Finite Group Databasebeta

← #33057514

Subgroups — #33057514

Order 18688.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618943C211yesmaximal, characteristic, complemented
2C4×#14556619C211yesmaximal, characteristic
3#22618666C211yesmaximal, characteristic, complemented
4#22608832C211yesmaximal, characteristic, complemented
5#22608820C211yesmaximal, characteristic, complemented
6#22608794C211yesmaximal, characteristic, complemented
7#22608787C211yesmaximal, characteristic, complemented
8#22608766C211yesmaximal, characteristic, complemented
9#22608759C211yesmaximal, characteristic, complemented
10#22608754C211yesmaximal, characteristic, complemented
11#22608725C211yesmaximal, characteristic, complemented
12#22608373C211yesmaximal, characteristic, complemented
13#22608368C211yesmaximal, characteristic, complemented
14#22608234C211yesmaximal, characteristic, complemented
15C73×#1569C211yesmaximal, characteristic, complemented
16C2×#14556619C411yescharacteristic
17C2×M16×D73—21no—
18C4×#11924028C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20#17988714C2211yescharacteristic, complemented
21#17988714C2211yescharacteristic, complemented
22#17988700C411yescharacteristic
23#17988700C411yescharacteristic
24#17988689C411yescharacteristic
25#17988689C411yescharacteristic
26C4×#11924017C2211yescharacteristic
27C4×#11924017C2211yescharacteristic
28C4×D584C2211yescharacteristic
29C4×D584C2211yescharacteristic
30#17988664C2211yescharacteristic
31#17988664C2211yescharacteristic
32#17988661—21no—
33#17988661—21no—
34#17988657—21no—
35#17988657—21no—
36#17988652C2211yescharacteristic
37#17988652C2211yescharacteristic
38#17988652C2211yescharacteristic, complemented
39#17988652C2211yescharacteristic, complemented
40#17988642—21no—
41#17988629C411yescharacteristic
42#17988593C411yescharacteristic
43#17988589C411yescharacteristic
44#17988550—21no—
45C4×#11924012C2211yescharacteristic
46C4×#11924012C2211yescharacteristic
47#97×D73—21no—
48#17988317C2211yescharacteristic, complemented
49#17988317C2211yescharacteristic, complemented
50#17988314C2211yescharacteristic
51#17988314C2211yescharacteristic
52#17988314C2211yescharacteristic
53#17988314C2211yescharacteristic
54#17988314C2211yescharacteristic, complemented
55#17988314C2211yescharacteristic, complemented
56#17988309C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60#432×C73C2211yescharacteristic
61#423×C73C2211yescharacteristic
62C4×M16×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic, complemented
64#335×C73C2211yescharacteristic, complemented
65#334×C73C2211yescharacteristic, complemented
66#334×C73C2211yescharacteristic, complemented
67C2×#11924028C2×C411yescharacteristic
68C22×D292C2×C411yescharacteristic
69C22×C4×D73—21no—
70#14556619—24no—
71C2×#11924017C2×C411yescharacteristic
72C2×#11924017C2×C411yescharacteristic
73C2×D584C2×C411yescharacteristic
74C2×D584C2×C411yescharacteristic
75C2×#11924012—21no—
76C2×#11924012D411yescharacteristic
77C2×#11924012D411yescharacteristic
78C2×#11924011—21no—
79C2×#11924011—21no—
80M16×D73—44no—
81C2×C8×D73—21no—
82C2×C8×D73—21no—
83C2×C8×D73—21no—
84C2×C8×D73—21no—
85C4×Dic146C2×C411yescharacteristic
86C4×Dic146C2311yescharacteristic
87C4×D292—21no—
88C4×D292—21no—
89C4×D292C2×C411yescharacteristic
90C4×D292C2×C411yescharacteristic
91C4×D292C2×C411yescharacteristic
92C4×D292C2311yescharacteristic
93C4×D292C2311yescharacteristic
94C4×D292C2311yescharacteristic
95C4×C22⋊D73—21no—
96C4×C22⋊D73—21no—
97#14556550C2×C411yescharacteristic
98#14556548—21no—
99C2×C4⋊Dic73C2×C411yescharacteristic
100C2×#11924005—21no—
101C42×D73—21no—
102#14556525C2×C411yescharacteristic
103#14556525C2×C411yescharacteristic
104#14556523C2×C411yescharacteristic
105#14556523C2×C411yescharacteristic
106#14556520—21no—
107#14556520—21no—
108#14556516C2×C411yescharacteristic
109#14556516C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556513C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114#14556513C2×C411yescharacteristic
115#14556513C2×C411yescharacteristic
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503—21no—
121#14556503—21no—
122#14556502—21no—
123#14556502D411yescharacteristic
124#14556502D411yescharacteristic
125#14556501—21no—
126#14556501—21no—
127C4×#10596831C2311yescharacteristic
128C4×#10596831C2311yescharacteristic
129C4×#10596831C2311yescharacteristic
130C4×#10596831C2311yescharacteristic
131C2×M16×C73—21no—
132C2×M16×C73C2×C411yescharacteristic
133C2×M16×C73C2×C411yescharacteristic
134C2×C42×C73C2311yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138C4⋊C8×C73C2311yescharacteristic
139C4⋊C8×C73C2311yescharacteristic
140C22⋊C8×C73—21no—
141#97×C73—21no—
142#97×C73C2×C411yescharacteristic
143#97×C73C2×C411yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C4×C8×C73C2311yescharacteristic
146C23×D73—21no—
147#11924028—22no—
148#11924028—41no—
149C2×Dic146C22×C411yescharacteristic
150C2×D292—22no—
151C2×D292—21no—
152C2×D292—41no—
153C2×D292C22×C411yescharacteristic
154C2×D292C22×C411yescharacteristic
155C2×D292C22×C411yescharacteristic
156C2×C22⋊D73—21no—
157C22×Dic73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—21no—
161C2×C4×D73—21no—
162C2×C4×D73—21no—
163C2×C4×D73—21no—
164C2×C4×D73—21no—
165C2×C4×D73—42no—
166C2×C4×D73—42no—
167#11924017—22no—
168#11924017—22no—
169D584—22no—
170D584M1614yescomplemented
171#11924012—22no—
172#11924012—41no—
173#11924012—42no—
174#11924011—42no—
175#11924011—42no—
176C2×#10596831—21no—
177C2×#10596831—21no—
178C2×#10596831C2×D411yescharacteristic
179C2×#10596831C2×D411yescharacteristic
180C2×#10596831C2×D411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C8×D73—42no—
183C8×D73—42no—
184C8×D73—42no—
185C8×D73—42no—
186C4⋊Dic73—21no—
187C4⋊Dic73C22×C411yescharacteristic
188C4⋊Dic73C22×C411yescharacteristic
189C4⋊Dic73C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191#11924007—21no—
192#11924007—21no—
193C22⋊Dic73—21no—
194#11924005—21no—
195#11924005—21no—
196#11924005—21no—
197#11924005—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C22×C4×C73C22×C411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73#4111yescharacteristic
206M16×C73—22no—
207M16×C73—41no—
208M16×C73—42no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73C22×C411yescharacteristic
214C2×C8×C73C22×C411yescharacteristic
215C2×C8×C73C22×C411yescharacteristic
216C2×C8×C73C22×C411yescharacteristic
217C42×C73C22×C411yescharacteristic
218C42×C73C2×D411yescharacteristic
219C42×C73#4111yescharacteristic
220C42×C73C2411yescharacteristic
221C23×C73C4×D411yescharacteristic
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—42no—
226C22×D73—42no—
227C22×D73—81no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—41no—
233C2×C4×C73C4×D411yescharacteristic
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C23×C411yescharacteristic
237C2×C4×C73C22×D411yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C22⋊D73—42no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—42no—
246D292—21no—
247D292—21no—
248D292—21no—
249D292—21no—
250D292—42no—
251D292C2×M1612yes—
252D292C2×M1612yescomplemented
253C4×D73—42no—
254C4×D73—42no—
255C4×D73—42no—
256C4×D73—41no—
257C4×D73—41no—
258C4×D73—41no—
259C4×D73—41no—
260C4×D73—41no—
261C4×D73—41no—
262C4×D73—81no—
263Dic146—21no—
264Dic146—21no—
265C8×C73—21no—
266C8×C73—21no—
267C8×C73—21no—
268C8×C73—41no—
269C8×C73—41no—
270C8×C73—41no—
271C8×C73—41no—
272C8×C73C2×M1612yescomplemented
273#10596831—22no—
274#10596831—21no—
275#10596831—21no—
276#10596831—21no—
277#10596831—41no—
278#10596831AGL(1,ℤ/8)12yescomplemented
279C22×C73—21no—
280C22×C73—21no—
281C22×C73—21no—
282C22×C73C2×C4×D411yescharacteristic
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—41no—
286C2×D73—41no—
287C2×D73—41no—
288C2×D73—41no—
289C2×D73—41no—
290C2×D73—41no—
291C2×D73—41no—
292C2×D73—84no—
293C4×C73—21no—
294C4×C73—21no—
295C4×C73—21no—
296C4×C73—21no—
297C4×C73C22×M1611yescharacteristic
298C4×C73#56811yescharacteristic
299C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C73#57511yescharacteristic
301Dic73—41no—
302Dic73—41no—
303Dic73—41no—
304Dic73—41no—
305Dic73—41no—
306#18046—731nomaximal, Sylow
307C2×C73—41no—
308C2×C73D4×M1611yescharacteristic
309C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
310C2×C73#295611yescharacteristic
311D73—42no—
312D73—42no—
313D73—81no—
314#2970—731no—
315C4×AGL(1,ℤ/8)—731no—
316D4×M16—731no—
317#1592—731no—
318#1590—731no—
319#1586—731no—
320#1585—731no—
321#1582—731no—
322#1581—731no—
323#1580—731no—
324C8×D8—731no—
325#1569D7311yescharacteristic, complemented
326#1490—731no—
327#1489—731no—
328#1455—731no—
329C73#1804611yescharacteristic, complemented, Sylow
330C2×AGL(1,ℤ/8)—731no—
331C22×M16—1461no—
332C4×#41—731no—
333C2×C4×D4—731no—
334#446—731no—
335#446—731no—
336#442—731no—
337#442—731no—
338#440—731no—
339#440—731no—
340C4×SD16—731no—
341C4×SD16—731no—
342C4×D8—731no—
343C4×D8—731no—
344#436—731no—
345#436—731no—
346#435—1461no—
347#435—1461no—
348#435—1461no—
349#435—1461no—
350C8×D4—731no—
351C8×D4—731no—
352C8×D4—731no—
353C8×D4—731no—
354#432—1461no—
355#432C2×D7311yescharacteristic
356#428—731no—
357#423C2×D7311yescharacteristic
358#418—731no—
359#417—731no—
360#407—1461no—
361C4×M16—731no—
362C4×M16—731no—
363C4×M16C2×D7311yescharacteristic
364C2×#97—1461no—
365#335C2×D7311yescharacteristic, complemented
366#335C2×D7311yescharacteristic, complemented
367#334C2×D7311yescharacteristic, complemented
368#334C2×D7311yescharacteristic, complemented
369Q8⋊C8—731no—
370Q8⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8—731no—
374D4⋊C8—731no—
375D4⋊C8—731no—
376D4⋊C8—731no—
377#322—731no—
378#322—731no—
379#322—731no—
380C82—731no—
381C2×#41—731no—
382C22×D4—731no—
383C23×C4—1461no—
384AGL(1,ℤ/8)—1464no—
385C2×SD16—731no—
386C2×SD16—731no—
387C2×D8—731no—
388C2×D8—731no—
389C2×M16—21no—
390C2×M16—2924no—
391C2×M16—731no—
392C2×M16—731no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16C4×D7311yescharacteristic
397C2×M16C4×D7311yescharacteristic
398C22×C8—1461no—
399C22×C8—1461no—
400C22×C8—1461no—
401C22×C8—1461no—
402C4×Q8—731no—
403C4×Q8—731no—
404C4×D4—731no—
405C4×D4—731no—
406C4×D4—731no—
407C4×D4—731no—
408C4×D4—731no—
409C4×D4—731no—
410C4×D4—1461no—
411C4×D4—1461no—
412C4×D4—1461no—
413C4×D4—1461no—
414#117—731no—
415#117—1461no—
416C2×C4⋊C4—731no—
417C2×C22⋊C4—1461no—
418C2×C42—1461no—
419C2×C42C22×D7311yescharacteristic
420#107—731no—
421#107—731no—
422#106—731no—
423#106—731no—
424C4⋊C8—21no—
425C4⋊C8—1461no—
426C4⋊C8—1461no—
427C4⋊C8C22×D7311yescharacteristic
428C4⋊C8C22×D7311yescharacteristic
429C4⋊C8C22×D7311yescharacteristic
430C4⋊C8C22×D7311yescharacteristic
431#103—731no—
432#103—731no—
433#102—731no—
434#102—731no—
435#102—731no—
436#102—731no—
437#102—731no—
438#102—731no—
439C22⋊C8—21no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444C22⋊C8—1461no—
445C22⋊C8—1461no—
446#97—21no—
447#97—731no—
448#97—731no—
449#97—1461no—
450#97—1461no—
451#97—1461no—
452#97C4×D7311yescharacteristic
453#97C4×D7311yescharacteristic
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—731no—
458C4×C8C22×D7311yescharacteristic
459C4×C8C22×D7311yescharacteristic
460C24—1461no—
461#41—2921no—
462#41—1462no—
463C2×Q8—731no—
464C2×D4—2921no—
465C2×D4—731no—
466C2×D4—731no—
467C2×D4—731no—
468C2×D4—1462no—
469C2×D4—1461no—
470C2×D4—1461no—
471C22×C4—2922no—
472C22×C4—2922no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4C2×C4×D7311yescharacteristic
482C22×C4D4×D7311yescharacteristic
483C22×C4#1192402911yescharacteristic
484SD16—1462no—
485SD16—1462no—
486D8—734no—
487D8—1462no—
488M16—22no—
489M16—41no—
490M16—42no—
491M16—2922no—
492M16—2922no—
493M16—2922no—
494M16—2921no—
495M16—1462no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—2922no—
501C2×C8—2922no—
502C2×C8—2922no—
503C2×C8—2922no—
504C2×C8—731no—
505C2×C8—731no—
506C2×C8—731no—
507C2×C8—731no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8C2×C4×D7311yescharacteristic
511C2×C8C2×C4×D7311yescharacteristic
512C2×C8C2×C4×D7311yescharacteristic
513C2×C8C2×C4×D7311yescharacteristic
514C4⋊C4—731no—
515C4⋊C4—731no—
516C4⋊C4—731no—
517C4⋊C4—731no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C22⋊C4—1461no—
522C22⋊C4—1461no—
523C22⋊C4—1461no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C22⋊C4—1461no—
527C22⋊C4—1461no—
528C42—1461no—
529C42—1461no—
530C42—1461no—
531C42C2×C4×D7311yescharacteristic
532C42D4×D7311yescharacteristic
533C42#1192402911yescharacteristic
534C42C23×D7311yescharacteristic
535C23—2922no—
536C23—2922no—
537C23—5841no—
538C23—1461no—
539C23—1461no—
540C23—1461no—
541C23#1455655311yescharacteristic
542Q8—1461no—
543Q8—1461no—
544D4—2922no—
545D4—2922no—
546D4—732no—
547D4—732no—
548D4—1461no—
549D4—1461no—
550D4—1461no—
551D4—1461no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—2922no—
558C2×C4—2922no—
559C2×C4—2922no—
560C2×C4—2922no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—5841no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4#1455655311yescharacteristic
574C2×C4#1455655311yescharacteristic
575C2×C4#1455655311yescharacteristic
576C2×C4C22×C4×D7311yescharacteristic
577C2×C4C2×D4×D7311yescharacteristic
578C2×C4C2×#1192402911yescharacteristic
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—41no—
585C8—41no—
586C8—2921no—
587C8—732no—
588C8—1462no—
589C8—1461no—
590C8—1461no—
591C8—1461no—
592C8M16×D7312yescomplemented
593C22—21no—
594C22—21no—
595C22—21no—
596C22—2921no—
597C22—2921no—
598C22—2921no—
599C22—2921no—
600C22—2921no—
601C22—2921no—
602C22—2921no—
603C22—5844no—
604C22—1462no—
605C22—1462no—
606C22C2×#1455655311yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—2921no—
612C4—2921no—
613C4—2921no—
614C4—2921no—
615C4—2921no—
616C4C2×M16×D7311yescharacteristic
617C4#1798935311yescharacteristic
618C4AGL(1,ℤ/8)×D7311yescharacteristic
619C4#1798938611yescharacteristic
620C2—41no—
621C2—2922no—
622C2—2922no—
623C2—5841no—
624C2#2261866611yescharacteristic
625C2#2261873811yescharacteristic
626C2#2261884011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1569C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556619C4
cosocleC42×C73C24
hypercenter#1569D73
hypocenterC73#18046
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
1. #22618943C2semidirect
3. #22618666C2semidirect
4. #22608832C2semidirect
5. #22608820C2semidirect
6. #22608794C2semidirect
7. #22608787C2semidirect
8. #22608766C2semidirect
9. #22608759C2semidirect
10. #22608754C2semidirect
11. #22608725C2semidirect
12. #22608373C2semidirect
13. #22608368C2semidirect
14. #22608234C2semidirect
15. C73×#1569C2semidirect
20. #17988714C22semidirect
21. #17988714C22semidirect
38. #17988652C22semidirect
39. #17988652C22semidirect
48. #17988317C22semidirect
49. #17988317C22semidirect
54. #17988314C22semidirect
55. #17988314C22semidirect
56. #17988309C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. C8×#10596831C22semidirect
63. #335×C73C22semidirect
64. #335×C73C22semidirect
65. #334×C73C22semidirect
66. #334×C73C22semidirect
170. D584M16semidirect
252. D292C2×M16semidirect
272. C8×C73C2×M16semidirect
278. #10596831AGL(1,ℤ/8)semidirect
325. #1569D73semidirect
329. C73#18046semidirect
365. #335C2×D73semidirect
366. #335C2×D73semidirect
367. #334C2×D73semidirect
368. #334C2×D73semidirect
592. C8M16×D73semidirect

Maximal subgroups

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

subgroupindex
#226189432
C4×#145566192
#226186662
#226088322
#226088202
#226087942
#226087872
#226087662
#226087592
#226087542
#226087252
#226083732
#226083682
#226082342
C73×#15692
#1804673