Finite Group Databasebeta

← #26422379

Subgroups — #26422379

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×#15689670C216yesmaximal, complemented
2#20251785C211yesmaximal, characteristic, complemented
3#20251782C211yesmaximal, characteristic, complemented
4#20251742C216yesmaximal, complemented
5#20251734C211yesmaximal, characteristic, complemented
6#20251732C216yesmaximal, complemented
7#20251730C211yesmaximal, characteristic, complemented
8C4×#13094021C211yesmaximal, characteristic, complemented
9C4×#13094020C216yesmaximal, complemented
10C43×D47C211yesmaximal, characteristic, complemented
11C4×#127×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2214yes—
13#15689670—248no—
14C2×#13094023C2216yes—
15C2×#13094021C2211yescharacteristic
16C2×#13094020C2216yes—
17C2×#13094020C2216yes—
18C2×#13094010C22112yes—
19C2×#13093999C2216yes—
20C2×#13093997C22112yes—
21C2×#13093995C2216yes—
22C2×C4×D188C2216yes—
23C2×C4×C22⋊D47C22112yes—
24C2×C42×D47C2211yescharacteristic
25C2×C42×D47C2213yes—
26C2×C42×D47C2213yes—
27#15689026C2216yes—
28#15689025C2216yes—
29#15689024C2216yes—
30#15689023C22124yes—
31#15688993C2211yescharacteristic
32#15688992C2216yes—
33C42×Dic47C2213yes—
34C42×Dic47C2211yescharacteristic
35C42×Dic47C2213yes—
36C2×#127×C47C2211yescharacteristic
37C2×C4×D4×C47C2216yes—
38C47×#390C2216yes—
39C47×#384C2211yescharacteristic
40C43×C47C2211yescharacteristic
41C2×#10625350—248no—
42C22×Dic94C2314yes—
43C22×D188C2316yes—
44C22×D188C2314yes—
45C22×C22⋊D47C23112yes—
46C22×C22⋊D47C2314yes—
47C22×C22⋊D47C2314yes—
48C22×C4×D47C2314yes—
49C22×C4×D47C2313yes—
50C22×C4×D47C2313yes—
51C22×C4×D47C2311yescharacteristic
52#13094023—224no—
53#13094021—24no—
54#13094020—224no—
55#13094020—224no—
56#13094010—248no—
57#13093999—224no—
58#13093997—248no—
59#13093995—224no—
60C4×D188—224no—
61C4×C22⋊D47—248no—
62C2×C4⋊Dic47C2316yes—
63C2×Dic47⋊C4C23112yes—
64C2×C22⋊Dic47C23112yes—
65C2×(C2×D47)⋊C4C23112yes—
66C2×(C2×D47)⋊C4C23112yes—
67C2×C4×Dic47C2316yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C2×C4×Dic47C2313yes—
71C2×C4×Dic47C2313yes—
72C2×C4×Dic47C2313yes—
73C42×D47—24no—
74C42×D47—212no—
75C42×D47D4124yescomplemented
76C22×D4×C47C2316yes—
77C23×C4×C47C2314yes—
78#127×C47—24no—
79C4×D4×C47—224no—
80C2×C4⋊C4×C47C2316yes—
81C2×C22⋊C4×C47C23112yes—
82C2×C42×C47C2313yes—
83C2×C42×C47C2313yes—
84C2×C42×C47C2311yescharacteristic
85C24×C47C2414yes—
86C23×D47C2414yes—
87C23×D47C2411yescharacteristic
88C2×D4×C47—212no—
89C2×D4×C47—212no—
90C2×D4×C47—212no—
91C22×C4×C47—224no—
92C22×C4×C47C2414yes—
93C22×C4×C47C2413yes—
94C22×C4×C47C2411yescharacteristic
95C22×C4×C47C2413yes—
96C2×C22⋊D47—224no—
97C2×C22⋊D47—224no—
98C2×C22⋊D47—224no—
99C2×C22⋊D47—248no—
100C22×Dic47C2414yes—
101C22×Dic47C2413yes—
102C22×Dic47C2411yescharacteristic
103C22×Dic47C2413yes—
104#10625350—464no—
105C2×D188—224no—
106C2×D188—212no—
107C2×D188—212no—
108C2×D188—212no—
109C2×C4×D47—26no—
110C2×C4×D47—26no—
111C2×C4×D47—212no—
112C2×C4×D47—224no—
113C2×C4×D47C2×D4112yes—
114C2×C4×D47C2×D4112yes—
115C2×C4×D47C2×D4112yes—
116C2×Dic94—224no—
117C4⋊C4×C47—212no—
118C22⋊C4×C47—224no—
119C42×C47—26no—
120C42×C47C2×D4112yes—
121C42×C47#4114yescomplemented
122C22⋊Dic47—224no—
123(C2×D47)⋊C4—224no—
124(C2×D47)⋊C4—224no—
125C4⋊Dic47—212no—
126Dic47⋊C4—224no—
127C4×Dic47—26no—
128C4×Dic47—212no—
129C4×Dic47C2×D4112yes—
130C4×Dic47C2×D4112yes—
131C4×Dic47C2×D4112yes—
132C4×Dic47#41112yescomplemented
133C23×C47—212no—
134C23×C47—212no—
135C23×C47—24no—
136C23×C47C2511yescharacteristic
137C22×D47—23no—
138C22×D47—21no—
139C22×D47—212no—
140C22×D47—24no—
141C22×D47—212no—
142C22×D47C22×D416yes—
143D4×C47—424no—
144C2×C4×C47—26no—
145C2×C4×C47—212no—
146C2×C4×C47—416no—
147C2×C4×C47C22×D416yes—
148C2×C4×C47C22×D416yes—
149C2×C4×C47C22×D416yes—
150C2×C4×C47C2×#4116yes—
151C2×C4×C47C2×#4116yes—
152C22⋊D47—416no—
153C22⋊D47—416no—
154C22⋊D47—448no—
155C2×Dic47—23no—
156C2×Dic47—23no—
157C2×Dic47—212no—
158C2×Dic47C22×D416yes—
159C2×Dic47C22×D416yes—
160C2×Dic47C22×D416yes—
161C2×Dic47C2×#41112yes—
162D188—416no—
163D188—424no—
164C4×D47—224no—
165C4×D47—224no—
166C4×D47—416no—
167C4×D47#127116yescomplemented
168Dic94—416no—
169#36644—471nomaximal, Sylow
170C22×C47—44no—
171C22×C47—412no—
172C22×C47—412no—
173C22×C47C23×D413yes—
174C22×C47C22×#4113yes—
175C22×C47C22×#4111yescharacteristic
176C2×D47—26no—
177C2×D47—26no—
178C2×D47—44no—
179C2×D47—412no—
180C2×D47—412no—
181C2×D47C2×#12714yes—
182C4×C47—26no—
183C4×C47—44no—
184C4×C47C2×#12714yes—
185C4×C47#532112yescomplemented
186Dic47—26no—
187Dic47—44no—
188Dic47C2×#12714yes—
189Dic47#532112yescomplemented
190C2×#532—476no—
191#2868—471no—
192#2867—471no—
193#2858—476no—
194#2856—476no—
195#2856—471no—
196#2856—471no—
197C4×#127—471no—
198C4×#127—476no—
199C4×#127D4711yescharacteristic, complemented
200C2×C43—471no—
201C2×C47—84no—
202C2×C47C22×#12711yescharacteristic
203C2×C47C2×#53213yes—
204C2×C47C2×#53213yes—
205D47—24no—
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—476no—
216C2×#121—476no—
217C2×#121—4712no—
218C2×C4×D4—4712no—
219C2×C4×D4—476no—
220C2×C4×D4C2×D4716yes—
221C22×C42—473no—
222C22×C42—473no—
223C22×C42—471no—
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—473no—
234C43—471no—
235C43C2×D4711yescharacteristic
236C47#3664411yescharacteristic, complemented, Sylow
237C2×#41—9448no—
238C22×Q8—474no—
239C22×D4—4712no—
240C22×D4—474no—
241C22×D4—474no—
242C22×D4—476no—
243C22×D4—474no—
244C22×D4C22×D4716yes—
245C23×C4—471no—
246C23×C4—473no—
247C23×C4—473no—
248C23×C4—474no—
249C23×C4C22×D4714yes—
250C4⋊Q8—9424no—
251#127—24no—
252#127—9424no—
253#127—9424no—
254#127—944no—
255#124—9448no—
256#121—9448no—
257#121—9424no—
258#121—9424no—
259C4×D4—224no—
260C4×D4—9448no—
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—473no—
270C2×C42—473no—
271C2×C42—473no—
272C2×C42—473no—
273C2×C42—4724no—
274C2×C42—476no—
275C2×C42—473no—
276C2×C42—9412no—
277C2×C42—944no—
278C2×C42C22×D4713yes—
279C2×C42C22×D4711yescharacteristic
280C2×C42C22×D4713yes—
281C24—471no—
282C24—474no—
283C24C23×D4714yes—
284#41—18864no—
285C2×Q8—9424no—
286C2×D4—212no—
287C2×D4—212no—
288C2×D4—212no—
289C2×D4—9424no—
290C2×D4—9424no—
291C2×D4—9424no—
292C2×D4—9448no—
293C2×D4—9424no—
294C2×D4—9412no—
295C2×D4—9412no—
296C2×D4—9412no—
297C22×C4—224no—
298C22×C4—473no—
299C22×C4—471no—
300C22×C4—4712no—
301C22×C4—473no—
302C22×C4—4712no—
303C22×C4—4712no—
304C22×C4—474no—
305C22×C4—9412no—
306C22×C4—946no—
307C22×C4—946no—
308C22×C4—9424no—
309C22×C4C23×D4714yes—
310C22×C4C23×D4713yes—
311C22×C4C23×D4713yes—
312C22×C4C23×D4711yescharacteristic
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—9412no—
326C42—946no—
327C42#1062535014yescomplemented
328C42D4×D47112yes—
329C23—212no—
330C23—212no—
331C23—24no—
332C23—476no—
333C23—943no—
334C23—941no—
335C23—9412no—
336C23—9412no—
337C23—944no—
338C23C24×D4711yescharacteristic
339Q8—18816no—
340D4—424no—
341D4—18816no—
342D4—18848no—
343D4—18816no—
344D4—18816no—
345D4—18824no—
346C2×C4—26no—
347C2×C4—212no—
348C2×C4—416no—
349C2×C4—476no—
350C2×C4—476no—
351C2×C4—4712no—
352C2×C4—4716no—
353C2×C4—476no—
354C2×C4—9424no—
355C2×C4—943no—
356C2×C4—943no—
357C2×C4—9424no—
358C2×C4—9412no—
359C2×C4—18816no—
360C2×C4C2×D4×D4716yes—
361C2×C4C2×D4×D4716yes—
362C2×C4C2×D4×D4716yes—
363C2×C4#41×D4716yes—
364C2×C4C2×#1062535016yes—
365C22—44no—
366C22—412no—
367C22—412no—
368C22—474no—
369C22—946no—
370C22—946no—
371C22—1884no—
372C22—18812no—
373C22—18812no—
374C22C22×D4×D4713yes—
375C22C2×#41×D4713yes—
376C22C22×#1062535011yescharacteristic
377C4—26no—
378C4—44no—
379C4—474no—
380C4—4712no—
381C4—946no—
382C4—1884no—
383C4#127×D4714yes—
384C4#15689670112yescomplemented
385C2—84no—
386C2—944no—
387C2—3764no—
388C2C2×#127×D4711yescharacteristic
389C2#532×D4713yes—
390C2C2×#1568967013yes—
subgroupG/H
centerC22×C4C23×D47
commutator, socle, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC4×#127×C47C2
hypercenterC4×#127D47
hypocenterC47#36644
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C2×#15689670C2semidirect
2. #20251785C2semidirect
3. #20251782C2semidirect
4. #20251742C2semidirect
5. #20251734C2semidirect
6. #20251732C2semidirect
7. #20251730C2semidirect
8. C4×#13094021C2semidirect
9. C4×#13094020C2semidirect
10. C43×D47C2semidirect
11. C4×#127×C47C2semidirect
75. C42×D47D4semidirect
121. C42×C47#41semidirect
132. C4×Dic47#41semidirect
167. C4×D47#127semidirect
185. C4×C47#532semidirect
189. Dic47#532semidirect
199. C4×#127D47semidirect
236. C47#36644semidirect
327. C42#10625350semidirect
384. C4#15689670semidirect

Maximal subgroups

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

subgroupindex
C2×#156896702
#202517852
#202517822
#202517422
#202517342
#202517322
#202517302
C4×#130940212
C4×#130940202
C43×D472
C4×#127×C472
#3664447