4C2×D18818yes—
299C22⋊C4—944no—
300C22⋊C4—944no—
301C22⋊C4—944no—
302C22⋊C4—9416no—
303C22⋊C4C2×D18818yes—
304C42—944no—
305C42—944no—
306C42—944no—
307C42C2×D18814yes—
308C42C23×D4714yes—
309C23—42no—
310C23—9416no—
311C23—948no—
312C23—944no—
313C23—3768no—
314C23—37616no—
315C23—1888no—
316C23#1309400414yes—
317C23C22×D18812yes—
318C23C22×D18811yescharacteristic
319Q8—3768no—
320D4—478no—
321D4—948no—
322D4—9416no—
323D4—944no—
324C2×C4—44no—
325C2×C4—944no—
326C2×C4—948no—
327C2×C4—37616no—
328C2×C4—3764no—
329C2×C4—1884no—
330C2×C4#1309400414yes—
331C2×C4#1309400418yes—
332C2×C4C22×D18818yes—
333C2×C4C22×D18811yescharacteristic
334C2×C4C22×D18812yes—
335C2×C4C2×Q8×D4712yes—
336C2×C4C2×Q8×D4714yes—
337C2×C4C2×#1062535214yes—
338C2×C4C2×#1062535212yes—
339C2×C4C24×D4711yescharacteristic
340C8—28no—
341C22—22no—
342C22—24no—
343C22—44no—
344C22—948no—
345C22—37616no—
346C22—18816no—
347C22—1888no—
348C22C2×#1309400414yes—
349C22C2×#1309400412yes—
350C22C23×D18811yescharacteristic
351C4—28no—
352C4—1884no—
353C4—1888no—
354C4C2×#1309400412yes—
355C4C2×#1309400414yes—
356C4C22×Q8×D4711yescharacteristic
357C4C22×#1062535211yescharacteristic
358C2—24no—
359C2—22no—
360C2—1888no—
361C2C22×#1309400411yescharacteristic
362C2#2025816912yes—
subgroupG/H
centerC22C23×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#2905C2
socleC22×C47C23×D4
prime radicalC23×D188C22
hypercenter#2905D47
hypocenterC47#60490
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. #20256793C2semidirect
3. #20253220C2semidirect
4. #20253187C2semidirect
5. #20251978C2semidirect
7. C47×#2905C2semidirect
10. #15689751C22semidirect
12. #15689680C22semidirect
21. #15689424C22semidirect
22. #15689414C22semidirect
23. #15689404C22semidirect
24. #15689165C22semidirect
25. #15689165C22semidirect
26. #15689161C22semidirect
32. C47×#424C22semidirect
33. C47×#423C22semidirect
38. C22×D188Q8semidirect
42. #13094024C23semidirect
44. #13094016C23semidirect
66. #13093958C23semidirect
67. #13093956C23semidirect
79. C4⋊C8×C47C23semidirect
182. #2905D47semidirect
210. #424C2×D47semidirect
211. #423C2×D47semidirect
215. C47#60490semidirect
260. C4⋊C8C22×D47semidirect

Maximal subgroups

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

subgroupindex
#202567932
C2×#156895972
#202532202
#202531872
#202519782
C2×#156891292
C47×#29052
#6049047