Finite Group Databasebeta

← #26105083

Subgroups — #26105083

Order 12032.

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

normal subgroups only

This group has 504 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#20244721C211yesmaximal, characteristic, complemented
2#20244529C211yesmaximal, characteristic, complemented
3#20244470C211yesmaximal, characteristic
4#20241214C211yesmaximal, characteristic, complemented
5#20241211C211yesmaximal, characteristic, complemented
6#20241209C211yesmaximal, characteristic
7C47×#2732C211yesmaximal, characteristic
8C22×#10625335—21no—
9C2×#13094024C2211yescharacteristic
10C2×#13093955—21no—
11C2×#13093948—21no—
12C2×#13093947—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—
28C2×#10625335—42no—
29C2×#10625335—42no—
30C2×#10625335—42no—
31C22×C47⋊C8—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34#13094024—42no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37#13093955—42no—
38#13093948—42no—
39#13093947—42no—
40C22×Q8×C47D411yescharacteristic
41C22×D4×C47D411yescharacteristic
42C23×C4×C47—21no—
43C22⋊Q8×C47—42no—
44C22⋊Q8×C47—42no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C22⋊C4×C47—21no—
57C2×C22⋊C4×C47—21no—
58C2×C42×C47C2311yescharacteristic
59#95×C47—21no—
60#95×C47—21no—
61#95×C47—21no—
62#95×C47—21no—
63C24×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47SD1612yescomplemented
68C2×D4×C47—21no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47SD1612yes—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—42no—
78C22×C4×C47—81no—
79C22×C4×C47—81no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×Dic47—41no—
84C2×Dic94—22no—
85C2×Dic94—41no—
86C2×Dic94—41no—
87C2×Dic94—41no—
88C2×Dic94—41no—
89C2×Dic94—41no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—42no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—81no—
100C22⋊C4×C47—42no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C22⋊C4×C47—41no—
104C22⋊C4×C47—41no—
105C22⋊C4×C47—81no—
106C22⋊C4×C47—81no—
107C42×C47—41no—
108#10625335—48no—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—41no—
112C4⋊Dic47—41no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8—41no—
119C2×C47⋊C8—41no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122C23×C47—22no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—41no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47#12011yescharacteristic
130Q8×C47—42no—
131Q8×C47—81no—
132D4×C47—42no—
133D4×C47—81no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—22no—
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—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—84no—
156C2×C4×C47C2×SD1611yescharacteristic
157C2×C4×C47C2×SD1611yescharacteristic
158C2×C4×C47#13511yescharacteristic
159C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C47#13711yescharacteristic
161C2×C4×C47#13711yescharacteristic
162C2×Dic47—42no—
163C2×Dic47—81no—
164C2×Dic47—81no—
165Dic94—42no—
166Dic94—42no—
167Dic94—81no—
168Dic94—81no—
169C47⋊C8—42no—
170C47⋊C8—42no—
171C47⋊C8—81no—
172#7753—471nomaximal, Sylow
173C22×C47—42no—
174C22×C47—42no—
175C22×C47—42no—
176C22×C47—81no—
177C22×C47—81no—
178C22×C47—81no—
179C22×C47—81no—
180C22×C47#44811yescharacteristic
181C22×C47#45011yescharacteristic
182C22×C47#45211yescharacteristic
183C22×C47#45311yescharacteristic
184C22×C47#45511yescharacteristic
185C22×C47#45611yescharacteristic
186C22×C47B(4,𝔽2)11yescharacteristic
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—42no—
194C4×C47—81no—
195C4×C47—81no—
196C4×C47—81no—
197C4×C47—81no—
198Dic47—82no—
199#2732D4711yescharacteristic
200#2058—471no—
201#2024—471no—
202#2013—471no—
203#1272—471no—
204#1271—471no—
205#1270—471no—
206C2×C47—82no—
207C2×C47#159611yescharacteristic
208C2×C47#160111yescharacteristic
209C2×C47#160311yescharacteristic
210C2×C47#163111yescharacteristic
211C2×C47#164311yescharacteristic
212C2×C47#165111yescharacteristic
213C2×C47#166111yescharacteristic
214C22×SD16—941no—
215C2×C4⋊Q8—471no—
216C2×C22⋊Q8—21no—
217C2×#106—941no—
218C2×#103—941no—
219C2×#103—941no—
220#399—21no—
221#398—21no—
222#396—21no—
223#391C2×D4711yescharacteristic
224#390C2×D4711yescharacteristic
225#389C2×D4711yescharacteristic
226#386—21no—
227#385—21no—
228#340—941no—
229#340—941no—
230#336—471no—
231#336—471no—
232#336—471no—
233C47#775311yescharacteristic, complemented, Sylow
234C22×Q8—941no—
235C22×Q8C22⋊D4711yescharacteristic
236C22×D4C22⋊D4711yescharacteristic
237C23×C4—21no—
238C2×SD16—1882no—
239C2×SD16—1882no—
240C2×SD16—1882no—
241C22×C8—941no—
242C22×C8—941no—
243C22×C8—941no—
244C4⋊Q8—1882no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4C22⋊D4711yescharacteristic
254C2×C4⋊C4C22⋊D4711yescharacteristic
255C2×C4⋊C4C22⋊D4711yescharacteristic
256C2×C4⋊C4C22⋊D4711yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42C22×D4711yescharacteristic
263#106—1882no—
264#103—1882no—
265#103—1882no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—942no—
275C2×Q8—1881no—
276C2×Q8—1881no—
277C2×Q8—1881no—
278C2×Q8—1881no—
279C2×Q8—1881no—
280C2×Q8#1062533512yescomplemented
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4#1062533612yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—1881no—
294C22×C4C2×C22⋊D4711yescharacteristic
295C22×C4C2×C22⋊D4711yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297SD16—1888no—
298C2×C8—942no—
299C2×C8—942no—
300C2×C8—1881no—
301C2×C8—1881no—
302C2×C8—1881no—
303C2×C8—1881no—
304C2×C8—1881no—
305C2×C8—1881no—
306C2×C8—1881no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—81no—
317C4⋊C4—942no—
318C4⋊C4—942no—
319C4⋊C4—1881no—
320C4⋊C4—1881no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C42—41no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1309399311yescharacteristic
337Q8—42no—
338Q8—81no—
339Q8—3761no—
340Q8—3761no—
341Q8—1882no—
342Q8—1882no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
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—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—84no—
367C2×C4—3761no—
368C2×C4—3761no—
369C2×C4—1882no—
370C2×C4C2×#1062533611yescharacteristic
371C2×C4C2×#1062533511yescharacteristic
372C2×C4#1309404411yescharacteristic
373C2×C4#1309405011yescharacteristic
374C2×C4#1309405411yescharacteristic
375C2×C4#1309405511yescharacteristic
376C8—3761no—
377C8—1882no—
378C8—1882no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1568926211yescharacteristic
387C22#1568927411yescharacteristic
388C22#1568928511yescharacteristic
389C22#1568929111yescharacteristic
390C22#1568930111yescharacteristic
391C22#1568930611yescharacteristic
392C22#1568931011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—3762no—
405C2—82no—
406C2#2024231111yescharacteristic
407C2#2024234011yescharacteristic
408C2#2024235111yescharacteristic
409C2#2024251511yescharacteristic
410C2#2024259511yescharacteristic
411C2#2024265111yescharacteristic
412C2#2024270511yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2732C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2732D47
hypocenterC47#7753
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. #20244721C2semidirect
2. #20244529C2semidirect
4. #20241214C2semidirect
5. #20241211C2semidirect
67. C2×Q8×C47SD16semidirect
233. C47#7753semidirect
280. C2×Q8#10625335semidirect

Maximal subgroups

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

subgroupindex
#202447212
#202445292
#202444702
#202412142
#202412112
#202412092
C47×#27322
#775347