Finite Group Databasebeta

← #27079854

Subgroups — #27079854

Order 12032.

4810 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094071C211yesmaximal, characteristic
2C2×#15689890C211yesmaximal, characteristic, complemented
3#20251918C2116yesmaximal, complemented
4#20251905C216yesmaximal, complemented
5C2×#15689131C216yesmaximal, complemented
6C47×#3428C211yesmaximal, characteristic, complemented
7C2×#13094071—28no—
8C2×#13094071—24no—
9C2×#13094071C4116yescomplemented
10C2×#13094071C2218yes—
11C22×#10625350C2213yes—
12C22×#10625352—23no—
13C23×Dic94C2213yes—
14C22×Q8×D47—21no—
15#15689890—28no—
16C2×#13094031C2218yescomplemented
17C22×#10625329C2216yes—
18#15689131C22112yes—
19#15689131C22124yescomplemented
20#15689127C2214yes—
21#15689122C22124yescomplemented
22#15689117C22124yescomplemented
23C2×#13093947C22124yescomplemented
24C22×#41×C47C2211yescharacteristic
25C47×#520C2218yescomplemented
26C47×#519—21no—
27C47×#518—23no—
28C4×#41×C47—24no—
29C2×#117×C47C2216yes—
30#13094071—264no—
31#13094071—232no—
32#13094071C2×C4164yescomplemented
33C2×#10625350—212no—
34C2×#10625350—26no—
35C2×#10625350C2×C4124yescomplemented
36C2×#10625350C23112yes—
37C2×#10625352—248no—
38C2×#10625352—224no—
39C22×Dic94—23no—
40C22×Dic94—212no—
41C22×Dic94—26no—
42C22×Dic94C2×C4124yescomplemented
43C22×Dic94C23112yes—
44C22×Dic94C2313yes—
45C2×Q8×D47—216no—
46C2×Q8×D47—28no—
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—23no—
49C23×Dic47—23no—
50C22×C4×D47—23no—
51#13094031—232no—
52C2×#10625329—224no—
53C2×#10625329C2314yes—
54C2×#10625329C23112yes—
55C22×C47⋊C8C23112yes—
56#13093947C23148yes—
57#13093944C23116yes—
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×C47C23112yes—
71C2×C4⋊C4×C47C23112yes—
72C2×C22⋊C4×C47—26no—
73C2×C42×C47—26no—
74C24×C47C2×D413yes—
75C23×D47—21no—
76#41×C47C22⋊C4132yes—
77#41×C47C22×C4116yes—
78#41×C47C2×D4116yes—
79C2×Q8×C47C22⋊C418yes—
80C2×Q8×C47C2×D414yes—
81C2×D4×C47C22⋊C4124yes—
82C2×D4×C47C2×D4112yes—
83C22×C4×C47—212no—
84C22×C4×C47C22×C4112yes—
85C22×C4×C47C2×D416yes—
86C22×C4×C47C2×D413yes—
87C22×C4×C47C2×D4112yes—
88C22×C4×C47C2416yes—
89C22×C4×C47C2411yescharacteristic
90C2×C22⋊D47—224no—
91C2×C22⋊D47—212no—
92C22×Dic47—224no—
93C22×Dic47—23no—
94C22×Dic47—212no—
95Q8×D47—264no—
96#10625352—2192no—
97#10625350—248no—
98#10625350—224no—
99#10625350C22×C4148yes—
100C2×D188—21no—
101C2×D188—22no—
102C2×D188C22×C414yes—
103C2×D188C2412yes—
104C2×C4×D47—224no—
105C2×C4×D47—212no—
106C2×Dic94—224no—
107C2×Dic94—248no—
108C2×Dic94—212no—
109C2×Dic94—23no—
110C2×Dic94—26no—
111C2×Dic94—224no—
112C2×Dic94C22×C4112yes—
113C2×Dic94C22×C4148yes—
114C2×Dic94C2416yes—
115C4⋊C4×C47—212no—
116C4⋊C4×C47C2418yes—
117C22⋊C4×C47—24no—
118C22⋊C4×C47—212no—
119C42×C47—24no—
120C42×C47—212no—
121#10625329—28no—
122#10625329—224no—
123C2×C47⋊C8—224no—
124C2×C47⋊C8C2418yes—
125C23×C47—46no—
126C23×C47C2×C22⋊C4112yes—
127C23×C47C22×D411yescharacteristic
128C23×C47C22×D416yes—
129C22×D47—24no—
130C22×D47—22no—
131C22×D47—44no—
132Q8×C47C2×C22⋊C4116yes—
133D4×C47C2×C22⋊C4148yes—
134C2×C4×C47—28no—
135C2×C4×C47—812no—
136C2×C4×C47C2×C22⋊C4112yes—
137C2×C4×C47C2×C22⋊C4124yes—
138C2×C4×C47C23×C412yes—
139C2×C4×C47C23×C4112yes—
140C2×C4×C47C22×D411yescharacteristic
141C2×C4×C47C22×D416yes—
142C2×C4×C47C22×D4112yes—
143C2×C4×C47C22×D412yes—
144C2×C4×C47C2511yescharacteristic
145C22⋊D47—248no—
146C2×Dic47—212no—
147C2×Dic47—248no—
148C2×Dic47—26no—
149D188—24no—
150D188—22no—
151D188C23×C414yes—
152C4×D47—248no—
153Dic94—212no—
154Dic94—248no—
155Dic94—26no—
156Dic94C23×C4112yes—
157C47⋊C8—28no—
158#60088—471nomaximal, Sylow
159C22×C47—212no—
160C22×C47—22no—
161C22×C47—412no—
162C22×C47C22×C22⋊C412yes—
163C22×C47C22×C22⋊C4112yes—
164C22×C47C23×D411yescharacteristic
165C2×D47—24no—
166C2×D47—48no—
167C2×D47—44no—
168C4×C47—48no—
169C4×C47C22×C22⋊C412yes—
170C4×C47C22×C22⋊C4112yes—
171C4×C47C24×C411yescharacteristic
172C4×C47C23×D411yescharacteristic
173Dic47—212no—
174C22×2−1+4—471no—
175C2×#568—471no—
176#3428D4711yescharacteristic, complemented
177#2898—4716no—
178#2896—476no—
179C2×#419—476no—
180C2×C47—22no—
181C2×C47—212no—
182C2×C47C23×C22⋊C411yescharacteristic
183C2×C47#358612yes—
184D47—44no—
185C2×2−1+4—4716no—
186C2×2−1+4—478no—
187C2×2−1+4—948no—
188C2×2−1+4—944no—
189C22×#41—473no—
190C22×#41—943no—
191C22×#41C2×D4711yescharacteristic
192C23×Q8—473no—
193C23×Q8—941no—
194#568—948no—
195C2×#131—478no—
196C22×M16—476no—
197#520C2×D4718yescomplemented
198#519—21no—
199#518—23no—
200C4×#41—24no—
201C2×#117C2×D4716yes—
202#419—4712no—
203#419—4724no—
204#418—474no—
205#417—4724no—
206#416—4724no—
207C2×#103—4724no—
208C47#6008811yescharacteristic, complemented, Sylow
2092−1+4—4764no—
2102−1+4—9464no—
2112−1+4—9432no—
212C2×#41—4724no—
213C2×#41—4712no—
214C2×#41—9448no—
215C2×#41—9424no—
216C2×#41—9412no—
217C2×#41—946no—
218C2×#41C4×D4718yes—
219C2×#41D18818yes—
220C2×#41C22⋊D4714yes—
221C2×#41C22×D4714yes—
222C22×Q8—4724no—
223C22×Q8—473no—
224C22×Q8—4712no—
225C22×Q8—9416no—
226C22×Q8—948no—
227C22×Q8—9412no—
228C22×Q8—943no—
229C22×Q8—946no—
230C22×Q8C22⋊D4711yescharacteristic
231C22×D4—471no—
232C22×D4—943no—
233C22×D4C22⋊D4713yes—
234C23×C4—943no—
235C23×C4—943no—
236C23×C4C22×D4713yes—
237#131—9432no—
238C2×M16—474no—
239C2×M16—4712no—
240C2×M16—9424no—
241C22×C8—4712no—
242C4×Q8—28no—
243C4×D4—224no—
244#117—26no—
245#117—26no—
246#117C22×D4714yes—
247#117C22×D47112yes—
248C2×C4⋊C4C22×D47112yes—
249C2×C22⋊C4—26no—
250C2×C42—26no—
251#103—4748no—
252#102—4716no—
253C24—941no—
254C24C2×C22⋊D4713yes—
255#41—4748no—
256#41—94192no—
257#41—9448no—
258#41—9424no—
259#41(C2×D47)⋊C4132yes—
260#41C2×C4×D47116yes—
261#41C2×D188116yes—
262C2×Q8—4712no—
263C2×Q8—4748no—
264C2×Q8—476no—
265C2×Q8—9464no—
266C2×Q8—9424no—
267C2×Q8—9448no—
268C2×Q8—943no—
269C2×Q8—9412no—
270C2×Q8—946no—
271C2×Q8—9424no—
272C2×Q8(C2×D47)⋊C418yes—
273C2×Q8C2×C22⋊D4714yes—
274C2×D4—474no—
275C2×D4—472no—
276C2×D4—9424no—
277C2×D4—9412no—
278C2×D4—941no—
279C2×D4—942no—
280C2×D4(C2×D47)⋊C4124yes—
281C2×D4C2×C22⋊D47112yes—
282C22×C4—212no—
283C22×C4—9424no—
284C22×C4—943no—
285C22×C4—9412no—
286C22×C4—9424no—
287C22×C4—9412no—
288C22×C4C2×C4×D47112yes—
289C22×C4C2×D188112yes—
290C22×C4C2×C22⋊D4716yes—
291C22×C4C2×C22⋊D4713yes—
292C22×C4C23×D4716yes—
293C22×C4C23×D4711yescharacteristic
294M16—948no—
295M16—9424no—
296C2×C8—478no—
297C2×C8—9424no—
298C4⋊C4—212no—
299C4⋊C4C23×D4718yes—
300C22⋊C4—24no—
301C22⋊C4—212no—
302C42—24no—
303C42—212no—
304C23—46no—
305C23—944no—
306C23—942no—
307C23—1884no—
308C23C2×(C2×D47)⋊C4112yes—
309C23C22×C22⋊D4711yescharacteristic
310C23C22×C22⋊D4716yes—
311Q8—4712no—
312Q8—9448no—
313Q8—9412no—
314Q8—946no—
315Q8C2×(C2×D47)⋊C4116yes—
316D4—474no—
317D4—9448no—
318D4—944no—
319D4—942no—
320D4C2×(C2×D47)⋊C4148yes—
321C2×C4—28no—
322C2×C4—812no—
323C2×C4—9412no—
324C2×C4—9448no—
325C2×C4—946no—
326C2×C4—9448no—
327C2×C4C2×(C2×D47)⋊C4112yes—
328C2×C4C2×(C2×D47)⋊C4124yes—
329C2×C4C22×C4×D4712yes—
330C2×C4C22×C4×D47112yes—
331C2×C4C22×C22⋊D4711yescharacteristic
332C2×C4C22×C22⋊D4716yes—
333C2×C4C22×D188112yes—
334C2×C4C22×D18812yes—
335C2×C4C24×D4711yescharacteristic
336C8—948no—
337C22—212no—
338C22—22no—
339C22—412no—
340C22—944no—
341C22—1888no—
342C22—1884no—
343C22C22×(C2×D47)⋊C412yes—
344C22C22×(C2×D47)⋊C4112yes—
345C22C23×C22⋊D4711yescharacteristic
346C4—48no—
347C4—9412no—
348C4C22×(C2×D47)⋊C4112yes—
349C4C22×(C2×D47)⋊C412yes—
350C4C23×C4×D4711yescharacteristic
351C4C23×D18811yescharacteristic
352C2—22no—
353C2—212no—
354C2—1884no—
355C2C23×(C2×D47)⋊C411yescharacteristic
356C2#2025817412yes—
subgroupG/H
centerC22C23×C22⋊D47
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC47×#3428C2
socleC22×C47C23×D4
prime radicalC22×#13094071C2
cosocleC2×C4×C47C25
hypercenter#3428D47
hypocenterC47#60088
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. #20251918C2semidirect
4. #20251905C2semidirect
5. C2×#15689131C2semidirect
6. C47×#3428C2semidirect
9. C2×#13094071C4semidirect
16. C2×#13094031C22semidirect
19. #15689131C22semidirect
21. #15689122C22semidirect
22. #15689117C22semidirect
23. C2×#13093947C22semidirect
25. C47×#520C22semidirect
32. #13094071C2×C4semidirect
35. C2×#10625350C2×C4semidirect
42. C22×Dic94C2×C4semidirect
176. #3428D47semidirect
197. #520C2×D47semidirect
208. C47#60088semidirect

Maximal subgroups

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

subgroupindex
C22×#130940712
C2×#156898902
#202519182
#202519052
C2×#156891312
C47×#34282
#6008847