Finite Group Databasebeta

← #34018696

Subgroups — #34018696

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626836C211yesmaximal, characteristic, complemented
2#22626779C211yesmaximal, characteristic
3#22625549C211yesmaximal, characteristic, complemented
4#22625548C211yesmaximal, characteristic, complemented
5#22625537C211yesmaximal, characteristic, complemented
6#22625536C211yesmaximal, characteristic, complemented
7#22625521C211yesmaximal, characteristic, complemented
8SD16⋊#10596833C211yesmaximal, characteristic, complemented
9#22625515C211yesmaximal, characteristic, complemented
10SD16×#10596833C211yesmaximal, characteristic, complemented
11#22625503C211yesmaximal, characteristic, complemented
12#22625271C211yesmaximal, characteristic
13#22625254C211yesmaximal, characteristic, complemented
14#22625149C211yesmaximal, characteristic, complemented
15#22619938C211yesmaximal, characteristic
16#17989639—21no—
17#17989639—21no—
18#17989638C2211yescharacteristic, complemented
19#17989638C2211yescharacteristic, complemented
20Q8×#10596833C2211yescharacteristic, complemented
21Q8×#10596833C2211yescharacteristic, complemented
22#17989615C2211yescharacteristic
23#17989615C2211yescharacteristic
24#17989614—21no—
25#17989614—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989612—21no—
29#17989612—21no—
30#17989609C2211yescharacteristic
31#17989606C2211yescharacteristic
32#17989605C2211yescharacteristic
33C2×#14556648—21no—
34#17989555—21no—
35C4×#11924038C2211yescharacteristic
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40Q8⋊#10596833C2211yescharacteristic, complemented
41Q8⋊#10596833C2211yescharacteristic, complemented
42Q8⋊#10596833C2211yescharacteristic, complemented
43Q8⋊#10596833C2211yescharacteristic, complemented
44#17989435C2211yescharacteristic
45#17989435C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50#17989428C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52#17989142C2211yescharacteristic
53C4×D4×D73C2211yescharacteristic
54#17988916C411yescharacteristic
55#17988916C411yescharacteristic
56#17988878C411yescharacteristic
57#17988877C411yescharacteristic
58#17988789C411yescharacteristic
59#17988736C411yescharacteristic
60#17988736C411yescharacteristic
61SD16×Dic73C2211yescharacteristic
62SD16×Dic73C2211yescharacteristic
63#17988672C2211yescharacteristic
64#17988672C2211yescharacteristic
65#17988639C2211yescharacteristic
66#461×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924036—21no—
69C2×#11924036—21no—
70C22×#10596833—21no—
71C22×#10596833—21no—
72#14556658—21no—
73#14556658—21no—
74#14556658D411yescharacteristic
75#14556658D411yescharacteristic
76C4⋊#10596833—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833C2311yescharacteristic
79C4⋊#10596833C2311yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82#14556648—21no—
83#14556648—21no—
84#14556648—21no—
85#14556648—21no—
86#14556648D411yescharacteristic
87#14556648D411yescharacteristic
88C22⋊#10596833—21no—
89C22⋊#10596833—21no—
90#14556646—21no—
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
101C2×D4×D73C2×C411yescharacteristic
102C22×C4×D73—21no—
103C2×#11924015C2×C411yescharacteristic
104C2×#11924015C2×C411yescharacteristic
105#14556591C2×C411yescharacteristic
106#14556582—21no—
107#14556571—21no—
108#14556567C2×C411yescharacteristic
109#14556567C2×C411yescharacteristic
110Q8×Dic73C2311yescharacteristic
111Q8×Dic73C2311yescharacteristic
112#14556556—21no—
113C4×D292C2×C411yescharacteristic
114D4×Dic73C2311yescharacteristic
115#14556553C2311yescharacteristic
116C4×C22⋊D73—21no—
117#14556551—21no—
118#14556549—21no—
119C4⋊C4×D73C2×C411yescharacteristic
120C22⋊C4×D73—21no—
121C42×D73C2311yescharacteristic
122#14556523C2×C411yescharacteristic
123#14556523C2×C411yescharacteristic
124#14556522C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126C4⋊#10596831C2×C411yescharacteristic
127#14556515C2×C411yescharacteristic
128#14556515C2×C411yescharacteristic
129#14556513C2×C411yescharacteristic
130#14556513C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556503C2×C411yescharacteristic
134#14556503C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×SD16×C73C2×C411yescharacteristic
138C2×SD16×C73C2×C411yescharacteristic
139C4⋊Q8×C73C2×C411yescharacteristic
140C4×D4×C73C2×C411yescharacteristic
141C4⋊C8×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#103×C73C2×C411yescharacteristic
144#11924038—42no—
145#11924036—81no—
146#11924036—81no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—81no—
154C2×#10596833—81no—
155C2×#10596833C2×D411yescharacteristic
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C23×D73—21no—
160C2×Dic146—21no—
161C2×D292C22×C411yescharacteristic
162C2×C22⋊D73—21no—
163D4×D73—22no—
164D4×D73—22no—
165C22×Dic73—21no—
166C2×C4×D73—21no—
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169C2×C4×D73—81no—
170C2×C4×D73—81no—
171C2×C4×D73C22×C411yescharacteristic
172C2×C4×D73C2×D411yescharacteristic
173C2×C4×D73#4111yescharacteristic
174#11924015—22no—
175#11924015—22no—
176C2×#10596831C22×C411yescharacteristic
177C2×#10596831C22×C411yescharacteristic
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C4×Dic73C2411yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C2×Q8×C73C22×C411yescharacteristic
199C2×D4×C73C22×C411yescharacteristic
200C22×C4×C73—21no—
201SD16×C73—22no—
202SD16×C73—22no—
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73C22×C411yescharacteristic
208C22⋊C4×C73—21no—
209C42×C73C22×C411yescharacteristic
210C23×C73—21no—
211C22×D73—21no—
212C22×D73—24no—
213C22×D73—81no—
214C22×D73—81no—
215C22×D73C4×D411yescharacteristic
216Q8×C73—21no—
217Q8×C73—21no—
218Q8×C73—21no—
219Q8×C73—21no—
220D4×C73—21no—
221D4×C73#13112yes—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—81no—
226C2×C4×C73C4×D411yescharacteristic
227C2×C4×C73C4×D411yescharacteristic
228C2×C4×C73C23×C411yescharacteristic
229C22⋊D73—24no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C22×D411yescharacteristic
237C2×Dic73C2×#4111yescharacteristic
238D292—21no—
239D292#13112yes—
240C4×D73—21no—
241C4×D73—21no—
242C4×D73—21no—
243C4×D73—21no—
244C4×D73—41no—
245C4×D73—41no—
246C4×D73—41no—
247C4×D73—81no—
248C4×D73—81no—
249Dic146—81no—
250#10596833—22no—
251#10596833—21no—
252#10596833—21no—
253#10596833—21no—
254#10596833—21no—
255#10596833—41no—
256#10596833—41no—
257#10596833—41no—
258#10596833—41no—
259#10596833—41no—
260C8×C73—21no—
261C8×C73—21no—
262#10596831—21no—
263#10596831—21no—
264C22×C73—22no—
265C22×C73—41no—
266C22×C73—41no—
267C22×C73C2×C4×D411yescharacteristic
268C2×D73—21no—
269C2×D73—22no—
270C2×D73—21no—
271C2×D73—21no—
272C2×D73—41no—
273C2×D73—41no—
274C2×D73—84no—
275C4×C73—21no—
276C4×C73—21no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73C2×#13111yescharacteristic
281C4×C73C2×#13111yescharacteristic
282Dic73—21no—
283Dic73—21no—
284Dic73—41no—
285Dic73—41no—
286Dic73—41no—
287Dic73#57711yescharacteristic
288Dic73#57711yescharacteristic
289#17924—731nomaximal, Sylow
290C2×C73—42no—
291C2×C73#292911yescharacteristic
292C2×C73#294911yescharacteristic
293C2×C73#295611yescharacteristic
294D73—41no—
295D73—42no—
296#3101—731no—
297#2985—731no—
298#2972—731no—
299#1585—731no—
300#1585—731no—
301#1582—731no—
302#1582—731no—
303SD16⋊C8—731no—
304SD16⋊C8—731no—
305C8×SD16—731no—
306C8×SD16—731no—
307#1572—731no—
308#1490—731no—
309#1485—731no—
310#1453—731no—
311C73#1792411yescharacteristic, complemented, Sylow
312#533—731no—
313C2×C4×D4—731no—
314#483—731no—
315#483—731no—
316#476—731no—
317#476—731no—
318#461—731no—
319#461#6440811yescharacteristic
320#450—731no—
321#450—731no—
322#446—731no—
323#446—731no—
324#446—1461no—
325#446—1461no—
326C8×Q8—731no—
327C8×Q8—731no—
328C4×SD16—731no—
329C4×SD16—731no—
330C4×SD16—731no—
331C4×SD16—731no—
332C8×D4—731no—
333C8×D4—731no—
334C8×D4—731no—
335C8×D4—731no—
336C8×D4—1461no—
337C8×D4—1461no—
338#433—1461no—
339#433—1461no—
340#432—731no—
341#431—731no—
342#431—731no—
343#431—731no—
344C2×C22⋊C8—1461no—
345C2×C22⋊C8—1461no—
346C4×M16—731no—
347#335—731no—
348#335—731no—
349#334—731no—
350#334—731no—
351Q8⋊C8—731no—
352Q8⋊C8—731no—
353Q8⋊C8—731no—
354Q8⋊C8—731no—
355D4⋊C8—731no—
356D4⋊C8—731no—
357D4⋊C8—731no—
358D4⋊C8—731no—
359#322—731no—
360#322—731no—
361#322—731no—
362C82—731no—
363C22×D4—731no—
364C23×C4—1461no—
365C2×SD16—731no—
366C2×SD16—731no—
367C2×SD16C2×#6440811yescharacteristic
368C2×SD16C2×#6440811yescharacteristic
369C2×M16—1461no—
370C22×C8—1461no—
371C22×C8—1461no—
372C22×C8—1461no—
373C22×C8—1461no—
374C4⋊Q8—731no—
375C4⋊Q8C2×#6440811yescharacteristic
376#125—1461no—
377C22⋊Q8—731no—
378C22⋊Q8—731no—
379C22⋊Q8—1461no—
380C4×Q8—731no—
381C4×Q8—731no—
382C4×Q8—1461no—
383C4×D4—731no—
384C4×D4—731no—
385C4×D4—731no—
386C4×D4—1461no—
387C4×D4—1461no—
388C4×D4C2×#6440811yescharacteristic
389#117—1461no—
390C2×C4⋊C4—731no—
391C2×C22⋊C4—1461no—
392C2×C42—731no—
393#106—731no—
394#106—731no—
395#106—731no—
396#106—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—731no—
401C4⋊C8—731no—
402C4⋊C8—731no—
403C4⋊C8—731no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8—1461no—
407C4⋊C8—1461no—
408C4⋊C8C2×#6440811yescharacteristic
409#103—731no—
410#103—731no—
411#103C2×#6440811yescharacteristic
412#103C2×#6440811yescharacteristic
413#102—731no—
414#102—731no—
415#102—731no—
416#102—731no—
417C22⋊C8—731no—
418C22⋊C8—731no—
419C22⋊C8—731no—
420C22⋊C8—731no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424C22⋊C8—1461no—
425C22⋊C8—1461no—
426C22⋊C8—1461no—
427#97—1461no—
428#97—1461no—
429#97—1461no—
430#97—1461no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—731no—
437C4×C8—1461no—
438C4×C8—1461no—
439C4×C8—1461no—
440C24—1461no—
441C2×Q8—1461no—
442C2×Q8C22×#6440811yescharacteristic
443C2×Q8C22×#6440811yescharacteristic
444C2×D4—731no—
445C2×D4—1461no—
446C2×D4—1462no—
447C2×D4—1462no—
448C2×D4C22×#6440811yescharacteristic
449C22×C4—21no—
450C22×C4—5841no—
451C22×C4—5841no—
452C22×C4—731no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459SD16—22no—
460SD16—22no—
461SD16—1462no—
462SD16—1462no—
463M16—2922no—
464C2×C8—5841no—
465C2×C8—5841no—
466C2×C8—5841no—
467C2×C8—5841no—
468C2×C8—731no—
469C2×C8—731no—
470C2×C8—731no—
471C2×C8—731no—
472C2×C8—731no—
473C2×C8—731no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8C22×#6440811yescharacteristic
481C2×C8C22×#6440811yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—731no—
485C4⋊C4—731no—
486C4⋊C4—731no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4C22×#6440811yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1461no—
496C22⋊C4—1461no—
497C22⋊C4—1461no—
498C22⋊C4—1461no—
499C22⋊C4—1461no—
500C42—731no—
501C42—731no—
502C42—731no—
503C42—1461no—
504C42—1461no—
505C42C22×#6440811yescharacteristic
506C23—21no—
507C23—5841no—
508C23—5841no—
509C23—731no—
510C23—1464no—
511C23—1461no—
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—21no—
516Q8—5841no—
517D4—21no—
518D4—732no—
519D4—1464no—
520D4—1461no—
521D4#1455667812yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—81no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—5841no—
532C2×C4—5841no—
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×C4—1461no—
543C2×C4D4×#6440811yescharacteristic
544C2×C4D4×#6440811yescharacteristic
545C2×C4C23×#6440811yescharacteristic
546C8—21no—
547C8—21no—
548C8—2921no—
549C8—2921no—
550C8—2921no—
551C8—2921no—
552C8—2921no—
553C8—1461no—
554C8—1461no—
555C8—1461no—
556C8—1461no—
557C8—1462no—
558C8—1461no—
559C8—1461no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—2921no—
564C22—2921no—
565C22—5844no—
566C22—1461no—
567C22—1462no—
568C22—1461no—
569C22—1461no—
570C22C2×D4×#6440811yescharacteristic
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—2921no—
577C4—2921no—
578C4—2921no—
579C4—731no—
580C4—731no—
581C4—1461no—
582C4—1461no—
583C4C2×#1455667811yescharacteristic
584C4D73⋊#13111yescharacteristic
585C2—42no—
586C2—2922no—
587C2—2921no—
588C2#2262664511yescharacteristic
589C2#2262669111yescharacteristic
590C2#2262670611yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#461×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×D4×D73C2×C4
cosocleC4×Dic73C24
hypercenter#461#64408
hypocenterC73#17924
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22626836C2semidirect
3. #22625549C2semidirect
4. #22625548C2semidirect
5. #22625537C2semidirect
6. #22625536C2semidirect
7. #22625521C2semidirect
8. SD16⋊#10596833C2semidirect
9. #22625515C2semidirect
10. SD16×#10596833C2semidirect
11. #22625503C2semidirect
13. #22625254C2semidirect
14. #22625149C2semidirect
18. #17989638C22semidirect
19. #17989638C22semidirect
20. Q8×#10596833C22semidirect
21. Q8×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. Q8⋊#10596833C22semidirect
41. Q8⋊#10596833C22semidirect
42. Q8⋊#10596833C22semidirect
43. Q8⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. #17989428C22semidirect
51. C8×#10596833C22semidirect
311. C73#17924semidirect

Maximal subgroups

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

subgroupindex
#226268362
#226267792
#226255492
#226255482
#226255372
#226255362
#226255212
SD16⋊#105968332
#226255152
SD16×#105968332
#226255032
#226252712
#226252542
#226251492
#226199382
#1792473