Finite Group Databasebeta

← #33057531

Subgroups — #33057531

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#22618946C211yesmaximal, characteristic
2C4×#14556622C211yesmaximal, characteristic
3#22608838C211yesmaximal, characteristic, complemented
4#22608826C211yesmaximal, characteristic, complemented
5#22608802C211yesmaximal, characteristic, complemented
6#22608786C211yesmaximal, characteristic, complemented
7#22608781C211yesmaximal, characteristic, complemented
8#22608765C211yesmaximal, characteristic, complemented
9#22608741C211yesmaximal, characteristic, complemented
10C8×#11924015C211yesmaximal, characteristic, complemented
11#22608696C211yesmaximal, characteristic, complemented
12#22608378C211yesmaximal, characteristic, complemented
13#22608368C211yesmaximal, characteristic
14#22608234C211yesmaximal, characteristic, complemented
15Q8×M16×C73C211yesmaximal, characteristic, complemented
16C2×#14556622C411yescharacteristic
17C4×#11924028C2211yescharacteristic
18#17988717C2211yescharacteristic
19#17988717C2211yescharacteristic
20#17988692C411yescharacteristic
21#17988692C411yescharacteristic
22#17988685C411yescharacteristic
23#17988685C411yescharacteristic
24C4×#11924018C2211yescharacteristic
25C4×#11924018C2211yescharacteristic
26C4×#11924015C2211yescharacteristic
27C4×#11924015C2211yescharacteristic
28#17988656—21no—
29#17988656—21no—
30C8×D292C2211yescharacteristic
31C8×D292C2211yescharacteristic
32#17988644C2211yescharacteristic
33#17988642—21no—
34#17988627C411yescharacteristic
35#17988596C411yescharacteristic
36#17988586C411yescharacteristic
37C4×#11924012C2211yescharacteristic
38C4×#11924012C2211yescharacteristic
39#97×D73—21no—
40#17988339C2211yescharacteristic
41#17988339C2211yescharacteristic
42#17988337C2211yescharacteristic
43#17988337C2211yescharacteristic
44Q8⋊#10596831C2211yescharacteristic
45Q8⋊#10596831C2211yescharacteristic
46Q8⋊#10596831C2211yescharacteristic
47Q8⋊#10596831C2211yescharacteristic
48#17988317C2211yescharacteristic
49#17988317C2211yescharacteristic
50#17988314C2211yescharacteristic
51#17988314C2211yescharacteristic
52#17988309C2211yescharacteristic
53#17988309C2211yescharacteristic
54#17988308C2211yescharacteristic
55C8×#10596831C2211yescharacteristic
56C2×C4×Q8×C73C2211yescharacteristic
57#446×C73—21no—
58#446×C73—21no—
59#446×C73C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C8×Q8×C73C2211yescharacteristic
62C8×Q8×C73C2211yescharacteristic
63#423×C73—21no—
64#423×C73C2211yescharacteristic
65C4×M16×C73—21no—
66C4×M16×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×C8×D73—21no—
77C2×C8×D73—21no—
78C4×Dic146C2×C411yescharacteristic
79C4×Dic146C2311yescharacteristic
80C4×D292C2×C411yescharacteristic
81C4×D292C2311yescharacteristic
82C4×C22⋊D73—21no—
83C4×C22⋊D73—21no—
84#14556550C2×C411yescharacteristic
85#14556548—21no—
86C42×D73—21no—
87#14556524C2×C411yescharacteristic
88#14556524C2×C411yescharacteristic
89#14556522C2×C411yescharacteristic
90#14556522C2×C411yescharacteristic
91C4⋊#10596831C2311yescharacteristic
92C4⋊#10596831C2311yescharacteristic
93#14556520—21no—
94#14556520—21no—
95#14556515C2×C411yescharacteristic
96#14556515C2×C411yescharacteristic
97#14556515C2×C411yescharacteristic
98#14556515C2×C411yescharacteristic
99#14556514C2×C411yescharacteristic
100#14556514C2×C411yescharacteristic
101#14556511C2×C411yescharacteristic
102#14556511C2×C411yescharacteristic
103C22⋊#10596831—21no—
104#14556503—21no—
105#14556503—21no—
106#14556502—21no—
107#14556502—21no—
108#14556502C2×C411yescharacteristic
109#14556502C2×C411yescharacteristic
110#14556501—21no—
111#14556501—21no—
112C4×#10596831C2311yescharacteristic
113C4×#10596831C2311yescharacteristic
114C4×#10596831C2311yescharacteristic
115C4×#10596831C2311yescharacteristic
116C22×Q8×C73C2×C411yescharacteristic
117C2×M16×C73—21no—
118C2×M16×C73D411yescharacteristic
119C2×M16×C73D411yescharacteristic
120C4×Q8×C73—21no—
121C4×Q8×C73—21no—
122C4×Q8×C73C2×C411yescharacteristic
123C4×Q8×C73C2×C411yescharacteristic
124C4×Q8×C73C2311yescharacteristic
125C4×Q8×C73C2311yescharacteristic
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73C2×C411yescharacteristic
128C2×C42×C73—21no—
129C2×C42×C73C2311yescharacteristic
130C4⋊C8×C73—21no—
131C4⋊C8×C73—21no—
132C4⋊C8×C73—21no—
133C4⋊C8×C73—21no—
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#97×C73—21no—
138#97×C73—21no—
139#97×C73D411yescharacteristic
140#97×C73D411yescharacteristic
141C4×C8×C73—21no—
142C4×C8×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—41no—
159#11924012—42no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831C22×C411yescharacteristic
165C2×#10596831C22×C411yescharacteristic
166C2×#10596831C22×C411yescharacteristic
167C2×#10596831C22×C411yescharacteristic
168C8×D73—42no—
169C8×D73—42no—
170C4⋊Dic73C22×C411yescharacteristic
171C4⋊Dic73C22×C411yescharacteristic
172#11924007—21no—
173#11924007—21no—
174C22⋊Dic73—21no—
175#11924005—21no—
176#11924005—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—22no—
182C2×Q8×C73—41no—
183C2×Q8×C73C22×C411yescharacteristic
184C2×Q8×C73C22×C411yescharacteristic
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73#4111yescharacteristic
190M16×C73—22no—
191M16×C73—41no—
192M16×C73—42no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73C2×D411yescharacteristic
196C2×C8×C73C2×D411yescharacteristic
197C2×C8×C73C2×D411yescharacteristic
198C2×C8×C73C2×D411yescharacteristic
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—21no—
251C8×C73—21no—
252C8×C73—21no—
253C8×C73—41no—
254C8×C73—41no—
255C8×C73#13712yescomplemented
256#10596831—22no—
257#10596831—21no—
258#10596831—21no—
259#10596831—21no—
260#10596831—41no—
261#10596831—41no—
262#10596831—41no—
263#10596831C2×M1612yes—
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×M16D7311yescharacteristic, complemented
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#1491—731no—
305#1489—731no—
306#1455—731no—
307C73#1804711yescharacteristic, complemented, Sylow
308C2×#137—731no—
309C4×#41—731no—
310C2×C4×Q8C2×D7311yescharacteristic
311#446—21no—
312#446—21no—
313#446—731no—
314#446—731no—
315#446C2×D7311yescharacteristic
316#446C2×D7311yescharacteristic
317C8×Q8C2×D7311yescharacteristic
318C8×Q8C2×D7311yescharacteristic
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—731no—
332#432—1461no—
333#428—731no—
334#423—21no—
335#423C2×D7311yescharacteristic
336#419—731no—
337#417—731no—
338C4×M16—21no—
339C4×M16—731no—
340C4×M16—731no—
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⋊C8—731no—
350Q8⋊C8—731no—
351Q8⋊C8—731no—
352Q8⋊C8—731no—
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—1461no—
370C2×M16C22⋊D7311yescharacteristic
371C2×M16C22⋊D7311yescharacteristic
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—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—731no—
403C4⋊C8—731no—
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—1461no—
417C22⋊C8—1461no—
418C22⋊C8—1461no—
419#97—21no—
420#97—21no—
421#97—731no—
422#97—731no—
423#97—1461no—
424#97—1461no—
425#97—1461no—
426#97—1461no—
427#97C22⋊D7311yescharacteristic
428#97C22⋊D7311yescharacteristic
429C4×C8—21no—
430C4×C8—21no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—731no—
434C4×C8—731no—
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×C4C2×C22⋊D7311yescharacteristic
454C22×C4#1192402811yescharacteristic
455Q16—1462no—
456Q16—1462no—
457SD16—734no—
458SD16—1462no—
459M16—22no—
460M16—41no—
461M16—42no—
462M16—2922no—
463M16—2921no—
464M16—1462no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—2922no—
468C2×C8—2922no—
469C2×C8—731no—
470C2×C8—731no—
471C2×C8—731no—
472C2×C8—731no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8C2×C22⋊D7311yescharacteristic
478C2×C8C2×C22⋊D7311yescharacteristic
479C2×C8C2×C22⋊D7311yescharacteristic
480C2×C8C2×C22⋊D7311yescharacteristic
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
503C42C2×C22⋊D7311yescharacteristic
504C42#1192402811yescharacteristic
505C42C23×D7311yescharacteristic
506C23—1461no—
507C23C4×C22⋊D7311yescharacteristic
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×C4C4×C22⋊D7311yescharacteristic
537C2×C4C4×C22⋊D7311yescharacteristic
538C2×C4C4×C22⋊D7311yescharacteristic
539C2×C4C22×C4×D7311yescharacteristic
540C2×C4C22×C22⋊D7311yescharacteristic
541C2×C4C2×#1192402811yescharacteristic
542C8—21no—
543C8—21no—
544C8—21no—
545C8—41no—
546C8—41no—
547C8—2921no—
548C8—2921no—
549C8—2921no—
550C8—732no—
551C8—1462no—
552C8—1461no—
553C8—1461no—
554C8—1461no—
555C8#1455662212yescomplemented
556C22—21no—
557C22—21no—
558C22—21no—
559C22—2921no—
560C22—2921no—
561C22—1462no—
562C22C2×C4×C22⋊D7311yescharacteristic
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
576C4C2×#1455662211yescharacteristic
577C4#1798938511yescharacteristic
578C2—41no—
579C2—2922no—
580C2M16×C22⋊D7311yescharacteristic
581C2C4×#1455662211yescharacteristic
582C2#2261883811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingQ8×M16×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C2×C4
cosocleC42×C73C24
hypercenterQ8×M16D73
hypocenterC73#18047
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. #22608838C2semidirect
4. #22608826C2semidirect
5. #22608802C2semidirect
6. #22608786C2semidirect
7. #22608781C2semidirect
8. #22608765C2semidirect
9. #22608741C2semidirect
10. C8×#11924015C2semidirect
11. #22608696C2semidirect
12. #22608378C2semidirect
14. #22608234C2semidirect
15. Q8×M16×C73C2semidirect
156. #11924015M16semidirect
255. C8×C73#137semidirect
293. Q8×M16D73semidirect
307. C73#18047semidirect
555. C8#14556622semidirect

Maximal subgroups

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

subgroupindex
#226189462
C4×#145566222
#226088382
#226088262
#226088022
#226087862
#226087812
#226087652
#226087412
C8×#119240152
#226086962
#226083782
#226083682
#226082342
Q8×M16×C732
#1804773