Finite Group Databasebeta

← #26201461

Subgroups — #26201461

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#20253181C211yesmaximal, characteristic
2C2×#15689400C211yesmaximal, characteristic
3#20252411C211yesmaximal, characteristic
4#20242836C214yesmaximal, complemented
5#20242750C214yesmaximal, complemented
6#20242163C211yesmaximal, characteristic, complemented
7#20242073C211yesmaximal, characteristic, complemented
8#20242045C211yesmaximal, characteristic, complemented
9C47×#3209C211yesmaximal, characteristic, complemented
10C2×C4×D188C2211yescharacteristic
11#15689420—22no—
12#15689410C2212yes—
13#15689400—22no—
14#15689400C2212yes—
15#15689400C2214yes—
16C8×D188C2214yes—
17#15689188—22no—
18#15689176C2211yescharacteristic
19#15689166—21no—
20C2×#13093957—21no—
21#15689147C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23#15689127—21no—
24C2×#13093944—21no—
25C22⋊C8×D47—22no—
26#15688876C2214yescomplemented
27#15688872C2212yescomplemented
28#15688869C2212yescomplemented
29#15688869C2214yescomplemented
30C47×#533C2211yescharacteristic
31C2×C4⋊Q8×C47C2211yescharacteristic
32C47×#494C2212yescomplemented
33C47×#487C2212yescomplemented
34C47×#462—22no—
35C47×#451—22no—
36C47×#431C2211yescharacteristic
37C22×D188D411yescharacteristic
38C22×C4×D47—21no—
39C2×#10625329—21no—
40C22×C47⋊C8—21no—
41C2×C8×D47—24no—
42C4×D188—21no—
43C4×D188—21no—
44C4×D188D412yes—
45C4×D188C2312yes—
46C2×C4⋊Dic47D411yescharacteristic
47C2×(C2×D47)⋊C4—21no—
48#13093957—24no—
49#13093957—22no—
50#13093955—22no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)—21no—
53C4⋊(C47⋊C8)C2312yes—
54C4⋊(C47⋊C8)C2312yes—
55C4⋊(C47⋊C8)C2312yes—
56#13093944—24no—
57#13093944—22no—
58#13093944—22no—
59C22⋊(C47⋊C8)—21no—
60(C2×D47)⋊C8—24no—
61C4×C47⋊C8—21no—
62C22×Q8×C47—21no—
63C2×Q16×C47—24no—
64C4⋊Q8×C47—21no—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47C2311yescharacteristic
68C4⋊Q8×C47C2312yes—
69C4⋊Q8×C47C2311yescharacteristic
70#125×C47—21no—
71C22⋊Q8×C47—21no—
72C22⋊Q8×C47—21no—
73C4×Q8×C47—21no—
74C4×Q8×C47—21no—
75#117×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C42×C47C2311yescharacteristic
79C4⋊C8×C47D412yes—
80#103×C47—24no—
81C22⋊C8×C47D412yes—
82C4×C8×C47C2312yes—
83C23×D47—21no—
84C2×Q8×C47—22no—
85C2×Q8×C47—21no—
86C2×Q8×C47—21no—
87C2×Q8×C47—41no—
88C2×Q8×C47—82no—
89C22×C4×C47—41no—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C22×Dic47—21no—
94C2×D188—21no—
95C2×D188—22no—
96C2×D188Q1614yescomplemented
97C2×D188C2×D412yes—
98C2×C4×D47—22no—
99C2×C4×D47—22no—
100C2×C4×D47—82no—
101Q16×C47—84no—
102C2×C8×C47C2×D414yes—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C22⋊C4×C47—41no—
114C42×C47—41no—
115C42×C47C2×D411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47#4111yescharacteristic
118C42×C47C2411yescharacteristic
119(C2×D47)⋊C4—22no—
120(C2×D47)⋊C4—22no—
121C4⋊Dic47—21no—
122C4⋊Dic47C2×D412yes—
123#10625329—42no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—42no—
128C8×D47—84no—
129C23×C47#12111yescharacteristic
130C22×D47—22no—
131C22×D47—24no—
132C22×D47—81no—
133C22×D47—81no—
134Q8×C47—42no—
135Q8×C47—44no—
136Q8×C47—81no—
137Q8×C47—82no—
138C2×C4×C47—22no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—42no—
143C2×C4×C47—82no—
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47C2×Q1612yes—
148C2×C4×C47#13512yes—
149C2×C4×C47C22×D411yescharacteristic
150C2×C4×C47C22×D411yescharacteristic
151C2×C4×C47C2×#4111yescharacteristic
152C2×Dic47—22no—
153C2×Dic47—81no—
154D188—22no—
155D188—24no—
156D188C2×Q1614yes—
157C4×D47—44no—
158C4×D47—82no—
159C8×C47—24no—
160C47⋊C8—42no—
161C47⋊C8—42no—
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—82no—
176C4×C47—81no—
177C4×C47—81no—
178C4×C47#46712yes—
179C4×C47C22×Q1611yescharacteristic
180C4×C47C2×#13511yescharacteristic
181C4×C47#57612yes—
182Dic47—42no—
183#3209D4711yescharacteristic, complemented
184#3077—471no—
185C2×#474—471no—
186#2972—471no—
187#1683—474no—
188#1669—474no—
189#1571—471no—
190#1549—471no—
191#1542—471no—
192C2×C47—22no—
193C2×C47#304111yescharacteristic
194C2×C47C2×#46711yescharacteristic
195C2×C47#306911yescharacteristic
196D47—44no—
197#533C2×D4711yescharacteristic
198C2×C4⋊Q8C2×D4711yescharacteristic
199C2×C4×D4—471no—
200#494C2×D4712yescomplemented
201#487C2×D4712yescomplemented
202#478—942no—
203#476—472no—
204#474—472no—
205#474—474no—
206#474—942no—
207#462—22no—
208#451—22no—
209C8×D4—474no—
210C8×D4—942no—
211#431—471no—
212#431C2×D4711yescharacteristic
213#428—941no—
214C2×#107—941no—
215#423—471no—
216C2×C4⋊C8—471no—
217#418—941no—
218C2×#102—941no—
219C2×C22⋊C8—942no—
220#335—474no—
221#333—472no—
222#332—472no—
223#332—474no—
224C47#1938711yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×D4—471no—
227C23×C4—941no—
228C2×Q16—24no—
229C2×M16—941no—
230C22×C8—941no—
231C22×C8—944no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C22×D4712yes—
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—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—471no—
251C2×C22⋊C4—941no—
252C2×C42C22×D4711yescharacteristic
253#107—944no—
254#107—942no—
255#106—942no—
256C4⋊C8—472no—
257C4⋊C8—472no—
258C4⋊C8—472no—
259C4⋊C8—941no—
260C4⋊C8—941no—
261C4⋊C8C22⋊D4712yes—
262#103—24no—
263#102—944no—
264#102—942no—
265#102—942no—
266C22⋊C8—941no—
267C22⋊C8—944no—
268C22⋊C8C22⋊D4712yes—
269C4×C8—941no—
270C4×C8C22×D4712yes—
271C24—941no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×D4—474no—
278C2×D4—472no—
279C2×D4—941no—
280C2×D4—942no—
281C22×C4—41no—
282C22×C4—941no—
283C22×C4—942no—
284C22×C4—942no—
285C22×C4—3762no—
286C22×C4D4×D4711yescharacteristic
287C22×C4#1062535211yescharacteristic
288C22×C4C2×C22⋊D4711yescharacteristic
289Q16—84no—
290M16—1882no—
291C2×C8—942no—
292C2×C8—942no—
293C2×C8—942no—
294C2×C8—3764no—
295C2×C8—1882no—
296C2×C8C2×C22⋊D4714yes—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—472no—
308C4⋊C4—941no—
309C22⋊C4—41no—
310C22⋊C4—942no—
311C22⋊C4—942no—
312C42—41no—
313C42D4×D4711yescharacteristic
314C42#1062535211yescharacteristic
315C42C2×C22⋊D4711yescharacteristic
316C42C23×D4711yescharacteristic
317C23—944no—
318C23—942no—
319C23—3761no—
320C23—3761no—
321C23#1309399511yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—474no—
327D4—944no—
328D4—942no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—942no—
336C2×C4—3762no—
337C2×C4—3761no—
338C2×C4—1884no—
339C2×C4#1309399511yescharacteristic
340C2×C4#1309399511yescharacteristic
341C2×C4#1309399511yescharacteristic
342C2×C4Q16×D4712yes—
343C2×C4#1309404712yes—
344C2×C4C2×D4×D4711yescharacteristic
345C2×C4C22×C22⋊D4711yescharacteristic
346C2×C4C2×#1062535211yescharacteristic
347C8—24no—
348C8—1882no—
349C8—1882no—
350C8—1882no—
351C22—21no—
352C22—21no—
353C22—944no—
354C22—3764no—
355C22—1882no—
356C22—1882no—
357C22#1568936112yes—
358C22C2×#1309399511yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—1882no—
366C4#1568936112yes—
367C4C2×Q16×D4711yescharacteristic
368C4C2×#1309404711yescharacteristic
369C4#1568992612yes—
370C2—22no—
371C2—1884no—
372C2#2025288711yescharacteristic
373C2C2×#1568936111yescharacteristic
374C2#2025312111yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3209C2
socleC22×C47C2×#121
prime radicalC22×D188D4
hypercenter#3209D47
hypocenterC47#19387
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #20242836C2semidirect
5. #20242750C2semidirect
6. #20242163C2semidirect
7. #20242073C2semidirect
8. #20242045C2semidirect
9. C47×#3209C2semidirect
26. #15688876C22semidirect
27. #15688872C22semidirect
28. #15688869C22semidirect
29. #15688869C22semidirect
32. C47×#494C22semidirect
33. C47×#487C22semidirect
96. C2×D188Q16semidirect
183. #3209D47semidirect
200. #494C2×D47semidirect
201. #487C2×D47semidirect
224. C47#19387semidirect

Maximal subgroups

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

subgroupindex
#202531812
C2×#156894002
#202524112
#202428362
#202427502
#202421632
#202420732
#202420452
C47×#32092
#1938747