Finite Group Databasebeta

← #26369735

Subgroups — #26369735

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256522C211yesmaximal, characteristic
2#20252306C211yesmaximal, characteristic, complemented
3#20245284C214yesmaximal, complemented
4#20245277C214yesmaximal, complemented
5C2×#15688945C211yesmaximal, characteristic, complemented
6#20245137C211yesmaximal, characteristic, complemented
7C2×#15688932C211yesmaximal, characteristic, complemented
8#20244993C211yesmaximal, characteristic, complemented
9C47×#2952C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11#15689889—21no—
12#15689564C2212yes—
13#15689563C412yescomplemented
14#15689559C2212yes—
15#15689558C412yescomplemented
16#15689553C414yescomplemented
17C2×#13093964C2211yescharacteristic
18C2×#13093964C2211yescharacteristic
19#15689246C2214yes—
20#15689241—22no—
21C2×#13093960C2211yescharacteristic
22#15689141—21no—
23#15688945C2214yescomplemented
24#15688939C2212yescomplemented
25#15688936C2212yescomplemented
26#15688932C2214yes—
27#15688901C2214yescomplemented
28C2×#135×C47C2211yescharacteristic
29C47×#520—21no—
30C47×#442C2212yescomplemented
31C47×#441C2212yescomplemented
32C4×SD16×C47—22no—
33C2×#106×C47C2211yescharacteristic
34C47×#417—21no—
35C47×#405C2211yescharacteristic
36C2×#10625350—21no—
37#13094046—21no—
38#13094046—21no—
39#13094046C2×C412yescomplemented
40#13094046C2312yes—
41C2×Dic188C2×C411yescharacteristic, complemented
42C2×#10625325—21no—
43C2×D376C2×C411yescharacteristic, complemented
44#13094032—42no—
45#13094030—24no—
46C2×#10625329—21no—
47M16×D47—42no—
48#13093970C2×C414yescomplemented
49#13093968C2×C414yescomplemented
50#13093967C2×C414yescomplemented
51#13093965C2×C414yescomplemented
52#13093964—21no—
53#13093964C2×C412yescomplemented
54#13093964C2312yes—
55#13093964C2312yes—
56C2×C47⋊C16—21no—
57C2×C47⋊C16C2×C412yescomplemented
58#13093960—21no—
59#13093960C2312yes—
60#13093952—24no—
61C2×#41×C47—21no—
62#135×C47—21no—
63#135×C47—21no—
64#135×C47C2×C412yes—
65#135×C47C2×C412yes—
66C2×Q16×C47C2311yescharacteristic
67C2×SD16×C47—21no—
68C2×D8×C47C2311yescharacteristic
69C2×M16×C47D411yescharacteristic
70C22×C8×C47C2311yescharacteristic
71C4×Q8×C47—22no—
72C4×D4×C47—22no—
73C4×D4×C47—42no—
74#117×C47—41no—
75#117×C47D411yescharacteristic
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—41no—
78#106×C47—21no—
79#106×C47C2312yes—
80#103×C47—22no—
81#102×C47—22no—
82#97×C47C2312yes—
83C4×C8×C47D412yes—
84#41×C47—22no—
85#41×C47—22no—
86#41×C47—41no—
87#41×C47—41no—
88C2×Q8×C47—21no—
89C2×D4×C47—21no—
90C2×D4×C47—41no—
91C22×C4×C47—42no—
92C22×C4×C47—41no—
93C22×C4×C47—41no—
94C22×C4×C47C2×D411yescharacteristic
95C2×C22⋊D47—41no—
96#10625350—22no—
97#10625350—22no—
98#10625350—41no—
99#10625350—41no—
100C2×D188—21no—
101C2×C4×D47—41no—
102C2×Dic94—21no—
103Q16×C47—21no—
104Q16×C47C22×C412yes—
105SD16×C47—22no—
106SD16×C47—22no—
107D8×C47—21no—
108D8×C47C22×C412yes—
109M16×C47—21no—
110M16×C47—21no—
111C2×C8×C47C22×C411yescharacteristic
112C2×C8×C47C22×C411yescharacteristic
113C2×C8×C47C2×D412yes—
114C2×C8×C47C2×D411yescharacteristic
115C2×C8×C47#4111yescharacteristic
116C2×C8×C47#4111yescharacteristic
117C2×C8×C47C2411yescharacteristic
118C4⋊C4×C47—21no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—42no—
121C4⋊C4×C47—41no—
122C22⋊C4×C47—21no—
123C22⋊C4×C47—42no—
124C42×C47—42no—
125C42×C47C2×D412yes—
126#10625329—22no—
127#10625329—41no—
128C2×C47⋊C8—41no—
129Dic188—21no—
130Dic188C22×C412yes—
131D376—21no—
132D376C22×C412yes—
133#10625325—22no—
134#10625325—22no—
135#10625324—44no—
136C8×D47—44no—
137C47⋊C16—22no—
138C47⋊C16C22×C414yes—
139C23×C47—41no—
140C23×C47AGL(1,ℤ/8)11yescharacteristic
141C22×D47—41no—
142Q8×C47—22no—
143Q8×C47—41no—
144D4×C47—22no—
145D4×C47—42no—
146D4×C47—42no—
147D4×C47—41no—
148C2×C4×C47—22no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—162no—
154C2×C4×C47—161no—
155C2×C4×C47C4×D411yescharacteristic
156C2×C4×C47C4×D411yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C47C22×D411yescharacteristic
161C22⋊D47—42no—
162C22⋊D47—42no—
163C2×Dic47—41no—
164D188—22no—
165D188—41no—
166C4×D47—42no—
167C4×D47—42no—
168Dic94—22no—
169Dic94—41no—
170C8×C47—22no—
171C8×C47C4×D411yescharacteristic
172C8×C47C4×D411yescharacteristic
173C8×C47C23×C411yescharacteristic
174C8×C47C2×#4111yescharacteristic
175C47⋊C8—42no—
176#33469—471nomaximal, Sylow
177C22×C47—41no—
178C22×C47—42no—
179C22×C47—81no—
180C22×C47—81no—
181C22×C47#44211yescharacteristic
182C22×C47#44211yescharacteristic
183C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
184C2×D47—42no—
185C2×D47—161no—
186C4×C47—42no—
187C4×C47—42no—
188C4×C47—82no—
189C4×C47#44211yescharacteristic
190C4×C47#44211yescharacteristic
191C4×C47C2×C4×D411yescharacteristic
192C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
193Dic47—42no—
194#3414—471no—
195#2957—471no—
196#2952D4711yescharacteristic, complemented
197AGL(1,ℤ/16)—474no—
198#2180—474no—
199C2×#365—471no—
200#2148—471no—
201C2×#360—471no—
202#2110—471no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—82no—
207C2×C47C2×#44211yescharacteristic
208D47—82no—
209C2×#135—471no—
210C2×#135C2×D4711yescharacteristic
211#568—941no—
212#520—21no—
213#510—472no—
214#510—472no—
215#509—472no—
216#509—472no—
217#508—474no—
218C2×M32—471no—
219C2×M32—471no—
220#444—474no—
221#443—942no—
222#442C2×D4712yescomplemented
223#441C2×D4712yescomplemented
224C4×SD16—22no—
225C2×#108—471no—
226C2×#106C2×D4711yescharacteristic
227#421—941no—
228#417—21no—
229#405C2×D4711yescharacteristic
230#365—474no—
231#362—472no—
232#361—472no—
233#360—474no—
234#347—474no—
235C47#3346911yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—941no—
238#135—21no—
239#135—21no—
240#135—472no—
241#135—472no—
242#135—941no—
243#135—941no—
244#135C4×D4712yes—
245#135C4×D4712yes—
246C2×Q16—471no—
247C2×Q16C22×D4711yescharacteristic
248C2×SD16—21no—
249C2×SD16—941no—
250C2×D8—471no—
251C2×D8C22×D4711yescharacteristic
252#131—944no—
253#131—1882no—
254C2×M16—941no—
255C2×M16—1882no—
256C2×M16C22⋊D4711yescharacteristic
257C22×C8C22×D4711yescharacteristic
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—42no—
261#117—41no—
262#117C22⋊D4711yescharacteristic
263C2×C4⋊C4—21no—
264C2×C4⋊C4—41no—
265Q32—474no—
266SD32—474no—
267SD32—474no—
268D16—474no—
269M32—472no—
270M32—472no—
271M32—472no—
272M32—941no—
273C2×C16—472no—
274C2×C16—941no—
275#108—472no—
276#108—941no—
277#106—21no—
278#106C22×D4712yes—
279#104—944no—
280#103—22no—
281#102—22no—
282#97C22×D4712yes—
283C4×C8C22⋊D4712yes—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—942no—
289#41—942no—
290#41—1881no—
291#41—1881no—
292C2×Q8—21no—
293C2×Q8—941no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—941no—
297C2×D4—1881no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—1881no—
302C22×C4C2×C22⋊D4711yescharacteristic
303Q16—21no—
304Q16—472no—
305Q16—941no—
306Q16C2×C4×D4712yes—
307SD16—22no—
308SD16—22no—
309SD16—942no—
310SD16—942no—
311D8—21no—
312D8—472no—
313D8—941no—
314D8C2×C4×D4712yes—
315M16—21no—
316M16—21no—
317M16—942no—
318M16—1884no—
319M16—1881no—
320C2×C8—1884no—
321C2×C8—1881no—
322C2×C8C2×C4×D4711yescharacteristic
323C2×C8C2×C4×D4711yescharacteristic
324C2×C8#1062535011yescharacteristic
325C2×C8#1062535011yescharacteristic
326C2×C8C2×C22⋊D4712yes—
327C2×C8C2×C22⋊D4711yescharacteristic
328C2×C8C23×D4711yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C22⋊C4—21no—
334C22⋊C4—42no—
335C42—42no—
336C42C2×C22⋊D4712yes—
337C16—474no—
338C16—942no—
339C23—41no—
340C23—1881no—
341C23#1309405011yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—942no—
345Q8—1881no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—942no—
351D4—1882no—
352D4—1882no—
353D4—1881no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—162no—
360C2×C4—161no—
361C2×C4—1881no—
362C2×C4—1882no—
363C2×C4—1882no—
364C2×C4C4×C22⋊D4711yescharacteristic
365C2×C4C4×C22⋊D4711yescharacteristic
366C2×C4#1309404911yescharacteristic
367C2×C4#1309404911yescharacteristic
368C2×C4#1309405011yescharacteristic
369C2×C4C22×C22⋊D4711yescharacteristic
370C8—22no—
371C8—1882no—
372C8C4×C22⋊D4711yescharacteristic
373C8C4×C22⋊D4711yescharacteristic
374C8C22×C4×D4711yescharacteristic
375C8C2×#1062535011yescharacteristic
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—1882no—
381C22—7521no—
382C22#1568923311yescharacteristic
383C22#1568923311yescharacteristic
384C22C2×#1309405011yescharacteristic
385C4—42no—
386C4—42no—
387C4—82no—
388C4—1882no—
389C4#1568923311yescharacteristic
390C4#1568923311yescharacteristic
391C4C2×C4×C22⋊D4711yescharacteristic
392C4C2×#1309404911yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—3762no—
398C2C2×#1568923311yescharacteristic
subgroupG/H
centerC2C2×#15689233
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2952C2
socleC2×C47C2×#442
prime radical#20256522C2
cosocleC2×C8×C47C24
hypercenter#2952D47
hypocenterC47#33469
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2952

Complements

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

NH ≅ G/Ntype
2. #20252306C2semidirect
3. #20245284C2semidirect
4. #20245277C2semidirect
5. C2×#15688945C2semidirect
6. #20245137C2semidirect
7. C2×#15688932C2semidirect
8. #20244993C2semidirect
9. C47×#2952C2semidirect
13. #15689563C4semidirect
15. #15689558C4semidirect
16. #15689553C4semidirect
23. #15688945C22semidirect
24. #15688939C22semidirect
25. #15688936C22semidirect
27. #15688901C22semidirect
30. C47×#442C22semidirect
31. C47×#441C22semidirect
39. #13094046C2×C4semidirect
41. C2×Dic188C2×C4semidirect
43. C2×D376C2×C4semidirect
48. #13093970C2×C4semidirect
49. #13093968C2×C4semidirect
50. #13093967C2×C4semidirect
51. #13093965C2×C4semidirect
53. #13093964C2×C4semidirect
57. C2×C47⋊C16C2×C4semidirect
196. #2952D47semidirect
222. #442C2×D47semidirect
223. #441C2×D47semidirect
235. C47#33469semidirect

Maximal subgroups

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

subgroupindex
#202565222
#202523062
#202452842
#202452772
C2×#156889452
#202451372
C2×#156889322
#202449932
C47×#29522
#3346947