Finite Group Databasebeta

← #26373123

Subgroups — #26373123

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20255757C211yesmaximal, characteristic, complemented
2#20245619C211yesmaximal, characteristic, complemented
3#20245583C211yesmaximal, characteristic, complemented
4#20245576C211yesmaximal, characteristic, complemented
5#20245545C211yesmaximal, characteristic, complemented
6#20245482C211yesmaximal, characteristic, complemented
7#20245477C211yesmaximal, characteristic, complemented
8#20245447C211yesmaximal, characteristic, complemented
9#20245328C211yesmaximal, characteristic, complemented
10#20245317C211yesmaximal, characteristic, complemented
11C4×#13093968C211yesmaximal, characteristic, complemented
12D4×C47⋊C16C211yesmaximal, characteristic, complemented
13#20245095C211yesmaximal, characteristic, complemented
14C2×#15688924C211yesmaximal, characteristic, complemented
15C47×#3280C211yesmaximal, characteristic
16#15689813—21no—
17C2×#13093970C2211yescharacteristic, complemented
18C2×#13093968C2211yescharacteristic
19C2×#13093968C2211yescharacteristic
20C22×C47⋊C16C2211yescharacteristic
21C22×C47⋊C16C2211yescharacteristic, complemented
22#15689525C2211yescharacteristic, complemented
23#15689465—21no—
24#15689454—21no—
25#15689431—21no—
26#15689364C2211yescharacteristic
27#15689364C2211yescharacteristic, complemented
28C4×Dic188C2211yescharacteristic, complemented
29#15689161C2211yescharacteristic, complemented
30C2×#13093958C2211yescharacteristic
31#15688950C2211yescharacteristic, complemented
32#15688950C2211yescharacteristic, complemented
33#15688948C2211yescharacteristic, complemented
34C4⋊(C47⋊C16)C2211yescharacteristic, complemented
35#15688927C2211yescharacteristic, complemented
36#15688927C2211yescharacteristic, complemented
37#15688927C2211yescharacteristic, complemented
38#15688926C2211yescharacteristic, complemented
39#15688924C2211yescharacteristic
40#15688924C2211yescharacteristic
41#15688924C2211yescharacteristic
42#15688924C2211yescharacteristic
43#15688924C2211yescharacteristic
44#15688924C2211yescharacteristic
45C22⋊(C47⋊C16)C2211yescharacteristic
46C22⋊(C47⋊C16)C2211yescharacteristic, complemented
47C4×C47⋊C16C2211yescharacteristic, complemented
48C2×#135×C47C2211yescharacteristic
49C22×D8×C47C2211yescharacteristic
50C47×#546—21no—
51C47×#495C2211yescharacteristic
52C47×#467C2211yescharacteristic
53C47×#466C2211yescharacteristic
54C47×#463—21no—
55C47×#450—21no—
56C47×#449—21no—
57C4×D8×C47C2211yescharacteristic
58C8×D4×C47C2211yescharacteristic
59C2×Dic188C2311yescharacteristic
60#13094017—41no—
61#13094016—21no—
62#13094014—41no—
63#13094005—41no—
64#13094003—21no—
65#13094003—21no—
66C4×Dic94—21no—
67#13093983—21no—
68#13093980—41no—
69C2×C4⋊Dic47—21no—
70#13093970—41no—
71#13093968—22no—
72#13093968—41no—
73C2×C47⋊C16—41no—
74C2×C47⋊C16—41no—
75C2×C47⋊C16C2311yescharacteristic
76C2×C47⋊C16C2311yescharacteristic
77C2×C47⋊C16C2311yescharacteristic
78C2×C47⋊C16C2311yescharacteristic
79#13093958—21no—
80#13093958C2311yescharacteristic
81#13093958C2311yescharacteristic
82#13093958C2311yescharacteristic
83#13093956—21no—
84#13093956—21no—
85#13093949—21no—
86#13093949—21no—
87#13093949—21no—
88C2×#41×C47—21no—
89C22×D4×C47—21no—
90#135×C47—22no—
91#135×C47—41no—
92C2×Q16×C47C2311yescharacteristic
93C2×SD16×C47—21no—
94C2×SD16×C47—21no—
95C2×D8×C47—22no—
96C2×D8×C47—21no—
97C2×D8×C47—41no—
98C2×D8×C47C2311yescharacteristic
99C2×D8×C47C2311yescharacteristic
100C22×C8×C47C2311yescharacteristic
101C22×C8×C47C2311yescharacteristic
102#124×C47—21no—
103#123×C47—41no—
104C22⋊Q8×C47—21no—
105#121×C47—21no—
106#121×C47—41no—
107#120×C47—41no—
108C4×D4×C47—21no—
109C4×D4×C47D411yescharacteristic
110C2×C22⋊C4×C47—41no—
111#107×C47C2311yescharacteristic
112C4⋊C8×C47D411yescharacteristic
113#103×C47—21no—
114#102×C47—21no—
115#102×C47—21no—
116C22⋊C8×C47D411yescharacteristic
117C22⋊C8×C47D411yescharacteristic
118C4×C8×C47C2311yescharacteristic
119C24×C47—41no—
120#41×C47—42no—
121#41×C47—81no—
122C2×Q8×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—42no—
126C2×D4×C47—41no—
127C2×D4×C47—41no—
128C2×D4×C47—41no—
129C2×D4×C47—81no—
130C2×D4×C47D811yescharacteristic
131C22×C4×C47—41no—
132C22×C4×C47—41no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×Dic47—41no—
136C2×Dic94—21no—
137Q16×C47—21no—
138Q16×C47—21no—
139SD16×C47—42no—
140SD16×C47—81no—
141D8×C47—21no—
142D8×C47—21no—
143D8×C47—21no—
144D8×C47—42no—
145D8×C47#4112yes—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47C2×D411yescharacteristic
151C2×C8×C47C2×D411yescharacteristic
152C2×C8×C47C2×D411yescharacteristic
153C2×C8×C47C2411yescharacteristic
154C4⋊C4×C47—21no—
155C4⋊C4×C47—41no—
156C4⋊C4×C47D811yescharacteristic
157C22⋊C4×C47—41no—
158C22⋊C4×C47—41no—
159C22⋊C4×C47—41no—
160C22⋊C4×C47—41no—
161C22⋊C4×C47—41no—
162C22⋊C4×C47D811yescharacteristic
163C22⋊C4×C47D811yescharacteristic
164C42×C47C2×D411yescharacteristic
165C22⋊Dic47—41no—
166C22⋊Dic47—41no—
167C22⋊Dic47—41no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—41no—
172Dic47⋊C4—41no—
173Dic47⋊C4—41no—
174Dic47⋊C4—41no—
175Dic47⋊C4—41no—
176Dic47⋊C4—41no—
177C4×Dic47—41no—
178C4×Dic47—41no—
179Dic188—21no—
180Dic188—21no—
181C47⋊C16—21no—
182C47⋊C16—21no—
183C47⋊C16—21no—
184C47⋊C16—21no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47—82no—
188C23×C47—82no—
189C23×C47—161no—
190C23×C47C2×D811yescharacteristic
191C23×C47C2×D811yescharacteristic
192Q8×C47—41no—
193Q8×C47—41no—
194D4×C47—22no—
195D4×C47—22no—
196D4×C47—41no—
197D4×C47—41no—
198D4×C47—41no—
199D4×C47—82no—
200D4×C47—82no—
201D4×C47—161no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—82no—
209C2×C4×C47—161no—
210C2×C4×C47C2×D811yescharacteristic
211C2×C4×C47C2×D811yescharacteristic
212C2×C4×C47C2×D811yescharacteristic
213C2×C4×C47C2×D811yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×Dic47—41no—
216C2×Dic47—41no—
217C2×Dic47—41no—
218C2×Dic47—41no—
219C2×Dic47—82no—
220Dic94—41no—
221Dic94—41no—
222C8×C47—21no—
223C8×C47—21no—
224C8×C47—21no—
225C8×C47C2×#4111yescharacteristic
226C8×C47H(5,𝔽2)11yescharacteristic
227#33863—471nomaximal, Sylow
228C22×C47—21no—
229C22×C47—21no—
230C22×C47—21no—
231C22×C47—21no—
232C22×C47—21no—
233C22×C47—42no—
234C22×C47—81no—
235C22×C47—81no—
236C22×C47—81no—
237C22×C47—81no—
238C22×C47—81no—
239C22×C47—164no—
240C22×C47#50812yes—
241C22×C47C22×D811yescharacteristic
242C4×C47—21no—
243C4×C47—21no—
244C4×C47—21no—
245C4×C47—21no—
246C4×C47—81no—
247C4×C47—81no—
248C4×C47#54611yescharacteristic
249C4×C47#57711yescharacteristic
250Dic47—81no—
251Dic47—81no—
252Dic47—81no—
253Dic47—81no—
254#3336—471no—
255#3280D4711yescharacteristic
256#2242—471no—
257#2235—471no—
258#2234—471no—
259#2228—471no—
260#2217—471no—
261#2216—471no—
262#2211—471no—
263#2189—471no—
264#2187—471no—
265C4×SD32—471no—
266D4×C16—471no—
267#2138—471no—
268C2×#357—471no—
269C2×C47—22no—
270C2×C47—41no—
271C2×C47—82no—
272C2×C47—161no—
273C2×C47#329411yescharacteristic
274C2×C47C2×#50811yescharacteristic
275C2×C47#341611yescharacteristic
276C2×#135C2×D4711yescharacteristic
277C22×D8C2×D4711yescharacteristic
278#552—941no—
279#546—21no—
280C2×Q32—471no—
281C2×SD32—471no—
282C2×SD32—471no—
283C22×C16—471no—
284C22×C16—471no—
285#499—471no—
286#495C2×D4711yescharacteristic
287#485—941no—
288#483—941no—
289#479—941no—
290#467—471no—
291#467—471no—
292#467C2×D4711yescharacteristic
293#466C2×D4711yescharacteristic
294#463—21no—
295#450—21no—
296#449—21no—
297C4×Q16—471no—
298C4×D8C2×D4711yescharacteristic
299C8×D4C2×D4711yescharacteristic
300#427—471no—
301C2×#107—471no—
302#367—471no—
303#367—471no—
304#366—471no—
305C4⋊C16—471no—
306#358—471no—
307#358—471no—
308#358—471no—
309#358—471no—
310#357—471no—
311#357—471no—
312#357—471no—
313#357—471no—
314#357—471no—
315#357—471no—
316C22⋊C16—471no—
317C22⋊C16—471no—
318C4×C16—471no—
319C47#3386311yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—21no—
322#135—22no—
323#135—41no—
324C2×Q16—471no—
325C2×Q16C22×D4711yescharacteristic
326C2×SD16—21no—
327C2×SD16—21no—
328C2×D8—22no—
329C2×D8—21no—
330C2×D8—41no—
331C2×D8C22×D4711yescharacteristic
332C2×D8C22×D4711yescharacteristic
333C22×C8C22×D4711yescharacteristic
334C22×C8C22×D4711yescharacteristic
335#126—1881no—
336#125—941no—
337#125—1881no—
338#124—21no—
339#123—41no—
340#123—1881no—
341C22⋊Q8—21no—
342C22⋊Q8—941no—
343C22⋊Q8—941no—
344#121—21no—
345#121—41no—
346#120—41no—
347C4×Q8—941no—
348C4×D4—21no—
349C4×D4C22⋊D4711yescharacteristic
350#117—941no—
351#117—1881no—
352C2×C4⋊C4—941no—
353C2×C22⋊C4—41no—
354Q32—1881no—
355SD32—942no—
356SD32—1881no—
357C2×C16—471no—
358C2×C16—471no—
359C2×C16—471no—
360C2×C16—471no—
361C2×C16—1881no—
362C2×C16—1881no—
363#107—471no—
364#107—471no—
365#107—471no—
366#107—941no—
367#107C22×D4711yescharacteristic
368#106—941no—
369#106—941no—
370C4⋊C8C22⋊D4711yescharacteristic
371#103—21no—
372#103—941no—
373#103—941no—
374#103—941no—
375#102—21no—
376#102—21no—
377C22⋊C8C22⋊D4711yescharacteristic
378C22⋊C8C22⋊D4711yescharacteristic
379C4×C8C22×D4711yescharacteristic
380C24—41no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—941no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—81no—
392C2×D4#1062533411yescharacteristic
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—1881no—
396C22×C4C2×C22⋊D4711yescharacteristic
397C22×C4C2×C22⋊D4711yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—941no—
401Q16—941no—
402SD16—42no—
403SD16—81no—
404D8—21no—
405D8—21no—
406D8—21no—
407D8—42no—
408D8#1062535212yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D4711yescharacteristic
414C2×C8C2×C22⋊D4711yescharacteristic
415C2×C8C2×C22⋊D4711yescharacteristic
416C2×C8C23×D4711yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—941no—
420C4⋊C4—941no—
421C4⋊C4—941no—
422C4⋊C4—1881no—
423C4⋊C4—1881no—
424C4⋊C4—1881no—
425C4⋊C4—1881no—
426C4⋊C4—1881no—
427C4⋊C4—1881no—
428C4⋊C4#1062533411yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—1881no—
435C22⋊C4—1881no—
436C22⋊C4—1881no—
437C22⋊C4#1062533411yescharacteristic
438C22⋊C4#1062533411yescharacteristic
439C42—1881no—
440C42—1881no—
441C42C2×C22⋊D4711yescharacteristic
442C16—941no—
443C16—941no—
444C16—941no—
445C16—941no—
446C23—41no—
447C23—41no—
448C23—82no—
449C23—82no—
450C23—161no—
451C23C2×#1062533411yescharacteristic
452C23C2×#1062533411yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—1881no—
456Q8—1881no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—161no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—161no—
473C2×C4—3762no—
474C2×C4—1881no—
475C2×C4—1881no—
476C2×C4—1881no—
477C2×C4—1881no—
478C2×C4C2×#1062533411yescharacteristic
479C2×C4C2×#1062533411yescharacteristic
480C2×C4C2×#1062533411yescharacteristic
481C2×C4C2×#1062533411yescharacteristic
482C2×C4C22×C22⋊D4711yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8C2×#1062535211yescharacteristic
487C8#1309406911yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—164no—
500C22#1568955312yes—
501C22C22×#1062533411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—3761no—
509C4—3761no—
510C4—3761no—
511C4—3761no—
512C4#1568976711yescharacteristic
513C4#1568993411yescharacteristic
514C2—22no—
515C2—41no—
516C2—82no—
517C2—161no—
518C2#2025520111yescharacteristic
519C2C2×#1568955311yescharacteristic
520C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
Fitting, prime radicalC47×#3280C2
socleC22×C47C22×D8
hypercenter#3280D47
hypocenterC47#33863
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #20255757C2semidirect
2. #20245619C2semidirect
3. #20245583C2semidirect
4. #20245576C2semidirect
5. #20245545C2semidirect
6. #20245482C2semidirect
7. #20245477C2semidirect
8. #20245447C2semidirect
9. #20245328C2semidirect
10. #20245317C2semidirect
11. C4×#13093968C2semidirect
12. D4×C47⋊C16C2semidirect
13. #20245095C2semidirect
14. C2×#15688924C2semidirect
17. C2×#13093970C22semidirect
21. C22×C47⋊C16C22semidirect
22. #15689525C22semidirect
27. #15689364C22semidirect
28. C4×Dic188C22semidirect
29. #15689161C22semidirect
31. #15688950C22semidirect
32. #15688950C22semidirect
33. #15688948C22semidirect
34. C4⋊(C47⋊C16)C22semidirect
35. #15688927C22semidirect
36. #15688927C22semidirect
37. #15688927C22semidirect
38. #15688926C22semidirect
46. C22⋊(C47⋊C16)C22semidirect
47. C4×C47⋊C16C22semidirect
319. C47#33863semidirect

Maximal subgroups

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

subgroupindex
#202557572
#202456192
#202455832
#202455762
#202455452
#202454822
#202454772
#202454472
#202453282
#202453172
C4×#130939682
D4×C47⋊C162
#202450952
C2×#156889242
C47×#32802
#3386347