Finite Group Databasebeta

← #33057535

Subgroups — #33057535

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
2M16×Dic146C211yesmaximal, characteristic, complemented
3C4×#14556624C211yesmaximal, characteristic
4#22608841C211yesmaximal, characteristic, complemented
5#22608829C211yesmaximal, characteristic, complemented
6#22608804C211yesmaximal, characteristic, complemented
7#22608790C211yesmaximal, characteristic, complemented
8#22608783C211yesmaximal, characteristic, complemented
9#22608769C211yesmaximal, characteristic, complemented
10#22608745C211yesmaximal, characteristic, complemented
11C8×#11924017C211yesmaximal, characteristic, complemented
12#22608698C211yesmaximal, characteristic, complemented
13#22608379C211yesmaximal, characteristic, complemented
14#22608371C211yesmaximal, characteristic
15C73×#1455C211yesmaximal, characteristic, complemented
16C2×#14556624C411yescharacteristic
17C4×#11924028C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19#17988717C2211yescharacteristic
20#17988717C2211yescharacteristic
21#17988717C2211yescharacteristic
22#17988717C2211yescharacteristic
23#17988715—21no—
24#17988715—21no—
25C8×Dic146C2211yescharacteristic
26C8×Dic146C2211yescharacteristic
27#17988695C411yescharacteristic
28#17988695C411yescharacteristic
29#17988689C411yescharacteristic
30#17988689C411yescharacteristic
31C4×Dic292C2211yescharacteristic
32C4×Dic292C2211yescharacteristic
33C4×#11924017C2211yescharacteristic
34C4×#11924017C2211yescharacteristic
35#17988656—21no—
36#17988656—21no—
37C8×D292C2211yescharacteristic
38C8×D292C2211yescharacteristic
39#17988644C2211yescharacteristic
40#17988642—21no—
41#17988629C411yescharacteristic
42#17988611C2211yescharacteristic
43#17988610—21no—
44#17988597C411yescharacteristic
45#17988589C411yescharacteristic
46M16×Dic73—21no—
47#97×D73—21no—
48#17988340C2211yescharacteristic
49#17988340C2211yescharacteristic
50#17988338C2211yescharacteristic
51#17988338C2211yescharacteristic
52#17988319C2211yescharacteristic
53#17988319C2211yescharacteristic
54#17988319C2211yescharacteristic
55#17988319C2211yescharacteristic
56#17988319C2211yescharacteristic
57#17988319C2211yescharacteristic
58#17988316C2211yescharacteristic
59#17988316C2211yescharacteristic
60C4×M16×C73C2211yescharacteristic
61C4×M16×C73C2211yescharacteristic
62C4×M16×C73C2211yescharacteristic
63#322×C73C2211yescharacteristic
64#322×C73C2211yescharacteristic
65#322×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C2×#11924028C2×C411yescharacteristic
68C22×Dic146C2×C411yescharacteristic
69#14556624—24no—
70C2×Dic292C2×C411yescharacteristic
71C2×Dic292C2×C411yescharacteristic
72C2×#11924017C2×C411yescharacteristic
73C2×#11924017C2×C411yescharacteristic
74C2×#11924012—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77C4×Dic146—21no—
78C4×Dic146—21no—
79C4×Dic146C2×C411yescharacteristic
80C4×Dic146C2×C411yescharacteristic
81C4×Dic146C2×C411yescharacteristic
82C4×Dic146C2311yescharacteristic
83C4×Dic146C2311yescharacteristic
84C4×Dic146C2311yescharacteristic
85C4×D292C2×C411yescharacteristic
86C4×D292C2311yescharacteristic
87C4×C22⋊D73—21no—
88C4×C22⋊D73—21no—
89#14556550C2×C411yescharacteristic
90#14556548—21no—
91C2×C4⋊Dic73C2×C411yescharacteristic
92C2×#11924007—21no—
93C2×C4×Dic73—21no—
94C42×D73—21no—
95#14556525C2×C411yescharacteristic
96#14556525C2×C411yescharacteristic
97#14556523C2×C411yescharacteristic
98#14556523C2×C411yescharacteristic
99C4⋊#10596831—21no—
100C4⋊#10596831C2311yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103C4⋊#10596831C2311yescharacteristic
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556516C2×C411yescharacteristic
111#14556516C2×C411yescharacteristic
112#14556516C2×C411yescharacteristic
113#14556516C2×C411yescharacteristic
114#14556516C2×C411yescharacteristic
115#14556516C2×C411yescharacteristic
116#14556513C2×C411yescharacteristic
117#14556513C2×C411yescharacteristic
118C22⋊#10596831—21no—
119#14556503—21no—
120#14556503—21no—
121#14556502—21no—
122#14556501—21no—
123#14556501—21no—
124#14556501—21no—
125#14556501—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C2×M16×C73—21no—
129C2×M16×C73C2×C411yescharacteristic
130C2×M16×C73C2×C411yescharacteristic
131C2×M16×C73D411yescharacteristic
132C2×M16×C73D411yescharacteristic
133C2×C42×C73C2311yescharacteristic
134#97×C73—21no—
135#97×C73C2×C411yescharacteristic
136#97×C73C2×C411yescharacteristic
137#97×C73D411yescharacteristic
138#97×C73D411yescharacteristic
139C4×C8×C73C2311yescharacteristic
140C4×C8×C73C2311yescharacteristic
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924028—22no—
146#11924028—41no—
147C2×Dic146—22no—
148C2×Dic146—21no—
149C2×Dic146—41no—
150C2×Dic146C22×C411yescharacteristic
151C2×Dic146C22×C411yescharacteristic
152C2×Dic146C22×C411yescharacteristic
153C2×D292C22×C411yescharacteristic
154C2×C22⋊D73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160Dic292—22no—
161Dic292—22no—
162#11924017—22no—
163#11924017M1614yescomplemented
164#11924012—42no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C8×D73—42no—
170C8×D73—42no—
171C4⋊Dic73—21no—
172C4⋊Dic73C22×C411yescharacteristic
173C4⋊Dic73C22×C411yescharacteristic
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C411yescharacteristic
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182C22⋊Dic73—21no—
183#11924005—21no—
184#11924005—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73#4111yescharacteristic
195M16×C73—22no—
196M16×C73—22no—
197M16×C73—41no—
198M16×C73—41no—
199M16×C73—42no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73C22×C411yescharacteristic
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C2×C8×C73C2×D411yescharacteristic
210C42×C73C22×C411yescharacteristic
211C42×C73C2×D411yescharacteristic
212C42×C73#4111yescharacteristic
213C42×C73C2411yescharacteristic
214C23×C73C4×D411yescharacteristic
215C22×D73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—41no—
221C2×C4×C73C4×D411yescharacteristic
222C2×C4×C73C4×D411yescharacteristic
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C23×C411yescharacteristic
225C2×C4×C73C22×D411yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C22⋊D73—42no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—42no—
236C2×Dic73—81no—
237D292—21no—
238D292C2×M1612yes—
239C4×D73—42no—
240C4×D73—41no—
241C4×D73—41no—
242C4×D73—81no—
243Dic146—21no—
244Dic146—21no—
245Dic146—21no—
246Dic146—21no—
247Dic146—21no—
248Dic146—42no—
249Dic146C2×M1612yes—
250C8×C73—22no—
251C8×C73—21no—
252C8×C73—21no—
253C8×C73—21no—
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—41no—
258C8×C73C2×M1612yes—
259C8×C73#13712yescomplemented
260#10596831—41no—
261#10596831—41no—
262#10596831—41no—
263#10596831—41no—
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—21no—
273C4×C73—21no—
274C4×C73—21no—
275C4×C73C22×M1611yescharacteristic
276C4×C73#56811yescharacteristic
277C4×C73C2×#13711yescharacteristic
278C4×C73#57511yescharacteristic
279Dic73—22no—
280Dic73—41no—
281Dic73—41no—
282Dic73—41no—
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#1491—731no—
305#1489—731no—
306#1455D7311yescharacteristic, complemented
307C73#1804711yescharacteristic, complemented, Sylow
308C2×#137—731no—
309C4×#41—731no—
310C2×C4×Q8—731no—
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—731no—
332#432—1461no—
333#428—731no—
334#423—731no—
335#423—1461no—
336#419—731no—
337#417—731no—
338C4×M16—1461no—
339C4×M16C2×D7311yescharacteristic
340C4×M16C2×D7311yescharacteristic
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#322C2×D7311yescharacteristic
356#322C2×D7311yescharacteristic
357#322C2×D7311yescharacteristic
358C82C2×D7311yescharacteristic
359C2×#41—731no—
360C22×Q8—731no—
361#137—1464no—
362C2×Q16—731no—
363C2×Q16—731no—
364C2×SD16—731no—
365C2×SD16—731no—
366C2×M16—21no—
367C2×M16—1461no—
368C2×M16C4×D7311yescharacteristic
369C2×M16C4×D7311yescharacteristic
370C2×M16D29211yescharacteristic
371C2×M16D29211yescharacteristic
372C22×C8—1461no—
373C22×C8—1461no—
374C4×Q8—731no—
375C4×Q8—731no—
376C4×Q8—731no—
377C4×Q8—731no—
378C4×Q8—731no—
379C4×Q8—731no—
380C4×Q8—1461no—
381C4×Q8—1461no—
382C4×D4—731no—
383C4×D4—731no—
384C4×D4—1461no—
385C4×D4—1461no—
386#117—731no—
387#117—1461no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C42—1461no—
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—731no—
400C4⋊C8—731no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8—1461no—
407C4⋊C8—1461no—
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—1461no—
421#97—1461no—
422#97—1461no—
423#97—1461no—
424#97—1461no—
425#97C4×D7311yescharacteristic
426#97C4×D7311yescharacteristic
427#97D29211yescharacteristic
428#97D29211yescharacteristic
429C4×C8—1461no—
430C4×C8—1461no—
431C4×C8C22×D7311yescharacteristic
432C4×C8C22×D7311yescharacteristic
433C4×C8C22×D7311yescharacteristic
434C4×C8C22×D7311yescharacteristic
435C4×C8C22×D7311yescharacteristic
436C4×C8C22×D7311yescharacteristic
437#41—2921no—
438#41—1462no—
439C2×Q8—2921no—
440C2×Q8—731no—
441C2×Q8—731no—
442C2×Q8—731no—
443C2×Q8—1462no—
444C2×Q8—1461no—
445C2×D4—731no—
446C2×D4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4C2×C4×D7311yescharacteristic
453C22×C4C2×D29211yescharacteristic
454C22×C4#1192402811yescharacteristic
455Q16—1462no—
456Q16—1462no—
457SD16—734no—
458SD16—1462no—
459M16—22no—
460M16—22no—
461M16—41no—
462M16—41no—
463M16—42no—
464M16—2922no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—2922no—
468C2×C8—2922no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8C2×C4×D7311yescharacteristic
474C2×C8C2×C4×D7311yescharacteristic
475C2×C8C2×C4×D7311yescharacteristic
476C2×C8C2×C4×D7311yescharacteristic
477C2×C8C2×D29211yescharacteristic
478C2×C8C2×D29211yescharacteristic
479C2×C8C2×D29211yescharacteristic
480C2×C8C2×D29211yescharacteristic
481C4⋊C4—731no—
482C4⋊C4—731no—
483C4⋊C4—731no—
484C4⋊C4—731no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C22⋊C4—1461no—
493C22⋊C4—1461no—
494C22⋊C4—1461no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C42—1461no—
500C42—1461no—
501C42—1461no—
502C42C2×C4×D7311yescharacteristic
503C42C2×D29211yescharacteristic
504C42#1192402811yescharacteristic
505C42C23×D7311yescharacteristic
506C23—1461no—
507C23C4×D29211yescharacteristic
508Q8—2922no—
509Q8—732no—
510Q8—1461no—
511Q8—1461no—
512Q8—1461no—
513Q8—1461no—
514Q8—1461no—
515D4—2922no—
516D4—732no—
517D4—1461no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—5841no—
528C2×C4—5841no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4C4×D29211yescharacteristic
537C2×C4C4×D29211yescharacteristic
538C2×C4C4×D29211yescharacteristic
539C2×C4C22×C4×D7311yescharacteristic
540C2×C4C22×D29211yescharacteristic
541C2×C4C2×#1192402811yescharacteristic
542C8—22no—
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—2921no—
551C8—2921no—
552C8—2921no—
553C8—2921no—
554C8M16×D7312yes—
555C8#1455662412yescomplemented
556C22—21no—
557C22—21no—
558C22—21no—
559C22—2921no—
560C22—2921no—
561C22—1462no—
562C22C2×C4×D29211yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—2921no—
568C4—2921no—
569C4—2921no—
570C4—2921no—
571C4—2921no—
572C4—2921no—
573C4—1462no—
574C4C2×M16×D7311yescharacteristic
575C4#1798935311yescharacteristic
576C4C2×#1455662411yescharacteristic
577C4#1798938711yescharacteristic
578C2—41no—
579C2—2922no—
580C2M16×D29211yescharacteristic
581C2C4×#1455662411yescharacteristic
582C2#2261884211yescharacteristic
subgroupG/H
centerC2×C4C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1455C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C2×C4
cosocleC42×C73C24
hypercenter#1455D73
hypocenterC73#18047
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1455

Complements

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

NH ≅ G/Ntype
2. M16×Dic146C2semidirect
4. #22608841C2semidirect
5. #22608829C2semidirect
6. #22608804C2semidirect
7. #22608790C2semidirect
8. #22608783C2semidirect
9. #22608769C2semidirect
10. #22608745C2semidirect
11. C8×#11924017C2semidirect
12. #22608698C2semidirect
13. #22608379C2semidirect
15. C73×#1455C2semidirect
163. #11924017M16semidirect
259. C8×C73#137semidirect
306. #1455D73semidirect
307. C73#18047semidirect
555. C8#14556624semidirect

Maximal subgroups

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

subgroupindex
#226189462
M16×Dic1462
C4×#145566242
#226088412
#226088292
#226088042
#226087902
#226087832
#226087692
#226087452
C8×#119240172
#226086982
#226083792
#226083712
C73×#14552
#1804773