Finite Group Databasebeta

← #34019462

Subgroups — #34019462

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#22626768C211yesmaximal, characteristic, complemented
2#22626671C211yesmaximal, characteristic, complemented
3#22625575C211yesmaximal, characteristic, complemented
4#22625565C211yesmaximal, characteristic, complemented
5#22625550C211yesmaximal, characteristic, complemented
6#22625549C211yesmaximal, characteristic, complemented
7#22625537C211yesmaximal, characteristic, complemented
8#22625530C211yesmaximal, characteristic, complemented
9#22625528C211yesmaximal, characteristic, complemented
10#22625511C211yesmaximal, characteristic, complemented
11#22625495C211yesmaximal, characteristic, complemented
12#22625272C211yesmaximal, characteristic, complemented
13#22625266C211yesmaximal, characteristic, complemented
14#22625154C211yesmaximal, characteristic, complemented
15#22618744C211yesmaximal, characteristic
16C2×D73⋊M16—21no—
17#17989638C2211yescharacteristic, complemented
18#17989638C2211yescharacteristic, complemented
19#17989621C2211yescharacteristic
20#17989621C2211yescharacteristic
21#17989618—21no—
22#17989618—21no—
23#17989617—21no—
24#17989617—21no—
25#17989615C2211yescharacteristic
26#17989615C2211yescharacteristic
27D4×#10596833C2211yescharacteristic, complemented
28D4×#10596833C2211yescharacteristic, complemented
29#17989608C2211yescharacteristic
30#17989607—21no—
31#17989590C2211yescharacteristic
32#17989559—21no—
33C4×#11924037C2211yescharacteristic
34C4×#11924037C2211yescharacteristic
35C2×#14556646—21no—
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40Q8⋊#10596833C2211yescharacteristic, complemented
41Q8⋊#10596833C2211yescharacteristic, complemented
42#17989435C2211yescharacteristic
43#17989435C2211yescharacteristic
44#17989435C2211yescharacteristic
45#17989435C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic, complemented
47D4⋊#10596833C2211yescharacteristic, complemented
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50#17989428C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52C2×#14556616C411yescharacteristic
53#41×Dic73C2211yescharacteristic
54C2×#14556553C2211yescharacteristic
55#17988696C411yescharacteristic
56#17988696C411yescharacteristic
57#17988690C411yescharacteristic
58#17988690C411yescharacteristic
59#17988672C2211yescharacteristic
60#17988672C2211yescharacteristic
61#17988667C2211yescharacteristic
62#17988667C2211yescharacteristic
63#17988627C411yescharacteristic
64#17988593C411yescharacteristic
65M16×Dic73C2211yescharacteristic
66#417×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69D73⋊M16—44no—
70C2×#11924037—21no—
71C2×#11924037—21no—
72C2×#11924037D411yescharacteristic
73C2×#11924037D411yescharacteristic
74C2×#11924036—21no—
75C2×#11924036—21no—
76C22×#10596833—21no—
77C22×#10596833—21no—
78#14556658—21no—
79#14556658—21no—
80#14556658—21no—
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C4⋊#10596833C2311yescharacteristic
85#14556648—21no—
86#14556648—21no—
87#14556648—21no—
88#14556648—21no—
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92#14556646—21no—
93#14556646—21no—
94#14556646—21no—
95#14556646—21no—
96#14556646D411yescharacteristic
97#14556646D411yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C4×#10596833C2311yescharacteristic
102C22×D292C2×C411yescharacteristic
103C22×C4×D73—21no—
104#14556616—24no—
105C2×#11924015C2×C411yescharacteristic
106C2×#11924015C2×C411yescharacteristic
107C2×#11924013C2×C411yescharacteristic
108C2×#11924013C2×C411yescharacteristic
109C2×#11924012C2×C411yescharacteristic
110Q8×Dic73C2×C411yescharacteristic
111Q8×Dic73C2311yescharacteristic
112D4×Dic73—21no—
113D4×Dic73—21no—
114D4×Dic73C2×C411yescharacteristic
115D4×Dic73C2311yescharacteristic
116#14556553—21no—
117#14556553—21no—
118#14556553C2×C411yescharacteristic
119#14556553C2×C411yescharacteristic
120#14556553C2311yescharacteristic
121#14556553C2311yescharacteristic
122#14556550—21no—
123#14556550C2×C411yescharacteristic
124C2×#11924005—21no—
125C2×C4×Dic73—21no—
126C2×C4×Dic73C2311yescharacteristic
127#14556524C2×C411yescharacteristic
128#14556524C2×C411yescharacteristic
129#14556522C2×C411yescharacteristic
130#14556522C2×C411yescharacteristic
131#14556513C2×C411yescharacteristic
132#14556513C2×C411yescharacteristic
133#14556513C2×C411yescharacteristic
134#14556513C2×C411yescharacteristic
135#14556501C2×C411yescharacteristic
136#14556501C2×C411yescharacteristic
137C8×Dic73C2311yescharacteristic
138C8×Dic73C2311yescharacteristic
139C2×#41×C73C2×C411yescharacteristic
140C2×M16×C73C2×C411yescharacteristic
141C2×C4⋊C4×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#103×C73C2×C411yescharacteristic
144#102×C73C2×C411yescharacteristic
145#102×C73C2×C411yescharacteristic
146#11924038—42no—
147#11924038—42no—
148#11924037—22no—
149#11924037—41no—
150#11924037—42no—
151#11924037—42no—
152#11924036—42no—
153#11924036—42no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—21no—
158C2×#10596833—21no—
159C2×#10596833—21no—
160C2×#10596833—42no—
161C2×#10596833—42no—
162C2×#10596833C2×D411yescharacteristic
163C2×#10596833C2×D411yescharacteristic
164C2×#10596833C2×D411yescharacteristic
165C2×#10596833C2×D411yescharacteristic
166C23×D73—21no—
167C2×D292—22no—
168C2×D292—21no—
169C2×D292—41no—
170C2×D292C22×C411yescharacteristic
171C2×D292C22×C411yescharacteristic
172C22×Dic73—21no—
173C22×Dic73—21no—
174C22×Dic73C2×D411yescharacteristic
175C22×Dic73#4111yescharacteristic
176C2×C4×D73—21no—
177C2×C4×D73—21no—
178C2×C4×D73—21no—
179C2×C4×D73—21no—
180C2×C4×D73—42no—
181C2×C4×D73—42no—
182#11924015—22no—
183#11924015—22no—
184#11924013—22no—
185#11924013M1614yescomplemented
186#11924012—22no—
187C2×#10596831C22×C411yescharacteristic
188C2×#10596831C22×C411yescharacteristic
189C4⋊Dic73—21no—
190C4⋊Dic73—21no—
191C4⋊Dic73C22×C411yescharacteristic
192C4⋊Dic73C22×C411yescharacteristic
193C22⋊Dic73—21no—
194C22⋊Dic73—21no—
195C22⋊Dic73—21no—
196#11924005—21no—
197#11924005—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73C22×C411yescharacteristic
204C4×Dic73C2×D411yescharacteristic
205C4×Dic73#4111yescharacteristic
206C4×Dic73C2411yescharacteristic
207#41×C73—22no—
208#41×C73—41no—
209C2×Q8×C73C22×C411yescharacteristic
210C2×D4×C73—21no—
211C2×D4×C73C22×C411yescharacteristic
212C22×C4×C73—21no—
213C22×C4×C73—21no—
214C22×C4×C73C22×C411yescharacteristic
215M16×C73—41no—
216C2×C8×C73C22×C411yescharacteristic
217C2×C8×C73C22×C411yescharacteristic
218C4⋊C4×C73—21no—
219C4⋊C4×C73C22×C411yescharacteristic
220C4⋊C4×C73C22×C411yescharacteristic
221C23×C73—21no—
222C23×C73C4×D411yescharacteristic
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—42no—
226C22×D73—42no—
227C22×D73—81no—
228Q8×C73—21no—
229Q8×C73—21no—
230D4×C73—21no—
231D4×C73—42no—
232D4×C73C2×M1612yescomplemented
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73C4×D411yescharacteristic
241C2×C4×C73C23×C411yescharacteristic
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—81no—
250C2×Dic73C4×D411yescharacteristic
251C2×Dic73C4×D411yescharacteristic
252C2×Dic73C22×D411yescharacteristic
253C2×Dic73C2×#4111yescharacteristic
254D292—21no—
255D292—21no—
256D292—21no—
257D292—42no—
258D292C2×M1612yes—
259C4×D73—42no—
260C4×D73—42no—
261C4×D73—41no—
262C4×D73—41no—
263C4×D73—41no—
264C4×D73—41no—
265#10596833—22no—
266#10596833—21no—
267#10596833—21no—
268#10596833—21no—
269#10596833—41no—
270#10596833—41no—
271#10596833—41no—
272#10596833—41no—
273#10596833—41no—
274#10596833AGL(1,ℤ/8)12yescomplemented
275C8×C73—21no—
276C8×C73—21no—
277#10596831—21no—
278#10596831C2×M1612yescomplemented
279C22×C73—21no—
280C22×C73—22no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73C2×C4×D411yescharacteristic
286C2×D73—22no—
287C2×D73—41no—
288C2×D73—41no—
289C2×D73—41no—
290C2×D73—41no—
291C2×D73—41no—
292C2×D73—84no—
293C4×C73—21no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73C22×M1611yescharacteristic
298C4×C73#56811yescharacteristic
299Dic73—21no—
300Dic73—21no—
301Dic73—21no—
302Dic73—41no—
303Dic73—41no—
304Dic73C2×AGL(1,ℤ/8)11yescharacteristic
305Dic73#57511yescharacteristic
306#18046—731nomaximal, Sylow
307C2×C73—41no—
308C2×C73—42no—
309C2×C73D4×M1611yescharacteristic
310C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
311C2×C73#295611yescharacteristic
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#1569—731no—
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—731no—
355#432—1461no—
356#428—731no—
357#423—731no—
358#418—731no—
359#417#6440811yescharacteristic
360#407—1461no—
361C4×M16—731no—
362C4×M16—731no—
363C4×M16—731no—
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⋊C8—731no—
374D4⋊C8—731no—
375D4⋊C8—731no—
376D4⋊C8—731no—
377#322—731no—
378#322—731no—
379#322—731no—
380C82—731no—
381C2×#41C2×#6440811yescharacteristic
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—2924no—
390C2×M16—731no—
391C2×M16—731no—
392C2×M16—731no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16—1461no—
397C2×M16C2×#6440811yescharacteristic
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⋊C4C2×#6440811yescharacteristic
417C2×C22⋊C4—1461no—
418C2×C42—731no—
419C2×C42—1461no—
420#107—731no—
421#107—731no—
422#106—731no—
423#106—731no—
424C4⋊C8—731no—
425C4⋊C8—731no—
426C4⋊C8—731no—
427C4⋊C8—731no—
428C4⋊C8—1461no—
429C4⋊C8—1461no—
430C4⋊C8—1461no—
431#103C2×#6440811yescharacteristic
432#103C2×#6440811yescharacteristic
433#102—731no—
434#102—731no—
435#102—731no—
436#102—731no—
437#102C2×#6440811yescharacteristic
438#102C2×#6440811yescharacteristic
439C22⋊C8—1461no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444C22⋊C8—1461no—
445C22⋊C8—1461no—
446#97—731no—
447#97—731no—
448#97—731no—
449#97—731no—
450#97—1461no—
451#97—1461no—
452#97—1461no—
453#97—1461no—
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—731no—
458C4×C8—731no—
459C4×C8—731no—
460C24—1461no—
461#41—22no—
462#41—41no—
463C2×Q8C22×#6440811yescharacteristic
464C2×D4—21no—
465C2×D4—2921no—
466C2×D4—731no—
467C2×D4—731no—
468C2×D4—1462no—
469C2×D4—1461no—
470C2×D4C22×#6440811yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—2922no—
474C22×C4—2922no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4C22×#6440811yescharacteristic
484SD16—1462no—
485SD16—1462no—
486D8—734no—
487D8—1462no—
488M16—41no—
489M16—2922no—
490M16—2922no—
491M16—2922no—
492M16—2922no—
493M16—2921no—
494M16—1462no—
495M16—1462no—
496C2×C8—2922no—
497C2×C8—2922no—
498C2×C8—2922no—
499C2×C8—2922no—
500C2×C8—731no—
501C2×C8—731no—
502C2×C8—731no—
503C2×C8—731no—
504C2×C8—731no—
505C2×C8—731no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8C22×#6440811yescharacteristic
513C2×C8C22×#6440811yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—731no—
516C4⋊C4—731no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4C22×#6440811yescharacteristic
520C4⋊C4C22×#6440811yescharacteristic
521C22⋊C4—1461no—
522C22⋊C4—1461no—
523C22⋊C4—1461no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C22⋊C4—1461no—
527C22⋊C4—1461no—
528C42—731no—
529C42—731no—
530C42—731no—
531C42—731no—
532C42—1461no—
533C42—1461no—
534C42—1461no—
535C23—21no—
536C23—2922no—
537C23—2922no—
538C23—5841no—
539C23—1461no—
540C23—1461no—
541C23D4×#6440811yescharacteristic
542Q8—21no—
543Q8—21no—
544D4—21no—
545D4—42no—
546D4—2922no—
547D4—732no—
548D4—1461no—
549D4—1461no—
550D4—1461no—
551D4D73⋊M1612yescomplemented
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—5841no—
569C2×C4—731no—
570C2×C4—731no—
571C2×C4—731no—
572C2×C4—731no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4D4×#6440811yescharacteristic
578C2×C4C23×#6440811yescharacteristic
579C8—21no—
580C8—21no—
581C8—2921no—
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—2921no—
586C8—732no—
587C8—732no—
588C8—1461no—
589C8—1461no—
590C8—1461no—
591C8—1462no—
592C8—1461no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—41no—
598C22—41no—
599C22—2921no—
600C22—2921no—
601C22—2921no—
602C22—2921no—
603C22—2921no—
604C22—5844no—
605C22—1462no—
606C22C2×D4×#6440811yescharacteristic
607C4—21no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—2921no—
612C4—2921no—
613C4—731no—
614C4—731no—
615C4—1461no—
616C4—1461no—
617C4—1461no—
618C4C2×D73⋊M1611yescharacteristic
619C4#1798966511yescharacteristic
620C2—41no—
621C2—42no—
622C2—2922no—
623C2—5841no—
624C2#2262666911yescharacteristic
625C2AGL(1,ℤ/8)×#6440811yescharacteristic
626C2#2262670611yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#417×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#14556616C4
cosocleC4×Dic73C24
hypercenter#417#64408
hypocenterC73#18046
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #22626768C2semidirect
2. #22626671C2semidirect
3. #22625575C2semidirect
4. #22625565C2semidirect
5. #22625550C2semidirect
6. #22625549C2semidirect
7. #22625537C2semidirect
8. #22625530C2semidirect
9. #22625528C2semidirect
10. #22625511C2semidirect
11. #22625495C2semidirect
12. #22625272C2semidirect
13. #22625266C2semidirect
14. #22625154C2semidirect
17. #17989638C22semidirect
18. #17989638C22semidirect
27. D4×#10596833C22semidirect
28. D4×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. Q8⋊#10596833C22semidirect
41. Q8⋊#10596833C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. #17989428C22semidirect
51. C8×#10596833C22semidirect
185. #11924013M16semidirect
232. D4×C73C2×M16semidirect
274. #10596833AGL(1,ℤ/8)semidirect
278. #10596831C2×M16semidirect
329. C73#18046semidirect
551. D4D73⋊M16semidirect

Maximal subgroups

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

subgroupindex
#226267682
#226266712
#226255752
#226255652
#226255502
#226255492
#226255372
#226255302
#226255282
#226255112
#226254952
#226252722
#226252662
#226251542
#226187442
#1804673