Finite Group Databasebeta

← #34019100

Subgroups — #34019100

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626726C211yesmaximal, characteristic, complemented
2D4×#11924037C211yesmaximal, characteristic
3#22625568C211yesmaximal, characteristic, complemented
4#22625567C211yesmaximal, characteristic, complemented
5#22625541C211yesmaximal, characteristic, complemented
6#22625540C211yesmaximal, characteristic, complemented
7#22625536C211yesmaximal, characteristic, complemented
8#22625534C211yesmaximal, characteristic, complemented
9#22625514C211yesmaximal, characteristic, complemented
10SD16×#10596833C211yesmaximal, characteristic, complemented
11#22625498C211yesmaximal, characteristic, complemented
12#22625279C211yesmaximal, characteristic, complemented
13C2×D4⋊#10596833C211yesmaximal, characteristic
14#22625132C211yesmaximal, characteristic, complemented
15#22619934C211yesmaximal, characteristic
16C22×#11924037—21no—
17#17989639—21no—
18#17989638C2211yescharacteristic, complemented
19#17989637—21no—
20#17989637C2211yescharacteristic, complemented
21#17989636C2211yescharacteristic, complemented
22Q8×#10596833C2211yescharacteristic, complemented
23#17989622C2211yescharacteristic
24#17989622C2211yescharacteristic
25#17989618—21no—
26#17989618—21no—
27D4×#10596833C2211yescharacteristic
28D4×#10596833C2211yescharacteristic
29#17989593—21no—
30#17989590C2211yescharacteristic
31C2×C4⋊#10596833C2211yescharacteristic
32#17989557—21no—
33#17989551—21no—
34C4×#11924037C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989449C2211yescharacteristic, complemented
37#17989449C2211yescharacteristic, complemented
38#17989448C2211yescharacteristic, 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
50C8×#10596833C2211yescharacteristic, 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—
69C2×#11924037—21no—
70C2×#11924037—44no—
71C2×#11924037D411yescharacteristic
72C2×#11924037D411yescharacteristic
73C22×#10596833—21no—
74C22×#10596833—21no—
75C22×#10596833—21no—
76C22×#10596833—21no—
77#14556658—21no—
78#14556658—21no—
79#14556658—21no—
80C4⋊#10596833—21no—
81C4⋊#10596833—21no—
82C4⋊#10596833—21no—
83C4⋊#10596833C2311yescharacteristic
84C4⋊#10596833C2311yescharacteristic
85C4⋊#10596833C2311yescharacteristic
86C4⋊#10596833C2311yescharacteristic
87C22⋊#10596833—21no—
88C22⋊#10596833—21no—
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91#14556646—21no—
92#14556646—21no—
93#14556646D411yescharacteristic
94#14556646D411yescharacteristic
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833—21no—
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C4×#10596833C2311yescharacteristic
102C23×Dic73—21no—
103C2×#11924016C2×C411yescharacteristic
104C2×#11924016C2×C411yescharacteristic
105#14556590C2×C411yescharacteristic
106#14556589C2×C411yescharacteristic
107#14556581—21no—
108#14556570C2×C411yescharacteristic
109#14556568—21no—
110Q8×Dic73C2311yescharacteristic
111#14556556—21no—
112#14556556C2311yescharacteristic
113D4×Dic73—21no—
114D4×Dic73—21no—
115D4×Dic73C2×C411yescharacteristic
116D4×Dic73C2×C411yescharacteristic
117D4×Dic73C2311yescharacteristic
118D4×Dic73C2311yescharacteristic
119#14556547—21no—
120C2×C4⋊Dic73C2×C411yescharacteristic
121C2×C22⋊Dic73—21no—
122C2×C4×Dic73C2311yescharacteristic
123#14556523C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556522C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556520C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556516C2×C411yescharacteristic
131#14556515C2×C411yescharacteristic
132#14556515C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134#14556512C2×C411yescharacteristic
135C22⋊#10596831C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C22×D4×C73C2×C411yescharacteristic
139C2×SD16×C73C2×C411yescharacteristic
140C2×SD16×C73C2×C411yescharacteristic
141C22⋊Q8×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144C22⋊C8×C73C2×C411yescharacteristic
145#11924037—22no—
146#11924037—42no—
147#11924037—42no—
148#11924037—42no—
149#11924037SD1614yescomplemented
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—42no—
157C2×#10596833—42no—
158C2×#10596833—42no—
159C2×#10596833—42no—
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C2×#10596833C2×D411yescharacteristic
163C2×#10596833C2×D411yescharacteristic
164C2×Dic146—21no—
165C2×Dic146C22×C411yescharacteristic
166C22×Dic73—21no—
167C22×Dic73—21no—
168C22×Dic73—21no—
169C22×Dic73—21no—
170C22×Dic73—21no—
171C22×Dic73—42no—
172C22×Dic73—42no—
173C22×Dic73C2×D411yescharacteristic
174C22×Dic73#4111yescharacteristic
175#11924016—22no—
176#11924016—41no—
177C2×#10596831C22×C411yescharacteristic
178C2×#10596831C22×C411yescharacteristic
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C22⋊Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73C22×C411yescharacteristic
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C4×Dic73C2411yescharacteristic
200C24×C73—21no—
201C2×Q8×C73C22×C411yescharacteristic
202C2×D4×C73—22no—
203C2×D4×C73—21no—
204C2×D4×C73—41no—
205C2×D4×C73C22×C411yescharacteristic
206C2×D4×C73C22×C411yescharacteristic
207C22×C4×C73C22×C411yescharacteristic
208SD16×C73—22no—
209SD16×C73—41no—
210C2×C8×C73C22×C411yescharacteristic
211C2×C8×C73C22×C411yescharacteristic
212C4⋊C4×C73—21no—
213C4⋊C4×C73C22×C411yescharacteristic
214C22⋊C4×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—42no—
218C23×C73—42no—
219C23×C73—81no—
220C23×C73C4×D411yescharacteristic
221Q8×C73—21no—
222Q8×C73—21no—
223D4×C73—21no—
224D4×C73—21no—
225D4×C73—21no—
226D4×C73—42no—
227D4×C73#13112yes—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73C4×D411yescharacteristic
233C2×C4×C73C23×C411yescharacteristic
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—41no—
239C2×Dic73—42no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73C4×D411yescharacteristic
247C2×Dic73C4×D411yescharacteristic
248C2×Dic73C2×SD1612yes—
249C2×Dic73#13512yes—
250C2×Dic73C22×D411yescharacteristic
251C2×Dic73C2×#4111yescharacteristic
252Dic146—21no—
253Dic146—21no—
254Dic146—81no—
255#10596833—22no—
256#10596833—21no—
257#10596833—21no—
258#10596833—41no—
259#10596833—41no—
260#10596833—41no—
261#10596833—41no—
262#10596833—41no—
263#10596833C2×SD1612yes—
264#10596833C2×SD1612yes—
265C8×C73—21no—
266C8×C73—21no—
267#10596831—21no—
268#10596831—21no—
269C22×C73—21no—
270C22×C73—21no—
271C22×C73—22no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—84no—
278C22×C73C4×SD1612yes—
279C22×C73C2×C4×D411yescharacteristic
280C4×C73—21no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73C2×#13111yescharacteristic
284C4×C73#56811yescharacteristic
285Dic73—21no—
286Dic73—21no—
287Dic73—41no—
288Dic73—41no—
289Dic73—41no—
290Dic73—41no—
291Dic73C4×SD1612yes—
292Dic73C22×SD1611yescharacteristic
293Dic73C2×#13511yescharacteristic
294#17995—731nomaximal, Sylow
295C2×C73—22no—
296C2×C73—42no—
297C2×C73—81no—
298C2×C73#293311yescharacteristic
299C2×C73C2×C4×SD1611yescharacteristic
300C2×C73#295611yescharacteristic
301#3101—731no—
302#2962—731no—
303D4×M16—731no—
304#1590—731no—
305#1590—731no—
306#1583—731no—
307#1583—731no—
308#1582—731no—
309#1582—731no—
310C8×SD16—731no—
311C8×SD16—731no—
312#1570—731no—
313#1493—731no—
314C2×D4⋊C8—731no—
315C8×M16—731no—
316C73#1799511yescharacteristic, complemented, Sylow
317C22×M16—1461no—
318#533—731no—
319C2×C4×D4—731no—
320#483—731no—
321#483—731no—
322#476—731no—
323#476—731no—
324#461—731no—
325#461—731no—
326#450—731no—
327#450#6440811yescharacteristic
328#446—731no—
329#446—731no—
330#446—1461no—
331#446—1461no—
332C8×Q8—731no—
333C8×Q8—731no—
334C4×SD16—731no—
335C4×SD16—731no—
336C4×SD16—731no—
337C4×SD16—731no—
338#436—731no—
339#436—731no—
340#435—1461no—
341#435—1461no—
342C8×D4—731no—
343C8×D4—731no—
344#431—731no—
345#424—1461no—
346#423—731no—
347C2×C4⋊C8—731no—
348#407—1461no—
349#405—1461no—
350C4×M16—731no—
351C2×C4×C8—731no—
352#334—731no—
353#334—731no—
354#334—731no—
355#334—731no—
356Q8⋊C8—731no—
357Q8⋊C8—731no—
358Q8⋊C8—731no—
359Q8⋊C8—731no—
360D4⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8—731no—
364#322—731no—
365#322—731no—
366C82—731no—
367C82—731no—
368C22×D4C2×#6440811yescharacteristic
369C23×C4—1461no—
370C2×SD16—731no—
371C2×SD16—731no—
372C2×SD16C2×#6440811yescharacteristic
373C2×SD16C2×#6440811yescharacteristic
374C2×M16—2924no—
375C2×M16—731no—
376C2×M16—731no—
377C2×M16—1461no—
378C2×M16—1461no—
379C2×M16—1461no—
380C22×C8—1461no—
381C22×C8—1461no—
382C22×C8—1461no—
383C22×C8—1461no—
384C4⋊Q8—731no—
385C4⋊Q8—731no—
386#125—1461no—
387C22⋊Q8—731no—
388C22⋊Q8—1461no—
389C22⋊Q8C2×#6440811yescharacteristic
390C4×Q8—731no—
391C4×Q8—731no—
392C4×Q8—1461no—
393C4×D4—731no—
394C4×D4—731no—
395C4×D4—731no—
396C4×D4—731no—
397C4×D4—1461no—
398C4×D4—1461no—
399#117—1461no—
400C2×C4⋊C4—731no—
401C2×C22⋊C4—1461no—
402C2×C42—731no—
403#106—731no—
404#106—731no—
405#106—731no—
406#106—731no—
407C4⋊C8—731no—
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—731no—
413C4⋊C8—1461no—
414C4⋊C8—1461no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8—1461no—
418C4⋊C8—1461no—
419#103—731no—
420#103—731no—
421#103—731no—
422#103—731no—
423#102—731no—
424#102—731no—
425#102C2×#6440811yescharacteristic
426#102C2×#6440811yescharacteristic
427C22⋊C8—731no—
428C22⋊C8—1461no—
429C22⋊C8—1461no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C22⋊C8C2×#6440811yescharacteristic
433#97—731no—
434#97—731no—
435#97—1461no—
436#97—1461no—
437C4×C8—731no—
438C4×C8—731no—
439C4×C8—731no—
440C4×C8—731no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—1461no—
444C4×C8—1461no—
445C4×C8—1461no—
446C24—21no—
447C2×Q8—731no—
448C2×Q8—1461no—
449C2×Q8C22×#6440811yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4C22×#6440811yescharacteristic
454C2×D4C22×#6440811yescharacteristic
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4C22×#6440811yescharacteristic
465SD16—22no—
466SD16—41no—
467SD16—2921no—
468SD16—1462no—
469M16—2922no—
470M16—2922no—
471M16—2922no—
472M16—734no—
473M16—1462no—
474C2×C8—2922no—
475C2×C8—2922no—
476C2×C8—2922no—
477C2×C8—2922no—
478C2×C8—731no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—731no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8C22×#6440811yescharacteristic
491C2×C8C22×#6440811yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—731no—
494C4⋊C4—731no—
495C4⋊C4—731no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4C22×#6440811yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C42—731no—
511C42—731no—
512C42—731no—
513C42—731no—
514C42—1461no—
515C42—1461no—
516C23—21no—
517C23—21no—
518C23—42no—
519C23—42no—
520C23—81no—
521C23D4×#6440811yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—5841no—
525Q8—1461no—
526Q8—1461no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—42no—
531D4#1455667812yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—2922no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2921no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—731no—
549C2×C4—731no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4D4×#6440811yescharacteristic
555C2×C4C23×#6440811yescharacteristic
556C8—21no—
557C8—21no—
558C8—2921no—
559C8—2921no—
560C8—2921no—
561C8—2921no—
562C8—2921no—
563C8—732no—
564C8—732no—
565C8—1461no—
566C8—1461no—
567C8—1462no—
568C8—1461no—
569C8—1461no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—84no—
579C22SD16×#6440812yes—
580C22C2×D4×#6440811yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—2921no—
585C4—2921no—
586C4—2921no—
587C4—2921no—
588C4—731no—
589C4—731no—
590C4—732no—
591C4—1461no—
592C4—1461no—
593C4C2×#1455667811yescharacteristic
594C4#1798966511yescharacteristic
595C2—22no—
596C2—42no—
597C2—81no—
598C2#2262666311yescharacteristic
599C2C2×SD16×#6440811yescharacteristic
600C2#2262670511yescharacteristic
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#17995
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. #22626726C2semidirect
3. #22625568C2semidirect
4. #22625567C2semidirect
5. #22625541C2semidirect
6. #22625540C2semidirect
7. #22625536C2semidirect
8. #22625534C2semidirect
9. #22625514C2semidirect
10. SD16×#10596833C2semidirect
11. #22625498C2semidirect
12. #22625279C2semidirect
14. #22625132C2semidirect
18. #17989638C22semidirect
20. #17989637C22semidirect
21. #17989636C22semidirect
22. Q8×#10596833C22semidirect
36. #17989449C22semidirect
37. #17989449C22semidirect
38. #17989448C22semidirect
39. #17989448C22semidirect
40. #17989437C22semidirect
41. #17989437C22semidirect
42. Q8⋊#10596833C22semidirect
43. Q8⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. C8×#10596833C22semidirect
51. C8×#10596833C22semidirect
149. #11924037SD16semidirect
316. C73#17995semidirect

Maximal subgroups

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

subgroupindex
#226267262
D4×#119240372
#226255682
#226255672
#226255412
#226255402
#226255362
#226255342
#226255142
SD16×#105968332
#226254982
#226252792
C2×D4⋊#105968332
#226251322
#226199342
#1799573