Finite Group Databasebeta

← #34031672

Subgroups — #34031672

Order 18688.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626777C211yesmaximal, characteristic, complemented
2#22626775C211yesmaximal, characteristic, complemented
3#22626764C213yesmaximal, complemented
4#22626763C213yesmaximal, complemented
5#22626747C213yesmaximal, complemented
6#22626745C213yesmaximal, complemented
7#22626737C211yesmaximal, characteristic, complemented
8#41×#10596833C211yesmaximal, characteristic, complemented
9Q8×#11924037C211yesmaximal, characteristic, complemented
10C2×Q8×#10596833C211yesmaximal, characteristic, complemented
11#22626673C213yesmaximal, complemented
12#22626654C213yesmaximal, complemented
13#22626626C213yesmaximal, complemented
14C4×#14556678C213yesmaximal, complemented
15#22623521C211yesmaximal, characteristic
16C2×#14556678C2213yes—
17C2×#14556678C2213yes—
18#17989638C2213yes—
19#17989638C2213yes—
20Q8×#10596833C2211yescharacteristic
21Q8×#10596833C2213yes—
22Q8×#10596833C2213yes—
23Q8×#10596833C2211yescharacteristic
24Q8×#10596833C2211yescharacteristic
25Q8×#10596833C2211yescharacteristic
26#17989622C2213yes—
27#17989622C2213yes—
28#17989621C2213yes—
29#17989621C2213yes—
30#17989620C2216yes—
31#17989619C2216yes—
32#17989616C2216yes—
33#17989615C2213yes—
34#17989615C2213yes—
35#17989614C2216yes—
36D4×#10596833C2213yes—
37D4×#10596833C2213yes—
38#17989612C2216yes—
39#17989611C2216yes—
40#17989609C2213yes—
41#17989608C2213yes—
42#17989606C2213yes—
43#17989603C2213yes—
44#17989590C2213yes—
45C2×C4⋊#10596833C2213yes—
46#17989560C2216yes—
47#17989552C2213yes—
48#17989552C2213yes—
49#17989551C2213yes—
50#17989551C2213yes—
51#17989551C2211yescharacteristic
52#17989551C2213yes—
53C4×#11924038C2213yes—
54C4×#11924037C2213yes—
55C4×#11924037C2213yes—
56C4×#11924036C2213yes—
57C2×C4×#10596833C2213yes—
58C2×C4×#10596833C2213yes—
59#17989319C413yes—
60#17989313C411yescharacteristic
61#17989308C413yes—
62#17989289C413yes—
63#17989283C413yes—
64#17989256C411yescharacteristic
65#17989248C411yescharacteristic
66#17989132C2213yes—
67#17989123C2213yes—
68#17989116C2213yes—
69#41×Dic73C2211yescharacteristic
70C4×#11924029C2211yescharacteristic
71C4×#11924027C2213yes—
72C2×Q8×Dic73C2211yescharacteristic
73#548×C73C411yescharacteristic
74#14556678—212no—
75#14556678—212no—
76C2×#11924038C2313yes—
77C2×#11924038C2313yes—
78C2×#11924037C2313yes—
79C2×#11924037C2313yes—
80C2×#11924037C2313yes—
81C2×#11924037C2311yescharacteristic
82C2×#11924036C2313yes—
83C2×#11924036C2313yes—
84C22×#10596833C2313yes—
85C22×#10596833C2313yes—
86C22×#10596833C2313yes—
87C22×#10596833C2311yescharacteristic
88#14556658C2316yes—
89C4⋊#10596833C2313yes—
90C4⋊#10596833C2313yes—
91C4⋊#10596833C2316yes—
92C4⋊#10596833C2313yes—
93C4⋊#10596833C2313yes—
94#14556648C2316yes—
95#14556648C2316yes—
96C22⋊#10596833C2316yes—
97C22⋊#10596833C2316yes—
98#14556646C2313yes—
99#14556646C2313yes—
100#14556646C2311yescharacteristic
101#14556646C2313yes—
102#14556646C2311yescharacteristic
103#14556646C2313yes—
104C4×#10596833C2313yes—
105C4×#10596833C2313yes—
106C4×#10596833C2313yes—
107C4×#10596833C2313yes—
108C4×#10596833C2313yes—
109C4×#10596833C2313yes—
110C4×#10596833C2311yescharacteristic
111C4×#10596833C2313yes—
112C4×#10596833C2311yescharacteristic
113C4×#10596833C2313yes—
114C2×#11924029C2×C411yescharacteristic
115C2×#11924027C2313yes—
116#14556590C2×C413yes—
117#14556587C2×C413yes—
118#14556586C2×C416yes—
119#14556584C2×C416yes—
120#14556583C2×C413yes—
121#14556582C2×C413yes—
122#14556580C2×C413yes—
123#14556577C2×C416yes—
124#14556575C2×C416yes—
125#14556573C2×C416yes—
126#14556567C2×C413yes—
127#14556565C2×C413yes—
128#14556565C2×C413yes—
129#14556563C2×C416yes—
130#14556562C2×C413yes—
131Q8×Dic73C2×C411yescharacteristic
132Q8×Dic73C2×C411yescharacteristic
133Q8×Dic73C2×C413yes—
134Q8×Dic73C2313yes—
135Q8×Dic73C2311yescharacteristic
136C4×Dic146C2×C413yes—
137#14556556C2×C413yes—
138C4×D292C2×C413yes—
139D4×Dic73C2×C413yes—
140D4×Dic73C2×C413yes—
141D4×Dic73C2313yes—
142#14556553C2313yes—
143C4×C22⋊D73C2×C413yes—
144C4×C22⋊D73C2×C413yes—
145#14556551C2×C413yes—
146#14556551C2×C413yes—
147#14556551C2316yes—
148#14556550C2×C413yes—
149#14556549C2×C413yes—
150#14556549C2313yes—
151#14556547C2311yescharacteristic
152#14556547C2313yes—
153C2×C4⋊Dic73C2×C413yes—
154C2×C4×Dic73C2313yes—
155C2×C4×Dic73C2313yes—
156C42×D73C2313yes—
157C2×#41×C73C2×C411yescharacteristic
158C22×Q8×C73C2×C411yescharacteristic
159#124×C73C2×C413yes—
160C22⋊Q8×C73C2×C413yes—
161#121×C73C2×C413yes—
162C4×Q8×C73C2×C411yescharacteristic
163C4×D4×C73C2×C413yes—
164#11924038—26no—
165#11924038—26no—
166#11924037—26no—
167#11924037—26no—
168#11924037—26no—
169#11924037#4114yescomplemented
170#11924036—26no—
171#11924036—26no—
172C2×#10596833—26no—
173C2×#10596833—26no—
174C2×#10596833—26no—
175C2×#10596833#4114yescomplemented
176C2×#10596833C2413yes—
177C2×#10596833C2413yes—
178C2×#10596833C2413yes—
179C2×#10596833C2413yes—
180C2×#10596833C2411yescharacteristic
181C2×#10596833C2411yescharacteristic
182C2×#10596833C2411yescharacteristic
183C2×#10596833C2411yescharacteristic
184#11924029—24no—
185#11924027—26no—
186#11924027—26no—
187C2×Dic146C22×C413yes—
188C2×D292C22×C413yes—
189C2×C22⋊D73C22×C413yes—
190C2×C22⋊D73C22×C413yes—
191C2×C22⋊D73C22×C416yes—
192C22×Dic73C22×C413yes—
193C22×Dic73C22×C413yes—
194C22×Dic73C2413yes—
195C22×Dic73C2411yescharacteristic
196C2×C4×D73C22×C413yes—
197C2×C4×D73C22×C413yes—
198C2×C4×D73C2413yes—
199C4⋊Dic73C22×C413yes—
200C4⋊Dic73C22×C413yes—
201C4⋊Dic73C22×C416yes—
202C4⋊Dic73C22×C413yes—
203C4⋊Dic73C22×C413yes—
204#11924007C22×C411yescharacteristic
205#11924007C22×C413yes—
206#11924007C22×C411yescharacteristic
207#11924007C22×C413yes—
208#11924007C22×C416yes—
209C22⋊Dic73C22×C411yescharacteristic
210C22⋊Dic73C22×C413yes—
211C22⋊Dic73C22×C416yes—
212#11924005C22×C416yes—
213#11924005C22×C413yes—
214#11924005C22×C413yes—
215#11924005C22×C416yes—
216C4×Dic73C22×C413yes—
217C4×Dic73C22×C413yes—
218C4×Dic73C22×C413yes—
219C4×Dic73C22×C413yes—
220C4×Dic73C22×C413yes—
221C4×Dic73C2411yescharacteristic
222C4×Dic73C2413yes—
223C4×Dic73C2411yescharacteristic
224C4×Dic73C2413yes—
225#41×C73—24no—
226C2×Q8×C73—24no—
227C2×Q8×C73C22×C413yes—
228C2×Q8×C73C22×C411yescharacteristic
229C2×D4×C73C22×C413yes—
230C2×D4×C73C22×C413yes—
231C22×C4×C73C22×C413yes—
232C22×C4×C73C22×C413yes—
233C4⋊C4×C73C22×C413yes—
234C4⋊C4×C73C22×C413yes—
235C22⋊C4×C73C22×C411yescharacteristic
236C22⋊C4×C73C22×C413yes—
237C22⋊C4×C73C22×C416yes—
238C42×C73C22×C413yes—
239C23×C73C23×C413yes—
240C23×C73C23×C411yescharacteristic
241C22×D73C23×C413yes—
242Q8×C73—26no—
243Q8×C73#13114yescomplemented
244D4×C73—26no—
245D4×C73—26no—
246C2×C4×C73—26no—
247C2×C4×C73—26no—
248C2×C4×C73C23×C411yescharacteristic
249C2×C4×C73C23×C413yes—
250C2×C4×C73C23×C413yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C23×C413yes—
253C22⋊D73—26no—
254C22⋊D73—26no—
255C22⋊D73—46no—
256C2×Dic73—23no—
257C2×Dic73—23no—
258C2×Dic73—43no—
259C2×Dic73—43no—
260C2×Dic73C23×C411yescharacteristic
261C2×Dic73C23×C413yes—
262C2×Dic73C23×C411yescharacteristic
263C2×Dic73C23×C413yes—
264C2×Dic73C23×C413yes—
265C2×Dic73C23×C411yescharacteristic
266C2×Dic73C2×#4112yes—
267C2×Dic73C2×#4112yes—
268C2×Dic73C2511yescharacteristic
269D292—26no—
270C4×D73—26no—
271C4×D73—23no—
272C4×D73—23no—
273C4×D73—43no—
274Dic146—26no—
275#10596833—23no—
276#10596833—23no—
277#10596833—23no—
278#10596833—23no—
279#10596833C2×#4112yes—
280#10596833C2×#4112yes—
281#10596833C2×#4112yes—
282#10596833C2×#4112yes—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—23no—
286C22×C73—23no—
287C22×C73—23no—
288C22×C73C4×#4112yes—
289C22×C73C24×C411yescharacteristic
290C2×D73—23no—
291C2×D73—23no—
292C2×D73—23no—
293C4×C73—23no—
294C4×C73—21no—
295C4×C73—23no—
296C4×C73—21no—
297C4×C73C2×#13116yes—
298Dic73—23no—
299Dic73—21no—
300Dic73—23no—
301Dic73—21no—
302Dic73—23no—
303Dic73C4×#4112yes—
304Dic73C22×#4111yescharacteristic
305Dic73C22×#4111yescharacteristic
306#60218—731nomaximal, Sylow
307C2×C73—22no—
308C2×C73—43no—
309C2×C73C2×C4×#4111yescharacteristic
310C2×C73C22×#13111yescharacteristic
311C2×C73#357311yescharacteristic
312D73—43no—
313#3480—731no—
314#2971—731no—
315#2971—731no—
316#2969—733no—
317#2969—733no—
318#2966—733no—
319#2966—733no—
320C8×#41—731no—
321C8×#41—731no—
322Q8×M16—731no—
323C2×C8×Q8—731no—
324#2935—733no—
325#2931—733no—
326#2925—733no—
327C4×#131—733no—
328C73#6021811yescharacteristic, complemented, Sylow
329C2×#131—733no—
330C2×#131—733no—
331#559—733no—
332#557—731no—
333#556—733no—
334#553—733no—
335#553—733no—
336#550—731no—
337#548—731no—
338#548#6440811yescharacteristic
339#532—733no—
340#529—733no—
341#529—733no—
342C4×#41—731no—
343C4×#41—731no—
344C4×#41—733no—
345C2×C4×Q8—731no—
346#446—733no—
347#446—733no—
348C8×Q8—731no—
349C8×Q8—733no—
350C8×Q8—733no—
351C8×Q8—731no—
352C8×Q8—731no—
353C8×Q8—731no—
354#436—736no—
355#436—733no—
356#436—733no—
357#436—733no—
358#436—733no—
359#435—736no—
360#435—736no—
361C8×D4—733no—
362C8×D4—733no—
363C8×D4—736no—
364C8×D4—733no—
365C8×D4—733no—
366#433—736no—
367#433—736no—
368#432—733no—
369#432—733no—
370#431—733no—
371#431—733no—
372#423—733no—
373C2×C4⋊C8—733no—
374#408—736no—
375#405—731no—
376#405—733no—
377#405—733no—
378#405—733no—
379#405—733no—
380#405—733no—
381C4×M16—733no—
382C4×M16—733no—
383C4×M16—733no—
384C2×C4×C8—733no—
385C2×C4×C8—733no—
386C2×C4×C8—733no—
387C2×#41—731no—
388C2×#41—733no—
389C2×#41C2×#6440811yescharacteristic
390C22×Q8C2×#6440811yescharacteristic
391#131—14612no—
392#131—14612no—
393C2×M16—731no—
394C2×M16—733no—
395C2×M16—733no—
396C2×M16—733no—
397C2×M16—733no—
398C2×M16—733no—
399C22×C8—731no—
400C22×C8—733no—
401C22×C8—733no—
402C22×C8—733no—
403C22×C8—733no—
404C22×C8—733no—
405C4⋊Q8—733no—
406#127—733no—
407#126—736no—
408#126—736no—
409#125—733no—
410#125—733no—
411#124—733no—
412#124—736no—
413#124C2×#6440813yes—
414#123—736no—
415#123—736no—
416C22⋊Q8—733no—
417C22⋊Q8C2×#6440813yes—
418#121—733no—
419#121—736no—
420#121—733no—
421#121—733no—
422#121C2×#6440813yes—
423C4×Q8—731no—
424C4×Q8—733no—
425C4×Q8—731no—
426C4×Q8—733no—
427C4×Q8—731no—
428C4×Q8—733no—
429C4×Q8—733no—
430C4×Q8C2×#6440811yescharacteristic
431C4×D4—733no—
432C4×D4—733no—
433C4×D4—733no—
434C4×D4—733no—
435C4×D4—733no—
436C4×D4—733no—
437C4×D4—733no—
438C4×D4—733no—
439C4×D4—736no—
440C4×D4—733no—
441C4×D4C2×#6440813yes—
442#117—733no—
443#117—733no—
444#117—733no—
445#117—731no—
446#117—733no—
447C2×C4⋊C4—733no—
448C2×C42—733no—
449C2×C42—733no—
450C2×C42—733no—
451C4⋊C8—733no—
452C4⋊C8—733no—
453C4⋊C8—736no—
454C4⋊C8—733no—
455C4⋊C8—733no—
456C4⋊C8—736no—
457C22⋊C8—736no—
458C22⋊C8—736no—
459C22⋊C8—736no—
460C22⋊C8—736no—
461#97—731no—
462#97—731no—
463#97—733no—
464#97—733no—
465#97—733no—
466#97—733no—
467C4×C8—731no—
468C4×C8—731no—
469C4×C8—733no—
470C4×C8—733no—
471C4×C8—733no—
472C4×C8—733no—
473C4×C8—733no—
474C4×C8—733no—
475C4×C8—733no—
476C4×C8—733no—
477#41—24no—
478#41—1466no—
479#41—1466no—
480#41—1464no—
481C2×Q8—24no—
482C2×Q8—733no—
483C2×Q8C22×#6440813yes—
484C2×Q8C22×#6440811yescharacteristic
485C2×D4—736no—
486C2×D4—733no—
487C2×D4—733no—
488C2×D4—733no—
489C2×D4C22×#6440813yes—
490C2×D4C22×#6440813yes—
491C22×C4—733no—
492C22×C4—733no—
493C22×C4—733no—
494C22×C4—731no—
495C22×C4—733no—
496C22×C4—733no—
497C22×C4—733no—
498C22×C4C22×#6440813yes—
499C22×C4C22×#6440813yes—
500M16—734no—
501M16—1466no—
502M16—1466no—
503M16—1466no—
504M16—1466no—
505M16—1466no—
506C2×C8—734no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—733no—
512C2×C8—733no—
513C2×C8—733no—
514C2×C8—733no—
515C2×C8—1466no—
516C2×C8—1466no—
517C2×C8—1466no—
518C2×C8—1466no—
519C2×C8—1466no—
520C4⋊C4—731no—
521C4⋊C4—731no—
522C4⋊C4—733no—
523C4⋊C4—733no—
524C4⋊C4—736no—
525C4⋊C4—733no—
526C4⋊C4—733no—
527C4⋊C4—736no—
528C4⋊C4—733no—
529C4⋊C4—733no—
530C4⋊C4C22×#6440813yes—
531C4⋊C4C22×#6440813yes—
532C22⋊C4—731no—
533C22⋊C4—733no—
534C22⋊C4—736no—
535C22⋊C4—733no—
536C22⋊C4—736no—
537C22⋊C4—733no—
538C22⋊C4—736no—
539C22⋊C4C22×#6440811yescharacteristic
540C22⋊C4C22×#6440813yes—
541C22⋊C4C22×#6440816yes—
542C42—733no—
543C42—733no—
544C42—733no—
545C42—733no—
546C42—733no—
547C42—731no—
548C42—731no—
549C42—733no—
550C42—733no—
551C42C22×#6440813yes—
552C23—733no—
553C23C23×#6440813yes—
554C23C23×#6440811yescharacteristic
555Q8—26no—
556Q8—1466no—
557Q8#1455667814yescomplemented
558D4—26no—
559D4—26no—
560D4—2926no—
561D4—1466no—
562D4—1466no—
563D4—1466no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—2923no—
567C2×C4—2923no—
568C2×C4—2923no—
569C2×C4—732no—
570C2×C4—732no—
571C2×C4—731no—
572C2×C4—731no—
573C2×C4—731no—
574C2×C4—733no—
575C2×C4—731no—
576C2×C4—733no—
577C2×C4—733no—
578C2×C4—1463no—
579C2×C4—1463no—
580C2×C4—1463no—
581C2×C4—1463no—
582C2×C4—1466no—
583C2×C4C23×#6440811yescharacteristic
584C2×C4C23×#6440813yes—
585C2×C4C23×#6440811yescharacteristic
586C2×C4C23×#6440813yes—
587C2×C4C23×#6440813yes—
588C8—732no—
589C8—732no—
590C8—732no—
591C8—732no—
592C8—1463no—
593C8—1463no—
594C8—1463no—
595C8—1463no—
596C22—21no—
597C22—21no—
598C22—23no—
599C22—23no—
600C22—23no—
601C22—1463no—
602C22—1463no—
603C22—1463no—
604C22#41×#6440812yes—
605C22C24×#6440811yescharacteristic
606C4—23no—
607C4—21no—
608C4—23no—
609C4—21no—
610C4—731no—
611C4—731no—
612C4—732no—
613C4—1463no—
614C4—1461no—
615C4—1463no—
616C4—1461no—
617C4—1463no—
618C4C2×#1455667816yes—
619C2—22no—
620C2—43no—
621C2—2923no—
622C2C2×#41×#6440811yescharacteristic
623C2C22×#1455667811yescharacteristic
624C2#2262692711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#548×C73C4
prime radical#22623521C2
cosocleC2×Dic73C25
hypercenter#548#64408
hypocenterC73#60218
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22626777C2semidirect
2. #22626775C2semidirect
3. #22626764C2semidirect
4. #22626763C2semidirect
5. #22626747C2semidirect
6. #22626745C2semidirect
7. #22626737C2semidirect
8. #41×#10596833C2semidirect
9. Q8×#11924037C2semidirect
10. C2×Q8×#10596833C2semidirect
11. #22626673C2semidirect
12. #22626654C2semidirect
13. #22626626C2semidirect
14. C4×#14556678C2semidirect
169. #11924037#41semidirect
175. C2×#10596833#41semidirect
243. Q8×C73#131semidirect
328. C73#60218semidirect
557. Q8#14556678semidirect

Maximal subgroups

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

subgroupindex
#226267772
#226267752
#226267642
#226267632
#226267472
#226267452
#226267372
#41×#105968332
Q8×#119240372
C2×Q8×#105968332
#226266732
#226266542
#226266262
C4×#145566782
#226235212
#6021873