Finite Group Databasebeta

← #34019332

Subgroups — #34019332

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626840C211yesmaximal, characteristic, complemented
2#22626787C211yesmaximal, characteristic
3#22625574C211yesmaximal, characteristic, complemented
4#22625568C211yesmaximal, characteristic, complemented
5#22625560C211yesmaximal, characteristic, complemented
6#22625546C211yesmaximal, characteristic, complemented
7#22625541C211yesmaximal, characteristic, complemented
8#22625536C211yesmaximal, characteristic, complemented
9#22625522C211yesmaximal, characteristic, complemented
10SD16×#10596833C211yesmaximal, characteristic, complemented
11#22625502C211yesmaximal, characteristic, complemented
12#22625293C211yesmaximal, characteristic, complemented
13#22625230C211yesmaximal, characteristic
14#22625152C211yesmaximal, characteristic, complemented
15#22619934C211yesmaximal, characteristic
16#17989639—21no—
17#17989638C2211yescharacteristic, complemented
18#17989637C2211yescharacteristic, complemented
19#17989636—21no—
20#17989636C2211yescharacteristic, complemented
21Q8×#10596833C2211yescharacteristic, complemented
22#17989622C2211yescharacteristic
23#17989622C2211yescharacteristic
24#17989618—21no—
25#17989618—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989611—21no—
29#17989611—21no—
30#17989608C2211yescharacteristic
31#17989604C2211yescharacteristic
32#17989603C2211yescharacteristic
33#17989603C2211yescharacteristic
34#17989557—21no—
35C2×C22⋊#10596833—21no—
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40#17989437C2211yescharacteristic, complemented
41#17989437C2211yescharacteristic, complemented
42Q8⋊#10596833C2211yescharacteristic, complemented
43Q8⋊#10596833C2211yescharacteristic, complemented
44D4⋊#10596833C2211yescharacteristic
45D4⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50#17989428C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52#17989139C2211yescharacteristic
53C2×D4×Dic73C2211yescharacteristic
54#17988918C411yescharacteristic
55#17988915C411yescharacteristic
56#17988875C411yescharacteristic
57#17988875C411yescharacteristic
58#17988790C411yescharacteristic
59#17988790C411yescharacteristic
60#17988738C411yescharacteristic
61#17988678C2211yescharacteristic
62#17988678C2211yescharacteristic
63SD16×Dic73C2211yescharacteristic
64SD16×Dic73C2211yescharacteristic
65#17988637C2211yescharacteristic
66#450×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C22×#10596833—21no—
70C22×#10596833—21no—
71#14556658—21no—
72#14556658—21no—
73#14556658D411yescharacteristic
74#14556658D411yescharacteristic
75C4⋊#10596833—21no—
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833C2311yescharacteristic
79C4⋊#10596833C2311yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C22⋊#10596833—21no—
83C22⋊#10596833—21no—
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833—21no—
89C22⋊#10596833D411yescharacteristic
90C22⋊#10596833D411yescharacteristic
91#14556646—21no—
92#14556646—21no—
93#14556646—21no—
94C4×#10596833—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C23×Dic73—21no—
102C2×#11924016C2×C411yescharacteristic
103C2×#11924016C2×C411yescharacteristic
104#14556590C2×C411yescharacteristic
105#14556589C2×C411yescharacteristic
106#14556581—21no—
107#14556570C2×C411yescharacteristic
108#14556568—21no—
109Q8×Dic73C2311yescharacteristic
110#14556556—21no—
111#14556556C2311yescharacteristic
112D4×Dic73—21no—
113D4×Dic73—21no—
114D4×Dic73C2×C411yescharacteristic
115D4×Dic73C2×C411yescharacteristic
116D4×Dic73C2311yescharacteristic
117D4×Dic73C2311yescharacteristic
118#14556547—21no—
119C2×C4⋊Dic73C2×C411yescharacteristic
120C2×C22⋊Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556523C2×C411yescharacteristic
123#14556523C2×C411yescharacteristic
124#14556522C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556516C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C22×D4×C73C2×C411yescharacteristic
138C2×SD16×C73C2×C411yescharacteristic
139C2×SD16×C73C2×C411yescharacteristic
140C22⋊Q8×C73C2×C411yescharacteristic
141#102×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924037—81no—
145#11924037—81no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—81no—
153C2×#10596833—81no—
154C2×#10596833C2×D411yescharacteristic
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×Dic146—21no—
159C2×Dic146C22×C411yescharacteristic
160C22×Dic73—21no—
161C22×Dic73—21no—
162C22×Dic73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73—81no—
166C22×Dic73—81no—
167C22×Dic73C2×D411yescharacteristic
168C22×Dic73#4111yescharacteristic
169#11924016—22no—
170#11924016—41no—
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C22×C411yescharacteristic
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178#11924007—21no—
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183C22⋊Dic73—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186C22⋊Dic73—21no—
187C22⋊Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73C22×C411yescharacteristic
191C4×Dic73C2×D411yescharacteristic
192C4×Dic73#4111yescharacteristic
193C4×Dic73C2411yescharacteristic
194C24×C73—21no—
195C2×Q8×C73C22×C411yescharacteristic
196C2×D4×C73—22no—
197C2×D4×C73—21no—
198C2×D4×C73—41no—
199C2×D4×C73C22×C411yescharacteristic
200C2×D4×C73C22×C411yescharacteristic
201C22×C4×C73C22×C411yescharacteristic
202SD16×C73—22no—
203SD16×C73—41no—
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C4⋊C4×C73—21no—
207C4⋊C4×C73C22×C411yescharacteristic
208C22⋊C4×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—42no—
212C23×C73—42no—
213C23×C73—81no—
214C23×C73C4×D411yescharacteristic
215Q8×C73—21no—
216Q8×C73—21no—
217D4×C73—21no—
218D4×C73—21no—
219D4×C73—21no—
220D4×C73—42no—
221D4×C73#13112yes—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73C4×D411yescharacteristic
227C2×C4×C73C23×C411yescharacteristic
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—81no—
240C2×Dic73—81no—
241C2×Dic73—81no—
242C2×Dic73C4×D411yescharacteristic
243C2×Dic73C4×D411yescharacteristic
244C2×Dic73C22×D411yescharacteristic
245C2×Dic73C2×#4111yescharacteristic
246Dic146—21no—
247Dic146—21no—
248Dic146—42no—
249#10596833—21no—
250#10596833—21no—
251#10596833—21no—
252#10596833—21no—
253#10596833—41no—
254#10596833—41no—
255#10596833—41no—
256#10596833—41no—
257#10596833—41no—
258#10596833—41no—
259C8×C73—21no—
260C8×C73—21no—
261#10596831—21no—
262#10596831—21no—
263C22×C73—21no—
264C22×C73—22no—
265C22×C73—21no—
266C22×C73—21no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—84no—
273C22×C73C2×C4×D411yescharacteristic
274C4×C73—21no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73C2×#13111yescharacteristic
278C4×C73#56811yescharacteristic
279Dic73—21no—
280Dic73—21no—
281Dic73—21no—
282Dic73—41no—
283Dic73—41no—
284Dic73—41no—
285Dic73—41no—
286Dic73#57711yescharacteristic
287Dic73#57711yescharacteristic
288#18028—731nomaximal, Sylow
289C2×C73—41no—
290C2×C73—42no—
291C2×C73—81no—
292C2×C73#293311yescharacteristic
293C2×C73#294911yescharacteristic
294C2×C73#295511yescharacteristic
295#3101—731no—
296#2986—731no—
297#2974—731no—
298#1592—731no—
299#1590—731no—
300#1588—731no—
301#1585—731no—
302#1583—731no—
303#1582—731no—
304SD16⋊C8—731no—
305C8×SD16—731no—
306#1572—731no—
307#1497—731no—
308#1478—731no—
309#1454—731no—
310C73#1802811yescharacteristic, complemented, Sylow
311#533—731no—
312C2×C4×D4—731no—
313#483—731no—
314#483—731no—
315#476—731no—
316#476—731no—
317#461—731no—
318#461—731no—
319#450—731no—
320#450#6440811yescharacteristic
321#446—731no—
322#446—731no—
323#446—1461no—
324C8×Q8—731no—
325C8×Q8—731no—
326C8×Q8—1461no—
327C4×SD16—731no—
328C4×SD16—731no—
329C4×SD16—731no—
330C4×SD16—731no—
331#436—731no—
332#436—731no—
333#435—1461no—
334#435—1461no—
335C8×D4—731no—
336C8×D4—731no—
337#433—1461no—
338#433—1461no—
339#432—731no—
340#431—731no—
341#431—731no—
342#431—731no—
343#431—731no—
344#407—1461no—
345C2×C22⋊C8—1461no—
346#335—731no—
347#335—731no—
348#334—731no—
349#334—731no—
350Q8⋊C8—731no—
351Q8⋊C8—731no—
352Q8⋊C8—731no—
353Q8⋊C8—731no—
354D4⋊C8—731no—
355D4⋊C8—731no—
356D4⋊C8—731no—
357D4⋊C8—731no—
358#322—731no—
359#322—731no—
360#322—731no—
361C82—731no—
362C22×D4C2×#6440811yescharacteristic
363C23×C4—1461no—
364C2×SD16—731no—
365C2×SD16—731no—
366C2×SD16C2×#6440811yescharacteristic
367C2×SD16C2×#6440811yescharacteristic
368C2×M16—1461no—
369C2×M16—1461no—
370C22×C8—1461no—
371C22×C8—1461no—
372C4⋊Q8—731no—
373C4⋊Q8—731no—
374#125—1461no—
375C22⋊Q8—731no—
376C22⋊Q8—1461no—
377C22⋊Q8C2×#6440811yescharacteristic
378C4×Q8—731no—
379C4×Q8—731no—
380C4×Q8—1461no—
381C4×D4—731no—
382C4×D4—731no—
383C4×D4—731no—
384C4×D4—731no—
385C4×D4—1461no—
386C4×D4—1461no—
387#117—1461no—
388C2×C4⋊C4—731no—
389C2×C22⋊C4—1461no—
390C2×C42—731no—
391#106—731no—
392#106—731no—
393#106—731no—
394#106—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—731no—
401C4⋊C8—731no—
402C4⋊C8—731no—
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#102—731no—
413#102—731no—
414#102C2×#6440811yescharacteristic
415#102C2×#6440811yescharacteristic
416C22⋊C8—731no—
417C22⋊C8—731no—
418C22⋊C8—731no—
419C22⋊C8—1461no—
420C22⋊C8—1461no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424C22⋊C8—1461no—
425C22⋊C8—1461no—
426C22⋊C8C2×#6440811yescharacteristic
427#97—1461no—
428#97—1461no—
429#97—1461no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—1461no—
437C4×C8—1461no—
438C4×C8—1461no—
439C24—21no—
440C2×Q8—731no—
441C2×Q8—1461no—
442C2×Q8C22×#6440811yescharacteristic
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—41no—
446C2×D4C22×#6440811yescharacteristic
447C2×D4C22×#6440811yescharacteristic
448C22×C4—5841no—
449C22×C4—5841no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4C22×#6440811yescharacteristic
458SD16—22no—
459SD16—41no—
460SD16—2921no—
461SD16—1462no—
462M16—5841no—
463M16—5841no—
464C2×C8—5841no—
465C2×C8—5841no—
466C2×C8—731no—
467C2×C8—731no—
468C2×C8—731no—
469C2×C8—731no—
470C2×C8—731no—
471C2×C8—731no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8C22×#6440811yescharacteristic
479C2×C8C22×#6440811yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—731no—
482C4⋊C4—731no—
483C4⋊C4—731no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4C22×#6440811yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—1461no—
494C22⋊C4—1461no—
495C22⋊C4—1461no—
496C22⋊C4—1461no—
497C22⋊C4—1461no—
498C42—731no—
499C42—731no—
500C42—731no—
501C42—731no—
502C42—1461no—
503C42—1461no—
504C23—21no—
505C23—21no—
506C23—42no—
507C23—42no—
508C23—81no—
509C23D4×#6440811yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—2922no—
513Q8—1461no—
514Q8—1461no—
515D4—21no—
516D4—21no—
517D4—21no—
518D4—42no—
519D4#1455667812yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—5841no—
530C2×C4—5841no—
531C2×C4—5841no—
532C2×C4—731no—
533C2×C4—731no—
534C2×C4—731no—
535C2×C4—731no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4D4×#6440811yescharacteristic
543C2×C4C23×#6440811yescharacteristic
544C8—21no—
545C8—21no—
546C8—2921no—
547C8—2921no—
548C8—2921no—
549C8—2921no—
550C8—2921no—
551C8—2921no—
552C8—1461no—
553C8—1461no—
554C8—1461no—
555C8—1461no—
556C8—1461no—
557C8—1461no—
558C22—21no—
559C22—22no—
560C22—21no—
561C22—21no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—84no—
568C22C2×D4×#6440811yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—2921no—
573C4—2921no—
574C4—2921no—
575C4—2921no—
576C4—731no—
577C4—731no—
578C4—1461no—
579C4—1461no—
580C4—1461no—
581C4C2×#1455667811yescharacteristic
582C4#1798966511yescharacteristic
583C2—41no—
584C2—42no—
585C2—81no—
586C2#2262666311yescharacteristic
587C2#2262669111yescharacteristic
588C2#2262670311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#450×C73C4
socleC22×C73C2×C4×D4
prime radicalC22×D4×C73C2×C4
cosocleC4×Dic73C24
hypercenter#450#64408
hypocenterC73#18028
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22626840C2semidirect
3. #22625574C2semidirect
4. #22625568C2semidirect
5. #22625560C2semidirect
6. #22625546C2semidirect
7. #22625541C2semidirect
8. #22625536C2semidirect
9. #22625522C2semidirect
10. SD16×#10596833C2semidirect
11. #22625502C2semidirect
12. #22625293C2semidirect
14. #22625152C2semidirect
17. #17989638C22semidirect
18. #17989637C22semidirect
20. #17989636C22semidirect
21. Q8×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. #17989437C22semidirect
41. #17989437C22semidirect
42. Q8⋊#10596833C22semidirect
43. Q8⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. #17989428C22semidirect
51. C8×#10596833C22semidirect
310. C73#18028semidirect

Maximal subgroups

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

subgroupindex
#226268402
#226267872
#226255742
#226255682
#226255602
#226255462
#226255412
#226255362
#226255222
SD16×#105968332
#226255022
#226252932
#226252302
#226251522
#226199342
#1802873