Finite Group Databasebeta

← #26224333

Subgroups — #26224333

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689480C211yesmaximal, characteristic
2C4×#13094012C211yesmaximal, characteristic
3#20244354C211yesmaximal, characteristic, complemented
4#20244197C211yesmaximal, characteristic
5#20244084C212yesmaximal
6#20244077C212yesmaximal, complemented
7#20243518C211yesmaximal, characteristic, complemented
8#20243221C212yesmaximal, complemented
9#20243216C212yesmaximal
10C4×#13093935C211yesmaximal, characteristic, complemented
11C4×#125×C47C211yesmaximal, characteristic, complemented
12C2×#13094012C2211yescharacteristic
13C2×C4×Dic94C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic
15#15689480C418yescomplemented
16C2×#13093947C2212yes—
17C2×#13093947C2212yes—
18C2×#13093944C2212yes—
19C2×#13093944C2212yes—
20C2×#13093935C2212yes—
21C2×#13093935C2212yes—
22C2×C4×C47⋊C8C2211yescharacteristic
23C2×C4×C47⋊C8C2211yescharacteristic
24#15689029C2211yescharacteristic
25#15689025C2211yescharacteristic
26#15689014—22no—
27#15688989C2211yescharacteristic
28C4×(C2×D47)⋊C4—22no—
29#15688885C2212yes—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33C2×#125×C47C2211yescharacteristic
34C47×#389C2212yes—
35C47×#387—22no—
36C47×#382C2211yescharacteristic
37C4×C4⋊C4×C47—22no—
38C4×C4⋊C4×C47C2212yes—
39C43×C47C2211yescharacteristic
40C22×Dic94C2311yescharacteristic
41C22×D188C2311yescharacteristic
42C22×C4×D47—21no—
43C22×C47⋊C8C2312yes—
44C22×C47⋊C8C2312yes—
45#13094012—22no—
46#13094012C2×C414yes—
47C4×Dic94—24no—
48C4×D188—24no—
49C2×C4⋊Dic47C2311yescharacteristic
50C2×C4⋊Dic47C2311yescharacteristic
51C2×Dic47⋊C4—21no—
52C2×(C2×D47)⋊C4—22no—
53C2×(C2×D47)⋊C4—22no—
54C2×(C2×D47)⋊C4—21no—
55C2×C4×Dic47—21no—
56#13093947—24no—
57#13093947C2×C418yescomplemented
58#13093944—24no—
59#13093944C2×C418yes—
60#13093935—24no—
61#13093935—24no—
62C4×C47⋊C8—22no—
63C4×C47⋊C8C2×C414yescomplemented
64C47⋊#95—22no—
65C47⋊#95—22no—
66#125×C47—22no—
67#125×C47C2×C414yescomplemented
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47C2312yes—
71C2×C4⋊C4×C47C2312yes—
72C2×C42×C47—22no—
73C2×C42×C47D412yes—
74C2×C42×C47D412yes—
75C2×C42×C47C2311yescharacteristic
76C2×C42×C47C2311yescharacteristic
77C2×C42×C47C2311yescharacteristic
78#95×C47—22no—
79#95×C47—22no—
80#95×C47—22no—
81C23×D47—21no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2411yescharacteristic
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—21no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—21no—
95C2×D188C22×C412yes—
96C2×C4×D47—44no—
97C2×C4×D47—42no—
98C2×Dic94—22no—
99C2×Dic94—22no—
100C2×Dic94—21no—
101C2×Dic94C22×C412yes—
102C4⋊C4×C47—28no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—44no—
107C4⋊C4×C47C22×C414yes—
108C42×C47—22no—
109C42×C47—42no—
110C42×C47—42no—
111C42×C47—44no—
112C42×C47C22×C412yes—
113C42×C47#4112yes—
114C42×C47#4114yes—
115(C2×D47)⋊C4—28no—
116(C2×D47)⋊C4—44no—
117(C2×D47)⋊C4—42no—
118C4⋊Dic47—22no—
119C4⋊Dic47—22no—
120Dic47⋊C4—42no—
121C4×Dic47—42no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8C22×C414yes—
127C2×C47⋊C8#4114yes—
128C23×C47C22×D411yescharacteristic
129C22×D47—22no—
130C22×D47—41no—
131C22×D47—41no—
132C22×D47—42no—
133C22×D47—42no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—24no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47C4×D414yes—
148C2×C4×C47#12412yes—
149C2×C4×C47#12412yes—
150C2×C4×C47C23×C411yescharacteristic
151C2×C4×C47C2×#4112yes—
152C2×C4×C47C2×#4111yescharacteristic
153C2×C4×C47C2×#4112yes—
154C2×Dic47—22no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—42no—
160C2×Dic47—42no—
161D188—24no—
162D188—42no—
163C4×D47—84no—
164Dic94—24no—
165Dic94—42no—
166C47⋊C8—24no—
167C47⋊C8—24no—
168#21648—471nomaximal, Sylow
169C22×C47C2×C4×D411yescharacteristic
170C22×C47C2×#12411yescharacteristic
171C22×C47#53211yescharacteristic
172C22×C47#57512yes—
173C22×C47#57512yes—
174C2×D47—44no—
175C2×D47—42no—
176C2×D47—81no—
177C2×D47—81no—
178C2×D47—82no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—24no—
182C4×C47—42no—
183C4×C47—44no—
184C4×C47—82no—
185C4×C47C4×#4112yes—
186C4×C47#52912yes—
187Dic47—42no—
188Dic47—81no—
189Dic47—81no—
190C2×#488—471no—
191C4×#125D4711yescharacteristic, complemented
192C4×#124—471no—
193#1991—471no—
194#1958—471no—
195#1936—472no—
196#1935—472no—
197#1835—471no—
198#1763—472no—
199#1762—472no—
200C4×#97—471no—
201C2×C47C4×#12411yescharacteristic
202C2×C47#294312yes—
203C2×C47#314011yescharacteristic
204C2×C47#314211yescharacteristic
205C2×C47#318312yes—
206D47—82no—
207C2×#125C2×D4711yescharacteristic
208C2×#124—471no—
209C2×C4×Q8—471no—
210C2×C4×D4—471no—
211#488—478no—
212C2×#103—472no—
213C2×#103—472no—
214C2×#102—472no—
215C2×#102—472no—
216C2×#97—472no—
217C2×#97—472no—
218C2×C4×C8—471no—
219C2×C4×C8—471no—
220#391—471no—
221#390—471no—
222#389C2×D4712yes—
223#388—942no—
224#387—22no—
225#382—471no—
226#382C2×D4711yescharacteristic
227C4×C4⋊C4—22no—
228C4×C4⋊C4C2×D4712yes—
229C4×C22⋊C4—942no—
230C43C2×D4711yescharacteristic
231#340—472no—
232#340—472no—
233#340—472no—
234#340—472no—
235C47#2164811yescharacteristic, complemented, Sylow
236C22×Q8—471no—
237C22×D4—471no—
238C23×C4—941no—
239C22×C8—472no—
240C22×C8—472no—
241#125—22no—
242#125C4×D4714yescomplemented
243#124—474no—
244#124—942no—
245C4×Q8—944no—
246C4×D4—944no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—471no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4C22×D4712yes—
253C2×C4⋊C4C22×D4712yes—
254C2×C22⋊C4—942no—
255C2×C22⋊C4—941no—
256C2×C22⋊C4—942no—
257C2×C42—22no—
258C2×C42—941no—
259C2×C42C22⋊D4712yes—
260C2×C42C22⋊D4712yes—
261C2×C42C22×D4711yescharacteristic
262C2×C42C22×D4711yescharacteristic
263C2×C42C22×D4711yescharacteristic
264#103—478no—
265#103—944no—
266#102—478no—
267#102—944no—
268#97—944no—
269#97—944no—
270C4×C8—474no—
271C4×C8—942no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—942no—
276#95—942no—
277C24—941no—
278C2×Q8—472no—
279C2×Q8—941no—
280C2×Q8—942no—
281C2×Q8—942no—
282C2×D4—472no—
283C2×D4—942no—
284C2×D4—941no—
285C2×D4—942no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—1884no—
292C22×C4—1882no—
293C22×C4C2×C22⋊D4712yes—
294C22×C4C2×C22⋊D4712yes—
295C22×C4C2×C22⋊D4711yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297C22×C4C23×D4711yescharacteristic
298C2×C8—474no—
299C2×C8—474no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8—942no—
303C2×C8—942no—
304C4⋊C4—28no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—44no—
309C4⋊C4—942no—
310C4⋊C4—942no—
311C4⋊C4—1882no—
312C4⋊C4C2×C4×D4714yes—
313C22⋊C4—948no—
314C22⋊C4—1884no—
315C22⋊C4—1882no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—1882no—
321C42C2×C4×D4712yes—
322C42#1062535012yes—
323C42#1062535414yes—
324C23—942no—
325C23—1881no—
326C23—1882no—
327C23—1882no—
328C23—1881no—
329C23C22×C22⋊D4711yescharacteristic
330Q8—944no—
331Q8—1882no—
332D4—944no—
333D4—1882no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—942no—
348C2×C4—3764no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4—1882no—
352C2×C4—1882no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4C4×C22⋊D4714yes—
356C2×C4#1309401312yes—
357C2×C4#1309401312yes—
358C2×C4C22×C4×D4711yescharacteristic
359C2×C4#41×D4712yes—
360C2×C4C2×#1062535011yescharacteristic
361C2×C4C2×#1062535412yes—
362C8—944no—
363C8—944no—
364C22—3761no—
365C22—3761no—
366C22—3762no—
367C22—1884no—
368C22—1882no—
369C22C2×C4×C22⋊D4711yescharacteristic
370C22C2×#1309401311yescharacteristic
371C22#1568966911yescharacteristic
372C22#1568992112yes—
373C22#1568992112yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—42no—
379C4—82no—
380C4—3761no—
381C4—3761no—
382C4—1882no—
383C4C4×#1062535412yes—
384C4#1568966012yes—
385C2—3762no—
386C2C4×#1309401311yescharacteristic
387C2#2025218612yes—
388C2#2025377711yescharacteristic
389C2#2025379111yescharacteristic
390C2#2025411112yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#125×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenterC4×#125D47
hypocenterC47#21648
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #20244354C2semidirect
6. #20244077C2semidirect
7. #20243518C2semidirect
8. #20243221C2semidirect
10. C4×#13093935C2semidirect
11. C4×#125×C47C2semidirect
15. #15689480C4semidirect
57. #13093947C2×C4semidirect
63. C4×C47⋊C8C2×C4semidirect
67. #125×C47C2×C4semidirect
191. C4×#125D47semidirect
235. C47#21648semidirect
242. #125C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894802
C4×#130940122
#202443542
#202441972
#202440842
#202440772
#202435182
#202432212
#202432162
C4×#130939352
C4×#125×C472
#2164847