Finite Group Databasebeta

← #26201468

Subgroups — #26201468

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254333C211yesmaximal, characteristic, complemented
2#20253181C211yesmaximal, characteristic
3C2×#15689404C211yesmaximal, characteristic
4#20252411C211yesmaximal, characteristic
5#20242842C214yesmaximal, complemented
6#20242752C214yesmaximal, complemented
7#20242162C211yesmaximal, characteristic, complemented
8#20242072C211yesmaximal, characteristic, complemented
9C47×#1542C211yesmaximal, characteristic, complemented
10C2×#13094024C2211yescharacteristic
11C2×C4×D188C2211yescharacteristic
12#15689502C2212yescomplemented
13#15689475C2212yescomplemented
14#15689420—22no—
15#15689410C2212yes—
16#15689404—22no—
17#15689404C2214yes—
18#15689400C2212yes—
19#15689333—22no—
20#15689278—22no—
21C8×D188—22no—
22#15689188C2214yes—
23#15689176C2211yescharacteristic
24#15689166—21no—
25C2×#13093958—21no—
26#15689147C2211yescharacteristic
27#15689127—21no—
28C2×#13093946—21no—
29C22⋊C8×D47—22no—
30#15688875C2214yescomplemented
31#15688870C2214yescomplemented
32C47×#533C2211yescharacteristic
33C47×#431C2211yescharacteristic
34C2×C4⋊C8×C47C2211yescharacteristic
35C47×#333C2212yescomplemented
36C47×#332C2212yescomplemented
37C22×Dic94—21no—
38C22×D188D411yescharacteristic
39C22×C4×D47—21no—
40C2×#10625337—24no—
41C2×#10625329—21no—
42C2×C8×D47—24no—
43#13094024—21no—
44#13094024—21no—
45#13094024—21no—
46#13094024C2312yes—
47C4×D188—21no—
48C4×D188—21no—
49C4×D188D412yes—
50C4×D188C2312yes—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47D411yescharacteristic
54C2×(C2×D47)⋊C4—21no—
55#13093958—24no—
56#13093957—22no—
57#13093955—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)D412yes—
60C4⋊(C47⋊C8)C2312yes—
61#13093947—24no—
62#13093946—24no—
63#13093944—22no—
64#13093944—22no—
65C22⋊(C47⋊C8)D412yes—
66(C2×D47)⋊C8—24no—
67C4×C47⋊C8C2312yes—
68C22×C8×C47—21no—
69C4⋊Q8×C47C2311yescharacteristic
70C4⋊Q8×C47C2311yescharacteristic
71#125×C47—21no—
72C22⋊Q8×C47—21no—
73C22⋊Q8×C47—21no—
74C4×Q8×C47—21no—
75C4×Q8×C47—21no—
76#117×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—21no—
85C2×Q8×C47—21no—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47#4111yescharacteristic
89C22×Dic47—21no—
90C22×Dic47—41no—
91C2×D188—21no—
92C2×D188—22no—
93C2×D188Q1614yescomplemented
94C2×D188C2×D412yes—
95C2×C4×D47—22no—
96C2×C4×D47—22no—
97C2×C4×D47—82no—
98C2×Dic94—22no—
99C2×Dic94—41no—
100C2×Dic94—82no—
101C2×C8×C47—22no—
102C2×C8×C47—22no—
103C2×C8×C47—42no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C42×C47—41no—
112C42×C47C2×D411yescharacteristic
113C42×C47C2×D411yescharacteristic
114C42×C47#4111yescharacteristic
115C42×C47C2411yescharacteristic
116#10625337—84no—
117(C2×D47)⋊C4—22no—
118(C2×D47)⋊C4—22no—
119C4⋊Dic47—21no—
120C4⋊Dic47—22no—
121C4⋊Dic47—22no—
122C4⋊Dic47—41no—
123C4⋊Dic47—41no—
124C4⋊Dic47C2×D412yes—
125#10625329—42no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8C2×D414yes—
128C8×D47—84no—
129C23×C47#12111yescharacteristic
130C22×D47—22no—
131C22×D47—24no—
132C22×D47—81no—
133C22×D47—81no—
134Q8×C47—42no—
135Q8×C47—81no—
136C2×C4×C47—22no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47C2×Q1612yes—
144C2×C4×C47#13512yes—
145C2×C4×C47C22×D411yescharacteristic
146C2×C4×C47C22×D411yescharacteristic
147C2×C4×C47C2×#4111yescharacteristic
148C2×Dic47—22no—
149C2×Dic47—42no—
150C2×Dic47—81no—
151C2×Dic47—82no—
152D188—22no—
153D188—24no—
154D188C2×Q1614yes—
155C4×D47—44no—
156C4×D47—82no—
157Dic94—44no—
158Dic94—82no—
159C8×C47—42no—
160C8×C47—42no—
161C47⋊C8—24no—
162C47⋊C8—42no—
163#19387—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47#46712yes—
167C22×C47C2×#12111yescharacteristic
168C2×D47—24no—
169C2×D47—42no—
170C2×D47—42no—
171C2×D47—84no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47#46712yes—
178C4×C47C22×Q1611yescharacteristic
179C4×C47C2×#13511yescharacteristic
180C4×C47#57612yes—
181Dic47—42no—
182Dic47—82no—
183#3209—471no—
184#3077—471no—
185C2×#474—471no—
186#2972—471no—
187#1683—474no—
188#1669—474no—
189#1571—471no—
190#1549—471no—
191#1542D4711yescharacteristic, complemented
192C2×C47—22no—
193C2×C47#304111yescharacteristic
194C2×C47C2×#46711yescharacteristic
195C2×C47#306911yescharacteristic
196D47—44no—
197#533C2×D4711yescharacteristic
198C2×C4⋊Q8—471no—
199C2×C4×D4—471no—
200#494—472no—
201#487—472no—
202#478—942no—
203#476—472no—
204#474—472no—
205#474—474no—
206#474—942no—
207#462—942no—
208#451—942no—
209C8×D4—474no—
210C8×D4—942no—
211#431—471no—
212#431C2×D4711yescharacteristic
213#428—941no—
214C2×#107—941no—
215#423—471no—
216C2×C4⋊C8C2×D4711yescharacteristic
217#418—941no—
218C2×#102—941no—
219C2×C22⋊C8—942no—
220#335—474no—
221#333C2×D4712yescomplemented
222#332—474no—
223#332C2×D4712yescomplemented
224C47#1938711yescharacteristic, complemented, Sylow
225C22×Q8—941no—
226C22×D4—471no—
227C23×C4—941no—
228C2×Q16—944no—
229C2×M16—941no—
230C22×C8—21no—
231C22×C8—944no—
232C4⋊Q8—472no—
233C4⋊Q8—941no—
234C4⋊Q8—941no—
235C4⋊Q8—941no—
236C4⋊Q8C22×D4711yescharacteristic
237C4⋊Q8C22×D4711yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—472no—
244C4×D4—472no—
245C4×D4—941no—
246C4×D4—941no—
247#117—21no—
248C2×C4⋊C4—471no—
249C2×C4⋊C4—941no—
250C2×C4⋊C4—941no—
251C2×C22⋊C4—941no—
252C2×C42C22×D4711yescharacteristic
253#107—942no—
254#107—944no—
255#106—942no—
256C4⋊C8—21no—
257C4⋊C8—472no—
258C4⋊C8—472no—
259C4⋊C8—941no—
260C4⋊C8C22×D4712yes—
261C4⋊C8C22×D4712yes—
262#103—944no—
263#102—942no—
264#102—942no—
265#102—944no—
266C22⋊C8—21no—
267C22⋊C8—472no—
268C22⋊C8—944no—
269C4×C8—21no—
270C4×C8—472no—
271C24—941no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—942no—
275C2×Q8—3762no—
276C2×Q8—1881no—
277C2×D4—474no—
278C2×D4—472no—
279C2×D4—941no—
280C2×D4—942no—
281C22×C4—941no—
282C22×C4—942no—
283C22×C4—942no—
284C22×C4—3762no—
285C22×C4—1881no—
286C22×C4#1062535011yescharacteristic
287C22×C4D4×D4711yescharacteristic
288C22×C4D4×D4711yescharacteristic
289Q16—3764no—
290M16—1882no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—474no—
295C2×C8—942no—
296C2×C8—3764no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—472no—
304C4⋊C4—941no—
305C4⋊C4—942no—
306C4⋊C4—942no—
307C4⋊C4—1881no—
308C4⋊C4—1881no—
309C22⋊C4—41no—
310C22⋊C4—942no—
311C22⋊C4—942no—
312C42—41no—
313C42#1062535011yescharacteristic
314C42D4×D4711yescharacteristic
315C42D4×D4711yescharacteristic
316C42C23×D4711yescharacteristic
317C23—944no—
318C23—942no—
319C23—3761no—
320C23—3761no—
321C23#1309399711yescharacteristic
322Q8—42no—
323Q8—81no—
324Q8—3762no—
325Q8—1884no—
326D4—474no—
327D4—944no—
328D4—942no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—942no—
334C2×C4—3762no—
335C2×C4—3762no—
336C2×C4—3761no—
337C2×C4—1882no—
338C2×C4—1884no—
339C2×C4#1309399711yescharacteristic
340C2×C4#1309399711yescharacteristic
341C2×C4#1309399711yescharacteristic
342C2×C4Q16×D4712yes—
343C2×C4#1309404512yes—
344C2×C4C2×D4×D4711yescharacteristic
345C2×C4C2×D4×D4711yescharacteristic
346C2×C4C2×#1062535011yescharacteristic
347C8—42no—
348C8—42no—
349C8—944no—
350C8—1882no—
351C22—21no—
352C22—21no—
353C22—944no—
354C22—3764no—
355C22—1882no—
356C22—1882no—
357C22#1568936312yes—
358C22C2×#1309399711yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—3762no—
365C4—1882no—
366C4#1568936312yes—
367C4C2×Q16×D4711yescharacteristic
368C4C2×#1309404511yescharacteristic
369C4#1568992512yes—
370C2—22no—
371C2—1884no—
372C2#2025289411yescharacteristic
373C2C2×#1568936311yescharacteristic
374C2#2025312811yescharacteristic
subgroupG/H
centerC22C2×#13093997
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#1542C2
socleC22×C47C2×#121
prime radicalC22×D188D4
hypercenter#1542D47
hypocenterC47#19387
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #20254333C2semidirect
5. #20242842C2semidirect
6. #20242752C2semidirect
7. #20242162C2semidirect
8. #20242072C2semidirect
9. C47×#1542C2semidirect
12. #15689502C22semidirect
13. #15689475C22semidirect
30. #15688875C22semidirect
31. #15688870C22semidirect
35. C47×#333C22semidirect
36. C47×#332C22semidirect
93. C2×D188Q16semidirect
191. #1542D47semidirect
221. #333C2×D47semidirect
223. #332C2×D47semidirect
224. C47#19387semidirect

Maximal subgroups

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

subgroupindex
#202543332
#202531812
C2×#156894042
#202524112
#202428422
#202427522
#202421622
#202420722
C47×#15422
#1938747