Finite Group Databasebeta

← #27079848

Subgroups — #27079848

Order 12032.

5154 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094070C211yesmaximal, characteristic
2C2×#15689890C211yesmaximal, characteristic, complemented
3#20251911C2116yesmaximal, complemented
4#20251905C216yesmaximal, complemented
5C2×#15689127C216yesmaximal, complemented
6C47×#3428C211yesmaximal, characteristic, complemented
7C2×#13094070—28no—
8C2×#13094070—24no—
9C2×#13094070C4116yescomplemented
10C2×#13094070C2218yes—
11C22×#10625354—21no—
12C22×#10625350C2213yes—
13C23×D188C2213yes—
14C22×D4×D47—23no—
15#15689890—28no—
16C2×#13094031C2218yescomplemented
17C22×#10625329C2216yescomplemented
18#15689131C2214yescomplemented
19#15689127C22112yescomplemented
20#15689127C22124yescomplemented
21#15689122C22124yescomplemented
22#15689117C22124yescomplemented
23C2×#13093944C22124yescomplemented
24C22×#41×C47C2211yescharacteristic
25C47×#520C2218yescomplemented
26C47×#519—21no—
27C47×#518—23no—
28C4×#41×C47—24no—
29C2×#117×C47C2216yescomplemented
30C24×D47—23no—
31#13094070—264no—
32#13094070—232no—
33#13094070C2×C4164yescomplemented
34C2×#10625354—216no—
35C2×#10625354—28no—
36C2×#10625350—212no—
37C2×#10625350—26no—
38C2×#10625350C2×C4124yescomplemented
39C2×#10625350C23112yes—
40C22×Dic94C2311yescharacteristic
41C22×D188—23no—
42C22×D188—212no—
43C22×D188—26no—
44C22×D188C2×C4124yescomplemented
45C22×D188C23112yes—
46C22×D188C2313yes—
47C22×C22⋊D47—23no—
48C2×D4×D47—248no—
49C2×D4×D47—224no—
50C22×C4×D47—23no—
51#13094031—232no—
52C2×#10625329—224no—
53C2×#10625329C2314yes—
54C2×#10625329C23112yescomplemented
55C22×C47⋊C8C23112yescomplemented
56#13093947C23116yescomplemented
57#13093944C23148yescomplemented
58C2×#41×C47C2×C418yes—
59C2×#41×C47D418yes—
60C2×#41×C47D414yes—
61C2×#41×C47C2314yes—
62C22×Q8×C47D411yescharacteristic
63C22×D4×C47D413yes—
64C23×C4×C47C2313yes—
65C4×Q8×C47—28no—
66C4×D4×C47—224no—
67#117×C47—26no—
68#117×C47—26no—
69#117×C47C2314yes—
70#117×C47C23112yescomplemented
71C2×C4⋊C4×C47C23112yescomplemented
72C2×C22⋊C4×C47—26no—
73C2×C42×C47—26no—
74C24×C47C2×D413yes—
75C23×D47—23no—
76C23×D47—224no—
77C23×D47—212no—
78C23×D47—812no—
79#41×C47C22⋊C4132yes—
80#41×C47C22×C4116yes—
81#41×C47C2×D4116yes—
82C2×Q8×C47C22⋊C418yes—
83C2×Q8×C47C2×D414yes—
84C2×D4×C47C22⋊C4124yes—
85C2×D4×C47C2×D4112yes—
86C22×C4×C47—212no—
87C22×C4×C47C22×C4112yes—
88C22×C4×C47C2×D416yes—
89C22×C4×C47C2×D413yes—
90C22×C4×C47C2×D4112yes—
91C22×C4×C47C2416yes—
92C22×C4×C47C2411yescharacteristic
93C2×C22⋊D47—224no—
94C2×C22⋊D47—212no—
95C22×Dic47—21no—
96#10625354—264no—
97D4×D47—2192no—
98#10625350—248no—
99#10625350—224no—
100#10625350C22×C4148yes—
101C2×D188—224no—
102C2×D188—248no—
103C2×D188—212no—
104C2×D188—23no—
105C2×D188—26no—
106C2×D188—224no—
107C2×D188C22×C4112yes—
108C2×D188C22×C4148yes—
109C2×D188C2416yes—
110C2×C4×D47—224no—
111C2×C4×D47—212no—
112C2×Dic94—21no—
113C2×Dic94—22no—
114C2×Dic94C22×C414yes—
115C2×Dic94C2412yes—
116C4⋊C4×C47—212no—
117C4⋊C4×C47C2418yes—
118C22⋊C4×C47—24no—
119C22⋊C4×C47—212no—
120C42×C47—24no—
121C42×C47—212no—
122#10625329—28no—
123#10625329—224no—
124C2×C47⋊C8—224no—
125C2×C47⋊C8C2418yes—
126C23×C47—46no—
127C23×C47C2×C22⋊C4112yes—
128C23×C47C22×D411yescharacteristic
129C23×C47C22×D416yes—
130C22×D47—212no—
131C22×D47—26no—
132C22×D47—248no—
133C22×D47—412no—
134C22×D47—824no—
135C22×D47—848no—
136Q8×C47C2×C22⋊C4116yes—
137D4×C47C2×C22⋊C4148yes—
138C2×C4×C47—28no—
139C2×C4×C47—812no—
140C2×C4×C47C2×C22⋊C4112yes—
141C2×C4×C47C2×C22⋊C4124yes—
142C2×C4×C47C23×C412yes—
143C2×C4×C47C23×C4112yes—
144C2×C4×C47C22×D411yescharacteristic
145C2×C4×C47C22×D416yes—
146C2×C4×C47C22×D4112yes—
147C2×C4×C47C22×D412yes—
148C2×C4×C47C2511yescharacteristic
149C22⋊D47—248no—
150C2×Dic47—24no—
151C2×Dic47—22no—
152D188—212no—
153D188—248no—
154D188—26no—
155D188C23×C4112yes—
156C4×D47—248no—
157Dic94—24no—
158Dic94—22no—
159Dic94C23×C414yes—
160C47⋊C8—28no—
161#60087—471nomaximal, Sylow
162C22×C47—212no—
163C22×C47—22no—
164C22×C47—412no—
165C22×C47C22×C22⋊C412yes—
166C22×C47C22×C22⋊C4112yes—
167C22×C47C23×D411yescharacteristic
168C2×D47—212no—
169C2×D47—424no—
170C2×D47—412no—
171C2×D47—848no—
172C4×C47—48no—
173C4×C47C22×C22⋊C412yes—
174C4×C47C22×C22⋊C4112yes—
175C4×C47C24×C411yescharacteristic
176C4×C47C23×D411yescharacteristic
177Dic47—24no—
178C22×H(5,𝔽2)—471no—
179C2×#568—471no—
180#3428D4711yescharacteristic, complemented
181#2897—4716no—
182#2896—476no—
183C2×#418—476no—
184C2×C47—22no—
185C2×C47—212no—
186C2×C47C23×C22⋊C411yescharacteristic
187C2×C47#358512yes—
188D47—412no—
189C2×H(5,𝔽2)—4716no—
190C2×H(5,𝔽2)—478no—
191C2×H(5,𝔽2)—948no—
192C2×H(5,𝔽2)—944no—
193C22×#41—473no—
194C22×#41—941no—
195C22×#41C2×D4711yescharacteristic
196C23×D4—473no—
197C23×D4—943no—
198#568—948no—
199C2×#131—478no—
200C22×M16—476no—
201#520C2×D4718yescomplemented
202#519—21no—
203#518—23no—
204C4×#41—24no—
205C2×#117C2×D4716yescomplemented
206#419—474no—
207#418—4712no—
208#418—4724no—
209#417—4724no—
210#416—4724no—
211C2×#102—4724no—
212C47#6008711yescharacteristic, complemented, Sylow
213C25—943no—
214H(5,𝔽2)—4764no—
215H(5,𝔽2)—9464no—
216H(5,𝔽2)—9432no—
217C2×#41—4724no—
218C2×#41—4712no—
219C2×#41—9416no—
220C2×#41—948no—
221C2×#41—9412no—
222C2×#41—946no—
223C2×#41C4×D4718yes—
224C2×#41D18818yes—
225C2×#41C22⋊D4714yes—
226C2×#41C22×D4714yes—
227C22×Q8—471no—
228C22×Q8C22⋊D4711yescharacteristic
229C22×D4—4724no—
230C22×D4—473no—
231C22×D4—4712no—
232C22×D4—943no—
233C22×D4—9448no—
234C22×D4—9424no—
235C22×D4—9412no—
236C22×D4—943no—
237C22×D4—946no—
238C22×D4C22⋊D4713yes—
239C23×C4—943no—
240C23×C4C22×D4713yes—
241#131—9432no—
242C2×M16—474no—
243C2×M16—4712no—
244C2×M16—9424no—
245C22×C8—4712no—
246C4×Q8—28no—
247C4×D4—224no—
248#117—26no—
249#117—26no—
250#117C22×D4714yes—
251#117C22×D47112yescomplemented
252C2×C4⋊C4C22×D47112yescomplemented
253C2×C22⋊C4—26no—
254C2×C42—26no—
255#103—4716no—
256#102—4748no—
257C24—9424no—
258C24—9412no—
259C24—943no—
260C24—37612no—
261C24C2×C22⋊D4713yes—
262#41—4748no—
263#41—9464no—
264#41—9448no—
265#41—9424no—
266#41(C2×D47)⋊C4132yes—
267#41C2×C4×D47116yes—
268#41C2×D188116yes—
269C2×Q8—474no—
270C2×Q8—472no—
271C2×Q8—941no—
272C2×Q8—942no—
273C2×Q8(C2×D47)⋊C418yes—
274C2×Q8C2×C22⋊D4714yes—
275C2×D4—4712no—
276C2×D4—4748no—
277C2×D4—476no—
278C2×D4—9424no—
279C2×D4—9412no—
280C2×D4—94192no—
281C2×D4—9424no—
282C2×D4—9448no—
283C2×D4—943no—
284C2×D4—9412no—
285C2×D4—946no—
286C2×D4—9424no—
287C2×D4(C2×D47)⋊C4124yes—
288C2×D4C2×C22⋊D47112yes—
289C22×C4—212no—
290C22×C4—941no—
291C22×C4—9424no—
292C22×C4—9412no—
293C22×C4C2×C4×D47112yes—
294C22×C4C2×D188112yes—
295C22×C4C2×C22⋊D4716yes—
296C22×C4C2×C22⋊D4713yes—
297C22×C4C23×D4716yes—
298C22×C4C23×D4711yescharacteristic
299M16—948no—
300M16—9424no—
301C2×C8—478no—
302C2×C8—9424no—
303C4⋊C4—212no—
304C4⋊C4C23×D4718yes—
305C22⋊C4—24no—
306C22⋊C4—212no—
307C42—24no—
308C42—212no—
309C23—46no—
310C23—9448no—
311C23—9412no—
312C23—946no—
313C23—37624no—
314C23—37648no—
315C23—18812no—
316C23C2×(C2×D47)⋊C4112yes—
317C23C22×C22⋊D4711yescharacteristic
318C23C22×C22⋊D4716yes—
319Q8—474no—
320Q8—944no—
321Q8—942no—
322Q8C2×(C2×D47)⋊C4116yes—
323D4—4712no—
324D4—9448no—
325D4—9448no—
326D4—9412no—
327D4—946no—
328D4C2×(C2×D47)⋊C4148yes—
329C2×C4—28no—
330C2×C4—812no—
331C2×C4—944no—
332C2×C4—942no—
333C2×C4—9448no—
334C2×C4C2×(C2×D47)⋊C4112yes—
335C2×C4C2×(C2×D47)⋊C4124yes—
336C2×C4C22×C4×D4712yes—
337C2×C4C22×C4×D47112yes—
338C2×C4C22×C22⋊D4711yescharacteristic
339C2×C4C22×C22⋊D4716yes—
340C2×C4C22×D188112yes—
341C2×C4C22×D18812yes—
342C2×C4C24×D4711yescharacteristic
343C8—948no—
344C22—212no—
345C22—22no—
346C22—412no—
347C22—9412no—
348C22—37648no—
349C22—18824no—
350C22—18812no—
351C22C22×(C2×D47)⋊C412yes—
352C22C22×(C2×D47)⋊C4112yes—
353C22C23×C22⋊D4711yescharacteristic
354C4—48no—
355C4—944no—
356C4C22×(C2×D47)⋊C4112yes—
357C4C22×(C2×D47)⋊C412yes—
358C4C23×C4×D4711yescharacteristic
359C4C23×D18811yescharacteristic
360C2—22no—
361C2—212no—
362C2—18812no—
363C2C23×(C2×D47)⋊C411yescharacteristic
364C2#2025816712yes—
subgroupG/H
centerC22C23×C22⋊D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC47×#3428C2
socleC22×C47C23×D4
prime radicalC22×#13094070C2
cosocleC2×C4×C47C25
hypercenter#3428D47
hypocenterC47#60087
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×#41 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. C2×#15689890C2semidirect
3. #20251911C2semidirect
4. #20251905C2semidirect
5. C2×#15689127C2semidirect
6. C47×#3428C2semidirect
9. C2×#13094070C4semidirect
16. C2×#13094031C22semidirect
17. C22×#10625329C22semidirect
18. #15689131C22semidirect
19. #15689127C22semidirect
20. #15689127C22semidirect
21. #15689122C22semidirect
22. #15689117C22semidirect
23. C2×#13093944C22semidirect
25. C47×#520C22semidirect
29. C2×#117×C47C22semidirect
33. #13094070C2×C4semidirect
38. C2×#10625350C2×C4semidirect
44. C22×D188C2×C4semidirect
54. C2×#10625329C23semidirect
55. C22×C47⋊C8C23semidirect
56. #13093947C23semidirect
57. #13093944C23semidirect
70. #117×C47C23semidirect
71. C2×C4⋊C4×C47C23semidirect
180. #3428D47semidirect
201. #520C2×D47semidirect
205. C2×#117C2×D47semidirect
212. C47#60087semidirect
251. #117C22×D47semidirect
252. C2×C4⋊C4C22×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940702
C2×#156898902
#202519112
#202519052
C2×#156891272
C47×#34282
#6008747