Finite Group Databasebeta

← #26121021

Subgroups — #26121021

Order 12032.

402 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244181C211yesmaximal, characteristic
2#20243515C211yesmaximal, characteristic, complemented
3#20243389C211yesmaximal, characteristic
4#20243385C211yesmaximal, characteristic
5#20241272C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C47×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9C2×C4⋊(C47⋊C8)C2211yescharacteristic
10C2×#13093946—21no—
11C2×#13093944—21no—
12C2×(C2×D47)⋊C8—21no—
13C2×(C2×D47)⋊C8—21no—
14C2×C4×C47⋊C8C2211yescharacteristic
15#15688993C2211yescharacteristic
16#15688886—21no—
17#15688885—21no—
18#15688885—21no—
19#15688885—21no—
20C2×C4⋊C8×C47C2211yescharacteristic
21C2×C4⋊C8×C47C2211yescharacteristic
22C47×#384C2211yescharacteristic
23C22×D188D411yescharacteristic
24C22×C4×D47—21no—
25C22×C47⋊C8—21no—
26C22×C47⋊C8D411yescharacteristic
27C22×C47⋊C8D411yescharacteristic
28C4×D188—21no—
29C4×D188—21no—
30C4×D188—21no—
31C4×D188—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47D411yescharacteristic
36C2×(C2×D47)⋊C4—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39#13093946—42no—
40#13093944—42no—
41(C2×D47)⋊C8—42no—
42(C2×D47)⋊C8—42no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8—21no—
45C22×C8×C47—21no—
46C22×C8×C47—21no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47—21no—
50C2×C42×C47C2311yescharacteristic
51C4⋊C8×C47—21no—
52C4⋊C8×C47—21no—
53C4⋊C8×C47—21no—
54C4⋊C8×C47—21no—
55C23×D47—21no—
56C22×C4×C47—41no—
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47#4111yescharacteristic
60C22×Dic47—21no—
61C22×Dic47—41no—
62C2×D188—21no—
63C2×D188—21no—
64C2×D188—21no—
65C2×D188—21no—
66C2×D188—21no—
67C2×D188Q1612yes—
68C2×C4×D47—41no—
69C2×C4×D47—41no—
70C2×C4×D47—41no—
71C2×C4×D47—41no—
72C2×C4×D47—41no—
73C2×C4×D47—41no—
74C2×C8×C47—41no—
75C2×C8×C47—41no—
76C2×C8×C47—41no—
77C2×C8×C47—41no—
78C2×C8×C47—41no—
79C2×C8×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—81no—
85C42×C47#4111yescharacteristic
86C42×C47#4111yescharacteristic
87C42×C47#4111yescharacteristic
88C42×C47#4111yescharacteristic
89(C2×D47)⋊C4—41no—
90(C2×D47)⋊C4—41no—
91C4⋊Dic47—21no—
92C4⋊Dic47—21no—
93C4⋊Dic47—41no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96C4⋊Dic47—41no—
97C4⋊Dic47—81no—
98C2×C47⋊C8—21no—
99C2×C47⋊C8—21no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8—21no—
103C2×C47⋊C8—21no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C2×C47⋊C8—41no—
107C23×C47#12111yescharacteristic
108C22×D47—22no—
109C22×D47—41no—
110C22×D47—41no—
111C22×D47—41no—
112C22×D47—41no—
113C22×D47—41no—
114C22×D47—41no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—81no—
118C2×C4×C47—81no—
119C2×C4×C47—81no—
120C2×C4×C47#12311yescharacteristic
121C2×C4×C47#12311yescharacteristic
122C2×C4×C47#12311yescharacteristic
123C2×C4×C47#12411yescharacteristic
124C2×C4×C47#12611yescharacteristic
125C2×C4×C47#12611yescharacteristic
126C2×C4×C47C2×Q1611yescharacteristic
127C2×C4×C47#13511yescharacteristic
128C2×C4×C47#13511yescharacteristic
129C2×C4×C47#13511yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—81no—
138C2×Dic47—81no—
139C2×Dic47—81no—
140D188—22no—
141D188—22no—
142D188—41no—
143D188—41no—
144C4×D47—81no—
145C4×D47—81no—
146C4×D47—81no—
147C4×D47—81no—
148C8×C47—81no—
149C8×C47—81no—
150C47⋊C8—41no—
151C47⋊C8—41no—
152C47⋊C8—81no—
153#10622—471nomaximal, Sylow
154C22×C47#39711yescharacteristic
155C22×C47#46311yescharacteristic
156C22×C47#46311yescharacteristic
157C22×C47#46711yescharacteristic
158C22×C47#46911yescharacteristic
159C22×C47#47111yescharacteristic
160C22×C47#47111yescharacteristic
161C2×D47—42no—
162C2×D47—42no—
163C2×D47—42no—
164C2×D47—81no—
165C2×D47—81no—
166C2×D47—81no—
167C2×D47—81no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—161no—
171C4×C47#48211yescharacteristic
172C4×C47#48211yescharacteristic
173C4×C47#48211yescharacteristic
174C4×C47#48211yescharacteristic
175C4×C47#48411yescharacteristic
176C4×C47#48411yescharacteristic
177C4×C47#48811yescharacteristic
178C4×C47#49011yescharacteristic
179Dic47—81no—
180Dic47—161no—
181#1954—471no—
182#1834—471no—
183#1802—471no—
184#1802—471no—
185#1296—471no—
186#1296D4711yescharacteristic, complemented
187#1277—471no—
188C2×C47#168311yescharacteristic
189C2×C47#168311yescharacteristic
190C2×C47#170111yescharacteristic
191C2×C47#170111yescharacteristic
192C2×C47#207311yescharacteristic
193C2×C47#207611yescharacteristic
194C2×C47#207911yescharacteristic
195D47—82no—
196C2×C4×D4—471no—
197C2×C4⋊C8—471no—
198C2×C4⋊C8C2×D4711yescharacteristic
199C2×C4⋊C8C2×D4711yescharacteristic
200C2×#102—941no—
201C2×#102—941no—
202C2×C22⋊C8—941no—
203C2×C22⋊C8—941no—
204C2×C4×C8—471no—
205#384—471no—
206#384C2×D4711yescharacteristic
207#340—941no—
208#340—941no—
209#340—941no—
210#340—941no—
211C47#1062211yescharacteristic, complemented, Sylow
212C22×D4—471no—
213C23×C4—941no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—471no—
217C22×C8—471no—
218C22×C8—941no—
219C4×D4—941no—
220C4×D4—941no—
221C4×D4—941no—
222C4×D4—941no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—471no—
227C2×C4⋊C4—941no—
228C2×C4⋊C4—941no—
229C2×C4⋊C4—941no—
230C2×C22⋊C4—941no—
231C2×C42C22×D4711yescharacteristic
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—941no—
237C4⋊C8—941no—
238#102—1882no—
239#102—1882no—
240C22⋊C8—1882no—
241C22⋊C8—1882no—
242C4×C8—941no—
243C4×C8—941no—
244C24—941no—
245C2×D4—472no—
246C2×D4—941no—
247C2×D4—941no—
248C2×D4—941no—
249C2×D4—941no—
250C2×D4—941no—
251C22×C4—41no—
252C22×C4—941no—
253C22×C4—1881no—
254C22×C4—1881no—
255C22×C4—1881no—
256C22×C4—1881no—
257C22×C4—1881no—
258C22×C4—1881no—
259C22×C4—1881no—
260C22×C4#1062535011yescharacteristic
261C22×C4D4×D4711yescharacteristic
262C22×C4D4×D4711yescharacteristic
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—941no—
270C2×C8—941no—
271C2×C8—941no—
272C2×C8—941no—
273C2×C8—941no—
274C2×C8—941no—
275C2×C8—1881no—
276C2×C8—1881no—
277C2×C8—1881no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—941no—
284C4⋊C4—941no—
285C4⋊C4—3761no—
286C4⋊C4—1881no—
287C4⋊C4—1881no—
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C22⋊C4—1881no—
291C22⋊C4—1881no—
292C42#1062535011yescharacteristic
293C42#1062535011yescharacteristic
294C42#1062535211yescharacteristic
295C42#1062535411yescharacteristic
296C23—942no—
297C23—1881no—
298C23—1881no—
299C23—1881no—
300C23—1881no—
301C23—1881no—
302C23—1881no—
303C23#1309399711yescharacteristic
304D4—942no—
305D4—942no—
306D4—1881no—
307D4—1881no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—3761no—
314C2×C4—3761no—
315C2×C4—3761no—
316C2×C4—3761no—
317C2×C4—3761no—
318C2×C4—3761no—
319C2×C4—3761no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4#1309400611yescharacteristic
324C2×C4#1309400911yescharacteristic
325C2×C4#1309400911yescharacteristic
326C2×C4#1309401011yescharacteristic
327C2×C4#1309401811yescharacteristic
328C2×C4#1309401911yescharacteristic
329C2×C4Q16×D4711yescharacteristic
330C2×C4#1309404511yescharacteristic
331C2×C4#1309404511yescharacteristic
332C2×C4#1309404711yescharacteristic
333C2×C4#1309404811yescharacteristic
334C2×C4#1309404811yescharacteristic
335C2×C4#1309405111yescharacteristic
336C2×C4#1309405111yescharacteristic
337C8—81no—
338C8—81no—
339C8—3761no—
340C8—1881no—
341C8—1881no—
342C22—3761no—
343C22—3761no—
344C22—3761no—
345C22—3761no—
346C22—1882no—
347C22—1882no—
348C22—1882no—
349C22#1568905211yescharacteristic
350C22#1568933711yescharacteristic
351C22#1568934111yescharacteristic
352C22#1568936311yescharacteristic
353C22#1568937511yescharacteristic
354C22#1568938511yescharacteristic
355C22#1568938911yescharacteristic
356C4—21no—
357C4—21no—
358C4—161no—
359C4—3761no—
360C4—7521no—
361C4#1568944611yescharacteristic
362C4#1568944911yescharacteristic
363C4#1568944911yescharacteristic
364C4#1568945011yescharacteristic
365C4#1568945711yescharacteristic
366C4#1568946011yescharacteristic
367C4#1568947911yescharacteristic
368C4#1568948611yescharacteristic
369C2—3762no—
370C2#2024283811yescharacteristic
371C2#2024284211yescharacteristic
372C2#2024294811yescharacteristic
373C2#2024295211yescharacteristic
374C2#2024480511yescharacteristic
375C2#2024482011yescharacteristic
376C2#2024483711yescharacteristic
subgroupG/H
centerC23#13093997
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenter#1296D47
hypocenterC47#10622
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20243515C2semidirect
5. #20241272C2semidirect
6. #20241222C2semidirect
7. C47×#1296C2semidirect
186. #1296D47semidirect
211. C47#10622semidirect

Maximal subgroups

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

subgroupindex
#202441812
#202435152
#202433892
#202433852
#202412722
#202412222
C47×#12962
#1062247