Finite Group Databasebeta

← #34019328

Subgroups — #34019328

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626882C211yesmaximal, characteristic, complemented
2#22626857C211yesmaximal, characteristic, complemented
3#22625575C211yesmaximal, characteristic, complemented
4#22625571C211yesmaximal, characteristic, complemented
5#22625549C211yesmaximal, characteristic, complemented
6#22625548C211yesmaximal, characteristic, complemented
7#22625544C211yesmaximal, characteristic, complemented
8#22625530C211yesmaximal, characteristic, complemented
9SD16⋊#10596833C211yesmaximal, characteristic, complemented
10#22625511C211yesmaximal, characteristic, complemented
11#22625500C211yesmaximal, characteristic, complemented
12#22625288C211yesmaximal, characteristic, complemented
13#22625226C211yesmaximal, characteristic, complemented
14#22625152C211yesmaximal, characteristic, complemented
15#22619950C211yesmaximal, characteristic
16#17989638C2211yescharacteristic
17#17989638C2211yescharacteristic
18#17989638C2211yescharacteristic
19Q8×#10596833C2211yescharacteristic
20#17989621C2211yescharacteristic
21#17989618—21no—
22#17989618—21no—
23#17989616—21no—
24#17989615C2211yescharacteristic
25#17989614—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28#17989612—21no—
29#17989607—21no—
30#17989604C2211yescharacteristic
31#17989603C2211yescharacteristic
32#17989594—21no—
33#17989590C2211yescharacteristic
34C2×#14556658—21no—
35C2×C4⋊#10596833C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989451C2211yescharacteristic
38#17989450C2211yescharacteristic
39#17989450C2211yescharacteristic
40Q8⋊#10596833C2211yescharacteristic
41Q8⋊#10596833C2211yescharacteristic
42Q8⋊#10596833C2211yescharacteristic
43Q8⋊#10596833C2211yescharacteristic
44#17989435C2211yescharacteristic
45#17989435C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic
47D4⋊#10596833C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51C8×#10596833C2211yescharacteristic
52#17989123C2211yescharacteristic
53#41×Dic73C2211yescharacteristic
54#17988925C411yescharacteristic
55#17988912C411yescharacteristic
56#17988892C411yescharacteristic
57#17988885C411yescharacteristic
58#17988811C411yescharacteristic
59#17988804C411yescharacteristic
60#17988732C411yescharacteristic
61Q16×Dic73C2211yescharacteristic
62SD16×Dic73C2211yescharacteristic
63#17988672C2211yescharacteristic
64#17988667C2211yescharacteristic
65#17988637C2211yescharacteristic
66#452×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924037—21no—
69C2×#11924036—21no—
70C22×#10596833—21no—
71C22×#10596833—21no—
72C22×#10596833—21no—
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658—21no—
77#14556658D411yescharacteristic
78#14556658D411yescharacteristic
79C4⋊#10596833—21no—
80C4⋊#10596833—21no—
81C4⋊#10596833—21no—
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C4⋊#10596833C2311yescharacteristic
85C4⋊#10596833C2311yescharacteristic
86#14556648—21no—
87#14556648—21no—
88C22⋊#10596833—21no—
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833D411yescharacteristic
92C22⋊#10596833D411yescharacteristic
93#14556646—21no—
94#14556646—21no—
95#14556646—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924015C2×C411yescharacteristic
102C2×#11924013C2×C411yescharacteristic
103#14556586—21no—
104#14556582C2×C411yescharacteristic
105#14556580C2×C411yescharacteristic
106#14556575—21no—
107#14556565C2×C411yescharacteristic
108Q8×Dic73C2×C411yescharacteristic
109Q8×Dic73C2311yescharacteristic
110Q8×Dic73C2311yescharacteristic
111D4×Dic73—21no—
112D4×Dic73—21no—
113D4×Dic73C2×C411yescharacteristic
114D4×Dic73C2311yescharacteristic
115#14556553C2311yescharacteristic
116#14556551—21no—
117#14556550—21no—
118#14556550C2×C411yescharacteristic
119#14556549—21no—
120C2×C4×Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556524C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556515C2×C411yescharacteristic
129#14556515C2×C411yescharacteristic
130#14556515C2×C411yescharacteristic
131#14556513C2×C411yescharacteristic
132#14556513C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C2×#41×C73C2×C411yescharacteristic
138C2×Q16×C73C2×C411yescharacteristic
139C2×SD16×C73C2×C411yescharacteristic
140C22⋊Q8×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924038—81no—
145#11924037—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×#10596833—81no—
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C2×#10596833C2×D411yescharacteristic
160C2×D292C22×C411yescharacteristic
161C2×C22⋊D73—21no—
162C22×Dic73—21no—
163C22×Dic73—21no—
164C22×Dic73C2×D411yescharacteristic
165C22×Dic73#4111yescharacteristic
166C2×C4×D73—21no—
167#11924015—41no—
168#11924013—22no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73C22×C411yescharacteristic
176C4⋊Dic73C22×C411yescharacteristic
177C4⋊Dic73C22×C411yescharacteristic
178#11924007—21no—
179C22⋊Dic73—21no—
180C22⋊Dic73—21no—
181C22⋊Dic73—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185#11924005—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73C22×C411yescharacteristic
191C4×Dic73C2×D411yescharacteristic
192C4×Dic73#4111yescharacteristic
193C4×Dic73C2411yescharacteristic
194#41×C73—22no—
195#41×C73—41no—
196C2×Q8×C73C22×C411yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C2×D4×C73—21no—
199C2×D4×C73C22×C411yescharacteristic
200C22×C4×C73—21no—
201C22×C4×C73C22×C411yescharacteristic
202Q16×C73—41no—
203SD16×C73—22no—
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C4⋊C4×C73—21no—
207C4⋊C4×C73C22×C411yescharacteristic
208C22⋊C4×C73—21no—
209C23×C73—21no—
210C23×C73C4×D411yescharacteristic
211C22×D73—21no—
212Q8×C73—21no—
213Q8×C73—21no—
214Q8×C73—21no—
215Q8×C73—21no—
216D4×C73—21no—
217D4×C73—42no—
218D4×C73#13112yes—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—42no—
225C2×C4×C73C4×D411yescharacteristic
226C2×C4×C73C23×C411yescharacteristic
227C22⋊D73—42no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—41no—
234C2×Dic73—42no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73C4×D411yescharacteristic
238C2×Dic73C4×D411yescharacteristic
239C2×Dic73C22×D411yescharacteristic
240C2×Dic73C2×#4111yescharacteristic
241D292—21no—
242D292—21no—
243C4×D73—41no—
244C4×D73—41no—
245#10596833—21no—
246#10596833—21no—
247#10596833—21no—
248#10596833—21no—
249#10596833—41no—
250#10596833—41no—
251#10596833—41no—
252#10596833—41no—
253#10596833—41no—
254#10596833—41no—
255C8×C73—21no—
256C8×C73—21no—
257#10596831—21no—
258#10596831—21no—
259C22×C73—21no—
260C22×C73—22no—
261C22×C73—21no—
262C22×C73—21no—
263C22×C73—41no—
264C22×C73—41no—
265C22×C73C2×C4×D411yescharacteristic
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—41no—
269C4×C73—21no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73C2×#13111yescharacteristic
274C4×C73#56811yescharacteristic
275Dic73—21no—
276Dic73—21no—
277Dic73—21no—
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281Dic73#57611yescharacteristic
282Dic73#57811yescharacteristic
283#18027—731nomaximal, Sylow
284C2×C73—41no—
285C2×C73—42no—
286C2×C73#293311yescharacteristic
287C2×C73#295111yescharacteristic
288C2×C73#295511yescharacteristic
289D73—81no—
290#3102—731no—
291#2994—731no—
292#2990—731no—
293#1592—731no—
294#1591—731no—
295#1585—731no—
296#1585—731no—
297#1584—731no—
298#1581—731no—
299SD16⋊C8—731no—
300C8×D8—731no—
301#1571—731no—
302#1495—731no—
303#1477—731no—
304#1454—731no—
305C73#1802711yescharacteristic, complemented, Sylow
306#529—731no—
307C4×#41—731no—
308#485—731no—
309#482—731no—
310#479—731no—
311#478—731no—
312#464—731no—
313#463—731no—
314#452#6440811yescharacteristic
315#449—731no—
316#446—731no—
317#446—731no—
318#446—731no—
319C8×Q8—731no—
320C4×Q16—731no—
321C4×SD16—731no—
322C4×SD16—731no—
323C4×D8—731no—
324#436—731no—
325#435—1461no—
326#435—1461no—
327#435—1461no—
328C8×D4—731no—
329C8×D4—731no—
330C8×D4—731no—
331C8×D4—1461no—
332#433—1461no—
333#432—1461no—
334#431—731no—
335#431—731no—
336#431—731no—
337#424—1461no—
338#423—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—1461no—
341#335—731no—
342#335—731no—
343#335—731no—
344#335—731no—
345Q8⋊C8—731no—
346Q8⋊C8—731no—
347Q8⋊C8—731no—
348Q8⋊C8—731no—
349D4⋊C8—731no—
350D4⋊C8—731no—
351D4⋊C8—731no—
352D4⋊C8—731no—
353#322—731no—
354#322—731no—
355#322—731no—
356C82—731no—
357C2×#41C2×#6440811yescharacteristic
358C2×Q16C2×#6440811yescharacteristic
359C2×SD16—731no—
360C2×SD16C2×#6440811yescharacteristic
361C2×D8—731no—
362C2×M16—1461no—
363C2×M16—1461no—
364C22×C8—1461no—
365C22×C8—1461no—
366C22×C8—1461no—
367C22×C8—1461no—
368#126—1461no—
369#125—731no—
370#124—731no—
371#123—1461no—
372C22⋊Q8C2×#6440811yescharacteristic
373#121—731no—
374C4×Q8—731no—
375C4×Q8—731no—
376C4×Q8—731no—
377C4×D4—731no—
378C4×D4—731no—
379C4×D4—731no—
380C4×D4—1461no—
381C4×D4—1461no—
382C4×D4—1461no—
383#117—731no—
384#117—1461no—
385#117—1461no—
386C2×C42—731no—
387C2×C42—1461no—
388#107—731no—
389#107—731no—
390#106—731no—
391#106—731no—
392C4⋊C8—731no—
393C4⋊C8—731no—
394C4⋊C8—731no—
395C4⋊C8—731no—
396C4⋊C8—731no—
397C4⋊C8—731no—
398C4⋊C8—731no—
399C4⋊C8—731no—
400C4⋊C8—1461no—
401C4⋊C8—1461no—
402C4⋊C8—1461no—
403C4⋊C8—1461no—
404C4⋊C8—1461no—
405C4⋊C8—1461no—
406C4⋊C8—1461no—
407#103—731no—
408#103—731no—
409#103—731no—
410#103C2×#6440811yescharacteristic
411#102—731no—
412#102—731no—
413#102—731no—
414#102C2×#6440811yescharacteristic
415C22⋊C8—731no—
416C22⋊C8—731no—
417C22⋊C8—731no—
418C22⋊C8—1461no—
419C22⋊C8—1461no—
420C22⋊C8—1461no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8C2×#6440811yescharacteristic
424#97—1461no—
425#97—1461no—
426#97—1461no—
427C4×C8—731no—
428C4×C8—731no—
429C4×C8—731no—
430C4×C8—731no—
431C4×C8—731no—
432C4×C8—731no—
433C4×C8—1461no—
434#41—22no—
435#41—41no—
436C2×Q8C22×#6440811yescharacteristic
437C2×Q8C22×#6440811yescharacteristic
438C2×D4—21no—
439C2×D4—731no—
440C2×D4—1461no—
441C2×D4C22×#6440811yescharacteristic
442C22×C4—21no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4C22×#6440811yescharacteristic
449Q16—41no—
450SD16—22no—
451SD16—2921no—
452D8—1462no—
453M16—5841no—
454M16—5841no—
455C2×C8—5841no—
456C2×C8—5841no—
457C2×C8—5841no—
458C2×C8—5841no—
459C2×C8—731no—
460C2×C8—731no—
461C2×C8—731no—
462C2×C8—731no—
463C2×C8—731no—
464C2×C8—731no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8C22×#6440811yescharacteristic
472C2×C8C22×#6440811yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—731no—
475C4⋊C4—731no—
476C4⋊C4—731no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4C22×#6440811yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1461no—
485C22⋊C4—1461no—
486C22⋊C4—1461no—
487C22⋊C4—1461no—
488C22⋊C4—1461no—
489C22⋊C4—1461no—
490C22⋊C4—1461no—
491C42—731no—
492C42—731no—
493C42—731no—
494C42—731no—
495C42—1461no—
496C42—1461no—
497C42—1461no—
498C42—1461no—
499C23—21no—
500C23—1461no—
501C23D4×#6440811yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506D4—21no—
507D4—42no—
508D4—2922no—
509D4—1461no—
510D4—1461no—
511D4#1455667812yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2922no—
523C2×C4—2921no—
524C2×C4—731no—
525C2×C4—731no—
526C2×C4—731no—
527C2×C4—731no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4D4×#6440811yescharacteristic
534C2×C4C23×#6440811yescharacteristic
535C8—21no—
536C8—21no—
537C8—2921no—
538C8—2921no—
539C8—2921no—
540C8—2921no—
541C8—2921no—
542C8—2921no—
543C8—1461no—
544C8—1461no—
545C8—1461no—
546C8—1461no—
547C8—1461no—
548C8—1461no—
549C22—21no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—2921no—
556C22—2921no—
557C22—2921no—
558C22C2×D4×#6440811yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—2921no—
564C4—2921no—
565C4—2921no—
566C4—731no—
567C4—731no—
568C4—1461no—
569C4—1461no—
570C4—1461no—
571C4C2×#1455667811yescharacteristic
572C4#1798966511yescharacteristic
573C2—41no—
574C2—42no—
575C2—5841no—
576C2#2262666311yescharacteristic
577C2#2262669311yescharacteristic
578C2#2262670411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#452×C73C4
socleC22×C73C2×C4×D4
prime radical#17988732C4
cosocleC4×Dic73C24
hypercenter#452#64408
hypocenterC73#18027
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #22626882C2semidirect
2. #22626857C2semidirect
3. #22625575C2semidirect
4. #22625571C2semidirect
5. #22625549C2semidirect
6. #22625548C2semidirect
7. #22625544C2semidirect
8. #22625530C2semidirect
9. SD16⋊#10596833C2semidirect
10. #22625511C2semidirect
11. #22625500C2semidirect
12. #22625288C2semidirect
13. #22625226C2semidirect
14. #22625152C2semidirect
305. C73#18027semidirect

Maximal subgroups

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

subgroupindex
#226268822
#226268572
#226255752
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