Finite Group Databasebeta

← #26112335

Subgroups — #26112335

Order 12032.

516 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249204C211yesmaximal, characteristic
2#20245990C211yesmaximal, characteristic
3#20241961C211yesmaximal, characteristic
4#20241959C211yesmaximal, characteristic
5C2×#15688848C211yesmaximal, characteristic
6#20241255C211yesmaximal, characteristic
7C47×#1289C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9C22×Dic47⋊C4C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11#15689063C412yescomplemented
12#15689053—22no—
13#15689044C412yescomplemented
14#15689010—22no—
15#15689004—22no—
16#15688998—21no—
17#15688984—21no—
18#15688984—21no—
19#15688983C2211yescharacteristic
20C22⋊C4×Dic47—21no—
21#15688861—22no—
22#15688859—22no—
23#15688848—22no—
24C2×C22⋊C8×C47C2211yescharacteristic
25C47×#380C2211yescharacteristic
26C47×#380C2211yescharacteristic
27C22×Dic94C2×C411yescharacteristic
28C23×Dic47D411yescharacteristic
29C23×Dic47D411yescharacteristic
30C22×C47⋊C8—21no—
31#13094003—21no—
32#13094003—21no—
33#13094003—21no—
34#13094003—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—22no—
37C2×C4⋊Dic47C2×C411yescharacteristic
38C2×Dic47⋊C4—21no—
39C2×Dic47⋊C4—21no—
40C2×Dic47⋊C4—21no—
41C2×Dic47⋊C4—21no—
42C2×Dic47⋊C4—21no—
43C2×Dic47⋊C4C2×C412yes—
44C2×Dic47⋊C4D412yes—
45C2×C22⋊Dic47—21no—
46C2×C22⋊Dic47—21no—
47C2×C4×Dic47—22no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C47⋊#95—22no—
51C47⋊#95—24no—
52C47⋊#95—22no—
53C47⋊#95—22no—
54C47⋊#95—22no—
55C47⋊#95D412yes—
56C23×C4×C47C2311yescharacteristic
57C22×C8×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C22⋊C4×C47—21no—
61C2×C22⋊C4×C47—21no—
62C22⋊C8×C47—21no—
63C22⋊C8×C47—21no—
64#95×C47C2×C412yes—
65#95×C47D412yes—
66C24×C47C2×D411yescharacteristic
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—81no—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47#4111yescharacteristic
75C22×C4×C47#4111yescharacteristic
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—22no—
87C22×Dic47—81no—
88C22×Dic47C22⋊C412yes—
89C22×Dic47C22⋊C412yes—
90C2×Dic94—21no—
91C2×Dic94—21no—
92C2×Dic94—81no—
93C2×C8×C47—41no—
94C2×C8×C47—81no—
95C4⋊C4×C47—81no—
96C4⋊C4×C47—81no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C22⋊Dic47—41no—
102C22⋊Dic47—41no—
103C22⋊Dic47—41no—
104C22⋊Dic47—41no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107C4⋊Dic47—82no—
108C4⋊Dic47—81no—
109Dic47⋊C4—22no—
110Dic47⋊C4—22no—
111Dic47⋊C4—22no—
112Dic47⋊C4—22no—
113Dic47⋊C4—41no—
114Dic47⋊C4—41no—
115C4×Dic47—82no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—81no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47C2×C22⋊C411yescharacteristic
121C23×C47#12011yescharacteristic
122C23×C47#12311yescharacteristic
123C23×C47#13511yescharacteristic
124C23×C47#13511yescharacteristic
125C23×C47#13711yescharacteristic
126C23×C47#13711yescharacteristic
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—42no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—42no—
160C2×Dic47—81no—
161Dic94—81no—
162Dic94—81no—
163C8×C47—81no—
164C47⋊C8—81no—
165#9202—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—41no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47#38611yescharacteristic
177C22×C47#41611yescharacteristic
178C22×C47#41911yescharacteristic
179C22×C47#45211yescharacteristic
180C22×C47#45211yescharacteristic
181C22×C47#48511yescharacteristic
182C22×C47#48511yescharacteristic
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—82no—
188C4×C47—82no—
189Dic47—42no—
190Dic47—42no—
191Dic47—82no—
192Dic47—81no—
193Dic47—81no—
194#2627—471no—
195#2313—471no—
196#1520—471no—
197#1520—471no—
198C2×#324—471no—
199#1289—471no—
200#1289D4711yescharacteristic
201C2×C47—21no—
202C2×C47—21no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—21no—
207C2×C47#187311yescharacteristic
208C2×C47#191211yescharacteristic
209C2×C47#191311yescharacteristic
210C2×C22⋊Q8—471no—
211C22×C4⋊C4—471no—
212C2×C22⋊C8—471no—
213C2×C22⋊C8C2×D4711yescharacteristic
214#400—472no—
215#398—942no—
216#395—472no—
217#387—942no—
218#387—942no—
219#385—941no—
220#381—941no—
221#381—941no—
222#380—471no—
223#380C2×D4711yescharacteristic
224#380C2×D4711yescharacteristic
225C4×C22⋊C4—941no—
226#328—942no—
227#328—942no—
228#324—942no—
229C47#920211yescharacteristic, complemented, Sylow
230C22×Q8—471no—
231C23×C4—471no—
232C23×C4—471no—
233C23×C4C22×D4711yescharacteristic
234C22×C8—21no—
235C22×C8—941no—
236C22⋊Q8—941no—
237C22⋊Q8—941no—
238C22⋊Q8—941no—
239C22⋊Q8—941no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—472no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4—471no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4—941no—
249C2×C4⋊C4—942no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4—941no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—941no—
255C2×C22⋊C4—941no—
256C2×C42—942no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—941no—
260C22⋊C8—941no—
261#95—472no—
262#95—942no—
263#95—942no—
264#95—942no—
265#95—942no—
266#95—944no—
267#95C4×D4712yes—
268#95D18812yes—
269C24D4×D4711yescharacteristic
270C2×Q8—941no—
271C2×Q8—941no—
272C2×Q8—3761no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—81no—
276C22×C4—472no—
277C22×C4—472no—
278C22×C4—941no—
279C22×C4—941no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—942no—
289C22×C4—3761no—
290C22×C4C2×C4×D4711yescharacteristic
291C22×C4C2×D18811yescharacteristic
292C22×C4#1062535011yescharacteristic
293C22×C4D4×D4711yescharacteristic
294C22×C4D4×D4711yescharacteristic
295C22×C4#1062535411yescharacteristic
296C2×C8—41no—
297C2×C8—81no—
298C2×C8—3761no—
299C2×C8—1881no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—942no—
303C4⋊C4—942no—
304C4⋊C4—942no—
305C4⋊C4—942no—
306C4⋊C4—941no—
307C4⋊C4—941no—
308C4⋊C4—3761no—
309C4⋊C4—3762no—
310C4⋊C4—1881no—
311C4⋊C4—1881no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1881no—
317C22⋊C4—1881no—
318C22⋊C4—1881no—
319C22⋊C4—1881no—
320C42—3762no—
321C23—41no—
322C23—41no—
323C23C22⋊C4×D4711yescharacteristic
324C23#1309399411yescharacteristic
325C23#1309400911yescharacteristic
326C23#1309404311yescharacteristic
327C23#1309404511yescharacteristic
328C23#1309405311yescharacteristic
329C23#1309405611yescharacteristic
330Q8—3761no—
331Q8—3761no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—942no—
345C2×C4—942no—
346C2×C4—942no—
347C2×C4—942no—
348C2×C4—3761no—
349C2×C4—1882no—
350C2×C4—1882no—
351C2×C4—1882no—
352C2×C4—1882no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4—1882no—
356C2×C4—1882no—
357C2×C4—1881no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1882no—
362C2×C4#1309398611yescharacteristic
363C2×C4C4×D18811yescharacteristic
364C2×C4#1309399711yescharacteristic
365C2×C4#1309399911yescharacteristic
366C8—81no—
367C8—3761no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1568900111yescharacteristic
379C22#1568911611yescharacteristic
380C22#1568913011yescharacteristic
381C22#1568928611yescharacteristic
382C22#1568928811yescharacteristic
383C22#1568946611yescharacteristic
384C22#1568946711yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—82no—
391C4—3762no—
392C4—3761no—
393C4—3761no—
394C4—1882no—
395C4—1882no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#2024369911yescharacteristic
403C2#2024392411yescharacteristic
404C2#2024392911yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1289C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1289D47
hypocenterC47#9202
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
11. #15689063C4semidirect
13. #15689044C4semidirect
229. C47#9202semidirect

Maximal subgroups

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

subgroupindex
#202492042
#202459902
#202419612
#202419592
C2×#156888482
#202412552
C47×#12892
#920247