Finite Group Databasebeta

← #26195053

Subgroups — #26195053

Order 12032.

1192 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254509C211yesmaximal, characteristic, complemented
2#20253388C211yesmaximal, characteristic
3#20252860C211yesmaximal, characteristic, complemented
4#20242637C214yesmaximal, complemented
5#20242491C214yesmaximal, complemented
6#20242060C211yesmaximal, characteristic, complemented
7#20241813C211yesmaximal, characteristic, complemented
8#20241783C211yesmaximal, characteristic, complemented
9C2×D4×Q8×C47C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12#15689526C2212yescomplemented
13#15689471C2212yescomplemented
14#15689436—22no—
15#15689436—21no—
16#15689403C2212yes—
17#15689400—22no—
18#15689345—22no—
19#15689333C2212yescomplemented
20#15689323C2212yescomplemented
21#15689318C2212yes—
22#15689257—21no—
23C4×#10625334—22no—
24#15689176C2211yescharacteristic
25#15689176C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689122—21no—
28#15688869C2214yescomplemented
29Q8⋊(C47⋊C8)C2214yescomplemented
30D4⋊(C47⋊C8)C2214yes—
31C23×Q8×C47—21no—
32D4×Q8×C47—22no—
33D4×Q8×C47—22no—
34D4×Q8×C47—22no—
35D4×Q8×C47C2214yes—
36C2×C4⋊Q8×C47—21no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×C22⋊Q8×C47—21no—
39C2×C22⋊Q8×C47—22no—
40C2×C4×Q8×C47C2211yescharacteristic
41C2×C4×D4×C47—21no—
42C2×C4×D4×C47C2211yescharacteristic
43C2×#10625350—21no—
44#13094055—44no—
45C2×#10625337—22no—
46C2×#10625336—22no—
47C2×#10625334—22no—
48C2×#10625329—21no—
49#13094024C2311yescharacteristic
50#13094021C2311yescharacteristic
51#13094012—21no—
52#13093998—21no—
53#13093998—21no—
54C4×D188—21no—
55C4×D188—21no—
56#13093957—24no—
57#13093957—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093947—22no—
63#13093945—22no—
64#13093944—22no—
65#13093944—22no—
66#13093944—24no—
67C22⋊(C47⋊C8)—21no—
68C22⋊(C47⋊C8)—21no—
69C4×C47⋊C8—21no—
70C4×C47⋊C8C2312yes—
71C22×Q8×C47—22no—
72C22×Q8×C47—24no—
73C22×Q8×C47—24no—
74C22×Q8×C47—21no—
75C22×Q8×C47—44no—
76C22×Q8×C47—44no—
77C22×Q8×C47D411yescharacteristic
78C22×D4×C47D411yescharacteristic
79C23×C4×C47—22no—
80C23×C4×C47—21no—
81C4⋊Q8×C47—24no—
82C4⋊Q8×C47—24no—
83C4⋊Q8×C47—21no—
84C4⋊Q8×C47—21no—
85C4⋊Q8×C47D412yes—
86C4⋊Q8×C47C2312yes—
87C22⋊Q8×C47—24no—
88C22⋊Q8×C47—28no—
89C22⋊Q8×C47—24no—
90C22⋊Q8×C47—28no—
91C4×Q8×C47—21no—
92C4×Q8×C47—21no—
93C4×Q8×C47D412yes—
94C4×Q8×C47C2312yes—
95C4×D4×C47—24no—
96C4×D4×C47—24no—
97C4×D4×C47—21no—
98C4×D4×C47—21no—
99C4×D4×C47D412yes—
100C4×D4×C47C2312yes—
101C2×C4⋊C4×C47—22no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C4⋊C4×C47D412yes—
107C2×C22⋊C4×C47—22no—
108C2×C22⋊C4×C47—21no—
109C2×C42×C47—21no—
110C2×C42×C47C2311yescharacteristic
111C24×C47—21no—
112C2×Q8×C47—28no—
113C2×Q8×C47—21no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47—48no—
118C2×Q8×C47—48no—
119C2×Q8×C47—416no—
120C2×Q8×C47—48no—
121C2×Q8×C47—42no—
122C2×Q8×C47—42no—
123C2×Q8×C47D814yescomplemented
124C2×Q8×C47C2×D412yes—
125C2×D4×C47—24no—
126C2×D4×C47—21no—
127C2×D4×C47C2×D412yes—
128C22×C4×C47—21no—
129C22×C4×C47—28no—
130C22×C4×C47—21no—
131C22×C4×C47—22no—
132C22×C4×C47—21no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—48no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C2×C22⋊D47—41no—
142#10625350—42no—
143#10625350—42no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×C4×D47—41no—
148C2×Dic94—21no—
149C4⋊C4×C47—28no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47C2×D412yes—
158C4⋊C4×C47C2×D412yes—
159C4⋊C4×C47C2×D414yes—
160C22⋊C4×C47—28no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C42×C47—22no—
164C42×C47—22no—
165C42×C47C2×D411yescharacteristic
166C42×C47C2×D411yescharacteristic
167C42×C47C2×D411yescharacteristic
168C42×C47C2411yescharacteristic
169#10625337—44no—
170#10625336—44no—
171#10625334—44no—
172(C2×D47)⋊C4—41no—
173(C2×D47)⋊C4—41no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176#10625329—42no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C23×C47—24no—
181C23×C47—22no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47#12011yescharacteristic
185C22×D47—41no—
186Q8×C47—22no—
187Q8×C47—24no—
188Q8×C47—416no—
189Q8×C47—42no—
190Q8×C47—42no—
191Q8×C47—44no—
192Q8×C47C2×D814yes—
193D4×C47—24no—
194D4×C47—22no—
195D4×C47#13714yescomplemented
196C2×C4×C47—22no—
197C2×C4×C47—24no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—24no—
201C2×C4×C47—22no—
202C2×C4×C47—41no—
203C2×C4×C47—416no—
204C2×C4×C47—42no—
205C2×C4×C47—44no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—44no—
210C2×C4×C47#12011yescharacteristic
211C2×C4×C47#12011yescharacteristic
212C2×C4×C47#12011yescharacteristic
213C2×C4×C47C2×D812yes—
214C2×C4×C47AGL(1,ℤ/8)12yes—
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C22⋊D47—82no—
219C2×Dic47—41no—
220D188—42no—
221D188—41no—
222D188—41no—
223D188—42no—
224C4×D47—82no—
225Dic94—41no—
226Dic94—41no—
227C47⋊C8—42no—
228C47⋊C8—42no—
229C47⋊C8—42no—
230#18776—471nomaximal, Sylow
231C22×C47—21no—
232C22×C47—21no—
233C22×C47—24no—
234C22×C47—48no—
235C22×C47—42no—
236C22×C47—42no—
237C22×C47#44712yes—
238C22×C47C2×#12011yescharacteristic
239C2×D47—81no—
240C2×D47—81no—
241C2×D47—81no—
242C4×C47—24no—
243C4×C47—21no—
244C4×C47—21no—
245C4×C47—21no—
246C4×C47—44no—
247C4×C47—42no—
248C4×C47—42no—
249C4×C47#44712yes—
250C4×C47C22×D811yescharacteristic
251C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C47C2×#13712yes—
253Dic47—81no—
254C2×D4×Q8D4711yescharacteristic, complemented
255#3226—471no—
256#3100—471no—
257#3037—471no—
258#1649—474no—
259#1627—474no—
260#1546—471no—
261#1486—471no—
262#1479—471no—
263C2×C47—22no—
264C2×C47—44no—
265C2×C47C2×#44711yescharacteristic
266C2×C47#300411yescharacteristic
267C2×C47#302011yescharacteristic
268D47—161no—
269C23×Q8—21no—
270C2×#137—941no—
271D4×Q8—22no—
272D4×Q8—22no—
273D4×Q8—22no—
274D4×Q8C2×D4714yes—
275#532—471no—
276C2×C4⋊Q8—21no—
277C2×C4⋊Q8C2×D4711yescharacteristic
278C2×C22⋊Q8—21no—
279C2×C22⋊Q8—22no—
280C2×C4×Q8C2×D4711yescharacteristic
281C2×C4×D4—21no—
282C2×C4×D4C2×D4711yescharacteristic
283#500—472no—
284#486—472no—
285#480—941no—
286#480—942no—
287#474—472no—
288#474—942no—
289#464—942no—
290#462—472no—
291#460—472no—
292#459—472no—
293#447—941no—
294C4×D8—942no—
295#431—471no—
296#431—471no—
297#423—471no—
298#417—941no—
299#332—474no—
300Q8⋊C8—474no—
301D4⋊C8—474no—
302C47#1877611yescharacteristic, complemented, Sylow
303C2×#41—941no—
304C22×Q8—22no—
305C22×Q8—24no—
306C22×Q8—24no—
307C22×Q8—21no—
308C22×Q8—44no—
309C22×Q8—44no—
310C22×Q8C22⋊D4711yescharacteristic
311C22×D4C22⋊D4711yescharacteristic
312C23×C4—22no—
313C23×C4—21no—
314#137—1884no—
315C2×Q16—942no—
316C2×SD16—942no—
317C2×D8—942no—
318C2×M16—941no—
319C4⋊Q8—24no—
320C4⋊Q8—24no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—471no—
324C4⋊Q8C22⋊D4712yes—
325C4⋊Q8C22×D4712yes—
326#127—471no—
327#124—941no—
328C22⋊Q8—24no—
329C22⋊Q8—28no—
330C22⋊Q8—24no—
331C22⋊Q8—28no—
332#121—941no—
333#121—941no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8C22⋊D4712yes—
337C4×Q8C22×D4712yes—
338C4×D4—24no—
339C4×D4—24no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—941no—
343C4×D4—941no—
344C4×D4C22⋊D4712yes—
345C4×D4C22×D4712yes—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D4711yescharacteristic
350C2×C4⋊C4C22⋊D4711yescharacteristic
351C2×C4⋊C4C22⋊D4712yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C42—21no—
355C2×C42C22×D4711yescharacteristic
356#107—942no—
357#107—944no—
358C4⋊C8—472no—
359C4⋊C8—472no—
360C4⋊C8—941no—
361C4⋊C8—941no—
362#103—942no—
363#102—942no—
364#102—942no—
365#102—942no—
366#102—944no—
367C22⋊C8—941no—
368C22⋊C8—941no—
369C4×C8—472no—
370C4×C8—941no—
371C24—21no—
372#41—1882no—
373#41—1882no—
374C2×Q8—28no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—48no—
380C2×Q8—416no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—941no—
386C2×Q8#1062533414yescomplemented
387C2×Q8C2×C22⋊D4712yes—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—941no—
391C2×D4—941no—
392C2×D4—941no—
393C2×D4—1881no—
394C2×D4C2×C22⋊D4712yes—
395C22×C4—21no—
396C22×C4—28no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—1881no—
406C22×C4C2×C22⋊D4711yescharacteristic
407C22×C4C2×C22⋊D4711yescharacteristic
408C22×C4C2×C22⋊D4711yescharacteristic
409Q16—1884no—
410SD16—1884no—
411D8—1884no—
412M16—1882no—
413C2×C8—942no—
414C2×C8—942no—
415C2×C8—942no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—941no—
425C4⋊C4—941no—
426C4⋊C4C2×C22⋊D4712yes—
427C4⋊C4C2×C22⋊D4712yes—
428C4⋊C4C2×C22⋊D4714yes—
429C22⋊C4—28no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—1881no—
433C22⋊C4—1881no—
434C42—22no—
435C42—22no—
436C42C2×C22⋊D4711yescharacteristic
437C42C2×C22⋊D4711yescharacteristic
438C42C2×C22⋊D4711yescharacteristic
439C42C23×D4711yescharacteristic
440C23—24no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23—1881no—
445C23#1309399311yescharacteristic
446Q8—24no—
447Q8—22no—
448Q8—416no—
449Q8—44no—
450Q8—42no—
451Q8—42no—
452Q8—1881no—
453Q8—1881no—
454Q8C2×#1062533414yes—
455D4—24no—
456D4—22no—
457D4—3762no—
458D4—1882no—
459D4—1881no—
460D4—1881no—
461D4—1882no—
462D4#1309405514yescomplemented
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—416no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—3762no—
478C2×C4—1881no—
479C2×C4#1309399311yescharacteristic
480C2×C4#1309399311yescharacteristic
481C2×C4#1309399311yescharacteristic
482C2×C4C2×#1062533412yes—
483C2×C4#1309405012yes—
484C2×C4C22×C22⋊D4711yescharacteristic
485C2×C4C22×C22⋊D4711yescharacteristic
486C2×C4C22×C22⋊D4711yescharacteristic
487C8—1882no—
488C8—1882no—
489C8—1882no—
490C22—21no—
491C22—21no—
492C22—24no—
493C22—48no—
494C22—42no—
495C22—42no—
496C22—3761no—
497C22—3761no—
498C22—3761no—
499C22#1568925712yes—
500C22C2×#1309399311yescharacteristic
501C4—21no—
502C4—21no—
503C4—24no—
504C4—21no—
505C4—42no—
506C4—44no—
507C4—42no—
508C4—3761no—
509C4#1568925712yes—
510C4C22×#1062533411yescharacteristic
511C4C2×#1309405011yescharacteristic
512C4C2×#1309405512yes—
513C2—22no—
514C2—44no—
515C2—7521no—
516C2C2×#1568925711yescharacteristic
517C2#2025262211yescharacteristic
518C2#2025275211yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×D4×Q8×C47C2
socleC22×C47C2×#120
prime radical#20253388C2
hypercenterC2×D4×Q8D47
hypocenterC47#18776
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20254509C2semidirect
3. #20252860C2semidirect
4. #20242637C2semidirect
5. #20242491C2semidirect
6. #20242060C2semidirect
7. #20241813C2semidirect
8. #20241783C2semidirect
9. C2×D4×Q8×C47C2semidirect
12. #15689526C22semidirect
13. #15689471C22semidirect
19. #15689333C22semidirect
20. #15689323C22semidirect
28. #15688869C22semidirect
29. Q8⋊(C47⋊C8)C22semidirect
123. C2×Q8×C47D8semidirect
195. D4×C47#137semidirect
254. C2×D4×Q8D47semidirect
302. C47#18776semidirect
386. C2×Q8#10625334semidirect
462. D4#13094055semidirect

Maximal subgroups

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

subgroupindex
#202545092
#202533882
#202528602
#202426372
#202424912
#202420602
#202418132
#202417832
C2×D4×Q8×C472
#1877647