Finite Group Databasebeta

← #33057510

Subgroups — #33057510

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×#14556617C211yesmaximal, characteristic
3D4×#11924012C211yesmaximal, characteristic, complemented
4#22608830C211yesmaximal, characteristic, complemented
5#22608824C211yesmaximal, characteristic, complemented
6#22608798C211yesmaximal, characteristic, complemented
7#22608785C211yesmaximal, characteristic, complemented
8#22608764C211yesmaximal, characteristic, complemented
9#22608763C211yesmaximal, characteristic, complemented
10#22608752C211yesmaximal, characteristic, complemented
11#22608723C211yesmaximal, characteristic, complemented
12#22608699C211yesmaximal, characteristic, complemented
13#22608371C211yesmaximal, characteristic, complemented
14#22608234C211yesmaximal, characteristic, complemented
15C73×#1490C211yesmaximal, characteristic, complemented
16C2×#14556617C411yescharacteristic
17C22×#11924012—21no—
18C4×#11924028C2211yescharacteristic
19#17988714C2211yescharacteristic, complemented
20#17988714C2211yescharacteristic, complemented
21#17988697C411yescharacteristic
22#17988697C411yescharacteristic
23#17988687C411yescharacteristic
24#17988687C411yescharacteristic
25C4×#11924016C2211yescharacteristic
26C4×#11924016C2211yescharacteristic
27C4×#11924013C2211yescharacteristic
28C4×#11924013C2211yescharacteristic
29#17988665C2211yescharacteristic
30#17988665C2211yescharacteristic
31#17988661—21no—
32#17988661—21no—
33#17988659—21no—
34#17988659—21no—
35D4×#10596831C2211yescharacteristic
36D4×#10596831C2211yescharacteristic
37#17988652C2211yescharacteristic, complemented
38#17988652C2211yescharacteristic, complemented
39#17988642—21no—
40#17988627C411yescharacteristic
41#17988611C2211yescharacteristic
42#17988592C411yescharacteristic
43#17988586C411yescharacteristic
44#17988551—21no—
45C4×#11924012C2211yescharacteristic
46C4×#11924012C2211yescharacteristic
47#97×D73—21no—
48#17988339C2211yescharacteristic, complemented
49#17988339C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988337C2211yescharacteristic, complemented
52#17988319C2211yescharacteristic, complemented
53#17988319C2211yescharacteristic, complemented
54#17988316C2211yescharacteristic, complemented
55#17988316C2211yescharacteristic, complemented
56#17988309C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C2×C4×D4×C73C2211yescharacteristic
61#432×C73C2211yescharacteristic
62C4×M16×C73C2211yescharacteristic
63D4⋊C8×C73C2211yescharacteristic
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66D4⋊C8×C73C2211yescharacteristic
67C2×#11924028C2×C411yescharacteristic
68#14556617—24no—
69C2×#11924016C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73C2×#11924012—21no—
74C2×#11924012—21no—
75C2×#11924012—21no—
76C2×#11924012—44no—
77C2×#11924012C2×C411yescharacteristic
78C2×#11924012C2×C411yescharacteristic
79C2×#11924012D411yescharacteristic
80C2×#11924012D411yescharacteristic
81C22×#10596831—21no—
82C22×#10596831—21no—
83C2×C8×D73—21no—
84C2×C8×D73—21no—
85C4×Dic146C2×C411yescharacteristic
86C4×Dic146C2311yescharacteristic
87C4×D292C2×C411yescharacteristic
88C4×D292C2311yescharacteristic
89C4×C22⋊D73—21no—
90C4×C22⋊D73—21no—
91#14556550C2×C411yescharacteristic
92#14556548—21no—
93C42×D73—21no—
94#14556524C2×C411yescharacteristic
95#14556524C2×C411yescharacteristic
96#14556522C2×C411yescharacteristic
97#14556522C2×C411yescharacteristic
98C4⋊#10596831—21no—
99C4⋊#10596831C2311yescharacteristic
100C4⋊#10596831C2311yescharacteristic
101#14556520—21no—
102#14556520—21no—
103#14556514C2×C411yescharacteristic
104#14556514C2×C411yescharacteristic
105#14556512C2×C411yescharacteristic
106#14556512C2×C411yescharacteristic
107#14556512C2×C411yescharacteristic
108#14556512C2×C411yescharacteristic
109#14556511C2×C411yescharacteristic
110#14556511C2×C411yescharacteristic
111C22⋊#10596831—21no—
112C22⋊#10596831—21no—
113C22⋊#10596831—21no—
114C22⋊#10596831—21no—
115#14556503—21no—
116#14556503—21no—
117#14556502C2×C411yescharacteristic
118#14556502C2×C411yescharacteristic
119#14556502D411yescharacteristic
120#14556502D411yescharacteristic
121#14556501—21no—
122#14556501—21no—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C22×D4×C73C2×C411yescharacteristic
128C23×C4×C73—21no—
129C2×M16×C73—21no—
130C4×D4×C73—21no—
131C4×D4×C73—21no—
132C4×D4×C73C2×C411yescharacteristic
133C4×D4×C73C2×C411yescharacteristic
134C4×D4×C73C2311yescharacteristic
135C4×D4×C73C2311yescharacteristic
136C2×C4⋊C4×C73C2×C411yescharacteristic
137C2×C22⋊C4×C73—21no—
138C2×C42×C73C2311yescharacteristic
139C4⋊C8×C73C2311yescharacteristic
140C4⋊C8×C73C2311yescharacteristic
141C22⋊C8×C73—21no—
142#97×C73—21no—
143#97×C73—21no—
144C4×C8×C73C2311yescharacteristic
145C4×C8×C73C2311yescharacteristic
146#11924028—22no—
147#11924028—41no—
148C2×Dic146C22×C411yescharacteristic
149C2×D292C22×C411yescharacteristic
150C2×C22⋊D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154#11924016—22no—
155#11924016—22no—
156#11924013—22no—
157#11924013M1614yescomplemented
158#11924012—22no—
159#11924012—22no—
160#11924012—41no—
161#11924012—41no—
162#11924012—42no—
163#11924012—42no—
164#11924012—42no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—42no—
168C2×#10596831—42no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C8×D73—42no—
178C8×D73—42no—
179C4⋊Dic73C22×C411yescharacteristic
180C4⋊Dic73C22×C411yescharacteristic
181#11924007—21no—
182#11924007—21no—
183C22⋊Dic73—21no—
184#11924005—21no—
185#11924005—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C24×C73—21no—
190C2×D4×C73—22no—
191C2×D4×C73—21no—
192C2×D4×C73—41no—
193C2×D4×C73C22×C411yescharacteristic
194C2×D4×C73C22×C411yescharacteristic
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—42no—
201C22×C4×C73—42no—
202C22×C4×C73C22×C411yescharacteristic
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73#4111yescharacteristic
205M16×C73—42no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73C22×C411yescharacteristic
212C4⋊C4×C73C22×C411yescharacteristic
213C22⋊C4×C73—21no—
214C22⋊C4×C73—21no—
215C22⋊C4×C73—21no—
216C22⋊C4×C73—21no—
217C42×C73C22×C411yescharacteristic
218C42×C73C2×D411yescharacteristic
219C42×C73#4111yescharacteristic
220C42×C73C2411yescharacteristic
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—42no—
224C23×C73—42no—
225C23×C73—81no—
226C23×C73C4×D411yescharacteristic
227C22×D73—21no—
228D4×C73—21no—
229D4×C73—21no—
230D4×C73—21no—
231D4×C73—42no—
232D4×C73C2×M1612yes—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—41no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73C4×D411yescharacteristic
248C2×C4×C73C4×D411yescharacteristic
249C2×C4×C73C4×D411yescharacteristic
250C2×C4×C73C23×C411yescharacteristic
251C2×C4×C73C22×D411yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C22⋊D73—42no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257D292—21no—
258D292C2×M1612yescomplemented
259C4×D73—42no—
260C4×D73—41no—
261C4×D73—41no—
262C4×D73—81no—
263Dic146—21no—
264Dic146—21no—
265C8×C73—22no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73—41no—
269#10596831—21no—
270#10596831—21no—
271#10596831—21no—
272#10596831—21no—
273#10596831—21no—
274#10596831—21no—
275#10596831—41no—
276#10596831—41no—
277#10596831C2×M1612yescomplemented
278#10596831AGL(1,ℤ/8)12yescomplemented
279C22×C73—21no—
280C22×C73—22no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—84no—
289C22×C73C2×C4×D411yescharacteristic
290C2×D73—22no—
291C2×D73—41no—
292C2×D73—41no—
293C4×C73—21no—
294C4×C73—21no—
295C4×C73—21no—
296C4×C73—21no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73C22×M1611yescharacteristic
300C4×C73#56811yescharacteristic
301C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C73#57511yescharacteristic
303Dic73—41no—
304Dic73—41no—
305Dic73—41no—
306#18046—731nomaximal, Sylow
307C2×C73—41no—
308C2×C73—42no—
309C2×C73—81no—
310C2×C73D4×M1611yescharacteristic
311C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
312C2×C73#295611yescharacteristic
313D73—42no—
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#1569—731no—
326#1490D7311yescharacteristic, complemented
327#1489—731no—
328#1455—731no—
329C73#1804611yescharacteristic, complemented, Sylow
330C2×AGL(1,ℤ/8)—731no—
331C22×M16—1461no—
332C4×#41—731no—
333C2×C4×D4C2×D7311yescharacteristic
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#423—731no—
358#418—731no—
359#417—731no—
360#407—1461no—
361C4×M16—731no—
362C4×M16—731no—
363C4×M16C2×D7311yescharacteristic
364C2×#97—1461no—
365#335—731no—
366#335—731no—
367#334—731no—
368#334—731no—
369Q8⋊C8—731no—
370Q8⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8C2×D7311yescharacteristic
374D4⋊C8C2×D7311yescharacteristic
375D4⋊C8C2×D7311yescharacteristic
376D4⋊C8C2×D7311yescharacteristic
377#322—731no—
378#322—731no—
379#322—731no—
380C82—731no—
381C2×#41—731no—
382C22×D4C4×D7311yescharacteristic
383C23×C4—21no—
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—731no—
394C2×M16—731no—
395C2×M16—1461no—
396C2×M16—1461no—
397C2×M16—1461no—
398C22×C8—1461no—
399C22×C8—1461no—
400C22×C8—1461no—
401C22×C8—1461no—
402C4×Q8—731no—
403C4×Q8—731no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—731no—
407C4×D4—731no—
408C4×D4—1461no—
409C4×D4—1461no—
410C4×D4C4×D7311yescharacteristic
411C4×D4C4×D7311yescharacteristic
412C4×D4C22×D7311yescharacteristic
413C4×D4C22×D7311yescharacteristic
414#117—731no—
415#117—1461no—
416C2×C4⋊C4C4×D7311yescharacteristic
417C2×C22⋊C4—21no—
418C2×C42—1461no—
419C2×C42C22×D7311yescharacteristic
420#107—731no—
421#107—731no—
422#106—731no—
423#106—731no—
424C4⋊C8—731no—
425C4⋊C8—731no—
426C4⋊C8—1461no—
427C4⋊C8—1461no—
428C4⋊C8—1461no—
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—21no—
448#97—731no—
449#97—731no—
450#97—731no—
451#97—731no—
452#97—1461no—
453#97—1461no—
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—731no—
458C4×C8C22×D7311yescharacteristic
459C4×C8C22×D7311yescharacteristic
460C24—21no—
461#41—2921no—
462#41—1462no—
463C2×Q8—731no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—41no—
467C2×D4—731no—
468C2×D4—1461no—
469C2×D4C2×C4×D7311yescharacteristic
470C2×D4C2×C4×D7311yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4C2×C4×D7311yescharacteristic
482C22×C4D4×D7311yescharacteristic
483C22×C4#1192402711yescharacteristic
484SD16—1462no—
485SD16—1462no—
486D8—734no—
487D8—1462no—
488M16—42no—
489M16—2922no—
490M16—2922no—
491M16—2922no—
492M16—2921no—
493M16—2921no—
494M16—1462no—
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—731no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—731no—
512C2×C8—1461no—
513C2×C8—1461no—
514C4⋊C4—21no—
515C4⋊C4—731no—
516C4⋊C4—731no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4C2×C4×D7311yescharacteristic
520C4⋊C4C2×C4×D7311yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—1461no—
526C22⋊C4—1461no—
527C22⋊C4—1461no—
528C42—1461no—
529C42—1461no—
530C42—1461no—
531C42C2×C4×D7311yescharacteristic
532C42D4×D7311yescharacteristic
533C42#1192402711yescharacteristic
534C42C23×D7311yescharacteristic
535C23—21no—
536C23—21no—
537C23—42no—
538C23—42no—
539C23—81no—
540C23—1461no—
541C23#1455655111yescharacteristic
542Q8—1461no—
543Q8—1461no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—42no—
548D4—2922no—
549D4—732no—
550D4—1461no—
551D4M16×D7312yes—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—2922no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—5841no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4#1455655111yescharacteristic
574C2×C4#1455655111yescharacteristic
575C2×C4#1455655111yescharacteristic
576C2×C4C22×C4×D7311yescharacteristic
577C2×C4C2×D4×D7311yescharacteristic
578C2×C4C2×#1192402711yescharacteristic
579C8—22no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—2921no—
584C8—2921no—
585C8—732no—
586C8—732no—
587C8—1461no—
588C8—1461no—
589C8—1461no—
590C8—1461no—
591C8—1461no—
592C8—1461no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—84no—
603C22—2921no—
604C22—2921no—
605C22—1462no—
606C22C2×#1455655111yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—41no—
612C4—41no—
613C4—2921no—
614C4—2921no—
615C4—2921no—
616C4C2×M16×D7311yescharacteristic
617C4#1798935311yescharacteristic
618C4AGL(1,ℤ/8)×D7311yescharacteristic
619C4#1798938311yescharacteristic
620C2—41no—
621C2—42no—
622C2—81no—
623C2—2922no—
624C2#2261866311yescharacteristic
625C2#2261873511yescharacteristic
626C2#2261883711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1490C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556617C4
cosocleC42×C73C24
hypercenter#1490D73
hypocenterC73#18046
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
1. #22618943C2semidirect
3. D4×#11924012C2semidirect
4. #22608830C2semidirect
5. #22608824C2semidirect
6. #22608798C2semidirect
7. #22608785C2semidirect
8. #22608764C2semidirect
9. #22608763C2semidirect
10. #22608752C2semidirect
11. #22608723C2semidirect
12. #22608699C2semidirect
13. #22608371C2semidirect
14. #22608234C2semidirect
15. C73×#1490C2semidirect
19. #17988714C22semidirect
20. #17988714C22semidirect
37. #17988652C22semidirect
38. #17988652C22semidirect
48. #17988339C22semidirect
49. #17988339C22semidirect
50. #17988337C22semidirect
51. #17988337C22semidirect
52. #17988319C22semidirect
53. #17988319C22semidirect
54. #17988316C22semidirect
55. #17988316C22semidirect
56. #17988309C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. C8×#10596831C22semidirect
157. #11924013M16semidirect
258. D292C2×M16semidirect
277. #10596831C2×M16semidirect
278. #10596831AGL(1,ℤ/8)semidirect
326. #1490D73semidirect
329. C73#18046semidirect

Maximal subgroups

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

subgroupindex
#226189432
C4×#145566172
D4×#119240122
#226088302
#226088242
#226087982
#226087852
#226087642
#226087632
#226087522
#226087232
#226086992
#226083712
#226082342
C73×#14902
#1804673