Finite Group Databasebeta

← #26369724

Subgroups — #26369724

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#20256522C211yesmaximal, characteristic
2C2×#15689245C211yesmaximal, characteristic, complemented
3#20252188C211yesmaximal, characteristic, complemented
4#20245283C214yesmaximal, complemented
5#20245276C214yesmaximal, complemented
6#20245160C211yesmaximal, characteristic, complemented
7#20245138C211yesmaximal, characteristic, complemented
8C2×#15688900C211yesmaximal, characteristic, complemented
9C47×#2145C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11C2×#13094031—21no—
12#15689591—21no—
13#15689564C412yescomplemented
14#15689563C2212yes—
15#15689559C412yescomplemented
16#15689558C2212yes—
17#15689553C414yescomplemented
18C2×#13093964C2211yescharacteristic
19#15689245—22no—
20#15689245C2214yes—
21#15689234C2212yescomplemented
22#15689229C2212yescomplemented
23#15689222—22no—
24C2×#13093960C2211yescharacteristic
25C2×#13093951—21no—
26#15689117—21no—
27C2×#13093934C2211yescharacteristic
28#15688945C2214yescomplemented
29#15688940C2212yescomplemented
30#15688937C2212yescomplemented
31#15688900C2214yescomplemented
32C2×#135×C47C2211yescharacteristic
33C2×M32×C47C2211yescharacteristic
34C47×#427C2211yescharacteristic
35C47×#360C2214yes—
36C2×#10625350—21no—
37#13094046—21no—
38#13094046—21no—
39#13094046C2×C412yes—
40#13094046C2×C412yes—
41C2×Dic188C2311yescharacteristic
42C2×#10625325—21no—
43C2×D376C2311yescharacteristic
44#13094031—24no—
45#13094031—42no—
46C2×#10625329—21no—
47C2×#10625329—41no—
48C22×C47⋊C8—41no—
49C22×C47⋊C8D411yescharacteristic
50#13093989—22no—
51#13093986—22no—
52(C22⋊D47)⋊C4—42no—
53#13093981—41no—
54C2×C4×Dic47D411yescharacteristic
55C42×D47—41no—
56#13093970C2×C414yescomplemented
57#13093968C2×C414yescomplemented
58#13093967C2×C414yescomplemented
59#13093965C2×C414yescomplemented
60#13093964C2×C412yescomplemented
61#13093964C2312yes—
62C2×C47⋊C16C2×C412yescomplemented
63#13093960—21no—
64#13093960C2312yes—
65#13093951—24no—
66#13093947—22no—
67#13093944—22no—
68#13093934D412yes—
69#13093934C2312yes—
70C2×#41×C47—21no—
71#135×C47—21no—
72#135×C47—21no—
73#135×C47C2×C412yescomplemented
74#135×C47C2312yes—
75C2×Q16×C47C2×C411yescharacteristic, complemented
76C2×SD16×C47—21no—
77C2×D8×C47C2×C411yescharacteristic, complemented
78C22×C8×C47C2311yescharacteristic
79#117×C47—21no—
80M32×C47—21no—
81M32×C47C2312yes—
82C2×C16×C47—21no—
83#107×C47—21no—
84#106×C47C2312yes—
85#41×C47—22no—
86#41×C47—22no—
87#41×C47—41no—
88#41×C47—41no—
89C2×Q8×C47—21no—
90C2×D4×C47—21no—
91C2×D4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47C2×D411yescharacteristic
94C2×C22⋊D47—41no—
95C22×Dic47—21no—
96#10625350—22no—
97#10625350—22no—
98#10625350—41no—
99#10625350—41no—
100C2×D188—21no—
101C2×C4×D47—41no—
102C2×C4×D47—42no—
103C2×Dic94—21no—
104Q16×C47—21no—
105Q16×C47C22×C412yes—
106SD16×C47—22no—
107SD16×C47—22no—
108D8×C47—21no—
109D8×C47C22×C412yes—
110C2×C8×C47C22×C411yescharacteristic
111C2×C8×C47C22×C411yescharacteristic
112C2×C8×C47C2×D411yescharacteristic
113C2×C8×C47#4111yescharacteristic
114C2×C8×C47#4111yescharacteristic
115C2×C8×C47C2411yescharacteristic
116C4⋊C4×C47—22no—
117C22⋊C4×C47—41no—
118C42×C47—41no—
119(C2×D47)⋊C4—42no—
120Dic47⋊C4—42no—
121C4×Dic47—21no—
122C4×Dic47—42no—
123C4×Dic47C2×D412yes—
124#10625329—22no—
125#10625329—44no—
126#10625329—41no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—44no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8C2×D412yes—
132Dic188—21no—
133Dic188C22×C412yes—
134D376—21no—
135D376C22×C412yes—
136#10625325—22no—
137#10625325—22no—
138C16×C47—22no—
139C47⋊C16C22×C414yes—
140C23×C47—41no—
141C23×C47AGL(1,ℤ/8)11yescharacteristic
142C22×D47—41no—
143Q8×C47—22no—
144Q8×C47—41no—
145D4×C47—22no—
146D4×C47—42no—
147D4×C47—42no—
148D4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47C4×D411yescharacteristic
154C2×C4×C47C4×D411yescharacteristic
155C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47C22×D411yescharacteristic
159C22⋊D47—42no—
160C22⋊D47—42no—
161C2×Dic47—22no—
162C2×Dic47—41no—
163C2×Dic47—81no—
164D188—22no—
165D188—41no—
166C4×D47—42no—
167C4×D47—42no—
168C4×D47—84no—
169Dic94—22no—
170Dic94—41no—
171C8×C47C4×D411yescharacteristic
172C8×C47C4×D411yescharacteristic
173C8×C47C23×C411yescharacteristic
174C8×C47C2×#4111yescharacteristic
175C47⋊C8—22no—
176C47⋊C8—42no—
177#33468—471nomaximal, Sylow
178C22×C47—41no—
179C22×C47—42no—
180C22×C47—82no—
181C22×C47#44211yescharacteristic
182C22×C47#44211yescharacteristic
183C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
184C2×D47—42no—
185C2×D47—82no—
186C4×C47—42no—
187C4×C47—82no—
188C4×C47#44211yescharacteristic
189C4×C47#44211yescharacteristic
190C4×C47C2×C4×D411yescharacteristic
191C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
192Dic47—42no—
193Dic47—42no—
194#3414—471no—
195C2×#444—471no—
196#2943—471no—
197AGL(1,ℤ/16)—474no—
198#2180—474no—
199#2155—471no—
200#2148—471no—
201#2145D4711yescharacteristic, complemented
202C2×#347—471no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—82no—
207C2×C47C2×#44211yescharacteristic
208D47—82no—
209C2×#135—471no—
210C2×#135C2×D4711yescharacteristic
211C2×#131—941no—
212C4×#41—941no—
213#510—472no—
214#510—472no—
215#509—472no—
216#509—472no—
217#508—474no—
218C2×M32—471no—
219C2×M32C2×D4711yescharacteristic
220#444—474no—
221#444—942no—
222#442—472no—
223#441—472no—
224#440—942no—
225C2×#108—471no—
226#427C2×D4711yescharacteristic
227C2×#104—941no—
228#416—941no—
229C2×#97—471no—
230#365—474no—
231#362—472no—
232#361—472no—
233#360C2×D4714yes—
234#347—474no—
235C47#3346811yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—941no—
238#135—21no—
239#135—21no—
240#135—472no—
241#135—472no—
242#135—941no—
243#135—941no—
244#135C4×D4712yescomplemented
245#135C22×D4712yes—
246C2×Q16—471no—
247C2×Q16C4×D4711yescharacteristic, complemented
248C2×SD16—21no—
249C2×SD16—941no—
250C2×D8—471no—
251C2×D8C4×D4711yescharacteristic, complemented
252#131—944no—
253#131—1882no—
254C2×M16—941no—
255C2×M16—1881no—
256C22×C8—471no—
257C22×C8—1881no—
258C22×C8C22×D4711yescharacteristic
259C4×Q8—942no—
260C4×D4—942no—
261C4×D4—1882no—
262#117—21no—
263#117—1881no—
264C2×C42—471no—
265C2×C42—1881no—
266Q32—474no—
267SD32—474no—
268SD32—474no—
269D16—474no—
270M32—21no—
271M32—472no—
272M32—472no—
273M32C22×D4712yes—
274C2×C16—21no—
275C2×C16—472no—
276#108—472no—
277#108—941no—
278#107—21no—
279#106C22×D4712yes—
280#104—944no—
281#103—942no—
282#102—942no—
283#97—472no—
284#97—472no—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289#41—942no—
290#41—942no—
291#41—1881no—
292#41—1881no—
293C2×Q8—21no—
294C2×Q8—941no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—941no—
298C2×D4—1881no—
299C22×C4—41no—
300C22×C4—941no—
301C22×C4—1882no—
302C22×C4—1881no—
303C22×C4D4×D4711yescharacteristic
304Q16—21no—
305Q16—472no—
306Q16—941no—
307Q16C2×C4×D4712yes—
308SD16—22no—
309SD16—22no—
310SD16—942no—
311SD16—942no—
312D8—21no—
313D8—472no—
314D8—941no—
315D8C2×C4×D4712yes—
316M16—942no—
317M16—1884no—
318M16—1881no—
319C2×C8—472no—
320C2×C8—941no—
321C2×C8—941no—
322C2×C8—1884no—
323C2×C8—1881no—
324C2×C8C2×C4×D4711yescharacteristic
325C2×C8C2×C4×D4711yescharacteristic
326C2×C8D4×D4711yescharacteristic
327C2×C8#1062535211yescharacteristic
328C2×C8#1062535211yescharacteristic
329C2×C8C23×D4711yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—1882no—
332C22⋊C4—41no—
333C22⋊C4—1882no—
334C42—41no—
335C42—472no—
336C42—941no—
337C42—1882no—
338C16—22no—
339C16—474no—
340C23—41no—
341C23—1881no—
342C23#1309404811yescharacteristic
343Q8—22no—
344Q8—41no—
345Q8—942no—
346Q8—1881no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—942no—
352D4—1882no—
353D4—1882no—
354D4—1881no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—942no—
360C2×C4—3764no—
361C2×C4—3761no—
362C2×C4—1881no—
363C2×C4—1882no—
364C2×C4—1882no—
365C2×C4(C22⋊D47)⋊C411yescharacteristic
366C2×C4(C22⋊D47)⋊C411yescharacteristic
367C2×C4#1309404811yescharacteristic
368C2×C4#1309405111yescharacteristic
369C2×C4#1309405111yescharacteristic
370C2×C4C2×D4×D4711yescharacteristic
371C8—942no—
372C8—1882no—
373C8(C22⋊D47)⋊C411yescharacteristic
374C8(C22⋊D47)⋊C411yescharacteristic
375C8C22×C4×D4711yescharacteristic
376C8C2×#1062535211yescharacteristic
377C22—41no—
378C22—42no—
379C22—82no—
380C22—3762no—
381C22—1882no—
382C22#1568923211yescharacteristic
383C22#1568923211yescharacteristic
384C22C2×#1309404811yescharacteristic
385C4—42no—
386C4—82no—
387C4—1882no—
388C4—1882no—
389C4#1568923211yescharacteristic
390C4#1568923211yescharacteristic
391C4C2×(C22⋊D47)⋊C411yescharacteristic
392C4C2×#1309405111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—3762no—
398C2C2×#1568923211yescharacteristic
subgroupG/H
centerC4#15689232
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2145C2
socleC2×C47C2×#442
prime radical#20256522C2
cosocleC2×C8×C47C24
hypercenter#2145D47
hypocenterC47#33468
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
2. C2×#15689245C2semidirect
3. #20252188C2semidirect
4. #20245283C2semidirect
5. #20245276C2semidirect
6. #20245160C2semidirect
7. #20245138C2semidirect
8. C2×#15688900C2semidirect
9. C47×#2145C2semidirect
13. #15689564C4semidirect
15. #15689559C4semidirect
17. #15689553C4semidirect
21. #15689234C22semidirect
22. #15689229C22semidirect
28. #15688945C22semidirect
29. #15688940C22semidirect
30. #15688937C22semidirect
31. #15688900C22semidirect
56. #13093970C2×C4semidirect
57. #13093968C2×C4semidirect
58. #13093967C2×C4semidirect
59. #13093965C2×C4semidirect
60. #13093964C2×C4semidirect
62. C2×C47⋊C16C2×C4semidirect
73. #135×C47C2×C4semidirect
75. C2×Q16×C47C2×C4semidirect
77. C2×D8×C47C2×C4semidirect
201. #2145D47semidirect
235. C47#33468semidirect
244. #135C4×D47semidirect
247. C2×Q16C4×D47semidirect
251. C2×D8C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202565222
C2×#156892452
#202521882
#202452832
#202452762
#202451602
#202451382
C2×#156889002
C47×#21452
#3346847