Finite Group Databasebeta

← #26201902

Subgroups — #26201902

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254539C211yesmaximal, characteristic, complemented
2#20253187C211yesmaximal, characteristic
3C2×#15689410C211yesmaximal, characteristic
4#20252428C211yesmaximal, characteristic
5#20242966C214yesmaximal, complemented
6#20242910C214yesmaximal, complemented
7#20242160C211yesmaximal, characteristic, complemented
8#20242043C211yesmaximal, characteristic, complemented
9C47×#1548C211yesmaximal, characteristic, complemented
10#15689680C2211yescharacteristic
11C2×C4×D188C2211yescharacteristic
12#15689526C2212yescomplemented
13#15689475C2212yescomplemented
14#15689458—22no—
15#15689424—22no—
16#15689415—22no—
17#15689414C2212yes—
18#15689410—22no—
19#15689410C2214yes—
20#15689404C2212yes—
21#15689202—22no—
22#15689200C2214yes—
23#15689176C2211yescharacteristic
24#15689165—21no—
25C2×#13093955—21no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27#15689129—21no—
28C2×#13093944—21no—
29#15689085—22no—
30#15688873C2214yescomplemented
31#15688871C2212yescomplemented
32#15688870C2212yescomplemented
33C2×C4⋊Q8×C47C2211yescharacteristic
34C47×#431C2211yescharacteristic
35C47×#423C2211yescharacteristic
36C47×#333C2214yescomplemented
37C22×D188D411yescharacteristic
38C22×C4×D47—21no—
39C2×#10625324—24no—
40C22×C47⋊C8—21no—
41#13094024C2311yescharacteristic
42#13094024C2311yescharacteristic
43#13094016—21no—
44#13094003—21no—
45#13094003—21no—
46C4×Dic94—21no—
47C4×Dic94—21no—
48C4×D188—21no—
49C4×D188—21no—
50C4×D188D412yes—
51C4×D188C2312yes—
52#13093983—21no—
53C2×C4⋊Dic47D411yescharacteristic
54C2×(C2×D47)⋊C4—21no—
55#13093958—22no—
56#13093957—24no—
57#13093956—22no—
58#13093955—24no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)D412yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093947—24no—
63#13093946—22no—
64#13093946—22no—
65#13093944—24no—
66C22⋊(C47⋊C8)D412yes—
67(C2×D47)⋊C8—24no—
68C4×C47⋊C8C2312yes—
69C22×Q8×C47—21no—
70C2×M16×C47—21no—
71C4⋊Q8×C47—21no—
72C4⋊Q8×C47—21no—
73C4⋊Q8×C47—21no—
74C4⋊Q8×C47C2312yes—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C42×C47C2311yescharacteristic
78C4⋊C8×C47—21no—
79C4⋊C8×C47C2312yes—
80C4⋊C8×C47C2312yes—
81C22⋊C8×C47—21no—
82C4×C8×C47—21no—
83C23×D47—21no—
84C2×Q8×C47—22no—
85C2×Q8×C47—41no—
86C2×Q8×C47—82no—
87C22×C4×C47—41no—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47#4111yescharacteristic
91C22×Dic47—21no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—21no—
95C2×D188C2×D412yes—
96C2×C4×D47—22no—
97C2×C4×D47—22no—
98C2×C4×D47—82no—
99C2×Dic94—21no—
100C2×Dic94—21no—
101M16×C47—42no—
102C2×C8×C47—22no—
103C2×C8×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47#4111yescharacteristic
111C42×C47C2411yescharacteristic
112C22⋊Dic47—41no—
113(C2×D47)⋊C4—22no—
114(C2×D47)⋊C4—22no—
115C4⋊Dic47—21no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118C4⋊Dic47—21no—
119C4⋊Dic47—21no—
120C4⋊Dic47C2×D412yes—
121Dic47⋊C4—41no—
122Dic47⋊C4—41no—
123C4×Dic47—41no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—42no—
126C2×C47⋊C8C2×D414yes—
127#10625324—84no—
128C23×C47#12111yescharacteristic
129C22×D47—22no—
130C22×D47—42no—
131C22×D47—81no—
132C22×D47—81no—
133Q8×C47—82no—
134Q8×C47—82no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47—42no—
138C2×C4×C47—82no—
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#13712yes—
143C2×C4×C47#13712yes—
144C2×C4×C47C22×D411yescharacteristic
145C2×C4×C47C22×D411yescharacteristic
146C2×C4×C47C2×#4111yescharacteristic
147C2×Dic47—22no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—81no—
151D188—22no—
152D188—22no—
153D188—42no—
154C4×D47—44no—
155C4×D47—82no—
156Dic94—42no—
157Dic94—81no—
158C8×C47—42no—
159C8×C47—42no—
160C47⋊C8—24no—
161C47⋊C8—42no—
162#19429—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47#47012yes—
166C22×C47C2×#12111yescharacteristic
167C2×D47—42no—
168C2×D47—42no—
169C2×D47—42no—
170C2×D47—84no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—82no—
175C4×C47#47012yes—
176C4×C47C2×#13711yescharacteristic
177C4×C47C2×#13711yescharacteristic
178C4×C47#57712yes—
179Dic47—42no—
180Dic47—81no—
181Dic47—81no—
182#3230—471no—
183#3077—471no—
184C2×#476—471no—
185#2975—471no—
186#1703—474no—
187#1695—474no—
188#1570—471no—
189#1548D4711yescharacteristic, complemented
190#1542—471no—
191C2×C47—22no—
192C2×C47#304211yescharacteristic
193C2×C47C2×#47011yescharacteristic
194C2×C47#306811yescharacteristic
195D47—82no—
196#533—471no—
197C2×C4⋊Q8C2×D4711yescharacteristic
198C2×C4×D4—471no—
199#500—472no—
200#487—472no—
201#484—942no—
202#478—942no—
203#477—942no—
204#476—474no—
205#476—472no—
206#476—942no—
207#474—472no—
208#436—474no—
209#436—942no—
210#431—471no—
211#431C2×D4711yescharacteristic
212#428—941no—
213C2×#106—941no—
214#423C2×D4711yescharacteristic
215C2×C4⋊C8—471no—
216#418—941no—
217C2×#102—941no—
218#407—942no—
219#334—474no—
220#333—472no—
221#333C2×D4714yescomplemented
222#332—472no—
223C47#1942911yescharacteristic, complemented, Sylow
224C22×Q8—21no—
225C22×D4—471no—
226C23×C4—941no—
227C2×M16—21no—
228C2×M16—944no—
229C22×C8—941no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—471no—
234C4⋊Q8—471no—
235C4⋊Q8C22×D4712yes—
236#125—941no—
237C22⋊Q8—941no—
238C22⋊Q8—941no—
239C4×Q8—941no—
240C4×Q8—941no—
241C4×D4—472no—
242C4×D4—472no—
243C4×D4—941no—
244C4×D4—941no—
245#117—941no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—471no—
249C2×C22⋊C4—941no—
250C2×C42C22×D4711yescharacteristic
251#107—944no—
252#107—942no—
253#106—944no—
254#106—942no—
255C4⋊C8—21no—
256C4⋊C8—472no—
257C4⋊C8—472no—
258C4⋊C8—941no—
259C4⋊C8C22×D4712yes—
260C4⋊C8C22×D4712yes—
261#103—944no—
262#102—944no—
263#102—942no—
264#102—942no—
265C22⋊C8—21no—
266C22⋊C8—472no—
267C22⋊C8—944no—
268C4×C8—21no—
269C4×C8—472no—
270C24—941no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—82no—
274C2×Q8—941no—
275C2×Q8—941no—
276C2×D4—472no—
277C2×D4—941no—
278C2×D4—942no—
279C2×D4—942no—
280C22×C4—41no—
281C22×C4—941no—
282C22×C4—942no—
283C22×C4—942no—
284C22×C4—3762no—
285C22×C4#1062535011yescharacteristic
286C22×C4D4×D4711yescharacteristic
287C22×C4D4×D4711yescharacteristic
288M16—42no—
289M16—3764no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—474no—
293C2×C8—942no—
294C2×C8—1882no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—472no—
300C4⋊C4—941no—
301C4⋊C4—941no—
302C4⋊C4—941no—
303C4⋊C4—941no—
304C4⋊C4—941no—
305C4⋊C4—1881no—
306C4⋊C4—1881no—
307C22⋊C4—942no—
308C22⋊C4—942no—
309C22⋊C4—1881no—
310C42—1881no—
311C42#1062535011yescharacteristic
312C42D4×D4711yescharacteristic
313C42D4×D4711yescharacteristic
314C42C23×D4711yescharacteristic
315C23—942no—
316C23—3761no—
317C23—3761no—
318C23—1882no—
319C23#1309399711yescharacteristic
320Q8—82no—
321Q8—82no—
322Q8—3761no—
323Q8—1882no—
324D4—942no—
325D4—942no—
326D4—1882no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—942no—
332C2×C4—3762no—
333C2×C4—3761no—
334C2×C4—1881no—
335C2×C4—1881no—
336C2×C4—1884no—
337C2×C4#1309399711yescharacteristic
338C2×C4#1309399711yescharacteristic
339C2×C4#1309399711yescharacteristic
340C2×C4#1309405312yes—
341C2×C4#1309405612yes—
342C2×C4C2×D4×D4711yescharacteristic
343C2×C4C2×D4×D4711yescharacteristic
344C2×C4C2×#1062535011yescharacteristic
345C8—42no—
346C8—42no—
347C8—944no—
348C8—1882no—
349C22—21no—
350C22—21no—
351C22—3764no—
352C22—1882no—
353C22—1882no—
354C22—1882no—
355C22#1568938012yes—
356C22C2×#1309399711yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—3761no—
362C4—3761no—
363C4—1882no—
364C4#1568938012yes—
365C4C2×#1309405311yescharacteristic
366C4C2×#1309405611yescharacteristic
367C4#1568993212yes—
368C2—22no—
369C2—3762no—
370C2#2025289911yescharacteristic
371C2C2×#1568938011yescharacteristic
372C2#2025311511yescharacteristic
subgroupG/H
centerC22C2×#13093997
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#1548C2
socleC22×C47C2×#121
prime radicalC22×D188D4
hypercenter#1548D47
hypocenterC47#19429
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #20254539C2semidirect
5. #20242966C2semidirect
6. #20242910C2semidirect
7. #20242160C2semidirect
8. #20242043C2semidirect
9. C47×#1548C2semidirect
12. #15689526C22semidirect
13. #15689475C22semidirect
30. #15688873C22semidirect
31. #15688871C22semidirect
32. #15688870C22semidirect
36. C47×#333C22semidirect
189. #1548D47semidirect
221. #333C2×D47semidirect
223. C47#19429semidirect

Maximal subgroups

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

subgroupindex
#202545392
#202531872
C2×#156894102
#202524282
#202429662
#202429102
#202421602
#202420432
C47×#15482
#1942947