Finite Group Databasebeta

← #26369730

Subgroups — #26369730

Order 12032.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256524C211yesmaximal, characteristic
2C2×#15689246C211yesmaximal, characteristic, complemented
3#20252190C211yesmaximal, characteristic, complemented
4#20245288C214yesmaximal, complemented
5#20245281C214yesmaximal, complemented
6#20245161C211yesmaximal, characteristic, complemented
7#20245138C211yesmaximal, characteristic, complemented
8#20245125C211yesmaximal, characteristic, complemented
9C2×C47×#347C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11C2×#13094046C2211yescharacteristic
12C2×#13094030—21no—
13C4×#10625350—21no—
14#15689566C412yescomplemented
15#15689566C2212yes—
16#15689561C412yescomplemented
17#15689561C2212yes—
18#15689555C414yescomplemented
19#15689246—22no—
20#15689246C2214yes—
21#15689236C2212yescomplemented
22#15689231C2212yescomplemented
23#15689225—22no—
24C2×#13093960C2211yescharacteristic
25#15689161C2211yescharacteristic
26C2×#13093952—21no—
27#15689120—21no—
28#15688946C2214yescomplemented
29#15688940C2212yescomplemented
30#15688937C2212yescomplemented
31#15688934C2214yes—
32C2×M32×C47C2211yescharacteristic
33C2×M32×C47C2211yescharacteristic
34C2×#97×C47C2211yescharacteristic
35C47×#347C2214yescomplemented
36C2×#10625350—21no—
37C2×#10625350—21no—
38#13094046—21no—
39#13094046—21no—
40#13094046—21no—
41#13094046—21no—
42#13094046C2×C412yes—
43#13094046C2×C412yes—
44#13094046C2×C412yescomplemented
45#13094046C2312yes—
46C2×Dic188C2×C411yescharacteristic, complemented
47C2×Dic188C2311yescharacteristic
48C2×#10625325—21no—
49C2×#10625325—21no—
50C2×D376C2×C411yescharacteristic, complemented
51C2×D376C2311yescharacteristic
52#13094030—24no—
53#13094030—42no—
54C2×#10625329—21no—
55C2×#10625324—41no—
56C2×C8×D47—41no—
57C4×Dic94—22no—
58C4×D188—22no—
59C4×C22⋊D47—42no—
60#13093983—21no—
61#13093981—41no—
62C42×D47—41no—
63Dic376C2×C414yescomplemented
64#13093969C2×C414yescomplemented
65#13093969C2×C414yescomplemented
66D752C2×C414yescomplemented
67#13093960—21no—
68#13093960C2312yes—
69#13093958—21no—
70#13093956C2312yes—
71#13093952—24no—
72#13093949—22no—
73#13093946—22no—
74C22×C8×C47D411yescharacteristic
75C22×C8×C47C2311yescharacteristic
76C2×C42×C47D411yescharacteristic
77M32×C47—21no—
78M32×C47C2×C412yescomplemented
79M32×C47C2312yes—
80M32×C47C2312yes—
81C2×C16×C47—21no—
82C2×C16×C47C2×C412yescomplemented
83#97×C47D412yes—
84#97×C47C2312yes—
85C22×C4×C47—21no—
86C22×C4×C47C2×D411yescharacteristic
87C2×C22⋊D47—41no—
88C2×C22⋊D47—41no—
89#10625350—22no—
90#10625350—22no—
91#10625350—22no—
92#10625350—22no—
93#10625350—41no—
94#10625350—41no—
95#10625350—41no—
96#10625350—41no—
97C2×D188—21no—
98C2×D188—21no—
99C2×C4×D47—41no—
100C2×C4×D47—41no—
101C2×C4×D47—42no—
102C2×Dic94—21no—
103C2×Dic94—21no—
104C2×C8×C47—21no—
105C2×C8×C47—21no—
106C2×C8×C47C22×C411yescharacteristic
107C2×C8×C47C22×C411yescharacteristic
108C2×C8×C47C2×D412yes—
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47#4111yescharacteristic
111C2×C8×C47#4111yescharacteristic
112C2×C8×C47C2411yescharacteristic
113C42×C47—21no—
114C42×C47C2×D412yes—
115C22⋊Dic47—41no—
116(C2×D47)⋊C4—42no—
117C4⋊Dic47—22no—
118Dic47⋊C4—42no—
119C4×Dic47—42no—
120C4×Dic47—41no—
121#10625329—22no—
122#10625329—41no—
123C2×C47⋊C8—41no—
124Dic188—21no—
125Dic188—21no—
126Dic188C22×C412yes—
127Dic188C22×C412yes—
128D376—21no—
129D376—21no—
130D376C22×C412yes—
131D376C22×C412yes—
132#10625325—22no—
133#10625325—22no—
134#10625325—22no—
135#10625325—22no—
136#10625324—44no—
137C8×D47—44no—
138C16×C47—22no—
139C16×C47C22×C414yes—
140C23×C47AGL(1,ℤ/8)11yescharacteristic
141C22×D47—41no—
142C22×D47—41no—
143C2×C4×C47—22no—
144C2×C4×C47—81no—
145C2×C4×C47C4×D411yescharacteristic
146C2×C4×C47C4×D411yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C47C22×D411yescharacteristic
151C22⋊D47—42no—
152C22⋊D47—42no—
153C22⋊D47—42no—
154C22⋊D47—42no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158D188—22no—
159D188—22no—
160D188—41no—
161D188—41no—
162C4×D47—42no—
163C4×D47—42no—
164C4×D47—42no—
165C4×D47—42no—
166C4×D47—84no—
167Dic94—22no—
168Dic94—22no—
169Dic94—41no—
170Dic94—41no—
171C8×C47—22no—
172C8×C47C4×D411yescharacteristic
173C8×C47C4×D411yescharacteristic
174C8×C47C23×C411yescharacteristic
175C8×C47C2×#4111yescharacteristic
176C47⋊C8—42no—
177#33468—471nomaximal, Sylow
178C22×C47—41no—
179C22×C47#44211yescharacteristic
180C22×C47#44211yescharacteristic
181C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
182C2×D47—42no—
183C2×D47—42no—
184C2×D47—82no—
185C2×D47—82no—
186C4×C47—42no—
187C4×C47#44211yescharacteristic
188C4×C47#44211yescharacteristic
189C4×C47C2×C4×D411yescharacteristic
190C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
191Dic47—42no—
192Dic47—42no—
193Dic47—82no—
194#3414—471no—
195C2×#444—471no—
196#2943—471no—
197AGL(1,ℤ/16)—474no—
198#2180—474no—
199#2155—471no—
200#2148—471no—
201#2145—471no—
202C2×#347D4711yescharacteristic, complemented
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47C2×#44211yescharacteristic
207D47—82no—
208D47—82no—
209C2×#135—471no—
210C2×#135—471no—
211C2×#131—941no—
212C4×#41—941no—
213#510—472no—
214#510—472no—
215#509—472no—
216#509—472no—
217#508—474no—
218C2×M32C2×D4711yescharacteristic
219C2×M32C2×D4711yescharacteristic
220#444—474no—
221#444—942no—
222#442—472no—
223#441—472no—
224#440—942no—
225C2×#108—471no—
226#427—471no—
227C2×#104—941no—
228#416—941no—
229C2×#97C2×D4711yescharacteristic
230#365—474no—
231#362—472no—
232#361—472no—
233#360—474no—
234#347C2×D4714yescomplemented
235C47#3346811yescharacteristic, complemented, Sylow
236C2×#41—941no—
237C2×#41—941no—
238#135—472no—
239#135—472no—
240#135—472no—
241#135—472no—
242#135—941no—
243#135—941no—
244#135—941no—
245#135—941no—
246C2×Q16—471no—
247C2×Q16—471no—
248C2×SD16—941no—
249C2×SD16—941no—
250C2×D8—471no—
251C2×D8—471no—
252#131—944no—
253#131—1882no—
254C2×M16—941no—
255C2×M16—1881no—
256C22×C8—1881no—
257C22×C8D18811yescharacteristic
258C22×C8C22×D4711yescharacteristic
259C4×Q8—942no—
260C4×D4—942no—
261C4×D4—1882no—
262#117—941no—
263#117—1881no—
264C2×C42—1881no—
265C2×C42D18811yescharacteristic
266Q32—474no—
267SD32—474no—
268SD32—474no—
269D16—474no—
270M32—21no—
271M32C4×D4712yescomplemented
272M32C22×D4712yes—
273M32C22×D4712yes—
274C2×C16—21no—
275C2×C16C4×D4712yescomplemented
276#108—472no—
277#108—941no—
278#107—941no—
279#106—472no—
280#104—944no—
281#103—942no—
282#102—942no—
283#97D18812yes—
284#97C22×D4712yes—
285#41—942no—
286#41—942no—
287#41—942no—
288#41—942no—
289#41—1881no—
290#41—1881no—
291#41—1881no—
292#41—1881no—
293C2×Q8—941no—
294C2×Q8—941no—
295C2×D4—941no—
296C2×D4—941no—
297C2×D4—1881no—
298C2×D4—1881no—
299C22×C4—21no—
300C22×C4—1882no—
301C22×C4—1881no—
302C22×C4—1881no—
303C22×C4C2×D18811yescharacteristic
304Q16—472no—
305Q16—472no—
306Q16—941no—
307Q16—941no—
308SD16—942no—
309SD16—942no—
310SD16—942no—
311SD16—942no—
312D8—472no—
313D8—472no—
314D8—941no—
315D8—941no—
316M16—942no—
317M16—1884no—
318M16—1881no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—1884no—
322C2×C8—1881no—
323C2×C8C2×C4×D4711yescharacteristic
324C2×C8C2×C4×D4711yescharacteristic
325C2×C8C2×D18812yes—
326C2×C8C2×D18811yescharacteristic
327C2×C8#1062535011yescharacteristic
328C2×C8#1062535011yescharacteristic
329C2×C8C23×D4711yescharacteristic
330C4⋊C4—942no—
331C4⋊C4—1882no—
332C22⋊C4—1882no—
333C22⋊C4—1881no—
334C42—21no—
335C42—1882no—
336C42—1881no—
337C42C2×D18812yes—
338C16—22no—
339C16C2×C4×D4714yes—
340C23—1881no—
341C23—1881no—
342C23#1309405211yescharacteristic
343Q8—942no—
344Q8—942no—
345Q8—1881no—
346Q8—1881no—
347D4—942no—
348D4—942no—
349D4—1882no—
350D4—1882no—
351D4—1882no—
352D4—1882no—
353D4—1881no—
354D4—1881no—
355C2×C4—22no—
356C2×C4—81no—
357C2×C4—3764no—
358C2×C4—1882no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4—1882no—
364C2×C4—1882no—
365C2×C4C4×D18811yescharacteristic
366C2×C4C4×D18811yescharacteristic
367C2×C4#1309405211yescharacteristic
368C2×C4#1309405211yescharacteristic
369C2×C4#1309405211yescharacteristic
370C2×C4C22×D18811yescharacteristic
371C8—22no—
372C8—1882no—
373C8C4×D18811yescharacteristic
374C8C4×D18811yescharacteristic
375C8C22×C4×D4711yescharacteristic
376C8C2×#1062535011yescharacteristic
377C22—41no—
378C22—3762no—
379C22—3762no—
380C22—1882no—
381C22—1882no—
382C22#1568923611yescharacteristic
383C22#1568923611yescharacteristic
384C22C2×#1309405211yescharacteristic
385C4—42no—
386C4—3762no—
387C4—1882no—
388C4—1882no—
389C4#1568923611yescharacteristic
390C4#1568923611yescharacteristic
391C4C2×C4×D18811yescharacteristic
392C4C2×#1309405211yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—3762no—
397C2—3762no—
398C2C2×#1568923611yescharacteristic
subgroupG/H
centerC4#15689236
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC2×C47×#347C2
socleC2×C47C2×#442
prime radical#20256524C2
cosocleC2×C8×C47C24
hypercenterC2×#347D47
hypocenterC47#33468
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#347

Complements

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

NH ≅ G/Ntype
2. C2×#15689246C2semidirect
3. #20252190C2semidirect
4. #20245288C2semidirect
5. #20245281C2semidirect
6. #20245161C2semidirect
7. #20245138C2semidirect
8. #20245125C2semidirect
9. C2×C47×#347C2semidirect
14. #15689566C4semidirect
16. #15689561C4semidirect
18. #15689555C4semidirect
21. #15689236C22semidirect
22. #15689231C22semidirect
28. #15688946C22semidirect
29. #15688940C22semidirect
30. #15688937C22semidirect
35. C47×#347C22semidirect
44. #13094046C2×C4semidirect
46. C2×Dic188C2×C4semidirect
50. C2×D376C2×C4semidirect
63. Dic376C2×C4semidirect
64. #13093969C2×C4semidirect
65. #13093969C2×C4semidirect
66. D752C2×C4semidirect
78. M32×C47C2×C4semidirect
82. C2×C16×C47C2×C4semidirect
202. C2×#347D47semidirect
234. #347C2×D47semidirect
235. C47#33468semidirect
271. M32C4×D47semidirect
275. C2×C16C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565242
C2×#156892462
#202521902
#202452882
#202452812
#202451612
#202451382
#202451252
C2×C47×#3472
#3346847