Finite Group Databasebeta

← #26105218

Subgroups — #26105218

Order 12032.

472 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 91 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#20244634C211yesmaximal, characteristic, complemented
2#20244529C211yesmaximal, characteristic, complemented
3#20244524C211yesmaximal, characteristic
4#20241214C211yesmaximal, characteristic, complemented
5#20241211C211yesmaximal, characteristic, complemented
6#20241209C211yesmaximal, characteristic
7C47×#2732C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9C2×#13093948—21no—
10C2×#13093947—21no—
11C2×#13093947—21no—
12C2×#13093945—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×C22⋊Q8×C47—21no—
19C47×#399—21no—
20C47×#398—21no—
21C47×#396—21no—
22C47×#391C2211yescharacteristic
23C47×#390C2211yescharacteristic
24C47×#389C2211yescharacteristic
25C47×#386—21no—
26C47×#385—21no—
27C22×Dic94—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—21no—
31#13094024—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34#13093948—42no—
35#13093947—42no—
36#13093947—42no—
37#13093945—42no—
38C22×Q8×C47D411yescharacteristic
39C22×D4×C47D411yescharacteristic
40C23×C4×C47—21no—
41C22⋊Q8×C47—42no—
42C22⋊Q8×C47—42no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C42×C47C2311yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61C24×C47—21no—
62C2×Q8×C47—21no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C2×Q8×C47SD1612yes—
66C2×D4×C47—21no—
67C2×D4×C47—41no—
68C2×D4×C47—41no—
69C2×D4×C47Q1612yes—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—42no—
76C22×C4×C47—81no—
77C22×C4×C47—81no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×Dic47—41no—
82C2×Dic94—22no—
83C2×Dic94—41no—
84C2×Dic94—41no—
85C2×Dic94—41no—
86C2×Dic94—41no—
87C2×Dic94—41no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—42no—
91C4⋊C4×C47—42no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—81no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C22⋊C4×C47—81no—
104C22⋊C4×C47—81no—
105C42×C47—41no—
106C4⋊Dic47—22no—
107C4⋊Dic47—22no—
108C4⋊Dic47—41no—
109C4⋊Dic47—41no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8—41no—
119C23×C47—22no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—41no—
126C23×C47#12011yescharacteristic
127Q8×C47—42no—
128Q8×C47—81no—
129D4×C47—42no—
130D4×C47—81no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—22no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—84no—
153C2×C4×C47C2×SD1611yescharacteristic
154C2×C4×C47C2×Q1611yescharacteristic
155C2×C4×C47#13511yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47#13711yescharacteristic
159C2×Dic47—42no—
160C2×Dic47—81no—
161C2×Dic47—81no—
162Dic94—42no—
163Dic94—42no—
164Dic94—81no—
165Dic94—81no—
166C47⋊C8—81no—
167C47⋊C8—81no—
168C47⋊C8—81no—
169#7774—471nomaximal, Sylow
170C22×C47—42no—
171C22×C47—42no—
172C22×C47—42no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47—81no—
176C22×C47—81no—
177C22×C47#44911yescharacteristic
178C22×C47#45011yescharacteristic
179C22×C47#45111yescharacteristic
180C22×C47#45411yescharacteristic
181C22×C47#45511yescharacteristic
182C22×C47#45511yescharacteristic
183C22×C47B(4,𝔽2)11yescharacteristic
184C4×C47—21no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—42no—
191C4×C47—81no—
192C4×C47—81no—
193C4×C47—81no—
194C4×C47—81no—
195Dic47—82no—
196#2732D4711yescharacteristic
197#2043—471no—
198#2024—471no—
199#2023—471no—
200#1272—471no—
201#1271—471no—
202#1270—471no—
203C2×C47—82no—
204C2×C47#159711yescharacteristic
205C2×C47#160111yescharacteristic
206C2×C47#160211yescharacteristic
207C2×C47#163711yescharacteristic
208C2×C47#164211yescharacteristic
209C2×C47#164411yescharacteristic
210C2×C47#166211yescharacteristic
211C2×C4⋊Q8—471no—
212C2×C22⋊Q8—21no—
213C2×#103—941no—
214C2×#103—941no—
215C2×#103—941no—
216C2×#102—941no—
217#399—21no—
218#398—21no—
219#396—21no—
220#391C2×D4711yescharacteristic
221#390C2×D4711yescharacteristic
222#389C2×D4711yescharacteristic
223#386—21no—
224#385—21no—
225#340—941no—
226#340—941no—
227#336—471no—
228#336—471no—
229#336—471no—
230C47#777411yescharacteristic, complemented, Sylow
231C22×Q8—941no—
232C22×Q8C22⋊D4711yescharacteristic
233C22×D4C22⋊D4711yescharacteristic
234C23×C4—21no—
235C22×C8—941no—
236C22×C8—941no—
237C22×C8—941no—
238C4⋊Q8—1882no—
239C22⋊Q8—42no—
240C22⋊Q8—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4C22⋊D4711yescharacteristic
248C2×C4⋊C4C22⋊D4711yescharacteristic
249C2×C4⋊C4C22⋊D4711yescharacteristic
250C2×C4⋊C4C22⋊D4711yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42C22×D4711yescharacteristic
257#103—1882no—
258#103—1882no—
259#103—1882no—
260#102—1882no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—942no—
270C2×Q8—1881no—
271C2×Q8—1881no—
272C2×Q8—1881no—
273C2×Q8—1881no—
274C2×Q8—1881no—
275C2×Q8#1062533512yes—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4#1062533712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—1881no—
289C22×C4C2×C22⋊D4711yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291C22×C4C2×C22⋊D4711yescharacteristic
292C2×C8—1881no—
293C2×C8—1881no—
294C2×C8—1881no—
295C2×C8—1881no—
296C2×C8—1881no—
297C2×C8—1881no—
298C2×C8—1881no—
299C2×C8—1881no—
300C2×C8—1881no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—81no—
311C4⋊C4—942no—
312C4⋊C4—942no—
313C4⋊C4—1881no—
314C4⋊C4—1881no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—41no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#1309399311yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—3761no—
334Q8—3761no—
335Q8—1882no—
336Q8—1882no—
337D4—42no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4—3761no—
362C2×C4—3761no—
363C2×C4—1882no—
364C2×C4C2×#1062533511yescharacteristic
365C2×C4C2×#1062533711yescharacteristic
366C2×C4#1309404411yescharacteristic
367C2×C4#1309404911yescharacteristic
368C2×C4#1309405011yescharacteristic
369C2×C4#1309405511yescharacteristic
370C8—3761no—
371C8—3761no—
372C8—3761no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22#1568926811yescharacteristic
381C22#1568927411yescharacteristic
382C22#1568927911yescharacteristic
383C22#1568929611yescharacteristic
384C22#1568930111yescharacteristic
385C22#1568930111yescharacteristic
386C22#1568931011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—3762no—
399C2—82no—
400C2#2024231711yescharacteristic
401C2#2024234011yescharacteristic
402C2#2024234511yescharacteristic
403C2#2024255311yescharacteristic
404C2#2024258811yescharacteristic
405C2#2024260211yescharacteristic
406C2#2024271111yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2732C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2732D47
hypocenterC47#7774
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
1. #20244634C2semidirect
2. #20244529C2semidirect
4. #20241214C2semidirect
5. #20241211C2semidirect
230. C47#7774semidirect

Maximal subgroups

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

subgroupindex
#202446342
#202445292
#202445242
#202412142
#202412112
#202412092
C47×#27322
#777447