Finite Group Databasebeta

← #34032659

Subgroups — #34032659

Order 18688.

1744 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626874C213yesmaximal, complemented
2#22626873C213yesmaximal, complemented
3#22626863C211yesmaximal, characteristic, complemented
4#22626862C213yesmaximal, complemented
5#22626840C213yesmaximal, complemented
6#22626825C211yesmaximal, characteristic, complemented
7#22626824C211yesmaximal, characteristic, complemented
8#22626763C213yesmaximal, complemented
9#22626762C213yesmaximal, complemented
10#22626741C213yesmaximal, complemented
11#41×#10596833C211yesmaximal, characteristic, complemented
12Q8×#11924038C211yesmaximal, characteristic, complemented
13Q8×#11924037C211yesmaximal, characteristic, complemented
14#22626719C213yesmaximal, complemented
15#22623547C211yesmaximal, characteristic
16#17989639C2213yes—
17#17989639C2213yes—
18#17989638C2213yes—
19#17989638C2213yes—
20#17989637C2213yes—
21#17989637C2213yes—
22#17989636C2213yes—
23#17989636C2213yes—
24#17989636C2216yes—
25Q8×#10596833C2211yescharacteristic
26Q8×#10596833C2213yes—
27Q8×#10596833C2211yescharacteristic
28Q8×#10596833C2213yes—
29Q8×#10596833C2211yescharacteristic
30Q8×#10596833C2211yescharacteristic
31#17989622C2216yes—
32#17989622C2213yes—
33#17989622C2213yes—
34#17989620C2213yes—
35#17989620C2216yes—
36#17989620C2211yescharacteristic
37#17989618C2213yes—
38#17989618C2213yes—
39#17989616C2213yes—
40#17989616C2213yes—
41D4×#10596833C2213yes—
42D4×#10596833C2213yes—
43#17989611C2216yes—
44#17989611C2216yes—
45#17989606C2213yes—
46#17989605C2213yes—
47#17989604C2213yes—
48#17989603C2213yes—
49#17989603C2213yes—
50#17989603C2213yes—
51#17989593C2216yes—
52#17989591C2213yes—
53#17989590C2213yes—
54#17989590C2213yes—
55#17989551C2216yes—
56C4×#11924038C2213yes—
57C4×#11924037C2213yes—
58C2×C4×#10596833C2213yes—
59#17989313C411yescharacteristic
60#17989306C413yes—
61#17989277C413yes—
62#17989271C413yes—
63Q8×C22⋊D73C411yescharacteristic
64#17989244C411yescharacteristic
65#17989237C413yes—
66#17989139C2213yes—
67#17989118C2213yes—
68#17989116C2213yes—
69#41×Dic73C2211yescharacteristic
70C4×#11924027C2213yes—
71C2×Q8×Dic73C2211yescharacteristic
72C4×Q8×D73C2211yescharacteristic
73#548×C73C411yescharacteristic
74C2×#11924038C2311yescharacteristic
75C2×#11924038C2313yes—
76C2×#11924037C2311yescharacteristic
77C2×#11924037C2313yes—
78C2×#11924037C2316yes—
79C22×#10596833C2313yes—
80C22×#10596833C2313yes—
81#14556658C2313yes—
82#14556658C2313yes—
83#14556658C2313yes—
84#14556658C2316yes—
85#14556658C2311yescharacteristic
86C4⋊#10596833C2313yes—
87C4⋊#10596833C2313yes—
88C4⋊#10596833C2313yes—
89C4⋊#10596833C2313yes—
90C4⋊#10596833C2313yes—
91C4⋊#10596833C2313yes—
92C4⋊#10596833C2316yes—
93#14556648C2311yescharacteristic
94#14556648C2313yes—
95C22⋊#10596833C2311yescharacteristic
96C22⋊#10596833C2313yes—
97C22⋊#10596833C2316yes—
98C22⋊#10596833C2313yes—
99C22⋊#10596833C2313yes—
100#14556646C2313yes—
101#14556646C2313yes—
102#14556646C2313yes—
103#14556646C2313yes—
104C4×#10596833C2313yes—
105C4×#10596833C2316yes—
106C4×#10596833C2313yes—
107C4×#10596833C2313yes—
108C4×#10596833C2313yes—
109C4×#10596833C2316yes—
110C4×#10596833C2313yes—
111C4×#10596833C2311yescharacteristic
112C2×#11924027C2×C413yes—
113C2×Q8×D73C2×C411yescharacteristic
114#14556590C2×C413yes—
115#14556589C2×C413yes—
116#14556584C2×C416yes—
117#14556584C2×C416yes—
118#14556583C2×C413yes—
119#14556582C2×C413yes—
120#14556581C2×C416yes—
121#14556578C2×C413yes—
122#14556573C2×C416yes—
123#14556572C2×C416yes—
124#14556570C2×C413yes—
125#14556570C2×C413yes—
126#14556568C2×C416yes—
127#14556567C2×C413yes—
128#14556562C2×C413yes—
129Q8×Dic73C2×C411yescharacteristic
130Q8×Dic73C2×C411yescharacteristic
131Q8×Dic73C2×C413yes—
132Q8×Dic73C2311yescharacteristic
133Q8×Dic73C2313yes—
134C4×Dic146C2×C413yes—
135C4×Dic146C2×C413yes—
136#14556556C2313yes—
137#14556556C2313yes—
138D4×Dic73C2×C413yes—
139D4×Dic73C2313yes—
140D4×Dic73C2313yes—
141C4×C22⋊D73C2×C413yes—
142C4×C22⋊D73C2×C413yes—
143#14556551C2311yescharacteristic
144#14556551C2313yes—
145#14556550C2×C413yes—
146#14556549C2×C413yes—
147#14556547C2×C413yes—
148#14556547C2×C413yes—
149#14556547C2316yes—
150C2×C4⋊Dic73C2×C413yes—
151C4⋊C4×D73C2×C413yes—
152C2×C4×Dic73C2313yes—
153C2×C4×Dic73C2313yes—
154C42×D73C2313yes—
155C2×#41×C73C2×C411yescharacteristic
156C22×Q8×C73C2×C411yescharacteristic
157#124×C73C2×C413yes—
158C22⋊Q8×C73C2×C413yes—
159#121×C73C2×C413yes—
160C4×Q8×C73C2×C411yescharacteristic
161C4×D4×C73C2×C413yes—
162#11924038—22no—
163#11924038—43no—
164#11924037—22no—
165#11924037—43no—
166#11924037—46no—
167C2×#10596833—26no—
168C2×#10596833—43no—
169C2×#10596833C2411yescharacteristic
170C2×#10596833C2413yes—
171C2×#10596833C2413yes—
172C2×#10596833C2413yes—
173C2×#10596833C2413yes—
174C2×#10596833C2411yescharacteristic
175C2×#10596833C2411yescharacteristic
176C2×#10596833C2411yescharacteristic
177#11924027—46no—
178C2×Dic146C22×C413yes—
179C2×Dic146C22×C413yes—
180Q8×D73—24no—
181C2×C22⋊D73C22×C411yescharacteristic
182C2×C22⋊D73C22×C413yes—
183C22×Dic73C22×C413yes—
184C22×Dic73C22×C413yes—
185C22×Dic73C2413yes—
186C22×Dic73C2411yescharacteristic
187C2×C4×D73C22×C413yes—
188C2×C4×D73C22×C413yes—
189C2×C4×D73C2411yescharacteristic
190C4⋊Dic73C22×C413yes—
191C4⋊Dic73C22×C413yes—
192C4⋊Dic73C22×C416yes—
193C4⋊Dic73C22×C413yes—
194C4⋊Dic73C22×C413yes—
195#11924007C22×C416yes—
196#11924007C22×C411yescharacteristic
197#11924007C22×C413yes—
198#11924007C22×C413yes—
199#11924007C22×C413yes—
200#11924007C22×C416yes—
201#11924007C22×C416yes—
202C22⋊Dic73C22×C413yes—
203C22⋊Dic73C22×C413yes—
204C22⋊Dic73C22×C416yes—
205C22⋊Dic73C22×C416yes—
206#11924005C22×C411yescharacteristic
207#11924005C22×C413yes—
208C4×Dic73C22×C413yes—
209C4×Dic73C22×C413yes—
210C4×Dic73C22×C413yes—
211C4×Dic73C22×C413yes—
212C4×Dic73C22×C413yes—
213C4×Dic73C2413yes—
214C4×Dic73C2413yes—
215C4×Dic73C2411yescharacteristic
216C4×Dic73C2413yes—
217#41×C73—24no—
218C2×Q8×C73—24no—
219C2×Q8×C73C22×C413yes—
220C2×Q8×C73C22×C411yescharacteristic
221C2×D4×C73C22×C413yes—
222C2×D4×C73C22×C413yes—
223C22×C4×C73C22×C413yes—
224C22×C4×C73C22×C413yes—
225C4⋊C4×C73C22×C413yes—
226C4⋊C4×C73C22×C413yes—
227C22⋊C4×C73C22×C411yescharacteristic
228C22⋊C4×C73C22×C413yes—
229C22⋊C4×C73C22×C416yes—
230C42×C73C22×C413yes—
231C23×C73C23×C413yes—
232C23×C73C23×C411yescharacteristic
233C22×D73C23×C411yescharacteristic
234Q8×C73—26no—
235Q8×C73#13114yes—
236D4×C73—26no—
237D4×C73—43no—
238C2×C4×C73—26no—
239C2×C4×C73—26no—
240C2×C4×C73C23×C411yescharacteristic
241C2×C4×C73C23×C413yes—
242C2×C4×C73C23×C413yes—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C23×C413yes—
245C22⋊D73—22no—
246C22⋊D73—43no—
247C2×Dic73—26no—
248C2×Dic73—23no—
249C2×Dic73—21no—
250C2×Dic73—23no—
251C2×Dic73—21no—
252C2×Dic73—43no—
253C2×Dic73C23×C411yescharacteristic
254C2×Dic73C23×C413yes—
255C2×Dic73C23×C413yes—
256C2×Dic73C23×C413yes—
257C2×Dic73C23×C413yes—
258C2×Dic73C23×C411yescharacteristic
259C2×Dic73C2511yescharacteristic
260C4×D73—26no—
261C4×D73—21no—
262C4×D73—21no—
263C4×D73—43no—
264Dic146—26no—
265Dic146—26no—
266#10596833—23no—
267#10596833—21no—
268#10596833—21no—
269#10596833—21no—
270#10596833—23no—
271#10596833—23no—
272#10596833—23no—
273#105968332−1+412yes—
274C22×C73—23no—
275C22×C73—21no—
276C22×C73—21no—
277C22×C73—23no—
278C22×C73—23no—
279C22×C73—21no—
280C22×C73C24×C411yescharacteristic
281C2×D73—21no—
282C2×D73—21no—
283C2×D73—21no—
284C4×C73—21no—
285C4×C73—21no—
286C4×C73—23no—
287C4×C73—23no—
288C4×C73C2×#13116yes—
289Dic73—23no—
290Dic73—21no—
291Dic73—23no—
292Dic73—23no—
293Dic73—23no—
294Dic73#56812yes—
295Dic73C2×2−1+411yescharacteristic
296Dic73#58511yescharacteristic
297#60345—731nomaximal, Sylow
298C2×C73—41no—
299C2×C73—43no—
300C2×C73C4×2−1+411yescharacteristic
301C2×C73C22×#13111yescharacteristic
302C2×C73C2×#56811yescharacteristic
303D73—41no—
304#3482—731no—
305#2993—733no—
306#2993—733no—
307#2991—733no—
308#2991—731no—
309#2986—733no—
310#2982—731no—
311#2982—731no—
312#2969—733no—
313#2969—733no—
314#2965—733no—
315C8×#41—731no—
316Q8×M16—731no—
317Q8×M16—731no—
318#2960—733no—
319C73#6034511yescharacteristic, complemented, Sylow
320#557—731no—
321#556—733no—
322#552—733no—
323#552—733no—
324D4×Q8—731no—
325#548—731no—
326#548#6440811yescharacteristic
327#547—733no—
328#533—733no—
329#529—733no—
330#529—733no—
331C4×#41—731no—
332C4×#41—733no—
333C2×C4×Q8—731no—
334C2×C4×Q8—731no—
335#446—733no—
336#446—733no—
337#446—733no—
338#446—733no—
339#446—733no—
340#446—733no—
341C8×Q8—733no—
342C8×Q8—731no—
343C8×Q8—731no—
344C8×Q8—731no—
345C8×Q8—731no—
346C8×Q8—733no—
347C8×Q8—733no—
348C8×Q8—736no—
349C8×Q8—733no—
350#436—731no—
351#436—733no—
352#436—736no—
353#436—733no—
354#436—733no—
355#436—736no—
356#435—733no—
357#435—733no—
358#435—733no—
359#435—733no—
360C8×D4—733no—
361C8×D4—733no—
362#433—736no—
363#433—736no—
364#431—733no—
365#431—733no—
366#431—733no—
367#431—733no—
368#431—733no—
369#431—733no—
370#424—736no—
371#423—733no—
372#423—733no—
373#423—733no—
374#405—736no—
375C4×M16—733no—
376C4×M16—733no—
377C2×C4×C8—733no—
378C2×#41—733no—
379C2×#41C2×#6440811yescharacteristic
380C22×Q8—731no—
381C22×Q8C2×#6440811yescharacteristic
382C2×M16—733no—
383C2×M16—736no—
384C2×M16—733no—
385C2×M16—731no—
386C2×M16—731no—
387C22×C8—733no—
388C22×C8—733no—
389C4⋊Q8—733no—
390C4⋊Q8—733no—
391#126—736no—
392#126—736no—
393#125—733no—
394#125—736no—
395#125—733no—
396#124—733no—
397#124C2×#6440813yes—
398#123—736no—
399#123—736no—
400C22⋊Q8—733no—
401C22⋊Q8—733no—
402C22⋊Q8—733no—
403C22⋊Q8—736no—
404C22⋊Q8C2×#6440813yes—
405#121—733no—
406#121C2×#6440813yes—
407C4×Q8—731no—
408C4×Q8—733no—
409C4×Q8—731no—
410C4×Q8—733no—
411C4×Q8—731no—
412C4×Q8—733no—
413C4×Q8—733no—
414C4×Q8—733no—
415C4×Q8—733no—
416C4×Q8C2×#6440811yescharacteristic
417C4×D4—733no—
418C4×D4—733no—
419C4×D4—733no—
420C4×D4—733no—
421C4×D4—733no—
422C4×D4—731no—
423C4×D4—733no—
424C4×D4C2×#6440813yes—
425#117—733no—
426#117—733no—
427#117—733no—
428#117—733no—
429#117—736no—
430C2×C4⋊C4—733no—
431C2×C4⋊C4—733no—
432C2×C42—733no—
433C2×C42—733no—
434C2×C42—733no—
435C4⋊C8—733no—
436C4⋊C8—733no—
437C4⋊C8—733no—
438C4⋊C8—733no—
439C4⋊C8—736no—
440C4⋊C8—736no—
441C4⋊C8—733no—
442C4⋊C8—733no—
443C4⋊C8—733no—
444C4⋊C8—731no—
445C4⋊C8—733no—
446C4⋊C8—733no—
447C22⋊C8—731no—
448C22⋊C8—731no—
449C22⋊C8—733no—
450C22⋊C8—733no—
451C22⋊C8—736no—
452C22⋊C8—733no—
453C22⋊C8—733no—
454#97—733no—
455#97—733no—
456#97—733no—
457#97—733no—
458C4×C8—731no—
459C4×C8—733no—
460C4×C8—733no—
461C4×C8—733no—
462C4×C8—733no—
463C4×C8—733no—
464C4×C8—736no—
465C4×C8—736no—
466#41—24no—
467#41—2926no—
468C2×Q8—24no—
469C2×Q8—733no—
470C2×Q8—733no—
471C2×Q8—1464no—
472C2×Q8C22×#6440813yes—
473C2×Q8C22×#6440811yescharacteristic
474C2×D4—733no—
475C2×D4—731no—
476C2×D4C22×#6440813yes—
477C2×D4C22×#6440813yes—
478C22×C4—733no—
479C22×C4—733no—
480C22×C4—733no—
481C22×C4—731no—
482C22×C4—733no—
483C22×C4—733no—
484C22×C4—731no—
485C22×C4C22×#6440813yes—
486C22×C4C22×#6440813yes—
487M16—2923no—
488M16—2926no—
489M16—2923no—
490M16—1462no—
491M16—1462no—
492C2×C8—2923no—
493C2×C8—731no—
494C2×C8—733no—
495C2×C8—731no—
496C2×C8—731no—
497C2×C8—731no—
498C2×C8—733no—
499C2×C8—733no—
500C2×C8—733no—
501C2×C8—1466no—
502C4⋊C4—736no—
503C4⋊C4—731no—
504C4⋊C4—733no—
505C4⋊C4—733no—
506C4⋊C4—733no—
507C4⋊C4—736no—
508C4⋊C4—736no—
509C4⋊C4—733no—
510C4⋊C4—733no—
511C4⋊C4—736no—
512C4⋊C4—733no—
513C4⋊C4—733no—
514C4⋊C4C22×#6440813yes—
515C4⋊C4C22×#6440813yes—
516C22⋊C4—733no—
517C22⋊C4—736no—
518C22⋊C4—733no—
519C22⋊C4—736no—
520C22⋊C4—731no—
521C22⋊C4—733no—
522C22⋊C4C22×#6440811yescharacteristic
523C22⋊C4C22×#6440813yes—
524C22⋊C4C22×#6440816yes—
525C42—733no—
526C42—733no—
527C42—733no—
528C42—733no—
529C42—733no—
530C42—731no—
531C42—733no—
532C42—733no—
533C42—733no—
534C42C22×#6440813yes—
535C23—731no—
536C23C23×#6440813yes—
537C23C23×#6440811yescharacteristic
538Q8—26no—
539Q8—1466no—
540Q8—1466no—
541Q8#1455667814yes—
542D4—26no—
543D4—43no—
544D4—2923no—
545D4—1462no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—2923no—
549C2×C4—2923no—
550C2×C4—731no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—733no—
554C2×C4—733no—
555C2×C4—733no—
556C2×C4—733no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1463no—
560C2×C4—1463no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1466no—
564C2×C4—1466no—
565C2×C4C23×#6440811yescharacteristic
566C2×C4C23×#6440811yescharacteristic
567C2×C4C23×#6440813yes—
568C2×C4C23×#6440813yes—
569C2×C4C23×#6440813yes—
570C8—732no—
571C8—1461no—
572C8—1463no—
573C8—1463no—
574C8—1463no—
575C8—1463no—
576C8—1461no—
577C8—1461no—
578C22—23no—
579C22—21no—
580C22—21no—
581C22—23no—
582C22—21no—
583C22—23no—
584C22—1461no—
585C22—1461no—
586C22—1461no—
587C22C24×#6440811yescharacteristic
588C4—21no—
589C4—21no—
590C4—23no—
591C4—23no—
592C4—731no—
593C4—731no—
594C4—732no—
595C4—1461no—
596C4—1463no—
597C4—1463no—
598C4—1463no—
599C4—1463no—
600C4C2×#1455667816yes—
601C2—41no—
602C2—43no—
603C2—2921no—
604C22−1+4×#6440811yescharacteristic
605C2C22×#1455667811yescharacteristic
606C2D73⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#548×C73C4
prime radical#22623547C2
cosocleC2×Dic73C25
hypercenter#548#64408
hypocenterC73#60345
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22626874C2semidirect
2. #22626873C2semidirect
3. #22626863C2semidirect
4. #22626862C2semidirect
5. #22626840C2semidirect
6. #22626825C2semidirect
7. #22626824C2semidirect
8. #22626763C2semidirect
9. #22626762C2semidirect
10. #22626741C2semidirect
11. #41×#10596833C2semidirect
12. Q8×#11924038C2semidirect
13. Q8×#11924037C2semidirect
14. #22626719C2semidirect
319. C73#60345semidirect

Maximal subgroups

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

subgroupindex
#226268742
#226268732
#226268632
#226268622
#226268402
#226268252
#226268242
#226267632
#226267622
#226267412
#41×#105968332
Q8×#119240382
Q8×#119240372
#226267192
#226235472
#6034573