Finite Group Databasebeta

← #33057525

Subgroups — #33057525

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618943C211yesmaximal, characteristic
2Q8×#11924012C211yesmaximal, characteristic, complemented
3C4×#14556622C211yesmaximal, characteristic
4#22608837C211yesmaximal, characteristic, complemented
5#22608825C211yesmaximal, characteristic, complemented
6#22608799C211yesmaximal, characteristic, complemented
7#22608791C211yesmaximal, characteristic, complemented
8#22608778C211yesmaximal, characteristic, complemented
9#22608770C211yesmaximal, characteristic, complemented
10#22608746C211yesmaximal, characteristic, complemented
11#22608728C211yesmaximal, characteristic, complemented
12#22608699C211yesmaximal, characteristic, complemented
13#22608371C211yesmaximal, characteristic
14#22608234C211yesmaximal, characteristic, complemented
15C73×#1491C211yesmaximal, characteristic, complemented
16C2×#14556622C411yescharacteristic
17C4×#11924028C2211yescharacteristic
18#17988718—21no—
19#17988718—21no—
20#17988718C2211yescharacteristic
21#17988718C2211yescharacteristic
22#17988714C2211yescharacteristic
23#17988714C2211yescharacteristic
24Q8×#10596831C2211yescharacteristic
25Q8×#10596831C2211yescharacteristic
26#17988692C411yescharacteristic
27#17988692C411yescharacteristic
28#17988685C411yescharacteristic
29#17988685C411yescharacteristic
30C4×#11924018C2211yescharacteristic
31C4×#11924018C2211yescharacteristic
32C4×#11924015C2211yescharacteristic
33C4×#11924015C2211yescharacteristic
34#17988661—21no—
35#17988661—21no—
36#17988652C2211yescharacteristic
37#17988652C2211yescharacteristic
38#17988642—21no—
39#17988627C411yescharacteristic
40#17988611—21no—
41#17988611C2211yescharacteristic
42#17988596C411yescharacteristic
43#17988586C411yescharacteristic
44C4×#11924012—21no—
45C4×#11924012C2211yescharacteristic
46C4×#11924012C2211yescharacteristic
47#97×D73—21no—
48#17988339C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988337C2211yescharacteristic
52#17988319C2211yescharacteristic
53#17988319C2211yescharacteristic
54#17988316C2211yescharacteristic
55#17988316C2211yescharacteristic
56#17988309C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988308C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C2×C4×Q8×C73C2211yescharacteristic
61#432×C73C2211yescharacteristic
62C4×M16×C73C2211yescharacteristic
63Q8⋊C8×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66Q8⋊C8×C73C2211yescharacteristic
67C2×#11924028C2×C411yescharacteristic
68#14556622—24no—
69C2×#11924018C2×C411yescharacteristic
70C2×#11924018C2×C411yescharacteristic
71C2×#11924015C2×C411yescharacteristic
72C2×#11924015C2×C411yescharacteristic
73C2×#11924012—21no—
74C2×#11924012C2×C411yescharacteristic
75C2×#11924012C2×C411yescharacteristic
76C2×#11924012D411yescharacteristic
77C2×#11924012D411yescharacteristic
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80C4×Dic146C2×C411yescharacteristic
81C4×Dic146C2311yescharacteristic
82C4×D292C2×C411yescharacteristic
83C4×D292C2311yescharacteristic
84C4×C22⋊D73—21no—
85C4×C22⋊D73—21no—
86#14556550C2×C411yescharacteristic
87#14556548—21no—
88C42×D73—21no—
89#14556524C2×C411yescharacteristic
90#14556524C2×C411yescharacteristic
91#14556522C2×C411yescharacteristic
92#14556522C2×C411yescharacteristic
93C4⋊#10596831—21no—
94C4⋊#10596831—21no—
95C4⋊#10596831—21no—
96C4⋊#10596831—21no—
97C4⋊#10596831—21no—
98C4⋊#10596831C2311yescharacteristic
99C4⋊#10596831C2311yescharacteristic
100#14556520—21no—
101#14556520—21no—
102#14556515C2×C411yescharacteristic
103#14556515C2×C411yescharacteristic
104#14556515C2×C411yescharacteristic
105#14556515C2×C411yescharacteristic
106#14556514C2×C411yescharacteristic
107#14556514C2×C411yescharacteristic
108#14556511C2×C411yescharacteristic
109#14556511C2×C411yescharacteristic
110#14556503—21no—
111#14556503—21no—
112#14556502—21no—
113#14556502—21no—
114#14556502C2×C411yescharacteristic
115#14556502C2×C411yescharacteristic
116#14556502D411yescharacteristic
117#14556502D411yescharacteristic
118#14556501—21no—
119#14556501—21no—
120C4×#10596831—21no—
121C4×#10596831—21no—
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C22×Q8×C73C2×C411yescharacteristic
127C2×M16×C73—21no—
128C4×Q8×C73—21no—
129C4×Q8×C73—21no—
130C4×Q8×C73C2×C411yescharacteristic
131C4×Q8×C73C2×C411yescharacteristic
132C4×Q8×C73C2311yescharacteristic
133C4×Q8×C73C2311yescharacteristic
134C2×C4⋊C4×C73—21no—
135C2×C4⋊C4×C73C2×C411yescharacteristic
136C2×C42×C73—21no—
137C2×C42×C73C2311yescharacteristic
138C4⋊C8×C73C2311yescharacteristic
139C4⋊C8×C73C2311yescharacteristic
140C22⋊C8×C73—21no—
141#97×C73—21no—
142#97×C73—21no—
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924028—22no—
146#11924028—41no—
147C2×Dic146C22×C411yescharacteristic
148C2×D292C22×C411yescharacteristic
149C2×C22⋊D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153#11924018—22no—
154#11924018—22no—
155#11924015—22no—
156#11924015M1614yescomplemented
157#11924012—22no—
158#11924012—22no—
159#11924012—41no—
160#11924012—41no—
161#11924012—42no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831C22×C411yescharacteristic
165C2×#10596831C22×C411yescharacteristic
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C2×#10596831C2×D411yescharacteristic
169C2×#10596831C2×D411yescharacteristic
170C2×#10596831C2×D411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C8×D73—42no—
173C8×D73—42no—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176#11924007—21no—
177#11924007—21no—
178C22⋊Dic73—21no—
179#11924005—21no—
180#11924005—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—22no—
186C2×Q8×C73—41no—
187C2×Q8×C73C22×C411yescharacteristic
188C2×Q8×C73C22×C411yescharacteristic
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73#4111yescharacteristic
194M16×C73—42no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C4⋊C4×C73C22×C411yescharacteristic
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73C22×C411yescharacteristic
211C42×C73C2×D411yescharacteristic
212C42×C73#4111yescharacteristic
213C42×C73C2411yescharacteristic
214C23×C73C4×D411yescharacteristic
215C22×D73—21no—
216Q8×C73—21no—
217Q8×C73—21no—
218Q8×C73—21no—
219Q8×C73—42no—
220Q8×C73C2×M1612yes—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—42no—
231C2×C4×C73—81no—
232C2×C4×C73C4×D411yescharacteristic
233C2×C4×C73C4×D411yescharacteristic
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C22⋊D73—42no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242D292—21no—
243D292C2×M1612yes—
244C4×D73—42no—
245C4×D73—41no—
246C4×D73—41no—
247C4×D73—81no—
248Dic146—21no—
249Dic146—21no—
250C8×C73—22no—
251C8×C73—41no—
252C8×C73—41no—
253C8×C73—41no—
254#10596831—21no—
255#10596831—21no—
256#10596831—21no—
257#10596831—21no—
258#10596831—21no—
259#10596831—21no—
260#10596831—41no—
261#10596831—41no—
262#10596831C2×M1612yes—
263#10596831#13712yescomplemented
264C22×C73—21no—
265C22×C73—21no—
266C22×C73—21no—
267C22×C73C2×C4×D411yescharacteristic
268C2×D73—22no—
269C2×D73—41no—
270C2×D73—41no—
271C4×C73—21no—
272C4×C73—22no—
273C4×C73—21no—
274C4×C73—21no—
275C4×C73—21no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73C22×M1611yescharacteristic
280C4×C73#56811yescharacteristic
281C4×C73C2×#13711yescharacteristic
282C4×C73#57511yescharacteristic
283Dic73—41no—
284Dic73—41no—
285Dic73—41no—
286#18047—731nomaximal, Sylow
287C2×C73—41no—
288C2×C73D4×M1611yescharacteristic
289C2×C73C4×#13711yescharacteristic
290C2×C73#295611yescharacteristic
291D73—42no—
292#2970—731no—
293Q8×M16—731no—
294C4×#137—731no—
295#1593—731no—
296#1591—731no—
297#1587—731no—
298#1585—731no—
299#1584—731no—
300#1582—731no—
301SD16⋊C8—731no—
302C8×SD16—731no—
303#1569—731no—
304#1491D7311yescharacteristic, complemented
305#1489—731no—
306#1455—731no—
307C73#1804711yescharacteristic, complemented, Sylow
308C2×#137—731no—
309C4×#41—731no—
310C2×C4×Q8C2×D7311yescharacteristic
311#446—731no—
312#446—731no—
313#446—731no—
314#446—731no—
315#446—1461no—
316#446—1461no—
317C8×Q8—731no—
318C8×Q8—731no—
319#441—731no—
320#441—731no—
321#440—731no—
322#440—731no—
323C4×Q16—731no—
324C4×Q16—731no—
325C4×SD16—731no—
326C4×SD16—731no—
327#435—1461no—
328#435—1461no—
329C8×D4—731no—
330C8×D4—731no—
331#432—1461no—
332#432C2×D7311yescharacteristic
333#428—731no—
334#423—731no—
335#423—1461no—
336#419—731no—
337#417—731no—
338C4×M16—731no—
339C4×M16—731no—
340C4×M16—1461no—
341C4×M16C2×D7311yescharacteristic
342C2×#97—1461no—
343#335—731no—
344#335—731no—
345#334—731no—
346#334—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349Q8⋊C8C2×D7311yescharacteristic
350Q8⋊C8C2×D7311yescharacteristic
351Q8⋊C8C2×D7311yescharacteristic
352Q8⋊C8C2×D7311yescharacteristic
353D4⋊C8—731no—
354D4⋊C8—731no—
355#322—731no—
356#322—731no—
357#322—731no—
358C82—731no—
359C2×#41—731no—
360C22×Q8C4×D7311yescharacteristic
361#137—1464no—
362C2×Q16—731no—
363C2×Q16—731no—
364C2×SD16—731no—
365C2×SD16—731no—
366C2×M16—21no—
367C2×M16—731no—
368C2×M16—731no—
369C2×M16—731no—
370C2×M16—731no—
371C2×M16—1461no—
372C22×C8—1461no—
373C22×C8—1461no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—731no—
377C4×Q8—731no—
378C4×Q8C4×D7311yescharacteristic
379C4×Q8C4×D7311yescharacteristic
380C4×Q8C22×D7311yescharacteristic
381C4×Q8C22×D7311yescharacteristic
382C4×D4—731no—
383C4×D4—731no—
384C4×D4—1461no—
385C4×D4—1461no—
386#117—731no—
387#117—1461no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4C4×D7311yescharacteristic
390C2×C42—21no—
391C2×C42—1461no—
392C2×C42C22×D7311yescharacteristic
393#107—731no—
394#107—731no—
395#106—731no—
396#106—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8C22×D7311yescharacteristic
407C4⋊C8C22×D7311yescharacteristic
408#103—731no—
409#103—731no—
410#103—731no—
411#103—731no—
412#103—731no—
413#103—731no—
414#102—731no—
415#102—731no—
416C22⋊C8—21no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—21no—
420#97—21no—
421#97—731no—
422#97—731no—
423#97—731no—
424#97—731no—
425#97—1461no—
426#97—1461no—
427#97—1461no—
428#97—1461no—
429C4×C8—731no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—1461no—
434C4×C8—1461no—
435C4×C8C22×D7311yescharacteristic
436C4×C8C22×D7311yescharacteristic
437#41—2921no—
438#41—1462no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—731no—
443C2×Q8C2×C4×D7311yescharacteristic
444C2×Q8C2×C4×D7311yescharacteristic
445C2×D4—731no—
446C2×D4—1461no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4C2×C4×D7311yescharacteristic
453C22×C4D4×D7311yescharacteristic
454C22×C4#1192402711yescharacteristic
455Q16—1462no—
456Q16—1462no—
457SD16—734no—
458SD16—1462no—
459M16—42no—
460M16—2922no—
461M16—2921no—
462M16—2921no—
463M16—1462no—
464M16—1462no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—2922no—
470C2×C8—2922no—
471C2×C8—731no—
472C2×C8—731no—
473C2×C8—731no—
474C2×C8—731no—
475C2×C8—731no—
476C2×C8—731no—
477C2×C8—731no—
478C2×C8—731no—
479C2×C8—1461no—
480C2×C8—1461no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—731no—
487C4⋊C4—731no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4C2×C4×D7311yescharacteristic
491C4⋊C4C2×C4×D7311yescharacteristic
492C22⋊C4—1461no—
493C22⋊C4—1461no—
494C22⋊C4—1461no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—1461no—
500C42—1461no—
501C42—1461no—
502C42C2×C4×D7311yescharacteristic
503C42D4×D7311yescharacteristic
504C42#1192402711yescharacteristic
505C42C23×D7311yescharacteristic
506C23—1461no—
507C23#1455655111yescharacteristic
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—42no—
512Q8—1461no—
513Q8—1461no—
514Q8M16×D7312yes—
515D4—2922no—
516D4—732no—
517D4—1461no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—2922no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—5841no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4#1455655111yescharacteristic
537C2×C4#1455655111yescharacteristic
538C2×C4#1455655111yescharacteristic
539C2×C4C22×C4×D7311yescharacteristic
540C2×C4C2×D4×D7311yescharacteristic
541C2×C4C2×#1192402711yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—2921no—
547C8—2921no—
548C8—732no—
549C8—732no—
550C8—1461no—
551C8—1461no—
552C8—1461no—
553C8—1461no—
554C8—1461no—
555C8—1461no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—2921no—
560C22—2921no—
561C22—1462no—
562C22C2×#1455655111yescharacteristic
563C4—21no—
564C4—22no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—2921no—
572C4—2921no—
573C4—2921no—
574C4C2×M16×D7311yescharacteristic
575C4#1798935311yescharacteristic
576C4#137×D7311yescharacteristic
577C4#1798938611yescharacteristic
578C2—41no—
579C2—2922no—
580C2#2261866311yescharacteristic
581C2#2261874611yescharacteristic
582C2#2261883711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1491C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C2×C4
cosocleC42×C73C24
hypercenter#1491D73
hypocenterC73#18047
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
2. Q8×#11924012C2semidirect
4. #22608837C2semidirect
5. #22608825C2semidirect
6. #22608799C2semidirect
7. #22608791C2semidirect
8. #22608778C2semidirect
9. #22608770C2semidirect
10. #22608746C2semidirect
11. #22608728C2semidirect
12. #22608699C2semidirect
14. #22608234C2semidirect
15. C73×#1491C2semidirect
156. #11924015M16semidirect
263. #10596831#137semidirect
304. #1491D73semidirect
307. C73#18047semidirect

Maximal subgroups

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

subgroupindex
#226189432
Q8×#119240122
C4×#145566222
#226088372
#226088252
#226087992
#226087912
#226087782
#226087702
#226087462
#226087282
#226086992
#226083712
#226082342
C73×#14912
#1804773