Finite Group Databasebeta

← #26120161

Subgroups — #26120161

Order 12032.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689404C211yesmaximal, characteristic
2#20244190C211yesmaximal, characteristic, complemented
3#20244182C211yesmaximal, characteristic
4#20243389C211yesmaximal, characteristic
5C2×#15688868C211yesmaximal, characteristic, complemented
6#20241272C211yesmaximal, characteristic, complemented
7C2×C47×#335C211yesmaximal, characteristic, complemented
8C2×#13094024C2211yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689404—22no—
11#15689404—22no—
12C2×#13093958—21no—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093949—21no—
15C2×#13093949—21no—
16C2×#13093946—21no—
17C2×#13093946—21no—
18C2×(C2×D47)⋊C8—21no—
19C2×(C2×D47)⋊C8—21no—
20#15688993C2211yescharacteristic
21#15688886—21no—
22#15688868—22no—
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×C4×C8×C47C2211yescharacteristic
26C47×#335—22no—
27C22×Dic94—21no—
28C22×D188D411yescharacteristic
29C22×C4×D47—21no—
30C22×C47⋊C8—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024D412yes—
35C4×D188—21no—
36C4×D188—21no—
37C4×D188—21no—
38C4×D188—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—21no—
44C2×C4⋊Dic47D411yescharacteristic
45C2×(C2×D47)⋊C4—21no—
46#13093958—24no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49#13093949—42no—
50#13093949—42no—
51#13093946—24no—
52#13093946—42no—
53(C2×D47)⋊C8—42no—
54(C2×D47)⋊C8—42no—
55C22×C8×C47—21no—
56C22×C8×C47—21no—
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C2×C42×C47C2311yescharacteristic
60C4⋊C8×C47—21no—
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47D412yes—
64C4×C8×C47—21no—
65C4×C8×C47—21no—
66C23×D47—21no—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C22×Dic47—21no—
71C22×Dic47—41no—
72C22×Dic47—41no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188—21no—
78C2×D188Q1612yes—
79C2×C4×D47—41no—
80C2×C4×D47—41no—
81C2×C4×D47—41no—
82C2×C4×D47—41no—
83C2×C4×D47—41no—
84C2×C4×D47—41no—
85C2×Dic94—22no—
86C2×Dic94—41no—
87C2×Dic94—41no—
88C2×Dic94—41no—
89C2×Dic94—41no—
90C2×Dic94—41no—
91C2×C8×C47—21no—
92C2×C8×C47—21no—
93C2×C8×C47—22no—
94C2×C8×C47—21no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—41no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C2×C8×C47—41no—
101C2×C8×C47—41no—
102C2×C8×C47D812yes—
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C42×C47#4111yescharacteristic
107(C2×D47)⋊C4—41no—
108(C2×D47)⋊C4—41no—
109C4⋊Dic47—22no—
110C4⋊Dic47—21no—
111C4⋊Dic47—22no—
112C4⋊Dic47—21no—
113C4⋊Dic47—41no—
114C4⋊Dic47—41no—
115C4⋊Dic47—41no—
116C4⋊Dic47—41no—
117C4⋊Dic47—41no—
118C4⋊Dic47—41no—
119C4⋊Dic47—81no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C23×C47#12111yescharacteristic
124C22×D47—22no—
125C22×D47—41no—
126C22×D47—41no—
127C22×D47—41no—
128C22×D47—41no—
129C22×D47—41no—
130C22×D47—41no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47#12011yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12311yescharacteristic
137C2×C4×C47#12311yescharacteristic
138C2×C4×C47#12411yescharacteristic
139C2×C4×C47C2×D811yescharacteristic
140C2×C4×C47C2×Q1611yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47#13511yescharacteristic
143C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C47#13711yescharacteristic
147C2×Dic47—42no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—81no—
152C2×Dic47—81no—
153C2×Dic47—81no—
154C2×Dic47—81no—
155C2×Dic47—81no—
156D188—22no—
157D188—22no—
158D188—41no—
159D188—41no—
160C4×D47—81no—
161C4×D47—81no—
162C4×D47—81no—
163C4×D47—81no—
164Dic94—42no—
165Dic94—42no—
166Dic94—81no—
167Dic94—81no—
168C8×C47—22no—
169C8×C47—42no—
170C8×C47—41no—
171C8×C47—81no—
172C47⋊C8—81no—
173#10485—471nomaximal, Sylow
174C22×C47#39411yescharacteristic
175C22×C47#45911yescharacteristic
176C22×C47#46311yescharacteristic
177C22×C47#46711yescharacteristic
178C22×C47#46811yescharacteristic
179C22×C47#47111yescharacteristic
180C22×C47#47311yescharacteristic
181C2×D47—42no—
182C2×D47—42no—
183C2×D47—42no—
184C2×D47—81no—
185C2×D47—81no—
186C2×D47—81no—
187C2×D47—81no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47#44911yescharacteristic
191C4×C47#45111yescharacteristic
192C4×C47#45911yescharacteristic
193C4×C47#46311yescharacteristic
194C4×C47#46611yescharacteristic
195C4×C47#46911yescharacteristic
196C4×C47#48411yescharacteristic
197C4×C47#48511yescharacteristic
198Dic47—82no—
199Dic47—81no—
200Dic47—161no—
201C2×#474—471no—
202#1956—471no—
203#1954—471no—
204#1802—471no—
205C2×#335D4711yescharacteristic, complemented
206C2×#332—471no—
207#1296—471no—
208C2×C47#166911yescharacteristic
209C2×C47#168311yescharacteristic
210C2×C47#168511yescharacteristic
211C2×C47#169911yescharacteristic
212C2×C47#203611yescharacteristic
213C2×C47#205211yescharacteristic
214C2×C47#205411yescharacteristic
215D47—82no—
216C2×C4⋊Q8—471no—
217C2×C4×D4—471no—
218#474—942no—
219#474—942no—
220C2×#107—941no—
221C2×C4⋊C8—471no—
222C2×C4⋊C8C2×D4711yescharacteristic
223C2×C4⋊C8C2×D4711yescharacteristic
224C2×#103—941no—
225C2×#103—941no—
226C2×#102—941no—
227C2×#102—941no—
228C2×C22⋊C8—941no—
229C2×C22⋊C8—941no—
230C2×C4×C8C2×D4711yescharacteristic
231#384—471no—
232#340—941no—
233#335—22no—
234#332—942no—
235C47#1048511yescharacteristic, complemented, Sylow
236C22×Q8—941no—
237C22×D4—471no—
238C23×C4—941no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—941no—
242C22×C8D18811yescharacteristic
243C22×C8D18811yescharacteristic
244C4⋊Q8—472no—
245C4⋊Q8—941no—
246C4⋊Q8—941no—
247C4⋊Q8—941no—
248C4×D4—941no—
249C4×D4—941no—
250C4×D4—941no—
251C4×D4—941no—
252C2×C4⋊C4—471no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4—941no—
255C2×C4⋊C4—941no—
256C2×C4⋊C4—941no—
257C2×C4⋊C4—941no—
258C2×C22⋊C4—941no—
259C2×C42C22×D4711yescharacteristic
260#107—944no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—941no—
265C4⋊C8—941no—
266C4⋊C8C22⋊D4712yes—
267#103—1882no—
268#103—1882no—
269#102—944no—
270#102—1882no—
271C22⋊C8—1882no—
272C22⋊C8—1882no—
273C4×C8—21no—
274C4×C8—21no—
275C24—941no—
276C2×Q8—942no—
277C2×Q8—1881no—
278C2×Q8—1881no—
279C2×Q8—1881no—
280C2×Q8—1881no—
281C2×Q8—1881no—
282C2×D4—472no—
283C2×D4—941no—
284C2×D4—941no—
285C2×D4—941no—
286C2×D4—941no—
287C2×D4—941no—
288C22×C4—941no—
289C22×C4—1881no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4—1881no—
295C22×C4—1881no—
296C22×C4—1881no—
297C22×C4C2×D18811yescharacteristic
298C22×C4D4×D4711yescharacteristic
299C22×C4#1062535411yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—1881no—
312C2×C8—1881no—
313C2×C8—1881no—
314C2×C8D37612yes—
315C4⋊C4—942no—
316C4⋊C4—941no—
317C4⋊C4—941no—
318C4⋊C4—942no—
319C4⋊C4—3761no—
320C4⋊C4—1881no—
321C4⋊C4—1881no—
322C4⋊C4—1881no—
323C4⋊C4—1881no—
324C4⋊C4—1881no—
325C4⋊C4—1881no—
326C22⋊C4—1881no—
327C22⋊C4—1881no—
328C42#1062535011yescharacteristic
329C42D4×D4711yescharacteristic
330C42#1062535411yescharacteristic
331C42C2×C22⋊D4711yescharacteristic
332C23—942no—
333C23—1881no—
334C23—1881no—
335C23—1881no—
336C23—1881no—
337C23—1881no—
338C23—1881no—
339C23#1309399911yescharacteristic
340Q8—3761no—
341Q8—3761no—
342Q8—1882no—
343Q8—1882no—
344D4—942no—
345D4—942no—
346D4—1881no—
347D4—1881no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—3761no—
351C2×C4—3761no—
352C2×C4—3761no—
353C2×C4—3761no—
354C2×C4—3761no—
355C2×C4—3761no—
356C2×C4—3761no—
357C2×C4—3761no—
358C2×C4—3761no—
359C2×C4—1881no—
360C2×C4—1882no—
361C2×C4—1881no—
362C2×C4—1881no—
363C2×C4#1309399211yescharacteristic
364C2×C4#1309399811yescharacteristic
365C2×C4#1309399911yescharacteristic
366C2×C4#1309400911yescharacteristic
367C2×C4#1309400911yescharacteristic
368C2×C4#1309401311yescharacteristic
369C2×C4C2×D37611yescharacteristic
370C2×C4Q16×D4711yescharacteristic
371C2×C4#1309404311yescharacteristic
372C2×C4#1309404611yescharacteristic
373C2×C4#1309404811yescharacteristic
374C2×C4#1309405211yescharacteristic
375C2×C4#1309405211yescharacteristic
376C2×C4#1309405611yescharacteristic
377C8—22no—
378C8—42no—
379C8—41no—
380C8—81no—
381C8—3761no—
382C22—3761no—
383C22—3761no—
384C22—3761no—
385C22—3761no—
386C22—1882no—
387C22—1882no—
388C22—1882no—
389C22#1568904011yescharacteristic
390C22#1568931911yescharacteristic
391C22#1568933911yescharacteristic
392C22#1568936511yescharacteristic
393C22#1568937011yescharacteristic
394C22#1568938711yescharacteristic
395C22#1568939911yescharacteristic
396C4—21no—
397C4—21no—
398C4—3761no—
399C4—3762no—
400C4—7521no—
401C4#1568926511yescharacteristic
402C4#1568927711yescharacteristic
403C4#1568931911yescharacteristic
404C4#1568933911yescharacteristic
405C4#1568935911yescharacteristic
406C4#1568937611yescharacteristic
407C4#1568946011yescharacteristic
408C4#1568946611yescharacteristic
409C2—3762no—
410C2#2024275411yescharacteristic
411C2#2024284011yescharacteristic
412C2#2024285411yescharacteristic
413C2#2024293611yescharacteristic
414C2#2024459311yescharacteristic
415C2#2024468911yescharacteristic
416C2#2024469911yescharacteristic
subgroupG/H
centerC23#13093999
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC2×C47×#335C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenterC2×#335D47
hypocenterC47#10485
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #20244190C2semidirect
5. C2×#15688868C2semidirect
6. #20241272C2semidirect
7. C2×C47×#335C2semidirect
205. C2×#335D47semidirect
235. C47#10485semidirect

Maximal subgroups

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

subgroupindex
C2×#156894042
#202441902
#202441822
#202433892
C2×#156888682
#202412722
C2×C47×#3352
#1048547