Finite Group Databasebeta

← #26374017

Subgroups — #26374017

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256363C211yesmaximal, characteristic
2#20245631C212yesmaximal, complemented
3#20245613C211yesmaximal, characteristic, complemented
4#20245542C212yesmaximal, complemented
5#20245522C211yesmaximal, characteristic, complemented
6#20245428C211yesmaximal, characteristic, complemented
7#20245428C212yesmaximal, complemented
8#20245297C211yesmaximal, characteristic, complemented
9#20245250C212yesmaximal, complemented
10C4×#13093967C211yesmaximal, characteristic, complemented
11C47×#3383C211yesmaximal, characteristic, complemented
12#15689795—21no—
13C2×#13093967C2212yes—
14C2×#13093967C2211yescharacteristic
15#15689500C2211yescharacteristic
16#15689500C2212yes—
17#15689319—21no—
18#15689319—22no—
19C4×D376C2212yes—
20C4×D376C2211yescharacteristic
21#15688950C2211yescharacteristic
22#15688948C2212yes—
23C4⋊(C47⋊C16)C2211yescharacteristic
24C4⋊(C47⋊C16)C2212yes—
25#15688927C2211yescharacteristic
26#15688923C2212yes—
27#15688923C2212yes—
28#15688923C2212yes—
29#15688923C2212yes—
30#15688923C2211yescharacteristic
31#15688899C2212yes—
32C4×C47⋊C16C2211yescharacteristic
33C47×#557—21no—
34C47×#500C2211yescharacteristic
35C47×#499C2212yes—
36C47×#479—22no—
37C47×#475—21no—
38C8×Q8×C47C2211yescharacteristic
39C4×Q16×C47C2212yes—
40C4×Q16×C47C2211yescharacteristic
41C2×#10625354—41no—
42C2×D376—22no—
43C2×D376—21no—
44C2×D376C2312yes—
45C2×D376C2311yescharacteristic
46#13094021—22no—
47#13094021—21no—
48#13093999—41no—
49#13093999—42no—
50C4×D188—22no—
51C4×D188—21no—
52#13093967—24no—
53#13093967—22no—
54C2×C47⋊C16C2311yescharacteristic
55C2×C47⋊C16C2312yes—
56C2×C47⋊C16C2311yescharacteristic
57#13093958C2311yescharacteristic
58#13093946—22no—
59#13093946—21no—
60C2×Q16×C47C2311yescharacteristic
61C4⋊Q8×C47—21no—
62C4⋊Q8×C47—41no—
63#125×C47—22no—
64#125×C47—42no—
65C4×Q8×C47—21no—
66C4×Q8×C47—22no—
67C4×Q8×C47—41no—
68C4×Q8×C47D411yescharacteristic
69#107×C47—21no—
70#107×C47C2311yescharacteristic
71#107×C47C2312yes—
72#106×C47—22no—
73C4⋊C8×C47D411yescharacteristic
74C4⋊C8×C47D412yes—
75#103×C47—22no—
76#103×C47—21no—
77C4×C8×C47C2312yes—
78C4×C8×C47C2311yescharacteristic
79C2×Q8×C47—21no—
80C2×Q8×C47—41no—
81C2×Q8×C47D811yescharacteristic
82#10625354—84no—
83C2×D188—22no—
84C2×D188—21no—
85C2×D188—42no—
86C2×D188—41no—
87C2×D188—42no—
88C2×D188—41no—
89C2×C4×D47—42no—
90C2×C4×D47—41no—
91Q16×C47—21no—
92Q16×C47#4112yescomplemented
93C2×C8×C47C2×D412yes—
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2411yescharacteristic
96C4⋊C4×C47—21no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—42no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47D811yescharacteristic
104C4⋊C4×C47D812yes—
105C42×C47—42no—
106C42×C47—41no—
107C42×C47C2×D412yes—
108C42×C47C2×D411yescharacteristic
109(C2×D47)⋊C4—42no—
110(C2×D47)⋊C4—41no—
111C4⋊Dic47—21no—
112D376—22no—
113D376—21no—
114D376—22no—
115D376—21no—
116D376—82no—
117D376—81no—
118C47⋊C16—21no—
119C47⋊C16—22no—
120C47⋊C16—21no—
121C22×D47—42no—
122C22×D47—41no—
123Q8×C47—22no—
124Q8×C47—22no—
125Q8×C47—41no—
126Q8×C47—82no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47C2×D812yes—
131C2×C4×C47C2×D811yescharacteristic
132C2×C4×C47C2×D812yes—
133C2×C4×C47C2×D811yescharacteristic
134C2×C4×C47C22×D411yescharacteristic
135C2×Dic47—41no—
136D188—42no—
137D188—41no—
138D188—41no—
139D188—42no—
140D188—84no—
141D188—82no—
142D188—162no—
143D188—161no—
144C4×D47—84no—
145C4×D47—82no—
146C8×C47—22no—
147C8×C47—21no—
148C8×C47C2×#4111yescharacteristic
149C8×C472−1+411yescharacteristic
150#33935—471nomaximal, Sylow
151C22×C47C22×D811yescharacteristic
152C2×D47—82no—
153C2×D47—81no—
154C2×D47—82no—
155C2×D47—81no—
156C2×D47—82no—
157C2×D47—81no—
158C4×C47—22no—
159C4×C47—22no—
160C4×C47—21no—
161C4×C47—42no—
162C4×C47—82no—
163C4×C47—81no—
164C4×C47#50912yes—
165C4×C47#54811yescharacteristic
166C4×C47#57711yescharacteristic
167Dic47—81no—
168#3391—471no—
169#3383D4711yescharacteristic, complemented
170#2244—472no—
171#2240—471no—
172#2228—472no—
173#2224—471no—
174#2207—472no—
175#2207—471no—
176#2183—471no—
177#2175—472no—
178C4×SD32—471no—
179C2×C47#335811yescharacteristic
180C2×C47C2×#50911yescharacteristic
181C2×C47#341611yescharacteristic
182D47—162no—
183D47—161no—
184#557—21no—
185#550—941no—
186C2×SD32—471no—
187C2×SD32—472no—
188#500C2×D4711yescharacteristic
189#499C2×D4712yes—
190#493—471no—
191#493—472no—
192#479—22no—
193#475—21no—
194#459—941no—
195#459—942no—
196C8×Q8C2×D4711yescharacteristic
197C4×Q16C2×D4711yescharacteristic
198C4×Q16C2×D4712yes—
199C4×D8—471no—
200C4×D8—472no—
201#367—471no—
202#366—472no—
203C4⋊C16—471no—
204C4⋊C16—472no—
205#358—471no—
206#357—471no—
207#357—472no—
208#357—472no—
209#357—472no—
210#357—472no—
211#346—472no—
212C4×C16—471no—
213C47#3393511yescharacteristic, complemented, Sylow
214C2×#41—1881no—
215C2×Q16C22×D4711yescharacteristic
216C2×D8—471no—
217C2×D8—472no—
218C2×D8—942no—
219C2×D8—941no—
220C4⋊Q8—21no—
221C4⋊Q8—41no—
222#127—942no—
223#127—941no—
224#125—22no—
225#125—42no—
226#121—1881no—
227#121—1882no—
228C4×Q8—21no—
229C4×Q8—22no—
230C4×Q8—41no—
231C4×Q8C22⋊D4711yescharacteristic
232C4×D4—942no—
233C4×D4—941no—
234SD32—942no—
235SD32—944no—
236C2×C16—471no—
237C2×C16—471no—
238C2×C16—472no—
239#107—21no—
240#107—471no—
241#107C22×D4712yes—
242#107C22×D4711yescharacteristic
243#106—22no—
244C4⋊C8C22⋊D4711yescharacteristic
245C4⋊C8C22⋊D4712yes—
246#103—22no—
247#103—21no—
248#102—942no—
249#102—941no—
250C4×C8C22×D4711yescharacteristic
251C4×C8C22×D4712yes—
252#41—3764no—
253C2×Q8—21no—
254C2×Q8—41no—
255C2×Q8#1062533411yescharacteristic
256C2×D4—942no—
257C2×D4—941no—
258C2×D4—1882no—
259C2×D4—1881no—
260C2×D4—1882no—
261C2×D4—1881no—
262C22×C4—1882no—
263C22×C4—1881no—
264Q16—21no—
265Q16#1062535412yescomplemented
266D8—941no—
267D8—942no—
268D8—941no—
269D8—942no—
270D8—3762no—
271D8—3761no—
272C2×C8C2×C22⋊D4712yes—
273C2×C8C2×C22⋊D4711yescharacteristic
274C2×C8C23×D4711yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—941no—
283C4⋊C4#1062533411yescharacteristic
284C4⋊C4#1062533412yes—
285C22⋊C4—1882no—
286C22⋊C4—1881no—
287C42—42no—
288C42—41no—
289C42C2×C22⋊D4712yes—
290C42C2×C22⋊D4711yescharacteristic
291C16—941no—
292C16—942no—
293C16—941no—
294C23—1882no—
295C23—1881no—
296Q8—22no—
297Q8—22no—
298Q8—41no—
299Q8—82no—
300D4—3764no—
301D4—3762no—
302D4—1881no—
303D4—1882no—
304D4—1881no—
305D4—1882no—
306D4—7522no—
307D4—7521no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—3764no—
312C2×C4—3762no—
313C2×C4—1881no—
314C2×C4C2×#1062533412yes—
315C2×C4C2×#1062533411yescharacteristic
316C2×C4C2×#1062533412yes—
317C2×C4C2×#1062533411yescharacteristic
318C2×C4C22×C22⋊D4711yescharacteristic
319C8—22no—
320C8—21no—
321C8C2×#1062535411yescharacteristic
322C8#1309407211yescharacteristic
323C22—3762no—
324C22—3761no—
325C22—3762no—
326C22—3761no—
327C22—3762no—
328C22—3761no—
329C22C22×#1062533411yescharacteristic
330C4—22no—
331C4—21no—
332C4—22no—
333C4—42no—
334C4—81no—
335C4—82no—
336C4—3761no—
337C4#1568955812yes—
338C4#1568978411yescharacteristic
339C4#1568993111yescharacteristic
340C2—7522no—
341C2—7521no—
342C2#2025602311yescharacteristic
343C2C2×#1568955811yescharacteristic
344C2#2025653211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3383C2
socleC22×C47C22×D8
prime radical#20256363C2
hypercenter#3383D47
hypocenterC47#33935
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
2. #20245631C2semidirect
3. #20245613C2semidirect
4. #20245542C2semidirect
5. #20245522C2semidirect
6. #20245428C2semidirect
7. #20245428C2semidirect
8. #20245297C2semidirect
9. #20245250C2semidirect
10. C4×#13093967C2semidirect
11. C47×#3383C2semidirect
92. Q16×C47#41semidirect
169. #3383D47semidirect
213. C47#33935semidirect
265. Q16#10625354semidirect

Maximal subgroups

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

subgroupindex
#202563632
#202456312
#202456132
#202455422
#202455222
#202454282
#202452972
#202452502
C4×#130939672
C47×#33832
#3393547