Finite Group Databasebeta

← #26371024

Subgroups — #26371024

Order 12032.

514 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#20252306C211yesmaximal, characteristic
2#20245561C214yesmaximal, complemented
3#20245556C214yesmaximal, complemented
4#20245160C211yesmaximal, characteristic, complemented
5C2×#15688945C211yesmaximal, characteristic, complemented
6#20245154C211yesmaximal, characteristic, complemented
7#20245123C211yesmaximal, characteristic
8C2×#15688932C211yesmaximal, characteristic
9C47×#3163C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11#15689889—21no—
12C2×#13093964C2211yescharacteristic
13C2×#13093964C2211yescharacteristic
14#15689246—22no—
15#15689241C2214yes—
16C2×#13093960C2211yescharacteristic
17#15689141—21no—
18#15688945C2212yescomplemented
19#15688945C2214yescomplemented
20#15688945C2212yescomplemented
21#15688944C2214yescomplemented
22#15688932C2212yes—
23#15688932C2214yes—
24#15688932C2212yes—
25C47×#544—21no—
26C47×#501—22no—
27C47×#499C2212yescomplemented
28C47×#498C2212yescomplemented
29C47×#428—21no—
30C47×#427C2211yescharacteristic
31C2×#106×C47C2211yescharacteristic
32C47×#405C2211yescharacteristic
33C2×#10625350—21no—
34#13094046—21no—
35#13094046—21no—
36#13094046Q812yescomplemented
37#13094046C2312yes—
38C2×Dic188Q811yescharacteristic, complemented
39C2×#10625325—21no—
40C2×D376Q811yescharacteristic, complemented
41#13094032—42no—
42#13094030—24no—
43C2×#10625329—21no—
44M16×D47—42no—
45#13093964—21no—
46#13093964—21no—
47#13093964C2312yes—
48#13093964C2312yes—
49C2×C47⋊C16—21no—
50C2×C47⋊C16—21no—
51#13093960—21no—
52#13093960C2312yes—
53#13093952—24no—
54C2×M16×C47D411yescharacteristic
55C22×C8×C47C2311yescharacteristic
56C4⋊Q8×C47—22no—
57C4⋊Q8×C47—41no—
58#125×C47—22no—
59#125×C47—41no—
60C22⋊Q8×C47—42no—
61#117×C47—21no—
62#117×C47D411yescharacteristic
63C2×C4⋊C4×C47—21no—
64#107×C47—22no—
65#107×C47—21no—
66#106×C47—22no—
67#106×C47—21no—
68#106×C47C2311yescharacteristic
69#106×C47C2312yes—
70#106×C47C2311yescharacteristic
71#97×C47D412yes—
72C4×C8×C47C2312yes—
73C2×Q8×C47—42no—
74C22×C4×C47—41no—
75C22×C4×C47C2×D411yescharacteristic
76C2×C22⋊D47—41no—
77#10625350—22no—
78#10625350—22no—
79#10625350—41no—
80#10625350—41no—
81C2×D188—21no—
82C2×C4×D47—41no—
83C2×Dic94—21no—
84M16×C47SD1612yes—
85M16×C47SD1612yes—
86C2×C8×C47C2×D411yescharacteristic
87C2×C8×C47C2×D412yes—
88C2×C8×C47C2×Q811yescharacteristic
89C2×C8×C47C2×Q811yescharacteristic
90C2×C8×C47#4111yescharacteristic
91C2×C8×C47#4111yescharacteristic
92C2×C8×C47C2411yescharacteristic
93C4⋊C4×C47—21no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—42no—
98C4⋊C4×C47—42no—
99C4⋊C4×C47—42no—
100C4⋊C4×C47—41no—
101C22⋊C4×C47—21no—
102C22⋊C4×C47—41no—
103C42×C47—41no—
104C42×C47C2×D412yes—
105#10625329—22no—
106#10625329—41no—
107C2×C47⋊C8—41no—
108Dic188—21no—
109Dic188C2×Q812yes—
110D376—21no—
111D376C2×Q812yes—
112#10625325—22no—
113#10625325—22no—
114#10625324—44no—
115C8×D47—44no—
116C47⋊C16—22no—
117C47⋊C16—22no—
118C23×C47AGL(1,ℤ/8)11yescharacteristic
119C22×D47—41no—
120Q8×C47—162no—
121C2×C4×C47—22no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—161no—
126C2×C4×C47C22⋊Q811yescharacteristic
127C2×C4×C47C22⋊Q811yescharacteristic
128C2×C4×C47C2×SD1611yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47C22×D411yescharacteristic
132C22⋊D47—42no—
133C22⋊D47—42no—
134C2×Dic47—41no—
135D188—22no—
136D188—41no—
137C4×D47—42no—
138C4×D47—42no—
139Dic94—22no—
140Dic94—41no—
141C8×C47C22⋊Q811yescharacteristic
142C8×C47C22⋊Q811yescharacteristic
143C8×C47C2×SD1614yes—
144C8×C47C22×Q811yescharacteristic
145C8×C47C2×#4111yescharacteristic
146C47⋊C8—42no—
147#33638—471nomaximal, Sylow
148C22×C47—41no—
149C22×C47#47611yescharacteristic
150C22×C47#47611yescharacteristic
151C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
152C2×D47—42no—
153C2×D47—161no—
154C4×C47—42no—
155C4×C47—81no—
156C4×C47—82no—
157C4×C47—81no—
158C4×C47#47611yescharacteristic
159C4×C47#47611yescharacteristic
160C4×C47C2×C22⋊Q811yescharacteristic
161C4×C47C22×SD1611yescharacteristic
162Dic47—42no—
163#3163D4711yescharacteristic, complemented
164#2957—471no—
165#2231—474no—
166#2230—474no—
167#2155—471no—
168C2×#365—471no—
169#2153—471no—
170#2145—471no—
171C2×#360—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—21no—
175C2×C47C2×#47611yescharacteristic
176D47—82no—
177C2×#135—471no—
178#568—941no—
179#544—21no—
180C2×M32—471no—
181C2×M32—471no—
182#501—22no—
183#499C2×D4712yescomplemented
184#498C2×D4712yescomplemented
185#444—942no—
186#443—474no—
187C2×#108—471no—
188#428—21no—
189#427C2×D4711yescharacteristic
190C2×#106C2×D4711yescharacteristic
191#421—941no—
192#405C2×D4711yescharacteristic
193#365—474no—
194#365—472no—
195#365—472no—
196#364—474no—
197#360—474no—
198#360—472no—
199#360—472no—
200C47#3363811yescharacteristic, complemented, Sylow
201C2×#41—941no—
202#135—472no—
203#135—472no—
204#135—941no—
205#135—941no—
206C2×Q16—471no—
207C2×SD16—941no—
208C2×D8—471no—
209#131—944no—
210#131—1882no—
211C2×M16—941no—
212C2×M16—1882no—
213C2×M16C22⋊D4711yescharacteristic
214C22×C8C22×D4711yescharacteristic
215C4⋊Q8—22no—
216C4⋊Q8—41no—
217#125—22no—
218#125—41no—
219C22⋊Q8—42no—
220#117—21no—
221#117C22⋊D4711yescharacteristic
222C2×C4⋊C4—21no—
223M32—472no—
224M32—472no—
225M32—941no—
226M32—941no—
227C2×C16—941no—
228C2×C16—941no—
229#108—472no—
230#108—941no—
231#107—21no—
232#107—22no—
233#106—21no—
234#106—22no—
235#106C22×D4712yes—
236#106C22×D4711yescharacteristic
237#106C22×D4711yescharacteristic
238#104—944no—
239#97C22⋊D4712yes—
240C4×C8C22×D4712yes—
241#41—942no—
242#41—942no—
243#41—1881no—
244#41—1881no—
245C2×Q8—42no—
246C2×Q8—941no—
247C2×D4—941no—
248C2×D4—1881no—
249C22×C4—41no—
250C22×C4—1881no—
251C22×C4C2×C22⋊D4711yescharacteristic
252Q16—472no—
253Q16—941no—
254SD16—942no—
255SD16—942no—
256D8—472no—
257D8—941no—
258M16—942no—
259M16—1881no—
260M16—1884no—
261M16#1062533612yes—
262M16#1062533612yes—
263C2×C8—1881no—
264C2×C8—1884no—
265C2×C8Q8×D4711yescharacteristic
266C2×C8Q8×D4711yescharacteristic
267C2×C8#1062535411yescharacteristic
268C2×C8#1062535411yescharacteristic
269C2×C8C2×C22⋊D4711yescharacteristic
270C2×C8C2×C22⋊D4712yes—
271C2×C8C23×D4711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C22⋊C4—21no—
281C22⋊C4—41no—
282C42—41no—
283C42C2×C22⋊D4712yes—
284C16—942no—
285C16—942no—
286C23—1881no—
287C23#1309405011yescharacteristic
288Q8—162no—
289Q8—942no—
290Q8—1881no—
291D4—942no—
292D4—1882no—
293D4—1882no—
294D4—1881no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—161no—
300C2×C4—1881no—
301C2×C4—1882no—
302C2×C4—1882no—
303C2×C4#1309400011yescharacteristic
304C2×C4#1309400011yescharacteristic
305C2×C4C2×#1062533611yescharacteristic
306C2×C4C2×#1062533611yescharacteristic
307C2×C4#1309405011yescharacteristic
308C2×C4C22×C22⋊D4711yescharacteristic
309C8—1882no—
310C8#1309400011yescharacteristic
311C8#1309400011yescharacteristic
312C8C2×#1062533614yes—
313C8C2×Q8×D4711yescharacteristic
314C8C2×#1062535411yescharacteristic
315C22—41no—
316C22—1882no—
317C22—7521no—
318C22#1568941011yescharacteristic
319C22#1568941011yescharacteristic
320C22C2×#1309405011yescharacteristic
321C4—42no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—1882no—
326C4#1568941011yescharacteristic
327C4#1568941011yescharacteristic
328C4C2×#1309400011yescharacteristic
329C4C22×#1062533611yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—3762no—
334C2C2×#1568941011yescharacteristic
subgroupG/H
centerC2C2×#15689410
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3163C2
socleC2×C47C2×#476
prime radicalC2×#13094046C22
hypercenter#3163D47
hypocenterC47#33638
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #3163

Complements

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

NH ≅ G/Ntype
2. #20245561C2semidirect
3. #20245556C2semidirect
4. #20245160C2semidirect
5. C2×#15688945C2semidirect
6. #20245154C2semidirect
9. C47×#3163C2semidirect
18. #15688945C22semidirect
19. #15688945C22semidirect
20. #15688945C22semidirect
21. #15688944C22semidirect
27. C47×#499C22semidirect
28. C47×#498C22semidirect
36. #13094046Q8semidirect
38. C2×Dic188Q8semidirect
40. C2×D376Q8semidirect
163. #3163D47semidirect
183. #499C2×D47semidirect
184. #498C2×D47semidirect
200. C47#33638semidirect

Maximal subgroups

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

subgroupindex
#202523062
#202455612
#202455562
#202451602
C2×#156889452
#202451542
#202451232
C2×#156889322
C47×#31632
#3363847