Finite Group Databasebeta

← #26185960

Subgroups — #26185960

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252468C211yesmaximal, characteristic, complemented
2#20252428C211yesmaximal, characteristic
3C2×C4×#10625325C211yesmaximal, characteristic
4#20242268C214yesmaximal, complemented
5#20242233C214yesmaximal, complemented
6#20242136C211yesmaximal, characteristic, complemented
7#20241784C211yesmaximal, characteristic, complemented
8#20241780C211yesmaximal, characteristic
9C47×#1452C211yesmaximal, characteristic, complemented
10C22×#10625325C411yescharacteristic
11C2×C4×Dic94C2211yescharacteristic
12C2×C4×D188C2211yescharacteristic
13C8×Dic94C2214yescomplemented
14#15689247—22no—
15C4×#10625325C412yes—
16C4×#10625325C412yes—
17C4×#10625325C2212yes—
18C4×#10625325C2212yes—
19#15689202C2214yes—
20#15689200—22no—
21#15689176C2211yescharacteristic
22#15689176C2211yescharacteristic
23#15689173—22no—
24C2×#13093956C411yescharacteristic
25C2×#13093949C411yescharacteristic
26C2×#13093946C411yescharacteristic
27#15689085—22no—
28#15688876C2212yescomplemented
29#15688874C2212yescomplemented
30#15688855C2214yescomplemented
31#15688852C2214yes—
32C47×#431C2211yescharacteristic
33C47×#431C2211yescharacteristic
34C2×C4×C8×C47C2211yescharacteristic
35C47×#322C2212yescomplemented
36C47×#322C2212yescomplemented
37C22×Dic94C2×C411yescharacteristic
38C22×D188C2×C411yescharacteristic
39C22×C4×D47—21no—
40C2×#10625325—24no—
41C2×#10625325C2×C412yes—
42C2×#10625325C2×C412yes—
43C2×#10625324—24no—
44C4×Dic94—21no—
45C4×Dic94—21no—
46C4×Dic94C2×C412yes—
47C4×Dic94C2312yes—
48C4×D188—21no—
49C4×D188—21no—
50C4×D188C2×C412yes—
51C4×D188C2312yes—
52C2×C4⋊Dic47C2×C411yescharacteristic
53C2×C4⋊Dic47C2×C411yescharacteristic
54C2×Dic47⋊C4—21no—
55C2×(C2×D47)⋊C4—21no—
56C2×C4×Dic47—21no—
57#13093956C2×C412yes—
58#13093956C2×C412yes—
59C4⋊(C47⋊C8)C2312yes—
60C4⋊(C47⋊C8)C2312yes—
61Dic47⋊C8—24no—
62#13093949C2×C412yes—
63#13093949C2×C412yes—
64#13093946C2×C412yes—
65#13093946C2×C412yes—
66C22⋊(C47⋊C8)—21no—
67C22⋊(C47⋊C8)—21no—
68(C2×D47)⋊C8—24no—
69C4×C47⋊C8—21no—
70C4×C47⋊C8—21no—
71C8×Dic47—24no—
72C22×C8×C47C2×C411yescharacteristic
73C22×C8×C47C2×C411yescharacteristic
74C2×C42×C47C2311yescharacteristic
75C4⋊C8×C47—21no—
76C4⋊C8×C47D412yes—
77C22⋊C8×C47—21no—
78C22⋊C8×C47D412yes—
79C4×C8×C47C2×C411yescharacteristic
80C4×C8×C47C2×C411yescharacteristic
81C4×C8×C47C2312yes—
82C4×C8×C47C2311yescharacteristic
83C4×C8×C47C2311yescharacteristic
84C4×C8×C47C2312yes—
85C23×D47—21no—
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47#4111yescharacteristic
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—21no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—21no—
95C2×D188C22×C412yes—
96C2×C4×D47—22no—
97C2×C4×D47—22no—
98C2×C4×D47—82no—
99C2×Dic94—21no—
100C2×Dic94—22no—
101C2×Dic94M1614yescomplemented
102C2×Dic94C22×C412yes—
103C2×C8×C47—22no—
104C2×C8×C47—22no—
105C2×C8×C47—22no—
106C2×C8×C47C22×C411yescharacteristic
107C2×C8×C47C22×C411yescharacteristic
108C2×C8×C47C22×C411yescharacteristic
109C2×C8×C47C22×C411yescharacteristic
110C2×C8×C47C2×D414yes—
111C42×C47C22×C411yescharacteristic
112C42×C47C2×D411yescharacteristic
113C42×C47#4111yescharacteristic
114C42×C47C2411yescharacteristic
115(C2×D47)⋊C4—22no—
116(C2×D47)⋊C4—22no—
117C4⋊Dic47—21no—
118C4⋊Dic47—21no—
119C4⋊Dic47C22×C412yes—
120C4⋊Dic47C22×C412yes—
121Dic47⋊C4—22no—
122Dic47⋊C4—22no—
123C4×Dic47—22no—
124C4×Dic47—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127#10625325—24no—
128#10625325—24no—
129#10625324—84no—
130C23×C47C4×D411yescharacteristic
131C22×D47—22no—
132C22×D47—42no—
133C22×D47—81no—
134C22×D47—81no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47C2×M1612yes—
141C2×C4×C47#13112yes—
142C2×C4×C47C23×C411yescharacteristic
143C2×C4×C47C22×D411yescharacteristic
144C2×C4×C47C2×#4111yescharacteristic
145C2×Dic47—22no—
146C2×Dic47—24no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—81no—
150C2×Dic47—81no—
151D188—22no—
152D188—22no—
153D188—42no—
154C4×D47—44no—
155C4×D47—82no—
156Dic94—22no—
157Dic94—24no—
158Dic94C2×M1614yes—
159C8×C47—21no—
160C8×C47—21no—
161C8×C47—21no—
162C8×C47—21no—
163C8×C47—24no—
164C8×C47—42no—
165C47⋊C8—42no—
166C47⋊C8—42no—
167#17878—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47#43512yes—
171C22×C47C2×C4×D411yescharacteristic
172C2×D47—42no—
173C2×D47—42no—
174C2×D47—42no—
175C2×D47—84no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47#43512yes—
180C4×C47C22×M1611yescharacteristic
181C4×C47C2×#13111yescharacteristic
182C4×C47#57712yes—
183Dic47—24no—
184Dic47—42no—
185Dic47—42no—
186#2982—471no—
187#2975—471no—
188C2×C4×SD16—471no—
189#1588—474no—
190#1583—474no—
191#1564—471no—
192#1479—471no—
193#1478—471no—
194#1452D4711yescharacteristic, complemented
195C2×C47—22no—
196C2×C47C2×#43511yescharacteristic
197C2×C47#294911yescharacteristic
198C2×C47#295711yescharacteristic
199D47—82no—
200C22×SD16—471no—
201C2×C4×Q8—471no—
202C2×C4×D4—471no—
203C8×Q8—474no—
204C8×Q8—942no—
205C4×SD16—472no—
206C4×SD16—472no—
207C4×SD16—472no—
208C4×SD16—472no—
209#436—474no—
210#436—942no—
211#431—471no—
212#431—471no—
213#431—942no—
214#431C2×D4711yescharacteristic
215#431C2×D4711yescharacteristic
216C2×#106—471no—
217C2×#103—471no—
218C2×#102—471no—
219#407—942no—
220C2×C4×C8C2×D4711yescharacteristic
221#335—472no—
222#334—472no—
223Q8⋊C8—474no—
224D4⋊C8—474no—
225#322C2×D4712yescomplemented
226#322C2×D4712yescomplemented
227C47#1787811yescharacteristic, complemented, Sylow
228C22×Q8—471no—
229C22×D4—471no—
230C23×C4—941no—
231C2×SD16—472no—
232C2×SD16—472no—
233C2×SD16—944no—
234C2×M16—944no—
235C22×C8C4×D4711yescharacteristic
236C22×C8C4×D4711yescharacteristic
237C4×Q8—472no—
238C4×Q8—472no—
239C4×Q8—941no—
240C4×Q8—941no—
241C4×D4—472no—
242C4×D4—472no—
243C4×D4—941no—
244C4×D4—941no—
245C2×C4⋊C4—471no—
246C2×C4⋊C4—471no—
247C2×C4⋊C4—941no—
248C2×C22⋊C4—941no—
249C2×C42—941no—
250C2×C42C22×D4711yescharacteristic
251#106—472no—
252#106—472no—
253C4⋊C8—21no—
254C4⋊C8—472no—
255C4⋊C8—472no—
256C4⋊C8—944no—
257C4⋊C8D18812yes—
258#103—472no—
259#103—472no—
260#102—472no—
261#102—472no—
262C22⋊C8—21no—
263C22⋊C8—941no—
264C22⋊C8—941no—
265C22⋊C8—944no—
266C22⋊C8D18812yes—
267C4×C8—944no—
268C4×C8—941no—
269C4×C8—941no—
270C4×C8C4×D4711yescharacteristic
271C4×C8C4×D4711yescharacteristic
272C4×C8C22×D4712yes—
273C4×C8C22×D4711yescharacteristic
274C4×C8C22×D4711yescharacteristic
275C4×C8C22×D4712yes—
276C24—941no—
277C2×Q8—474no—
278C2×Q8—472no—
279C2×Q8—941no—
280C2×Q8—942no—
281C2×D4—472no—
282C2×D4—941no—
283C2×D4—942no—
284C2×D4—942no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—942no—
289C22×C4—942no—
290C22×C4—3762no—
291C22×C4C2×C4×D4711yescharacteristic
292C22×C4C2×D18811yescharacteristic
293C22×C4#1062535011yescharacteristic
294SD16—944no—
295SD16—944no—
296M16—3764no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8C2×C4×D4711yescharacteristic
303C2×C8C2×C4×D4711yescharacteristic
304C2×C8C2×C4×D4711yescharacteristic
305C2×C8C2×C4×D4711yescharacteristic
306C2×C8C2×D18814yes—
307C4⋊C4—472no—
308C4⋊C4—472no—
309C4⋊C4—942no—
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—942no—
313C22⋊C4—942no—
314C22⋊C4—942no—
315C42—942no—
316C42—942no—
317C42C2×C4×D4711yescharacteristic
318C42C2×D18811yescharacteristic
319C42#1062535011yescharacteristic
320C42C23×D4711yescharacteristic
321C23—942no—
322C23—3761no—
323C23—3761no—
324C23—1882no—
325C23C4×D18811yescharacteristic
326Q8—474no—
327Q8—942no—
328Q8—944no—
329D4—942no—
330D4—942no—
331D4—1882no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—942no—
335C2×C4—944no—
336C2×C4—942no—
337C2×C4—942no—
338C2×C4—3762no—
339C2×C4—3761no—
340C2×C4—3761no—
341C2×C4—1884no—
342C2×C4C4×D18811yescharacteristic
343C2×C4C4×D18811yescharacteristic
344C2×C4C4×D18811yescharacteristic
345C2×C4M16×D4712yes—
346C2×C4#1309403212yes—
347C2×C4C22×C4×D4711yescharacteristic
348C2×C4C22×D18811yescharacteristic
349C2×C4C2×#1062535011yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—24no—
355C8—42no—
356C8—1882no—
357C8—1882no—
358C22—21no—
359C22—21no—
360C22—3764no—
361C22—1882no—
362C22—1882no—
363C22—1882no—
364C22#1568919612yes—
365C22C2×C4×D18811yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—944no—
370C4—1882no—
371C4—1882no—
372C4#1568919612yes—
373C4C2×M16×D4711yescharacteristic
374C4C2×#1309403211yescharacteristic
375C4#1568993512yes—
376C2—22no—
377C2—3762no—
378C2C2×#1568919611yescharacteristic
379C2#2025224911yescharacteristic
380C2#2025230611yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1452C2
socleC22×C47C2×C4×D4
prime radicalC22×D188C2×C4
cosocleC42×C47C24
hypercenter#1452D47
hypocenterC47#17878
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1452

Complements

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

NH ≅ G/Ntype
1. #20252468C2semidirect
4. #20242268C2semidirect
5. #20242233C2semidirect
6. #20242136C2semidirect
7. #20241784C2semidirect
9. C47×#1452C2semidirect
13. C8×Dic94C22semidirect
28. #15688876C22semidirect
29. #15688874C22semidirect
30. #15688855C22semidirect
35. C47×#322C22semidirect
36. C47×#322C22semidirect
101. C2×Dic94M16semidirect
194. #1452D47semidirect
225. #322C2×D47semidirect
226. #322C2×D47semidirect
227. C47#17878semidirect

Maximal subgroups

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

subgroupindex
#202524682
#202524282
C2×C4×#106253252
#202422682
#202422332
#202421362
#202417842
#202417802
C47×#14522
#1787847