Finite Group Databasebeta

← #26120090

Subgroups — #26120090

Order 12032.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689323C211yesmaximal, characteristic
2#20244194C211yesmaximal, characteristic
3#20243393C211yesmaximal, characteristic, complemented
4C2×#15688874C211yesmaximal, characteristic, complemented
5C2×#15688866C211yesmaximal, characteristic
6#20241272C211yesmaximal, characteristic, complemented
7C47×#1984C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—22no—
11#15689323—22no—
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093948—21no—
15C2×#13093946—21no—
16C2×#13093946—21no—
17C2×#13093944—21no—
18#15688993C2211yescharacteristic
19#15688885—21no—
20#15688874—22no—
21#15688866—22no—
22C2×C4×Q8×C47C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×C4×C8×C47C2211yescharacteristic
25C47×#336—21no—
26C47×#336—21no—
27C22×D188—21no—
28C22×D188—21no—
29C22×D188—21no—
30C2×#10625336—24no—
31C2×#10625336—42no—
32C2×#10625336—42no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35#13094021—21no—
36#13094021—21no—
37#13094021—21no—
38#13094021D412yes—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)D412yes—
46#13093948—42no—
47#13093946—42no—
48#13093946—42no—
49#13093944—24no—
50C22×Q8×C47D411yescharacteristic
51C22×C8×C47—21no—
52C22×C8×C47D411yescharacteristic
53C22×C8×C47D411yescharacteristic
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47D411yescharacteristic
60C2×C42×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4×C8×C47—21no—
65C4×C8×C47—21no—
66C23×D47—41no—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47SD1612yes—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C22×Dic47—41no—
79C2×D188—22no—
80C2×D188—22no—
81C2×D188—22no—
82C2×D188—41no—
83C2×D188—41no—
84C2×D188—41no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—41no—
90C2×D188—41no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—81no—
94C2×D188—81no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C2×C8×C47—41no—
101C2×C8×C47—41no—
102C2×C8×C47—41no—
103C2×C8×C47SD1612yescomplemented
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C42×C47—41no—
109C42×C47—41no—
110C42×C47C2×D411yescharacteristic
111C42×C47C2×D411yescharacteristic
112C42×C47#4111yescharacteristic
113C42×C47#4111yescharacteristic
114#10625336—48no—
115C4⋊Dic47—41no—
116C4⋊Dic47—41no—
117C4⋊Dic47—41no—
118C4⋊Dic47—41no—
119C4⋊Dic47—81no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C23×C47#12111yescharacteristic
127C22×D47—42no—
128C22×D47—81no—
129C22×D47—81no—
130C22×D47—81no—
131C22×D47—81no—
132C22×D47—81no—
133C22×D47—81no—
134Q8×C47—22no—
135Q8×C47—22no—
136Q8×C47—41no—
137Q8×C47—41no—
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#12011yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#12111yescharacteristic
149C2×C4×C47#12311yescharacteristic
150C2×C4×C47#12311yescharacteristic
151C2×C4×C47#12411yescharacteristic
152C2×C4×C47C2×SD1611yescharacteristic
153C2×C4×C47C2×SD1611yescharacteristic
154C2×C4×C47#13511yescharacteristic
155C2×C4×C47#13511yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C47#13711yescharacteristic
160C2×Dic47—81no—
161C2×Dic47—81no—
162C2×Dic47—81no—
163D188—42no—
164D188—42no—
165D188—42no—
166D188—42no—
167D188—82no—
168D188—82no—
169D188—81no—
170D188—81no—
171D188—81no—
172D188—81no—
173C8×C47—22no—
174C8×C47—41no—
175C8×C47—81no—
176C47⋊C8—42no—
177C47⋊C8—81no—
178#10475—471nomaximal, Sylow
179C22×C47#39411yescharacteristic
180C22×C47#46011yescharacteristic
181C22×C47#46411yescharacteristic
182C22×C47#46511yescharacteristic
183C22×C47#46911yescharacteristic
184C22×C47#47111yescharacteristic
185C22×C47#47211yescharacteristic
186C2×D47—82no—
187C2×D47—82no—
188C2×D47—82no—
189C2×D47—161no—
190C2×D47—161no—
191C2×D47—161no—
192C2×D47—161no—
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—42no—
196C4×C47—81no—
197C4×C47#44911yescharacteristic
198C4×C47#45011yescharacteristic
199C4×C47#46111yescharacteristic
200C4×C47#46311yescharacteristic
201C4×C47#46511yescharacteristic
202C4×C47#46811yescharacteristic
203C4×C47#48111yescharacteristic
204C4×C47#48211yescharacteristic
205Dic47—161no—
206C2×#460—471no—
207#1984D4711yescharacteristic, complemented
208#1957—471no—
209#1803—471no—
210C2×#334—471no—
211C2×#331—471no—
212#1296—471no—
213C2×C47#166611yescharacteristic
214C2×C47#167811yescharacteristic
215C2×C47#168811yescharacteristic
216C2×C47#170211yescharacteristic
217C2×C47#203411yescharacteristic
218C2×C47#205111yescharacteristic
219C2×C47#205511yescharacteristic
220D47—162no—
221C22×SD16—941no—
222C2×#127—471no—
223C2×C4×Q8C2×D4711yescharacteristic
224#460—942no—
225#460—942no—
226C2×C4⋊C8—471no—
227C2×C4⋊C8—471no—
228C2×C4⋊C8C2×D4711yescharacteristic
229C2×#103—941no—
230C2×#102—941no—
231C2×#102—941no—
232C2×#102—941no—
233C2×C4×C8C2×D4711yescharacteristic
234#384—471no—
235#340—941no—
236#336—21no—
237#336—21no—
238#334—942no—
239#331—942no—
240C47#1047511yescharacteristic, complemented, Sylow
241C22×Q8D18811yescharacteristic
242C22×D4—941no—
243C22×D4—941no—
244C22×D4—941no—
245C2×SD16—944no—
246C2×SD16—1882no—
247C2×SD16—1882no—
248C22×C8—21no—
249C22×C8—941no—
250C22×C8—941no—
251C22×C8C22⋊D4711yescharacteristic
252C22×C8C22⋊D4711yescharacteristic
253#127—472no—
254#127—941no—
255#127—941no—
256#127—941no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—941no—
263C2×C4⋊C4—941no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4D18811yescharacteristic
266C2×C42—21no—
267C2×C42C22×D4711yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—472no—
271C4⋊C8—941no—
272C4⋊C8—941no—
273C4⋊C8—941no—
274#103—1882no—
275#102—944no—
276#102—1882no—
277#102—1882no—
278C4×C8—21no—
279C4×C8—21no—
280C24—1881no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8#1062532512yes—
287C2×D4—942no—
288C2×D4—942no—
289C2×D4—942no—
290C2×D4—3761no—
291C2×D4—3761no—
292C2×D4—1881no—
293C2×D4—1881no—
294C2×D4—1881no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1881no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4—1881no—
302C2×D4—1881no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—1881no—
306C22×C4C2×D18811yescharacteristic
307C22×C4#1062535011yescharacteristic
308C22×C4C2×C22⋊D4711yescharacteristic
309SD16—1888no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—942no—
319C2×C8—1881no—
320C2×C8—1881no—
321C2×C8—1881no—
322C2×C8—1881no—
323C2×C8—1881no—
324C2×C8#1062533612yescomplemented
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—3761no—
330C4⋊C4—1881no—
331C4⋊C4—1881no—
332C4⋊C4—1881no—
333C4⋊C4—1881no—
334C42—41no—
335C42—41no—
336C42D4×D4711yescharacteristic
337C42D4×D4711yescharacteristic
338C42#1062535211yescharacteristic
339C42#1062535211yescharacteristic
340C23—3761no—
341C23—3761no—
342C23—3761no—
343C23—3761no—
344C23—3761no—
345C23—3761no—
346C23—1882no—
347C23#1309399811yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—41no—
351Q8—41no—
352D4—3762no—
353D4—3762no—
354D4—3761no—
355D4—3761no—
356D4—3761no—
357D4—3761no—
358D4—1882no—
359D4—1882no—
360D4—1882no—
361D4—1882no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—3761no—
371C2×C4—3761no—
372C2×C4—3761no—
373C2×C4#1309399411yescharacteristic
374C2×C4#1309399511yescharacteristic
375C2×C4#1309399611yescharacteristic
376C2×C4#1309400611yescharacteristic
377C2×C4#1309400811yescharacteristic
378C2×C4#1309401011yescharacteristic
379C2×C4C2×#1062533611yescharacteristic
380C2×C4C2×#1062532511yescharacteristic
381C2×C4#1309404411yescharacteristic
382C2×C4#1309404611yescharacteristic
383C2×C4#1309404911yescharacteristic
384C2×C4#1309405211yescharacteristic
385C2×C4#1309405211yescharacteristic
386C2×C4#1309405511yescharacteristic
387C8—22no—
388C8—41no—
389C8—81no—
390C8—3761no—
391C8—1882no—
392C22—3762no—
393C22—3762no—
394C22—3762no—
395C22—7521no—
396C22—7521no—
397C22—7521no—
398C22—7521no—
399C22#1568903911yescharacteristic
400C22#1568932311yescharacteristic
401C22#1568934511yescharacteristic
402C22#1568935211yescharacteristic
403C22#1568937611yescharacteristic
404C22#1568938611yescharacteristic
405C22#1568939311yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—7521no—
411C4#1568927011yescharacteristic
412C4#1568927611yescharacteristic
413C4#1568932511yescharacteristic
414C4#1568933511yescharacteristic
415C4#1568935411yescharacteristic
416C4#1568937111yescharacteristic
417C4#1568944211yescharacteristic
418C4#1568944811yescharacteristic
419C2—7522no—
420C2#2024273611yescharacteristic
421C2#2024280411yescharacteristic
422C2#2024287211yescharacteristic
423C2#2024295611yescharacteristic
424C2#2024457811yescharacteristic
425C2#2024467911yescharacteristic
426C2#2024470311yescharacteristic
subgroupG/H
centerC23#13093998
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1984C2
socleC23×C47#121
prime radicalC2×#13094021C22
hypercenter#1984D47
hypocenterC47#10475
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
3. #20243393C2semidirect
4. C2×#15688874C2semidirect
6. #20241272C2semidirect
7. C47×#1984C2semidirect
103. C2×C8×C47SD16semidirect
207. #1984D47semidirect
240. C47#10475semidirect
324. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156893232
#202441942
#202433932
C2×#156888742
C2×#156888662
#202412722
C47×#19842
#1047547