Finite Group Databasebeta

← #26373914

Subgroups — #26373914

Order 12032.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20255811C211yesmaximal, characteristic
2#20245661C211yesmaximal, characteristic, complemented
3#20245593C212yesmaximal, complemented
4#20245568C212yesmaximal, complemented
5#20245540C211yesmaximal, characteristic, complemented
6#20245532C211yesmaximal, characteristic, complemented
7#20245503C212yesmaximal, complemented
8C4×#13093967C211yesmaximal, characteristic, complemented
9#20245220C211yesmaximal, characteristic, complemented
10#20245174C212yesmaximal, complemented
11C47×#3338C211yesmaximal, characteristic, complemented
12#15689838—21no—
13C2×#13093967C2211yescharacteristic
14C2×#13093964C2212yes—
15#15689528C2211yescharacteristic
16#15689448—22no—
17#15689414—21no—
18#15689359C2212yes—
19C4×D376C2211yescharacteristic
20#15689161C2212yes—
21#15688950C2212yes—
22#15688950C2211yescharacteristic
23#15688948C2212yes—
24#15688948C2212yes—
25#15688948C2212yes—
26C4⋊(C47⋊C16)C2211yescharacteristic
27#15688927C2212yes—
28#15688927C2211yescharacteristic
29#15688923C2212yes—
30#15688923C2211yescharacteristic
31C22⋊(C47⋊C16)C2212yes—
32C4×C47⋊C16C2211yescharacteristic
33C47×#554—21no—
34C47×#500C2211yescharacteristic
35C47×#484—22no—
36C47×#477—21no—
37C47×#467C2212yes—
38C4×Q16×C47C2211yescharacteristic
39C8×D4×C47C2211yescharacteristic
40C2×#107×C47C2212yes—
41C2×D376C2311yescharacteristic
42#13094024—21no—
43#13094010—42no—
44#13093998—22no—
45#13093995—41no—
46C4×D188—21no—
47#13093983—22no—
48#13093982—41no—
49#13093967—22no—
50#13093964—24no—
51C2×C47⋊C16C2312yes—
52C2×C47⋊C16C2311yescharacteristic
53C2×C47⋊C16C2311yescharacteristic
54#13093958—21no—
55#13093958C2312yes—
56#13093958C2311yescharacteristic
57#13093956—22no—
58#13093946—22no—
59#13093946—21no—
60C2×Q16×C47C2311yescharacteristic
61C22×C8×C47C2312yes—
62C4⋊Q8×C47—21no—
63#125×C47—41no—
64C22⋊Q8×C47—22no—
65C22⋊Q8×C47—42no—
66C4×Q8×C47—21no—
67C4×D4×C47—41no—
68C4×D4×C47D411yescharacteristic
69C2×C4⋊C4×C47—22no—
70#107×C47—22no—
71#107×C47—21no—
72#107×C47C2312yes—
73#107×C47C2311yescharacteristic
74C4⋊C8×C47D411yescharacteristic
75#103×C47—22no—
76#103×C47—21no—
77C22⋊C8×C47D412yes—
78C4×C8×C47C2311yescharacteristic
79C2×Q8×C47—21no—
80C2×Q8×C47—41no—
81C2×D4×C47D811yescharacteristic
82C22×C4×C47—42no—
83C22×C4×C47C2×D412yes—
84C2×C22⋊D47—42no—
85C2×D188—21no—
86C2×C4×D47—41no—
87C2×Dic94—41no—
88Q16×C47—21no—
89Q16×C47—21no—
90C2×C8×C47—22no—
91C2×C8×C47—22no—
92C2×C8×C47C2×D412yes—
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2411yescharacteristic
95C4⋊C4×C47—21no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—42no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47D811yescharacteristic
102C22⋊C4×C47—42no—
103C22⋊C4×C47D812yes—
104C42×C47—41no—
105C42×C47C2×D411yescharacteristic
106C22⋊Dic47—42no—
107(C2×D47)⋊C4—42no—
108(C2×D47)⋊C4—41no—
109C4⋊Dic47—21no—
110C4⋊Dic47—22no—
111C4⋊Dic47—41no—
112C4×Dic47—42no—
113D376—21no—
114D376—21no—
115C47⋊C16—22no—
116C47⋊C16—21no—
117C47⋊C16#4112yes—
118C23×C47C2×D812yes—
119C22×D47—41no—
120Q8×C47—41no—
121Q8×C47—41no—
122Q8×C47—161no—
123D4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—162no—
130C2×C4×C47C2×D812yes—
131C2×C4×C47C2×D811yescharacteristic
132C2×C4×C47C2×D811yescharacteristic
133C2×C4×C47C22×D411yescharacteristic
134C22⋊D47—162no—
135C2×Dic47—41no—
136C2×Dic47—42no—
137D188—41no—
138D188—41no—
139C4×D47—82no—
140Dic94—161no—
141C8×C47—22no—
142C8×C47—21no—
143C8×C47C2×#4111yescharacteristic
144C8×C472−1+411yescharacteristic
145#33924—471nomaximal, Sylow
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47C22×D811yescharacteristic
150C2×D47—81no—
151C2×D47—81no—
152C2×D47—81no—
153C4×C47—22no—
154C4×C47—21no—
155C4×C47—81no—
156C4×C47—81no—
157C4×C47—82no—
158C4×C47#51012yes—
159C4×C47#54711yescharacteristic
160C4×C47#57811yescharacteristic
161Dic47—81no—
162Dic47—82no—
163#3341—471no—
164#3338D4711yescharacteristic, complemented
165#2252—471no—
166#2237—472no—
167#2232—472no—
168#2227—471no—
169#2226—471no—
170#2220—472no—
171C4×SD32—471no—
172#2169—471no—
173#2159—472no—
174C2×C47—42no—
175C2×C47#333411yescharacteristic
176C2×C47C2×#51011yescharacteristic
177C2×C47#341511yescharacteristic
178D47—161no—
179#555—941no—
180#554—21no—
181C2×SD32—471no—
182C2×M32—472no—
183#500—471no—
184#500C2×D4711yescharacteristic
185#484—22no—
186#482—942no—
187#477—21no—
188#476—941no—
189#467C2×D4712yes—
190#466—472no—
191C4×Q16C2×D4711yescharacteristic
192C4×D8—471no—
193C8×D4C2×D4711yescharacteristic
194#427—472no—
195C2×#107C2×D4712yes—
196#367—471no—
197#367—472no—
198#366—472no—
199#366—472no—
200#366—472no—
201C4⋊C16—471no—
202#358—471no—
203#358—472no—
204#357—471no—
205#357—472no—
206C22⋊C16—472no—
207C4×C16—471no—
208C47#3392411yescharacteristic, complemented, Sylow
209C2×Q16C22×D4711yescharacteristic
210C2×D8—471no—
211C22×C8C22×D4712yes—
212C4⋊Q8—21no—
213C4⋊Q8—941no—
214#125—41no—
215#124—1882no—
216C22⋊Q8—22no—
217C22⋊Q8—42no—
218#121—942no—
219#121—1881no—
220C4×Q8—21no—
221C4×D4—41no—
222C4×D4—941no—
223C4×D4C22⋊D4711yescharacteristic
224#117—942no—
225#117—1881no—
226C2×C4⋊C4—22no—
227SD32—942no—
228M32—944no—
229C2×C16—471no—
230C2×C16—471no—
231C2×C16—472no—
232#107—22no—
233#107—21no—
234#107—472no—
235#107—471no—
236#107—941no—
237#107C22×D4711yescharacteristic
238#107C22×D4712yes—
239#106—942no—
240C4⋊C8C22⋊D4711yescharacteristic
241#103—21no—
242#103—22no—
243#102—941no—
244#102—942no—
245C22⋊C8C22⋊D4712yes—
246C4×C8C22×D4711yescharacteristic
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—1881no—
250C2×D4—941no—
251C2×D4—1882no—
252C2×D4#1062533411yescharacteristic
253C22×C4—42no—
254C22×C4—1881no—
255C22×C4C2×C22⋊D4712yes—
256Q16—21no—
257Q16—21no—
258D8—941no—
259D8—941no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8C2×C22⋊D4711yescharacteristic
263C2×C8C2×C22⋊D4712yes—
264C2×C8C23×D4711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—941no—
272C4⋊C4—942no—
273C4⋊C4—1881no—
274C4⋊C4#1062533411yescharacteristic
275C22⋊C4—42no—
276C22⋊C4—1882no—
277C22⋊C4—1882no—
278C22⋊C4—1881no—
279C22⋊C4#1062533412yes—
280C42—41no—
281C42—1882no—
282C42C2×C22⋊D4711yescharacteristic
283C16—472no—
284C16—942no—
285C16—941no—
286C23—1881no—
287C23C2×#1062533412yes—
288Q8—41no—
289Q8—41no—
290Q8—161no—
291Q8—7521no—
292D4—22no—
293D4—1881no—
294D4—1881no—
295D4—7522no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—162no—
302C2×C4—3762no—
303C2×C4—1881no—
304C2×C4—1882no—
305C2×C4C2×#1062533412yes—
306C2×C4C2×#1062533411yescharacteristic
307C2×C4C2×#1062533411yescharacteristic
308C2×C4C22×C22⋊D4711yescharacteristic
309C8—22no—
310C8—21no—
311C8#41×D4711yescharacteristic
312C8#1309407111yescharacteristic
313C22—22no—
314C22—22no—
315C22—22no—
316C22—3761no—
317C22—3761no—
318C22—3761no—
319C22C22×#1062533411yescharacteristic
320C4—21no—
321C4—22no—
322C4—81no—
323C4—82no—
324C4—81no—
325C4—3761no—
326C4—3762no—
327C4#1568956412yes—
328C4#1568977211yescharacteristic
329C4#1568993811yescharacteristic
330C2—42no—
331C2—7521no—
332C2#2025573111yescharacteristic
333C2C2×#1568956411yescharacteristic
334C2#2025652711yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3338C2
socleC22×C47C22×D8
prime radical#20255811C2
hypercenter#3338D47
hypocenterC47#33924
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #20245661C2semidirect
3. #20245593C2semidirect
4. #20245568C2semidirect
5. #20245540C2semidirect
6. #20245532C2semidirect
7. #20245503C2semidirect
8. C4×#13093967C2semidirect
9. #20245220C2semidirect
10. #20245174C2semidirect
11. C47×#3338C2semidirect
164. #3338D47semidirect
208. C47#33924semidirect

Maximal subgroups

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

subgroupindex
#202558112
#202456612
#202455932
#202455682
#202455402
#202455322
#202455032
C4×#130939672
#202452202
#202451742
C47×#33382
#3392447