Finite Group Databasebeta

← #26422377

Subgroups — #26422377

Order 12032.

4086 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×#15689668C216yesmaximal, complemented
2#20251785C211yesmaximal, characteristic, complemented
3#20251782C211yesmaximal, characteristic, complemented
4#20251742C216yesmaximal, complemented
5#20251733C211yesmaximal, characteristic, complemented
6#20251731C216yesmaximal, complemented
7#127×Dic47C211yesmaximal, characteristic, complemented
8#20245952C216yesmaximal, complemented
9#20245950C211yesmaximal, characteristic, complemented
10C2×C42×Dic47C211yesmaximal, characteristic, complemented
11C47×#2856C211yesmaximal, characteristic, complemented
12C22×#10625352C2214yes—
13#15689668—248no—
14C2×#13094023C2216yes—
15C2×#13094021C2211yescharacteristic
16C2×#13094020C2216yes—
17C2×#13094020C2216yes—
18C2×#13094010C22112yes—
19C2×#13093998C2216yes—
20C2×#13093996C22112yes—
21C2×D4×Dic47C2216yes—
22C2×#13093986C2216yes—
23C2×(C22⋊D47)⋊C4C22112yes—
24C22×C4×Dic47C2213yes—
25C22×C4×Dic47C2213yes—
26#15689026C2216yes—
27#15689025C2216yes—
28#15689024C2216yes—
29#15689023C22124yes—
30#15688993C2211yescharacteristic
31#15688992C2216yes—
32C42×Dic47C2213yes—
33C42×Dic47C2211yescharacteristic
34C42×Dic47C2211yescharacteristic
35C42×Dic47C2213yes—
36C2×#127×C47C2211yescharacteristic
37C2×#121×C47C2216yes—
38C22×C42×C47C2211yescharacteristic
39C47×#390C2216yes—
40C47×#384C2211yescharacteristic
41C2×#10625352—248no—
42C22×Dic94C2314yes—
43C22×D188C2316yes—
44C22×C22⋊D47C23112yes—
45C22×C22⋊D47C2314yes—
46C22×C22⋊D47C2314yes—
47C23×Dic47C2314yes—
48C23×Dic47C2313yes—
49C23×Dic47C2311yescharacteristic
50C22×C4×D47C2314yes—
51#13094023—224no—
52#13094021—24no—
53#13094020—224no—
54#13094020—224no—
55#13094010—248no—
56#13093998—224no—
57#13093996—248no—
58D4×Dic47—224no—
59#13093986—224no—
60(C22⋊D47)⋊C4—248no—
61C2×C4⋊Dic47C2316yes—
62C2×Dic47⋊C4C23112yes—
63C2×C22⋊Dic47C23112yes—
64C2×(C2×D47)⋊C4C23112yes—
65C2×(C2×D47)⋊C4C23112yes—
66C2×C4×Dic47—212no—
67C2×C4×Dic47D4124yescomplemented
68C2×C4×Dic47C2316yes—
69C2×C4×Dic47C2313yes—
70C2×C4×Dic47C2313yes—
71C2×C4×Dic47C2313yes—
72C2×C4×Dic47C2313yes—
73C2×C4×Dic47C2313yes—
74C2×C4×Dic47C2313yes—
75C22×D4×C47C2316yes—
76C22×D4×C47C2314yes—
77C23×C4×C47C2313yes—
78#127×C47—24no—
79#121×C47—224no—
80C2×C4⋊C4×C47C2316yes—
81C2×C22⋊C4×C47C23112yes—
82C2×C42×C47—24no—
83C2×C42×C47C2313yes—
84C2×C42×C47C2311yescharacteristic
85C24×C47C2414yes—
86C24×C47C2411yescharacteristic
87C23×D47C2414yes—
88C2×D4×C47—224no—
89C2×D4×C47—212no—
90C2×D4×C47—212no—
91C2×D4×C47—212no—
92C22×C4×C47—26no—
93C22×C4×C47—26no—
94C22×C4×C47C2×D4112yes—
95C22×C4×C47C2414yes—
96C22×C4×C47C2413yes—
97C22×C4×C47C2413yes—
98C2×C22⋊D47—224no—
99C2×C22⋊D47—224no—
100C2×C22⋊D47—224no—
101C2×C22⋊D47—248no—
102C22×Dic47—212no—
103C22×Dic47—224no—
104C22×Dic47C2×D4112yes—
105C22×Dic47C2×D4112yes—
106C22×Dic47C2414yes—
107C22×Dic47C2413yes—
108C22×Dic47C2411yescharacteristic
109C22×Dic47C2411yescharacteristic
110C22×Dic47C2413yes—
111#10625352—464no—
112C2×D188—212no—
113C2×D188—212no—
114C2×D188—212no—
115C2×C4×D47—224no—
116C2×Dic94—224no—
117C4⋊C4×C47—212no—
118C22⋊C4×C47—224no—
119C42×C47—26no—
120C42×C47#4114yescomplemented
121C22⋊Dic47—224no—
122(C2×D47)⋊C4—224no—
123(C2×D47)⋊C4—224no—
124C4⋊Dic47—212no—
125Dic47⋊C4—224no—
126C4×Dic47—26no—
127C4×Dic47—212no—
128C4×Dic47C2×D4112yes—
129C4×Dic47C2×D4112yes—
130C4×Dic47C2×D4112yes—
131C4×Dic47C2×D4112yes—
132C4×Dic47#41112yescomplemented
133C23×C47—23no—
134C23×C47—21no—
135C23×C47—212no—
136C23×C47—212no—
137C23×C47—24no—
138C23×C47C22×D416yes—
139C23×C47C2511yescharacteristic
140C22×D47—212no—
141C22×D47—24no—
142C22×D47—212no—
143D4×C47—416no—
144D4×C47—424no—
145C2×C4×C47—23no—
146C2×C4×C47—23no—
147C2×C4×C47—224no—
148C2×C4×C47—212no—
149C2×C4×C47C22×D416yes—
150C2×C4×C47C22×D416yes—
151C2×C4×C47C2×#4116yes—
152C2×C4×C47C2×#4116yes—
153C22⋊D47—416no—
154C22⋊D47—416no—
155C22⋊D47—448no—
156C2×Dic47—26no—
157C2×Dic47—224no—
158C2×Dic47—212no—
159C2×Dic47—416no—
160C2×Dic47#127116yescomplemented
161C2×Dic47C22×D416yes—
162C2×Dic47C22×D416yes—
163C2×Dic47C22×D416yes—
164C2×Dic47C22×D416yes—
165C2×Dic47C2×#41112yes—
166D188—424no—
167C4×D47—416no—
168Dic94—416no—
169#36644—471nomaximal, Sylow
170C22×C47—26no—
171C22×C47—26no—
172C22×C47—44no—
173C22×C47—412no—
174C22×C47—412no—
175C22×C47C2×#12714yes—
176C22×C47C23×D413yes—
177C22×C47C22×#4113yes—
178C22×C47C22×#4111yescharacteristic
179C2×D47—44no—
180C2×D47—412no—
181C2×D47—412no—
182C4×C47—26no—
183C4×C47—44no—
184C4×C47#532112yescomplemented
185Dic47—26no—
186Dic47—44no—
187Dic47C2×#12714yes—
188Dic47C2×#12714yes—
189Dic47#532112yescomplemented
190C2×#532—476no—
191#2868—471no—
192#2867—471no—
193#2858—476no—
194#2856—471no—
195#2856—476no—
196#2856D4711yescharacteristic, complemented
197C4×#127—476no—
198C4×#127—471no—
199C4×#127—471no—
200C2×C43—471no—
201C2×C47—24no—
202C2×C47—84no—
203C2×C47C22×#12711yescharacteristic
204C2×C47C2×#53213yes—
205C2×C47C2×#53213yes—
206D47—84no—
207C22×#41—474no—
208#532—9448no—
209C2×C4⋊Q8—476no—
210C2×#127—476no—
211C2×#127—476no—
212C2×#127—471no—
213C2×#127C2×D4711yescharacteristic
214C2×#124—4712no—
215C2×#121—4712no—
216C2×#121—476no—
217C2×#121C2×D4716yes—
218C2×C4×D4—476no—
219C2×C4×D4—476no—
220C2×C4×D4—4712no—
221C22×C42—473no—
222C22×C42—473no—
223C22×C42C2×D4711yescharacteristic
224#390—476no—
225#390—476no—
226#390—4724no—
227#390—476no—
228#390C2×D4716yes—
229#384—476no—
230#384—471no—
231#384C2×D4711yescharacteristic
232C43—473no—
233C43—471no—
234C43—473no—
235C43—471no—
236C47#3664411yescharacteristic, complemented, Sylow
237C2×#41—9448no—
238C22×Q8—474no—
239C22×D4—4712no—
240C22×D4—474no—
241C22×D4—474no—
242C22×D4—476no—
243C22×D4C22×D4716yes—
244C22×D4C22×D4714yes—
245C23×C4—471no—
246C23×C4—473no—
247C23×C4—474no—
248C23×C4—474no—
249C23×C4C22×D4713yes—
250C4⋊Q8—9424no—
251#127—24no—
252#127—9424no—
253#127—9424no—
254#127—944no—
255#124—9448no—
256#121—224no—
257#121—9448no—
258#121—9424no—
259C4×D4—9448no—
260C4×D4—9424no—
261C4×D4—9424no—
262C2×C4⋊C4—4712no—
263C2×C4⋊C4—476no—
264C2×C4⋊C4C22×D4716yes—
265C2×C22⋊C4—4712no—
266C2×C22⋊C4—4712no—
267C2×C22⋊C4—4712no—
268C2×C22⋊C4C22×D47112yes—
269C2×C42—24no—
270C2×C42—4724no—
271C2×C42—473no—
272C2×C42—473no—
273C2×C42—473no—
274C2×C42—473no—
275C2×C42—473no—
276C2×C42—476no—
277C2×C42—473no—
278C2×C42—9412no—
279C2×C42C22×D4713yes—
280C2×C42C22×D4711yescharacteristic
281C24—474no—
282C24C23×D4714yes—
283C24C23×D4711yescharacteristic
284#41—18864no—
285C2×Q8—9424no—
286C2×D4—224no—
287C2×D4—212no—
288C2×D4—212no—
289C2×D4—212no—
290C2×D4—9424no—
291C2×D4—9424no—
292C2×D4—9448no—
293C2×D4—9424no—
294C2×D4—9412no—
295C2×D4—9412no—
296C2×D4—9412no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—4712no—
300C22×C4—473no—
301C22×C4—471no—
302C22×C4—471no—
303C22×C4—4712no—
304C22×C4—473no—
305C22×C4—474no—
306C22×C4—9424no—
307C22×C4—9412no—
308C22×C4—9424no—
309C22×C4D4×D47112yes—
310C22×C4C23×D4714yes—
311C22×C4C23×D4713yes—
312C22×C4C23×D4713yes—
313C4⋊C4—212no—
314C4⋊C4—9424no—
315C4⋊C4—9412no—
316C22⋊C4—224no—
317C22⋊C4—9424no—
318C22⋊C4—9424no—
319C22⋊C4—9424no—
320C42—26no—
321C42—4712no—
322C42—4712no—
323C42—4712no—
324C42—4712no—
325C42—4712no—
326C42—9412no—
327C42—946no—
328C42#1062535214yescomplemented
329C23—23no—
330C23—21no—
331C23—212no—
332C23—212no—
333C23—24no—
334C23—9412no—
335C23—9412no—
336C23—944no—
337C23C2×D4×D4716yes—
338C23C24×D4711yescharacteristic
339Q8—18816no—
340D4—416no—
341D4—424no—
342D4—18848no—
343D4—18816no—
344D4—18816no—
345D4—18824no—
346C2×C4—23no—
347C2×C4—23no—
348C2×C4—224no—
349C2×C4—212no—
350C2×C4—4716no—
351C2×C4—476no—
352C2×C4—476no—
353C2×C4—476no—
354C2×C4—4712no—
355C2×C4—476no—
356C2×C4—9424no—
357C2×C4—946no—
358C2×C4—9412no—
359C2×C4—18816no—
360C2×C4—18816no—
361C2×C4C2×D4×D4716yes—
362C2×C4C2×D4×D4716yes—
363C2×C4#41×D4716yes—
364C2×C4C2×#1062535216yes—
365C22—26no—
366C22—26no—
367C22—44no—
368C22—412no—
369C22—412no—
370C22—1884no—
371C22—18812no—
372C22—18812no—
373C22#127×D4714yes—
374C22C22×D4×D4713yes—
375C22C2×#41×D4713yes—
376C22C22×#1062535211yescharacteristic
377C4—26no—
378C4—44no—
379C4—474no—
380C4—474no—
381C4—4712no—
382C4—946no—
383C4—1884no—
384C4#15689668112yescomplemented
385C2—24no—
386C2—84no—
387C2—3764no—
388C2C2×#127×D4711yescharacteristic
389C2#532×D4713yes—
390C2C2×#1568966813yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC47×#2856C2
hypercenter#2856D47
hypocenterC47#36644
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#15689668C2semidirect
2. #20251785C2semidirect
3. #20251782C2semidirect
4. #20251742C2semidirect
5. #20251733C2semidirect
6. #20251731C2semidirect
7. #127×Dic47C2semidirect
8. #20245952C2semidirect
9. #20245950C2semidirect
10. C2×C42×Dic47C2semidirect
11. C47×#2856C2semidirect
67. C2×C4×Dic47D4semidirect
120. C42×C47#41semidirect
132. C4×Dic47#41semidirect
160. C2×Dic47#127semidirect
184. C4×C47#532semidirect
189. Dic47#532semidirect
196. #2856D47semidirect
236. C47#36644semidirect
328. C42#10625352semidirect
384. C4#15689668semidirect

Maximal subgroups

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

subgroupindex
C2×#156896682
#202517852
#202517822
#202517422
#202517332
#202517312
#127×Dic472
#202459522
#202459502
C2×C42×Dic472
C47×#28562
#3664447