Finite Group Databasebeta

← #26369622

Subgroups — #26369622

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689561C211yesmaximal, characteristic
2#20252265C211yesmaximal, characteristic, complemented
3#20252241C211yesmaximal, characteristic, complemented
4#20245265C212yesmaximal, complemented
5#20245254C212yesmaximal, complemented
6C4×#13093969C212yesmaximal, complemented
7C4×D752C212yesmaximal, complemented
8#20245176C211yesmaximal, characteristic, complemented
9#20245096C211yesmaximal, characteristic, complemented
10C2×#15688925C211yesmaximal, characteristic, complemented
11C47×#2108C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13C22×D376C2211yescharacteristic
14#15689603—21no—
15#15689597—21no—
16#15689561C418yescomplemented
17C2×#13093969C2212yes—
18C2×D752C2212yes—
19#15689236—22no—
20#15689225—22no—
21C4×Dic188C2212yescomplemented
22C4×D376C2212yes—
23C4×D376C2212yes—
24C4×D376C2212yescomplemented
25#15689161C2211yescharacteristic
26C2×#13093958C2211yescharacteristic
27#15689129—21no—
28#15689125—21no—
29#15688951C2212yescomplemented
30#15688949C2212yescomplemented
31#15688928C2211yescharacteristic, complemented
32#15688928C2211yescharacteristic, complemented
33#15688925C2212yes—
34#15688925C2212yes—
35#15688925C2211yescharacteristic, complemented
36#15688925C2211yescharacteristic, complemented
37C2×M32×C47C2211yescharacteristic
38C22×C16×C47C2211yescharacteristic
39C47×#405C2211yescharacteristic
40C47×#346C2212yescomplemented
41C4×C16×C47C2212yescomplemented
42C2×#10625350—21no—
43C22×D188—21no—
44#13094046—22no—
45#13094046C2×C414yescomplemented
46C2×Dic188C2311yescharacteristic
47C2×#10625325—21no—
48C2×D376—21no—
49C2×D376—22no—
50C2×D376C2×C414yescomplemented
51C2×D376C2311yescharacteristic
52C2×D376C2312yes—
53C4×Dic94—22no—
54C4×D188—22no—
55C4×D188—22no—
56C4×D188—22no—
57#13093986—41no—
58#13093986—41no—
59(C22⋊D47)⋊C4—42no—
60#13093983—21no—
61#13093982—41no—
62C2×C4⋊Dic47—21no—
63C4⋊C4×D47—41no—
64C22⋊C4×D47—42no—
65#13093969C2×C418yescomplemented
66D752C2×C418yescomplemented
67#13093958—21no—
68#13093958C2312yes—
69#13093958C2311yescharacteristic
70#13093958C2311yescharacteristic
71#13093956—21no—
72#13093949—22no—
73#13093946—22no—
74#13093946—22no—
75#13093946—22no—
76C2×M16×C47D411yescharacteristic
77C22×C8×C47C2311yescharacteristic
78#117×C47D411yescharacteristic
79M32×C47C2×C414yescomplemented
80C2×C16×C47—22no—
81C2×C16×C47C2311yescharacteristic
82C2×C16×C47C2311yescharacteristic
83C2×C16×C47C2312yes—
84#97×C47D412yes—
85C4×C8×C47C2312yes—
86C23×D47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C2×C22⋊D47—41no—
89C22×Dic47—41no—
90#10625350—24no—
91#10625350—42no—
92C2×D188—24no—
93C2×D188—22no—
94C2×D188—21no—
95C2×D188—41no—
96C2×D188—42no—
97C2×C4×D47—41no—
98C2×C4×D47—42no—
99C2×C4×D47—42no—
100C2×C4×D47—42no—
101C2×Dic94—21no—
102M16×C47—22no—
103C2×C8×C47C22×C412yes—
104C2×C8×C47C2×D412yes—
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47#4112yes—
107C2×C8×C47C2411yescharacteristic
108C4⋊C4×C47D811yescharacteristic
109C4⋊C4×C47D811yescharacteristic
110C22⋊C4×C47D812yes—
111C42×C47C2×D412yes—
112C22⋊Dic47—41no—
113(C2×D47)⋊C4—42no—
114(C2×D47)⋊C4—42no—
115(C2×D47)⋊C4—42no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118C4⋊Dic47—22no—
119C4⋊Dic47—41no—
120Dic47⋊C4—42no—
121C4×Dic47—42no—
122C4×Dic47—41no—
123Dic188—21no—
124Dic188C22×C412yes—
125D376—24no—
126D376—22no—
127D376—21no—
128D376C22×C412yes—
129D376C22×C414yes—
130#10625325—24no—
131C16×C47—21no—
132C16×C47—21no—
133C16×C47C22×C414yes—
134C23×C47C2×D811yescharacteristic
135C22×D47—41no—
136C22×D47—42no—
137C22×D47—44no—
138C22×D47—161no—
139C22×D47—161no—
140C2×C4×C47C4×D412yes—
141C2×C4×C47C2×D812yes—
142C2×C4×C47C2×D812yes—
143C2×C4×C47C2×D811yescharacteristic
144C2×C4×C47#13512yes—
145C2×C4×C47C22×D411yescharacteristic
146C22⋊D47—44no—
147C2×Dic47—41no—
148C2×Dic47—42no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—161no—
152D188—22no—
153D188—24no—
154D188—42no—
155D188—41no—
156D188—44no—
157C4×D47—44no—
158C4×D47—162no—
159C4×D47—162no—
160C4×D47—162no—
161Dic94—22no—
162Dic94—41no—
163C8×C47—22no—
164C8×C47C4×D412yes—
165C8×C47C23×C411yescharacteristic
166C8×C47C2×#4111yescharacteristic
167#33454—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47C4×D812yes—
171C22×C47C22×D811yescharacteristic
172C2×D47—42no—
173C2×D47—44no—
174C2×D47—81no—
175C2×D47—82no—
176C2×D47—82no—
177C2×D47—81no—
178C2×D47—164no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47C4×D812yes—
182C4×C47C2×C4×D411yescharacteristic
183C4×C47C2×#13511yescharacteristic
184Dic47—42no—
185Dic47—81no—
186Dic47—82no—
187Dic47—81no—
188C2×#509—471no—
189#2951—471no—
190#2948—471no—
191#2177—472no—
192#2175—472no—
193C4×SD32—472no—
194C4×D16—472no—
195#2159—471no—
196#2138—471no—
197C2×#357—471no—
198#2108D4711yescharacteristic, complemented
199C2×C47—22no—
200C2×C47C2×C4×D811yescharacteristic
201C2×C47#341511yescharacteristic
202C2×C47#341611yescharacteristic
203D47—82no—
204D47—84no—
205C2×#135—471no—
206C22×D8—471no—
207#520—941no—
208#518—941no—
209#509—478no—
210C2×SD32—472no—
211C2×D16—472no—
212C2×M32C2×D4711yescharacteristic
213C22×C16C2×D4711yescharacteristic
214#442—942no—
215#440—942no—
216C4×Q16—472no—
217C4×D8—472no—
218C4×D8—472no—
219C4×D8—472no—
220#427—471no—
221C2×#107—471no—
222#418—941no—
223#417—941no—
224#405C2×D4711yescharacteristic
225#367—472no—
226#366—472no—
227#358—471no—
228#358—471no—
229#357—472no—
230#357—472no—
231#357—471no—
232#357—471no—
233#346C2×D4712yescomplemented
234C4×C16C2×D4712yescomplemented
235C47#3345411yescharacteristic, complemented, Sylow
236C2×#41—941no—
237C22×D4—941no—
238#135—474no—
239#135—942no—
240C2×Q16—471no—
241C2×SD16—941no—
242C2×D8—474no—
243C2×D8—472no—
244C2×D8—471no—
245C2×D8—941no—
246C2×D8—942no—
247C2×M16D18811yescharacteristic
248C22×C8C22×D4711yescharacteristic
249C4×Q8—942no—
250C4×D4—942no—
251C4×D4—942no—
252C4×D4—942no—
253C4×D4—1881no—
254C4×D4—1881no—
255C4×D4—1882no—
256#117—941no—
257#117—1881no—
258#117D18811yescharacteristic
259C2×C4⋊C4—941no—
260C2×C4⋊C4—1881no—
261C2×C22⋊C4—1882no—
262SD32—478no—
263D16—478no—
264M32C4×D4714yescomplemented
265C2×C16—22no—
266C2×C16C22×D4711yescharacteristic
267C2×C16C22×D4711yescharacteristic
268C2×C16C22×D4712yes—
269#107—472no—
270#107—471no—
271#107—471no—
272#107—941no—
273#106—941no—
274#103—942no—
275#102—942no—
276#102—942no—
277#102—942no—
278#97D18812yes—
279C4×C8C22×D4712yes—
280C24—1881no—
281#41—944no—
282#41—1882no—
283C2×Q8—941no—
284C2×D4—944no—
285C2×D4—942no—
286C2×D4—941no—
287C2×D4—1881no—
288C2×D4—1881no—
289C2×D4—1882no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1882no—
293C22×C4—1882no—
294C22×C4—1882no—
295C22×C4C2×D18811yescharacteristic
296Q16—472no—
297Q16—941no—
298SD16—944no—
299D8—472no—
300D8—474no—
301D8—944no—
302D8—942no—
303D8—941no—
304M16—22no—
305C2×C8C2×C4×D4712yes—
306C2×C8C2×D18812yes—
307C2×C8C2×D18811yescharacteristic
308C2×C8#1062535012yes—
309C2×C8C23×D4711yescharacteristic
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—942no—
313C4⋊C4—1882no—
314C4⋊C4—1881no—
315C4⋊C4D37611yescharacteristic
316C4⋊C4D37611yescharacteristic
317C22⋊C4—1881no—
318C22⋊C4—1882no—
319C22⋊C4—1882no—
320C22⋊C4—1882no—
321C22⋊C4D37612yes—
322C42—1881no—
323C42—1882no—
324C42C2×D18812yes—
325C16—21no—
326C16—21no—
327C16C2×C4×D4714yes—
328C23—1881no—
329C23—1882no—
330C23—1884no—
331C23—7521no—
332C23—7521no—
333C23C2×D37611yescharacteristic
334Q8—942no—
335Q8—1881no—
336D4—944no—
337D4—942no—
338D4—1884no—
339D4—1882no—
340D4—1881no—
341D4—1884no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1882no—
345C2×C4—1881no—
346C2×C4—1884no—
347C2×C4—7522no—
348C2×C4—7522no—
349C2×C4—7522no—
350C2×C4—7521no—
351C2×C4C4×D18812yes—
352C2×C4C2×D37612yes—
353C2×C4C2×D37612yes—
354C2×C4C2×D37611yescharacteristic
355C2×C4#1309404612yes—
356C2×C4C22×D18811yescharacteristic
357C8—22no—
358C8C4×D18812yes—
359C8C22×C4×D4711yescharacteristic
360C8C2×#1062535011yescharacteristic
361C22—21no—
362C22—21no—
363C22—3761no—
364C22—3762no—
365C22—3762no—
366C22—3761no—
367C22—1882no—
368C22—1884no—
369C22—7524no—
370C22C4×D37612yes—
371C22C22×D37611yescharacteristic
372C4—22no—
373C4—22no—
374C4—3761no—
375C4—3762no—
376C4—3761no—
377C4—1882no—
378C4C4×D37612yes—
379C4C2×C4×D18811yescharacteristic
380C4C2×#1309404611yescharacteristic
381C2—22no—
382C2—3764no—
383C2—3762no—
384C2C2×C4×D37611yescharacteristic
385C2#2025652911yescharacteristic
386C2#2025653611yescharacteristic
subgroupG/H
centerC22C22×D376
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2108C2
socleC22×C47C22×D8
prime radicalC2×#15689561C2
cosocleC2×C8×C47C24
hypercenter#2108D47
hypocenterC47#33454
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
2. #20252265C2semidirect
3. #20252241C2semidirect
4. #20245265C2semidirect
5. #20245254C2semidirect
6. C4×#13093969C2semidirect
7. C4×D752C2semidirect
8. #20245176C2semidirect
9. #20245096C2semidirect
10. C2×#15688925C2semidirect
11. C47×#2108C2semidirect
16. #15689561C4semidirect
21. C4×Dic188C22semidirect
24. C4×D376C22semidirect
29. #15688951C22semidirect
30. #15688949C22semidirect
31. #15688928C22semidirect
32. #15688928C22semidirect
35. #15688925C22semidirect
36. #15688925C22semidirect
40. C47×#346C22semidirect
41. C4×C16×C47C22semidirect
45. #13094046C2×C4semidirect
50. C2×D376C2×C4semidirect
65. #13093969C2×C4semidirect
66. D752C2×C4semidirect
79. M32×C47C2×C4semidirect
198. #2108D47semidirect
233. #346C2×D47semidirect
234. C4×C16C2×D47semidirect
235. C47#33454semidirect
264. M32C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895612
#202522652
#202522412
#202452652
#202452542
C4×#130939692
C4×D7522
#202451762
#202450962
C2×#156889252
C47×#21082
#3345447