Finite Group Databasebeta

← #33108063

Subgroups — #33108063

Order 18688.

708 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619715C211yesmaximal, characteristic
2#22618875C211yesmaximal, characteristic, complemented
3#22618497C211yesmaximal, characteristic
4#22610906C211yesmaximal, characteristic, complemented
5#22610705C211yesmaximal, characteristic, complemented
6#22610684C211yesmaximal, characteristic, complemented
7#22610683C211yesmaximal, characteristic, complemented
8#22610530C211yesmaximal, characteristic, complemented
9#22610529C211yesmaximal, characteristic, complemented
10#22610208C211yesmaximal, characteristic
11#22610206C211yesmaximal, characteristic
12#22609785C211yesmaximal, characteristic, complemented
13#22609776C211yesmaximal, characteristic
14#104×Dic73C211yesmaximal, characteristic
15C73×#1765C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17#17989140C2211yescharacteristic
18#41×Dic73C2211yescharacteristic
19#17988896C411yescharacteristic, complemented
20#17988892C411yescharacteristic, complemented
21#17988889C411yescharacteristic, complemented
22#17988885C411yescharacteristic, complemented
23#17988880C411yescharacteristic, complemented
24#17988875C411yescharacteristic, complemented
25#17988870C411yescharacteristic, complemented
26#17988865C411yescharacteristic, complemented
27#17988717—21no—
28#17988715—21no—
29#17988715—21no—
30C8×Dic146—21no—
31C2×#14556527C2211yescharacteristic
32C2×#14556526C2211yescharacteristic
33#17988625C2211yescharacteristic
34#17988623C2211yescharacteristic
35#17988615—21no—
36#17988615C2211yescharacteristic
37#17988614C2211yescharacteristic
38#17988613—21no—
39#17988613C2211yescharacteristic
40#17988613C2211yescharacteristic
41#17988610—21no—
42#17988610C2211yescharacteristic
43C2×#14556520—21no—
44C2×#14556520C2211yescharacteristic
45C2×#14556520C2211yescharacteristic
46C2×#14556518C2211yescharacteristic
47#17988583C2211yescharacteristic
48#17988545C2211yescharacteristic
49M16×Dic73C2211yescharacteristic
50C2×#14556502C2211yescharacteristic
51C42×Dic73C2211yescharacteristic
52#17988351C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988346C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#131×C73C2211yescharacteristic
61C2×#104×C73C2211yescharacteristic
62#416×C73C2211yescharacteristic
63#405×C73C2211yescharacteristic
64C2×#97×C73C2211yescharacteristic
65#341×C73C2211yescharacteristic
66#339×C73C2211yescharacteristic
67#14556598—22no—
68#14556598—41no—
69C2×#11924012—21no—
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C22×#10596831—21no—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75#14556590C2×C411yescharacteristic
76#14556590C2×C411yescharacteristic
77#14556582C2×C411yescharacteristic
78#14556582C2×C411yescharacteristic
79#14556568—21no—
80#14556568—21no—
81Q8×Dic73C2×C411yescharacteristic
82Q8×Dic73C2×C411yescharacteristic
83C4×Dic146—21no—
84C4×Dic146—21no—
85D4×Dic73—21no—
86D4×Dic73—21no—
87D4×Dic73C2×C411yescharacteristic
88D4×Dic73C2×C411yescharacteristic
89#14556550—21no—
90#14556550C2311yescharacteristic
91#14556550C2311yescharacteristic
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73C2311yescharacteristic
96C2×C4×Dic73C2311yescharacteristic
97#14556527—41no—
98#14556526—41no—
99#14556525C2×C411yescharacteristic
100#14556525C2×C411yescharacteristic
101#14556524C2×C411yescharacteristic
102#14556524C2×C411yescharacteristic
103#14556523C2×C411yescharacteristic
104#14556523C2×C411yescharacteristic
105#14556522C2×C411yescharacteristic
106#14556522C2×C411yescharacteristic
107C4⋊#10596831—21no—
108C4⋊#10596831—21no—
109#14556520—21no—
110#14556520—21no—
111#14556520—21no—
112#14556520—21no—
113#14556520—21no—
114#14556520—21no—
115#14556520—21no—
116#14556520—21no—
117#14556520C2×C411yescharacteristic
118#14556520C2×C411yescharacteristic
119#14556520C2×C411yescharacteristic
120#14556520C2×C411yescharacteristic
121#14556520D411yescharacteristic
122#14556520D411yescharacteristic
123#14556520D411yescharacteristic
124#14556520D411yescharacteristic
125#14556518—22no—
126#14556515C2×C411yescharacteristic
127#14556515C2×C411yescharacteristic
128#14556512C2×C411yescharacteristic
129#14556512C2×C411yescharacteristic
130#14556502—41no—
131#14556501—21no—
132#14556501—21no—
133C8×Dic73—21no—
134C8×Dic73—21no—
135C2×#41×C73C2311yescharacteristic
136#131×C73—22no—
137#131×C73—41no—
138C2×M16×C73—21no—
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73—21no—
142C22×C8×C73C2311yescharacteristic
143C22×C8×C73C2311yescharacteristic
144#117×C73C2311yescharacteristic
145C2×C42×C73C2311yescharacteristic
146#104×C73Q814yescomplemented
147#103×C73C2×C411yescharacteristic
148#103×C73C2×C411yescharacteristic
149#102×C73C2×C411yescharacteristic
150#102×C73C2×C411yescharacteristic
151#97×C73—21no—
152#97×C73—41no—
153C4×C8×C73—21no—
154C2×Dic146—21no—
155C2×Dic146—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73#4111yescharacteristic
160C22×Dic73#4111yescharacteristic
161#11924012—21no—
162#11924012—21no—
163#11924012—21no—
164#11924012—21no—
165#11924012—42no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—42no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—21no—
199C4×Dic73—42no—
200C4×Dic73—42no—
201C4×Dic73C22×C411yescharacteristic
202C4×Dic73C22×C411yescharacteristic
203C4×Dic73C2×D411yescharacteristic
204C4×Dic73C2×D411yescharacteristic
205#41×C73—21no—
206#41×C73—21no—
207#41×C73C2×Q812yes—
208#41×C73#4112yes—
209C2×Q8×C73C22×C411yescharacteristic
210C2×D4×C73—21no—
211C2×D4×C73C22×C411yescharacteristic
212C22×C4×C73—21no—
213C22×C4×C73—21no—
214C22×C4×C73C2411yescharacteristic
215M16×C73—21no—
216M16×C73—21no—
217M16×C73—21no—
218M16×C73—42no—
219M16×C73C2×Q812yes—
220C2×C8×C73—21no—
221C2×C8×C73—21no—
222C2×C8×C73—21no—
223C2×C8×C73—21no—
224C2×C8×C73—21no—
225C2×C8×C73—21no—
226C2×C8×C73—42no—
227C2×C8×C73C22×C411yescharacteristic
228C2×C8×C73C22×C411yescharacteristic
229C2×C8×C73C2×D411yescharacteristic
230C2×C8×C73C2×D411yescharacteristic
231C4⋊C4×C73C22×C411yescharacteristic
232C4⋊C4×C73C22×C411yescharacteristic
233C22⋊C4×C73—21no—
234C42×C73—21no—
235C42×C73—21no—
236C42×C73C2×Q812yes—
237C23×C73—21no—
238C23×C73C2×#4111yescharacteristic
239Q8×C73—21no—
240Q8×C73C4×Q812yes—
241D4×C73—22no—
242D4×C73—22no—
243D4×C73—21no—
244D4×C73C4×Q812yes—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73C23×C411yescharacteristic
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C22×Q811yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73H(5,𝔽2)11yescharacteristic
258C2×C4×C732−1+411yescharacteristic
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—42no—
269C2×Dic73—81no—
270C2×Dic73C4×D411yescharacteristic
271C2×Dic73C4×D411yescharacteristic
272C2×Dic73C4×D411yescharacteristic
273C2×Dic73C4×D411yescharacteristic
274Dic146—42no—
275Dic146—81no—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—21no—
279C8×C73—21no—
280C8×C73—22no—
281C8×C73—41no—
282#10596831—21no—
283#10596831—21no—
284#10596831—21no—
285#10596831—21no—
286#10596831—41no—
287#10596831—41no—
288C22×C73—21no—
289C22×C73—21no—
290C22×C73—22no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73C2×C4×D411yescharacteristic
294C22×C73#55211yescharacteristic
295C22×C73#55411yescharacteristic
296C4×C73—22no—
297C4×C73—42no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73C2×C4×Q811yescharacteristic
301C4×C73#52011yescharacteristic
302C4×C73#54611yescharacteristic
303C4×C73D4×Q811yescharacteristic
304Dic73—21no—
305Dic73—21no—
306Dic73—21no—
307Dic73—42no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—41no—
312Dic73#44412yes—
313#22983—731nomaximal, Sylow
314C2×C73—21no—
315C2×C73—21no—
316C2×C73—42no—
317C2×C73#236011yescharacteristic
318C2×C73C2×#44411yescharacteristic
319C2×C73#295611yescharacteristic
320#3076—731no—
321#2962—731no—
322#2906—731no—
323#1994—731no—
324#1950—731no—
325#1946—731no—
326#1946—731no—
327#1922—731no—
328#1922—731no—
329#1866—731no—
330#1866—731no—
331#1767—731no—
332#1765—731no—
333#1765D7311yescharacteristic
334C4×#104—731no—
335C73#2298311yescharacteristic, complemented, Sylow
336C2×#131—731no—
337C2×#131C2×D7311yescharacteristic
338#533—731no—
339C4×#41—731no—
340#479—731no—
341#479—731no—
342#478—731no—
343#478—731no—
344#477—731no—
345#476—731no—
346#475—731no—
347#474—731no—
348#446—1461no—
349#446—1461no—
350#446—1461no—
351C8×Q8—1461no—
352C2×#108—731no—
353C2×#108—731no—
354#427—731no—
355#427—731no—
356#424—731no—
357#424—731no—
358#424—731no—
359#424—731no—
360#424—1461no—
361#424—1461no—
362#423—731no—
363#423—1461no—
364C2×C4⋊C8—731no—
365C2×C4⋊C8—731no—
366C2×C4⋊C8—1461no—
367C2×#104—731no—
368C2×#104C2×D7311yescharacteristic
369#416—731no—
370#416C2×D7311yescharacteristic
371#405—731no—
372#405C2×D7311yescharacteristic
373C4×M16—731no—
374C2×#97—731no—
375C2×#97C2×D7311yescharacteristic
376C43—731no—
377#341—731no—
378#341C2×D7311yescharacteristic
379#339—731no—
380#339—731no—
381#339—731no—
382#339—731no—
383#339—731no—
384#339C2×D7311yescharacteristic
385#336—731no—
386#336—731no—
387C2×#41C22×D7311yescharacteristic
388#131—22no—
389#131—41no—
390#131—2921no—
391#131—1462no—
392C2×M16—21no—
393C2×M16—731no—
394C2×M16—731no—
395C2×M16—1461no—
396C2×M16C22×D7311yescharacteristic
397C2×M16C22×D7311yescharacteristic
398C22×C8—21no—
399C22×C8—731no—
400C22×C8—731no—
401C22×C8—1461no—
402C22×C8C22×D7311yescharacteristic
403C22×C8C22×D7311yescharacteristic
404C4⋊Q8—731no—
405C4⋊Q8—731no—
406#125—731no—
407#125—731no—
408C22⋊Q8—1461no—
409C22⋊Q8—1461no—
410C4×Q8—731no—
411C4×Q8—731no—
412C4×Q8—1461no—
413C4×Q8—1461no—
414C4×D4—731no—
415C4×D4—731no—
416C4×D4—1461no—
417C4×D4—1461no—
418#117—731no—
419#117—731no—
420#117—1461no—
421#117C22×D7311yescharacteristic
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—1461no—
425C2×C42—1461no—
426C2×C42—1461no—
427C2×C42C22×D7311yescharacteristic
428#108—2921no—
429#108—2921no—
430#107—731no—
431#107—731no—
432#107—731no—
433#107—731no—
434#106—731no—
435#106—731no—
436#106—731no—
437#106—731no—
438C4⋊C8—731no—
439C4⋊C8—731no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—731no—
443C4⋊C8—731no—
444C4⋊C8—731no—
445C4⋊C8—731no—
446C4⋊C8—1461no—
447C4⋊C8—1461no—
448C4⋊C8—1461no—
449C4⋊C8—1461no—
450C4⋊C8—1461no—
451C4⋊C8—1461no—
452C4⋊C8—1461no—
453C4⋊C8—1461no—
454C4⋊C8—1461no—
455C4⋊C8—1461no—
456#104—1462no—
457#104Dic14614yescomplemented
458#103—731no—
459#103—731no—
460#103C4×D7311yescharacteristic
461#103C4×D7311yescharacteristic
462#102—731no—
463#102—731no—
464#102C4×D7311yescharacteristic
465#102C4×D7311yescharacteristic
466#97—21no—
467#97—41no—
468#97—2921no—
469#97—1461no—
470#97—1461no—
471C4×C8—21no—
472C4×C8—1461no—
473C4×C8—1461no—
474#41—21no—
475#41—21no—
476#41C2×Dic14612yes—
477#41#1192402812yes—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8C2×C4×D7311yescharacteristic
481C2×D4—21no—
482C2×D4C2×C4×D7311yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4C23×D7311yescharacteristic
491M16—21no—
492M16—21no—
493M16—21no—
494M16—42no—
495M16—2922no—
496M16—1461no—
497M16—1461no—
498M16—1461no—
499M16—1461no—
500M16C2×Dic14612yes—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—42no—
508C2×C8—2922no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—731no—
512C2×C8—731no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8C2×C4×D7311yescharacteristic
520C2×C8C2×C4×D7311yescharacteristic
521C2×C8D4×D7311yescharacteristic
522C2×C8D4×D7311yescharacteristic
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—731no—
526C4⋊C4—731no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4C2×C4×D7311yescharacteristic
536C4⋊C4C2×C4×D7311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C22⋊C4—1461no—
541C22⋊C4—1461no—
542C42—21no—
543C42—21no—
544C42—2922no—
545C42—2922no—
546C42—731no—
547C42—731no—
548C42—731no—
549C42—731no—
550C42—1461no—
551C42—1461no—
552C42—1461no—
553C42—1461no—
554C42—1461no—
555C42—1461no—
556C42C2×Dic14612yes—
557C23—21no—
558C23#41×D7311yescharacteristic
559Q8—21no—
560Q8—2922no—
561Q8—5841no—
562Q8C4×Dic14612yes—
563D4—22no—
564D4—22no—
565D4—21no—
566D4C4×Dic14612yes—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2922no—
578C2×C4—2921no—
579C2×C4—5841no—
580C2×C4—731no—
581C2×C4—731no—
582C2×C4—731no—
583C2×C4—731no—
584C2×C4—1462no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1462no—
590C2×C4C22×C4×D7311yescharacteristic
591C2×C4C2×D4×D7311yescharacteristic
592C2×C4C22×Dic14611yescharacteristic
593C2×C4C2×#1192402811yescharacteristic
594C2×C4#1455663611yescharacteristic
595C2×C4#1455663811yescharacteristic
596C8—21no—
597C8—21no—
598C8—22no—
599C8—21no—
600C8—21no—
601C8—41no—
602C8—2921no—
603C8—2921no—
604C8—1461no—
605C8—1461no—
606C8—1461no—
607C8—1461no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22C4×D4×D7311yescharacteristic
614C22#1798927011yescharacteristic
615C22#1798929611yescharacteristic
616C4—22no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—2922no—
621C4—2921no—
622C4—2921no—
623C4—2921no—
624C4—2921no—
625C4—732no—
626C4—1461no—
627C4—1461no—
628C4—1461no—
629C4C2×C4×Dic14611yescharacteristic
630C4#1798906611yescharacteristic
631C4#1798922711yescharacteristic
632C4D4×Dic14611yescharacteristic
633C2—21no—
634C2—21no—
635C2—42no—
636C2#2261293711yescharacteristic
637C2#444×D7311yescharacteristic
638C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1765C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1765D73
hypocenterC73#22983
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #22618875C2semidirect
4. #22610906C2semidirect
5. #22610705C2semidirect
6. #22610684C2semidirect
7. #22610683C2semidirect
8. #22610530C2semidirect
9. #22610529C2semidirect
12. #22609785C2semidirect
19. #17988896C4semidirect
20. #17988892C4semidirect
21. #17988889C4semidirect
22. #17988885C4semidirect
23. #17988880C4semidirect
24. #17988875C4semidirect
25. #17988870C4semidirect
26. #17988865C4semidirect
146. #104×C73Q8semidirect
335. C73#22983semidirect
457. #104Dic146semidirect

Maximal subgroups

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

subgroupindex
#226197152
#226188752
#226184972
#226109062
#226107052
#226106842
#226106832
#226105302
#226105292
#226102082
#226102062
#226097852
#226097762
#104×Dic732
C73×#17652
#2298373