Finite Group Databasebeta

← #26369731

Subgroups — #26369731

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×#15689244C211yesmaximal, characteristic, complemented
3#20252189C211yesmaximal, characteristic, complemented
4#20245289C214yesmaximal, complemented
5#20245282C214yesmaximal, complemented
6#20245162C211yesmaximal, characteristic, complemented
7#20245125C211yesmaximal, characteristic, complemented
8C2×#15688900C211yesmaximal, characteristic, complemented
9C47×#2148C211yesmaximal, characteristic, complemented
10C2×#13094046C2211yescharacteristic
11C2×#13094032—21no—
12#41×Dic47—21no—
13#15689564C2212yes—
14#15689563C412yescomplemented
15#15689559C2212yes—
16#15689558C412yescomplemented
17#15689553C414yescomplemented
18C2×#13093964C2211yescharacteristic
19#15689244—22no—
20#15689244C2214yes—
21#15689235C2212yescomplemented
22#15689230C2212yescomplemented
23#15689224—22no—
24#15689161C2211yescharacteristic
25C2×#13093950—21no—
26#15689119—21no—
27C2×#13093934C2211yescharacteristic
28#15688947C2214yescomplemented
29#15688934C2214yes—
30#15688900C2214yescomplemented
31C2×#135×C47C2211yescharacteristic
32C2×M32×C47C2211yescharacteristic
33C2×#108×C47C2211yescharacteristic
34C47×#362C2212yescomplemented
35C47×#361C2212yescomplemented
36C2×#10625350—21no—
37#13094046—21no—
38#13094046—21no—
39#13094046C2×C412yescomplemented
40#13094046C2312yes—
41C2×Dic188C2×C411yescharacteristic, complemented
42C2×#10625325—21no—
43C2×D376C2×C411yescharacteristic, complemented
44#13094032—24no—
45#13094032—42no—
46C2×#10625324—41no—
47C22×C47⋊C8D411yescharacteristic
48C2×C8×D47—41no—
49Q8×Dic47—22no—
50D4×Dic47—22no—
51D4×Dic47—42no—
52#13093983—21no—
53#13093983—41no—
54C2×C4×Dic47—41no—
55C2×C4×Dic47D411yescharacteristic
56#13093970C2×C414yescomplemented
57#13093968C2×C414yescomplemented
58#13093967C2×C414yescomplemented
59#13093965C2×C414yescomplemented
60#13093964C2×C412yescomplemented
61#13093964C2312yes—
62C2×C47⋊C16C2×C412yescomplemented
63#13093958—21no—
64#13093956C2312yes—
65#13093950—24no—
66#13093948—22no—
67#13093945—22no—
68#13093934D412yes—
69#13093934C2312yes—
70C2×#41×C47—21no—
71#135×C47—21no—
72#135×C47—21no—
73#135×C47C2×C412yes—
74#135×C47C2×C412yes—
75C2×Q16×C47C2311yescharacteristic
76C2×SD16×C47—21no—
77C2×D8×C47C2311yescharacteristic
78C2×M16×C47—21no—
79C22×C8×C47C2311yescharacteristic
80M32×C47—21no—
81M32×C47C2312yes—
82C2×C16×C47—21no—
83#108×C47—21no—
84#108×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—
96C22×Dic47—42no—
97#10625350—22no—
98#10625350—22no—
99#10625350—41no—
100#10625350—41no—
101C2×D188—21no—
102C2×C4×D47—41no—
103C2×Dic94—21no—
104Q16×C47—21no—
105Q16×C47C22×C412yes—
106SD16×C47—22no—
107SD16×C47—22no—
108D8×C47—21no—
109D8×C47C22×C412yes—
110M16×C47—22no—
111M16×C47—41no—
112C2×C8×C47—41no—
113C2×C8×C47C22×C411yescharacteristic
114C2×C8×C47C22×C411yescharacteristic
115C2×C8×C47C2×D411yescharacteristic
116C2×C8×C47#4111yescharacteristic
117C2×C8×C47#4111yescharacteristic
118C2×C8×C47C2411yescharacteristic
119C22⋊Dic47—42no—
120C22⋊Dic47—41no—
121C4⋊Dic47—22no—
122C4⋊Dic47—42no—
123C4×Dic47—21no—
124C4×Dic47—42no—
125C4×Dic47—41no—
126C4×Dic47C2×D412yes—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8C2×D412yes—
130Dic188—21no—
131Dic188C22×C412yes—
132D376—21no—
133D376C22×C412yes—
134#10625325—22no—
135#10625325—22no—
136#10625324—44no—
137C8×D47—44no—
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×C47C4×D411yescharacteristic
153C2×C4×C47C4×D411yescharacteristic
154C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C47C22×D411yescharacteristic
158C22⋊D47—42no—
159C22⋊D47—42no—
160C2×Dic47—22no—
161C2×Dic47—41no—
162C2×Dic47—42no—
163C2×Dic47—81no—
164C2×Dic47—84no—
165D188—22no—
166D188—41no—
167C4×D47—42no—
168C4×D47—42no—
169Dic94—22no—
170Dic94—41no—
171C8×C47—42no—
172C8×C47C4×D411yescharacteristic
173C8×C47C4×D411yescharacteristic
174C8×C47C23×C411yescharacteristic
175C8×C47C2×#4111yescharacteristic
176C47⋊C8—22no—
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#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#2148D4711yescharacteristic, complemented
201#2145—471no—
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×#108C2×D4711yescharacteristic
226#427—471no—
227C2×#104—941no—
228#416—941no—
229C2×#97—471no—
230#365—474no—
231#362C2×D4712yescomplemented
232#361C2×D4712yescomplemented
233#360—474no—
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×D4712yes—
245#135C4×D4712yes—
246C2×Q16—471no—
247C2×Q16C22×D4711yescharacteristic
248C2×SD16—21no—
249C2×SD16—941no—
250C2×D8—471no—
251C2×D8C22×D4711yescharacteristic
252#131—944no—
253#131—1882no—
254C2×M16—21no—
255C2×M16—1881no—
256C22×C8—471no—
257C22×C8—1881no—
258C22×C8C22×D4711yescharacteristic
259C4×Q8—942no—
260C4×D4—942no—
261C4×D4—1882no—
262#117—941no—
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—21no—
277#108C22×D4712yes—
278#107—941no—
279#106—472no—
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—22no—
317M16—41no—
318M16—1884no—
319C2×C8—41no—
320C2×C8—472no—
321C2×C8—941no—
322C2×C8—941no—
323C2×C8—1884no—
324C2×C8C2×C4×D4711yescharacteristic
325C2×C8C2×C4×D4711yescharacteristic
326C2×C8D4×D4711yescharacteristic
327C2×C8#1062535211yescharacteristic
328C2×C8#1062535211yescharacteristic
329C2×C8C23×D4711yescharacteristic
330C4⋊C4—942no—
331C4⋊C4—1882no—
332C22⋊C4—1882no—
333C22⋊C4—1881no—
334C42—472no—
335C42—941no—
336C42—1882no—
337C42—1881no—
338C16—22no—
339C16—474no—
340C23—41no—
341C23—1881no—
342C23#1309405111yescharacteristic
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—942no—
359C2×C4—3764no—
360C2×C4—3761no—
361C2×C4—1882no—
362C2×C4—1881no—
363C2×C4—1882no—
364C2×C4—1882no—
365C2×C4(C22⋊D47)⋊C411yescharacteristic
366C2×C4(C22⋊D47)⋊C411yescharacteristic
367C2×C4#1309404811yescharacteristic
368C2×C4#1309404811yescharacteristic
369C2×C4#1309405111yescharacteristic
370C2×C4C2×D4×D4711yescharacteristic
371C8—42no—
372C8—942no—
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×#1309405111yescharacteristic
385C4—42no—
386C4—3762no—
387C4—1882no—
388C4—1882no—
389C4#1568923211yescharacteristic
390C4#1568923211yescharacteristic
391C4C2×(C22⋊D47)⋊C411yescharacteristic
392C4C2×#1309404811yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—3762no—
398C2C2×#1568923211yescharacteristic
subgroupG/H
centerC4#15689232
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2148C2
socleC2×C47C2×#442
prime radical#20256522C2
cosocleC2×C8×C47C24
hypercenter#2148D47
hypocenterC47#33468
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#15689244C2semidirect
3. #20252189C2semidirect
4. #20245289C2semidirect
5. #20245282C2semidirect
6. #20245162C2semidirect
7. #20245125C2semidirect
8. C2×#15688900C2semidirect
9. C47×#2148C2semidirect
14. #15689563C4semidirect
16. #15689558C4semidirect
17. #15689553C4semidirect
21. #15689235C22semidirect
22. #15689230C22semidirect
28. #15688947C22semidirect
30. #15688900C22semidirect
34. C47×#362C22semidirect
35. C47×#361C22semidirect
39. #13094046C2×C4semidirect
41. C2×Dic188C2×C4semidirect
43. C2×D376C2×C4semidirect
56. #13093970C2×C4semidirect
57. #13093968C2×C4semidirect
58. #13093967C2×C4semidirect
59. #13093965C2×C4semidirect
60. #13093964C2×C4semidirect
62. C2×C47⋊C16C2×C4semidirect
200. #2148D47semidirect
231. #362C2×D47semidirect
232. #361C2×D47semidirect
235. C47#33468semidirect

Maximal subgroups

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

subgroupindex
#202565222
C2×#156892442
#202521892
#202452892
#202452822
#202451622
#202451252
C2×#156889002
C47×#21482
#3346847