Finite Group Databasebeta

← #26371614

Subgroups — #26371614

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253928C211yesmaximal, characteristic, complemented
2#20253875C211yesmaximal, characteristic
3#20245628C212yesmaximal, complemented
4#20245626C212yesmaximal, complemented
5#20245621C212yesmaximal, complemented
6#20245610C212yesmaximal, complemented
7#20245103C211yesmaximal, characteristic, complemented
8#20245100C211yesmaximal, characteristic, complemented
9C2×#15688925C211yesmaximal, characteristic, complemented
10#20244984C211yesmaximal, characteristic, complemented
11C47×#2138C211yesmaximal, characteristic, complemented
12C2×#13094049—21no—
13C22×D376C2211yescharacteristic
14#15689751—21no—
15#15689687—21no—
16C2×#13093964C2211yescharacteristic
17#15689530—22no—
18#15689527C2212yescomplemented
19#15689524C2212yescomplemented
20#15689509—22no—
21#15689508C2212yes—
22#15689498C2212yes—
23#15689166—21no—
24C2×#13093958C2211yescharacteristic
25#15689078C2211yescharacteristic
26#15688927C2212yescomplemented
27#15688925C2212yes—
28#15688925C2212yes—
29#15688924C2212yescomplemented
30#15688923C2212yes—
31#15688923C2212yes—
32#15688898C2212yescomplemented
33C16×Dic47C2212yescomplemented
34C2×#135×C47C2211yescharacteristic
35C22×C16×C47C2211yescharacteristic
36C47×#427C2211yescharacteristic
37C47×#358C2211yescharacteristic, complemented
38C47×#358C2211yescharacteristic, complemented
39C47×#357C2211yescharacteristic, complemented
40C47×#357C2211yescharacteristic, complemented
41C22×D188—21no—
42#13094049—44no—
43C2×#10625336—22no—
44C2×D376—22no—
45C2×D376—22no—
46C2×D376—21no—
47C2×D376C2312yes—
48C2×#10625334—22no—
49C2×#10625329D411yescharacteristic
50#13094024—41no—
51#13094023—22no—
52#13094020—22no—
53#13094016—41no—
54#13094015—22no—
55#13094013—22no—
56#13094001—42no—
57#13093998—41no—
58#13093998—41no—
59#13093992—42no—
60#13093983D411yescharacteristic
61C2×C4⋊Dic47—21no—
62#13093964—22no—
63C2×C47⋊C16C2312yes—
64#13093958—21no—
65#13093958C2312yes—
66#13093957—22no—
67#13093955—22no—
68#13093934D412yes—
69C8×Dic47C2312yes—
70C2×#41×C47—21no—
71#135×C47—22no—
72#135×C47—22no—
73C2×Q16×C47C2311yescharacteristic
74C2×SD16×C47—21no—
75C2×D8×C47C2311yescharacteristic
76C22×C8×C47C2311yescharacteristic
77#117×C47—21no—
78C2×C16×C47—22no—
79C2×C16×C47C2311yescharacteristic
80C2×C16×C47C2311yescharacteristic
81#107×C47C2311yescharacteristic
82#107×C47C2311yescharacteristic
83#106×C47—21no—
84C23×D47—41no—
85#41×C47—42no—
86#41×C47—42no—
87C2×Q8×C47—21no—
88C2×D4×C47—21no—
89C2×D4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47C2×D411yescharacteristic
92C2×C22⋊D47—42no—
93C2×C22⋊D47—42no—
94C22×Dic47—41no—
95C2×D188—22no—
96C2×D188—42no—
97C2×D188—42no—
98C2×D188—41no—
99C2×Dic94—42no—
100Q16×C47—21no—
101Q16×C47—21no—
102SD16×C47—42no—
103D8×C47—21no—
104D8×C47—21no—
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47#4112yes—
107C2×C8×C47#4112yes—
108C2×C8×C47C2411yescharacteristic
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C22⋊C4×C47—41no—
112C42×C47—41no—
113C22⋊Dic47D812yes—
114#10625336—44no—
115#10625334—44no—
116(C2×D47)⋊C4—42no—
117(C2×D47)⋊C4—42no—
118C4⋊Dic47—22no—
119C4⋊Dic47—41no—
120C4⋊Dic47—42no—
121C4⋊Dic47D811yescharacteristic
122C4⋊Dic47D811yescharacteristic
123Dic47⋊C4—42no—
124Dic47⋊C4—42no—
125C4×Dic47C2×D412yes—
126#10625329SD1614yes—
127C2×C47⋊C8C2×D412yes—
128D376—22no—
129D376—22no—
130D376—42no—
131C16×C47—21no—
132C16×C47—21no—
133C47⋊C16—22no—
134C23×C47—41no—
135C23×C47C2×D811yescharacteristic
136C22×D47—42no—
137C22×D47—82no—
138C22×D47—161no—
139C22×D47—161no—
140Q8×C47—41no—
141Q8×C47—41no—
142D4×C47—41no—
143D4×C47—41no—
144D4×C47—82no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—82no—
149C2×C4×C47#12412yes—
150C2×C4×C47C2×D811yescharacteristic
151C2×C4×C47C2×SD1612yes—
152C2×C4×C47C22×D411yescharacteristic
153C22⋊D47—162no—
154C22⋊D47—162no—
155C2×Dic47—42no—
156C2×Dic47—161no—
157C2×Dic47C2×D812yes—
158C2×Dic47C2×D812yes—
159D188—42no—
160D188—42no—
161D188—82no—
162Dic94—162no—
163C8×C47#12412yes—
164C8×C47C2×#4111yescharacteristic
165C8×C47C2×#4111yescharacteristic
166C47⋊C8C2×SD1614yes—
167#33702—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47#48612yes—
174C22×C47C22×D811yescharacteristic
175C2×D47—82no—
176C2×D47—82no—
177C2×D47—82no—
178C2×D47—164no—
179C4×C47—81no—
180C4×C47—81no—
181C4×C47—81no—
182C4×C47#48612yes—
183C4×C47C2×#12411yescharacteristic
184C4×C47C22×SD1611yescharacteristic
185Dic47—22no—
186Dic47—22no—
187Dic47—82no—
188#3164—471no—
189#3155—471no—
190#2244—472no—
191#2243—472no—
192#2242—472no—
193#2240—472no—
194#2140—471no—
195#2139—471no—
196#2138D4711yescharacteristic, complemented
197C2×#357—471no—
198#2108—471no—
199C2×C47—22no—
200C2×C47—161no—
201C2×C47C2×#48611yescharacteristic
202C2×C47#341511yescharacteristic
203C2×C47#341611yescharacteristic
204D47—162no—
205C2×AGL(1,ℤ/8)—941no—
206C2×#135C2×D4711yescharacteristic
207C22×D8—471no—
208#544—941no—
209#535—941no—
210C2×M32—471no—
211C22×C16C2×D4711yescharacteristic
212#501—942no—
213#500—472no—
214#499—472no—
215#496—942no—
216#495—472no—
217#493—472no—
218#428—941no—
219#427C2×D4711yescharacteristic
220C2×#107—471no—
221#405—471no—
222#358—472no—
223#358C2×D4711yescharacteristic, complemented
224#358C2×D4711yescharacteristic, complemented
225#357—472no—
226#357—472no—
227#357—472no—
228#357—472no—
229#357—472no—
230#357C2×D4711yescharacteristic, complemented
231#357C2×D4711yescharacteristic, complemented
232#346—472no—
233C4×C16—472no—
234C47#3370211yescharacteristic, complemented, Sylow
235C2×#41—21no—
236C22×D4—941no—
237AGL(1,ℤ/8)—1884no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D4711yescharacteristic
241C2×SD16—21no—
242C2×SD16—942no—
243C2×D8—472no—
244C2×D8—942no—
245C2×D8—942no—
246C2×D8—941no—
247C2×D8—942no—
248C2×D8C22×D4711yescharacteristic
249C2×M16—471no—
250C22×C8C22×D4711yescharacteristic
251C4⋊Q8—942no—
252C4⋊Q8—1881no—
253#127—942no—
254#125—942no—
255#125—1881no—
256#124—942no—
257C22⋊Q8—1882no—
258#121—1881no—
259#121—1881no—
260#120—1882no—
261#117—21no—
262#117—471no—
263C2×C4⋊C4—941no—
264M32—942no—
265C2×C16—22no—
266C2×C16—472no—
267C2×C16C22×D4711yescharacteristic
268C2×C16C22×D4711yescharacteristic
269#107—472no—
270#107—942no—
271#107—941no—
272#107C22×D4711yescharacteristic
273#107C22×D4711yescharacteristic
274#106—21no—
275#106—942no—
276#97—472no—
277C4×C8—472no—
278C24—1881no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—1882no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—942no—
286C2×D4—1882no—
287C2×D4—1882no—
288C2×D4—1882no—
289C2×D4—1881no—
290C2×D4—1882no—
291C22×C4—41no—
292C22×C4—1881no—
293C22×C4D4×D4711yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—1884no—
298D8—21no—
299D8—21no—
300D8—942no—
301D8—942no—
302D8—1884no—
303D8—1882no—
304M16—474no—
305C2×C8—472no—
306C2×C8#1062535012yes—
307C2×C8D4×D4711yescharacteristic
308C2×C8#1062535212yes—
309C2×C8C23×D4711yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—471no—
313C4⋊C4—471no—
314C4⋊C4—942no—
315C4⋊C4—1882no—
316C4⋊C4—1882no—
317C4⋊C4—1882no—
318C4⋊C4—1881no—
319C22⋊C4—41no—
320C22⋊C4—472no—
321C22⋊C4—1882no—
322C22⋊C4—1882no—
323C42—41no—
324C42—472no—
325C16—21no—
326C16—21no—
327C16—942no—
328C23—41no—
329C23—3762no—
330C23—1882no—
331C23—7521no—
332C23—7521no—
333C23D8×D4711yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—7522no—
337D4—41no—
338D4—41no—
339D4—82no—
340D4—3762no—
341D4—1882no—
342D4—1882no—
343D4—7522no—
344D4—7522no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—472no—
350C2×C4—472no—
351C2×C4—1882no—
352C2×C4—7521no—
353C2×C4#1309401012yes—
354C2×C4D8×D4711yescharacteristic
355C2×C4SD16×D4712yes—
356C2×C4C2×D4×D4711yescharacteristic
357C8—474no—
358C8#1309401012yes—
359C8C2×#1062535211yescharacteristic
360C8C2×#1062535011yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—3762no—
367C22—3762no—
368C22—3762no—
369C22—7524no—
370C22#1568946812yes—
371C22C2×D8×D4711yescharacteristic
372C4—81no—
373C4—81no—
374C4—81no—
375C4—942no—
376C4—942no—
377C4—3762no—
378C4#1568946812yes—
379C4C2×#1309401011yescharacteristic
380C4C2×SD16×D4711yescharacteristic
381C2—22no—
382C2—161no—
383C2—7522no—
384C2C2×#1568946811yescharacteristic
385C2#2025652611yescharacteristic
386C2#2025653111yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2138C2
socleC22×C47C22×D8
prime radical#20253875C2
hypercenter#2138D47
hypocenterC47#33702
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #20253928C2semidirect
3. #20245628C2semidirect
4. #20245626C2semidirect
5. #20245621C2semidirect
6. #20245610C2semidirect
7. #20245103C2semidirect
8. #20245100C2semidirect
9. C2×#15688925C2semidirect
10. #20244984C2semidirect
11. C47×#2138C2semidirect
18. #15689527C22semidirect
19. #15689524C22semidirect
26. #15688927C22semidirect
29. #15688924C22semidirect
32. #15688898C22semidirect
33. C16×Dic47C22semidirect
37. C47×#358C22semidirect
38. C47×#358C22semidirect
39. C47×#357C22semidirect
40. C47×#357C22semidirect
196. #2138D47semidirect
223. #358C2×D47semidirect
224. #358C2×D47semidirect
230. #357C2×D47semidirect
231. #357C2×D47semidirect
234. C47#33702semidirect

Maximal subgroups

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

subgroupindex
#202539282
#202538752
#202456282
#202456262
#202456212
#202456102
#202451032
#202451002
C2×#156889252
#202449842
C47×#21382
#3370247