Finite Group Databasebeta

← #34020509

Subgroups — #34020509

Order 18688.

998 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989630C212yesmaximal
2#22626503C211yesmaximal, characteristic, complemented
3#22626502C211yesmaximal, characteristic, complemented
4C4×#14556661C211yesmaximal, characteristic, complemented
5#22625734C212yesmaximal, complemented
6#22625712C212yesmaximal, complemented
7#22625710C212yesmaximal, complemented
8#22625695C211yesmaximal, characteristic, complemented
9#22625694C211yesmaximal, characteristic, complemented
10#22625587C211yesmaximal, characteristic, complemented
11C4×#14556612C211yesmaximal, characteristic
12C2×#14556679C2212yes—
13C2×#14556678C2212yes—
14#17989630C4116yescomplemented
15C2×#14556661C2212yes—
16C2×#14556656C2212yes—
17C2×#14556655C2212yes—
18#17989552—22no—
19#17989552C2212yes—
20#17989552C2212yes—
21#17989551—22no—
22C4×#11924038—21no—
23C4×#11924038C2211yescharacteristic
24C4×#11924038C2211yescharacteristic
25C4×#11924037—21no—
26#17989546—21no—
27#17989546C2211yescharacteristic
28C2×#14556646—21no—
29C2×#14556645C2212yes—
30C2×C8×#64408C2212yes—
31#17989523C2211yescharacteristic
32#17989463C2212yes—
33#17989457C2212yes—
34#17989457C2212yes—
35C2×#14556612C2211yescharacteristic
36C4×#11924029C2211yescharacteristic
37C4×#11924027C2211yescharacteristic
38C4×#11924018C411yescharacteristic, complemented
39#17988679C411yescharacteristic, complemented
40C4×#11924017C411yescharacteristic, complemented
41SD16×Dic73C411yescharacteristic, complemented
42#17988674C411yescharacteristic, complemented
43C4×#11924013C411yescharacteristic, complemented
44#17988667C411yescharacteristic, complemented
45#17988622C2211yescharacteristic
46#17988582C2211yescharacteristic
47#17988580C2211yescharacteristic
48C4×C8×D73C2211yescharacteristic
49C4×SD16×C73C411yescharacteristic, complemented
50#14556679—24no—
51#14556679C2×C418yescomplemented
52#14556678—24no—
53#14556678C2×C418yescomplemented
54C2×#11924038—22no—
55C2×#11924038C2312yes—
56C2×#11924038C2312yes—
57C2×#11924037—22no—
58C2×#11924036—22no—
59C2×#11924036C2312yes—
60C22×#10596833—22no—
61C2×C4×#64408C2312yes—
62#14556661C2×C418yescomplemented
63#14556656C2×C418yescomplemented
64#14556655C2×C418yescomplemented
65#14556646—22no—
66#14556646—22no—
67#14556646—21no—
68#14556646—21no—
69#14556646D411yescharacteristic
70#14556646D411yescharacteristic
71#14556646D412yes—
72#14556645—24no—
73C4×#10596833—22no—
74C4×#10596833—22no—
75C4×#10596833—21no—
76C4×#10596833—21no—
77C8×#64408C2×C418yescomplemented
78(C2×#64408)⋊C4—21no—
79(C2×#64408)⋊C4—21no—
80C2×#11924029C2311yescharacteristic
81C2×#11924027C2311yescharacteristic
82#14556612C2×C418yes—
83C2×#11924018C2×C411yescharacteristic
84C2×#11924017C2×C411yescharacteristic
85C2×#11924013C2×C411yescharacteristic
86C2×C8×D73C2311yescharacteristic
87C2×C8×D73C2311yescharacteristic
88Q8×Dic73C2×C411yescharacteristic
89C4×Dic146C2×C411yescharacteristic
90#14556556C2×C411yescharacteristic
91C4×D292—21no—
92C4×D292C2×C411yescharacteristic
93D4×Dic73C2×C411yescharacteristic
94#14556553—21no—
95#14556553C2×C411yescharacteristic
96C4×C22⋊D73—21no—
97#14556551—21no—
98#14556549—21no—
99#14556549C2311yescharacteristic
100#14556549C2311yescharacteristic
101#14556547—21no—
102C2×C4×Dic73—21no—
103C42×D73—21no—
104C42×D73C2311yescharacteristic
105#14556524C2×C411yescharacteristic
106#14556524C2×C411yescharacteristic
107#14556523C2×C411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556515C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114C4×#10596831C2×C411yescharacteristic
115C8×Dic73C2×C411yescharacteristic
116C8×Dic73C2×C411yescharacteristic
117C2×SD16×C73C2×C411yescharacteristic
118C4×Q8×C73C2×C411yescharacteristic
119C4×D4×C73C2×C411yescharacteristic
120#106×C73C2×C411yescharacteristic
121#103×C73C2×C411yescharacteristic
122#102×C73C2×C411yescharacteristic
123C4×C8×C73C2×C411yescharacteristic
124C22×#64408—22no—
125#11924038—22no—
126#11924038—22no—
127#11924038—28no—
128#11924038C22×C414yes—
129#11924038C22×C414yes—
130#11924037—28no—
131#11924036—28no—
132#11924036C22×C414yes—
133#11924036#4114yes—
134C2×#10596833—22no—
135C2×#10596833—22no—
136C2×#10596833—28no—
137C2×#10596833C2×D412yes—
138C2×#10596833C2×D412yes—
139C4×#64408—22no—
140C4×#64408C22×C414yes—
141#11924029—22no—
142#11924029C22×C414yes—
143#11924027—22no—
144#11924027C22×C414yes—
145C2×Dic146C22×C411yescharacteristic
146C2×D292—21no—
147C2×D292C22×C411yescharacteristic
148C2×C22⋊D73—21no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73#4111yescharacteristic
155C2×C4×D73#4111yescharacteristic
156C2×C4×D73C2411yescharacteristic
157#11924018C4214yes—
158#11924017C4214yes—
159#11924013C4214yes—
160C2×#10596831C22×C411yescharacteristic
161C2×#10596831C22×C411yescharacteristic
162C8×D73—22no—
163C8×D73C22×C414yes—
164C4⋊Dic73—21no—
165C4⋊Dic73C22×C411yescharacteristic
166C4⋊Dic73C22×C411yescharacteristic
167#11924007—21no—
168#11924007—21no—
169C22⋊Dic73—21no—
170#11924005—21no—
171#11924005—21no—
172#11924005—21no—
173#11924005—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73C22×C411yescharacteristic
180C4×Dic73C2×D411yescharacteristic
181C4×Dic73C2×D411yescharacteristic
182C2×Q8×C73C22×C411yescharacteristic
183C2×D4×C73C22×C411yescharacteristic
184C22×C4×C73—21no—
185SD16×C73C4214yes—
186C2×C8×C73C22×C411yescharacteristic
187C2×C8×C73C22×C411yescharacteristic
188C4⋊C4×C73—21no—
189C4⋊C4×C73C22×C411yescharacteristic
190C4⋊C4×C73C22×C411yescharacteristic
191C22⋊C4×C73—21no—
192C42×C73—21no—
193C42×C73C22×C411yescharacteristic
194C23×C73—21no—
195C22×D73—21no—
196C22×D73C2×#4111yescharacteristic
197C2×#64408—24no—
198C2×#64408—42no—
199C2×#64408—42no—
200Q8×C73—21no—
201Q8×C73C2×C4212yes—
202D4×C73—21no—
203D4×C73C2×C4212yes—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—81no—
208C2×C4×C73C4×D411yescharacteristic
209C2×C4×C73C4×D411yescharacteristic
210C2×C4×C73C23×C411yescharacteristic
211C22⋊D73—24no—
212C2×Dic73—21no—
213C2×Dic73—24no—
214C2×Dic73—21no—
215C2×Dic73—21no—
216C2×Dic73—81no—
217C2×Dic73C4×D411yescharacteristic
218C2×Dic73C4×D411yescharacteristic
219C2×Dic73C22×D411yescharacteristic
220D292—24no—
221D292—21no—
222D292C2×C4212yes—
223C4×D73—24no—
224C4×D73—41no—
225C4×D73—41no—
226C4×D73—81no—
227C4×D73—81no—
228C4×D73C23×C412yes—
229C4×D73C2×#4112yes—
230Dic146—21no—
231Dic146C2×C4212yes—
232#10596833—24no—
233#10596833—24no—
234#10596833C4×D414yes—
235#10596833C4×D414yes—
236C8×C73—21no—
237C8×C73C2×C4212yes—
238#10596831—21no—
239#10596831C2×C4212yes—
240C22×C73—22no—
241C22×C73—41no—
242C22×C73—41no—
243C22×C73C2×C4×D411yescharacteristic
244C2×D73—21no—
245C2×D73—21no—
246C2×D73—22no—
247C2×D73—41no—
248C2×D73—41no—
249C2×D73C4×#4112yes—
250#64408—44no—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—22no—
254C4×C73—41no—
255C4×C73—41no—
256C4×C73C22×C4211yescharacteristic
257C4×C73C4×#4111yescharacteristic
258Dic73—21no—
259Dic73—21no—
260Dic73—22no—
261Dic73—41no—
262Dic73—41no—
263Dic73C2×C4×D412yes—
264#20514—731nomaximal, Sylow
265C2×C73—42no—
266C2×C73C42×D411yescharacteristic
267C2×C73#295711yescharacteristic
268C2×C73#295711yescharacteristic
269D73—22no—
270D73—42no—
271C2×#443—732no—
272C4×#135—731no—
273#2875—731no—
274#2875—731no—
275C4×#108—731no—
276#1772—732no—
277#1765—732no—
278#1765—732no—
279#1759—731no—
280#1759—731no—
281#1726—731no—
282C73#2051411yescharacteristic, complemented, Sylow
283C2×#135—731no—
284C2×#131—732no—
285C2×#131—732no—
286C4×#41—731no—
287C4×#41—731no—
288#443—7316no—
289C4×Q16—731no—
290C4×Q16—731no—
291C4×SD16—731no—
292C4×SD16—731no—
293C4×SD16—731no—
294C4×SD16#6440811yescharacteristic, complemented
295C4×D8—731no—
296C4×D8—731no—
297C2×#108—732no—
298#427—731no—
299C2×#104—732no—
300C2×#104—732no—
301#416—731no—
302#416—731no—
303#405—732no—
304#405—732no—
305#405—1462no—
306#405—1462no—
307C4×M16—731no—
308C4×M16—731no—
309C4×M16—1461no—
310C4×M16—1461no—
311C2×#97—731no—
312C2×#97—732no—
313C2×#97—1461no—
314C2×#97—1461no—
315C2×C4×C8—732no—
316C2×C4×C8—731no—
317#376—731no—
318#341—732no—
319#339—732no—
320#339—732no—
321C2×#41—731no—
322C2×#41—731no—
323#135—738no—
324C2×Q16—731no—
325C2×SD16—731no—
326C2×SD16C2×#6440811yescharacteristic
327C2×D8—731no—
328#131—738no—
329#131—738no—
330#131—1464no—
331#131—1464no—
332C2×M16—732no—
333C2×M16—732no—
334C2×M16—1462no—
335C2×M16—1462no—
336C22×C8—732no—
337C22×C8—731no—
338C22×C8—731no—
339C22×C8—1462no—
340C22×C8—1462no—
341C4×Q8—731no—
342C4×Q8—731no—
343C4×Q8—731no—
344C4×Q8C2×#6440811yescharacteristic
345C4×D4—731no—
346C4×D4—731no—
347C4×D4—731no—
348C4×D4—1461no—
349C4×D4—1461no—
350C4×D4—1461no—
351C4×D4—1461no—
352C4×D4C2×#6440811yescharacteristic
353#117—731no—
354#117—731no—
355#117—1461no—
356#117—1461no—
357C2×C42—731no—
358C2×C42—732no—
359C2×C42—1461no—
360C2×C42—1461no—
361#108—738no—
362#107—731no—
363#107—731no—
364#106—731no—
365#106C2×#6440811yescharacteristic
366#104—738no—
367#104—738no—
368#103—731no—
369#103—731no—
370#103—731no—
371#103C2×#6440811yescharacteristic
372#102—731no—
373#102—731no—
374#102—731no—
375#102C2×#6440811yescharacteristic
376#97—731no—
377#97—731no—
378#97—732no—
379#97—1461no—
380#97—1461no—
381#97—1464no—
382#97—1462no—
383#97—1462no—
384C4×C8—738no—
385C4×C8—731no—
386C4×C8—731no—
387C4×C8—731no—
388C4×C8—1461no—
389C4×C8—1461no—
390C4×C8—1462no—
391C4×C8—1462no—
392C4×C8C2×#6440811yescharacteristic
393#95—1461no—
394#95—1461no—
395#41—734no—
396#41—734no—
397#41—1462no—
398#41—1462no—
399C2×Q8—731no—
400C2×Q8C22×#6440811yescharacteristic
401C2×D4—731no—
402C2×D4—1461no—
403C2×D4—1461no—
404C2×D4C22×#6440811yescharacteristic
405C22×C4—21no—
406C22×C4—731no—
407C22×C4—731no—
408C22×C4—731no—
409C22×C4—1461no—
410C22×C4—1461no—
411C22×C4—1462no—
412C22×C4—1461no—
413C22×C4—1461no—
414C22×C4—1461no—
415Q16—734no—
416SD16—734no—
417SD16C4×#6440814yes—
418D8—734no—
419M16—734no—
420M16—734no—
421M16—1468no—
422M16—1468no—
423M16—1462no—
424M16—1462no—
425C2×C8—732no—
426C2×C8—732no—
427C2×C8—734no—
428C2×C8—734no—
429C2×C8—731no—
430C2×C8—734no—
431C2×C8—731no—
432C2×C8—1462no—
433C2×C8—1468no—
434C2×C8—1468no—
435C2×C8—1462no—
436C2×C8—1462no—
437C2×C8C22×#6440811yescharacteristic
438C2×C8C22×#6440811yescharacteristic
439C4⋊C4—21no—
440C4⋊C4—731no—
441C4⋊C4—731no—
442C4⋊C4—1461no—
443C4⋊C4—1461no—
444C4⋊C4—1461no—
445C4⋊C4C22×#6440811yescharacteristic
446C4⋊C4C22×#6440811yescharacteristic
447C22⋊C4—21no—
448C22⋊C4—1461no—
449C22⋊C4—1461no—
450C22⋊C4—1461no—
451C22⋊C4—1461no—
452C22⋊C4—1461no—
453C42—21no—
454C42—731no—
455C42—731no—
456C42—731no—
457C42—734no—
458C42—1461no—
459C42—1461no—
460C42—1461no—
461C42—1461no—
462C42—1462no—
463C42—1461no—
464C42C22×#6440811yescharacteristic
465C23—21no—
466C23—731no—
467C23—1461no—
468Q8—21no—
469Q8—732no—
470Q8—1461no—
471Q8C2×C4×#6440812yes—
472D4—21no—
473D4—732no—
474D4—1464no—
475D4—1464no—
476D4—1461no—
477D4C2×C4×#6440812yes—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—81no—
482C2×C4—2921no—
483C2×C4—2921no—
484C2×C4—2922no—
485C2×C4—2922no—
486C2×C4—5841no—
487C2×C4—5841no—
488C2×C4—5841no—
489C2×C4—731no—
490C2×C4—732no—
491C2×C4—732no—
492C2×C4—731no—
493C2×C4—731no—
494C2×C4—1464no—
495C2×C4—1464no—
496C2×C4—1461no—
497C2×C4—1461no—
498C2×C4—1464no—
499C2×C4—1461no—
500C2×C4D4×#6440811yescharacteristic
501C2×C4D4×#6440811yescharacteristic
502C2×C4C23×#6440811yescharacteristic
503C8—21no—
504C8—734no—
505C8—734no—
506C8—732no—
507C8—1464no—
508C8—1464no—
509C8—1461no—
510C8C2×C4×#6440812yes—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—2921no—
515C22—2921no—
516C22—732no—
517C22—1462no—
518C22—1461no—
519C22—1461no—
520C22C2×D4×#6440811yescharacteristic
521C4—22no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—41no—
526C4—2924no—
527C4—2921no—
528C4—2921no—
529C4—732no—
530C4—1462no—
531C4—1461no—
532C4—1461no—
533C4C22×C4×#6440811yescharacteristic
534C4#41×#6440811yescharacteristic
535C2—42no—
536C2—2922no—
537C2—1462no—
538C2C4×D4×#6440811yescharacteristic
539C2#2262670911yescharacteristic
540C2#2262671011yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC4×C73C22×C42
FrattiniC2×C4C23×#64408
FittingC4×SD16×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#14556612C22
cosocleC2×C4×D73C24
hypercenterC4×SD16#64408
hypocenterC73#20514
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #22626503C2semidirect
3. #22626502C2semidirect
4. C4×#14556661C2semidirect
5. #22625734C2semidirect
6. #22625712C2semidirect
7. #22625710C2semidirect
8. #22625695C2semidirect
9. #22625694C2semidirect
10. #22625587C2semidirect
14. #17989630C4semidirect
38. C4×#11924018C4semidirect
39. #17988679C4semidirect
40. C4×#11924017C4semidirect
41. SD16×Dic73C4semidirect
42. #17988674C4semidirect
43. C4×#11924013C4semidirect
44. #17988667C4semidirect
49. C4×SD16×C73C4semidirect
51. #14556679C2×C4semidirect
53. #14556678C2×C4semidirect
62. #14556661C2×C4semidirect
63. #14556656C2×C4semidirect
64. #14556655C2×C4semidirect
77. C8×#64408C2×C4semidirect
282. C73#20514semidirect
294. C4×SD16#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179896302
#226265032
#226265022
C4×#145566612
#226257342
#226257122
#226257102
#226256952
#226256942
#226255872
C4×#145566122
#2051473