Finite Group Databasebeta

← #34019460

Subgroups — #34019460

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
2D4×#11924037C211yesmaximal, characteristic, complemented
3#22625574C211yesmaximal, characteristic, complemented
4#22625567C211yesmaximal, characteristic, complemented
5#22625552C211yesmaximal, characteristic, complemented
6#22625548C211yesmaximal, characteristic, complemented
7#22625536C211yesmaximal, characteristic, complemented
8#22625532C211yesmaximal, characteristic, complemented
9#22625527C211yesmaximal, characteristic, complemented
10D8×#10596833C211yesmaximal, characteristic, complemented
11#22625495C211yesmaximal, characteristic, complemented
12#22625270C211yesmaximal, characteristic, complemented
13#22625266C211yesmaximal, characteristic, complemented
14#22625154C211yesmaximal, characteristic, complemented
15AGL(1,ℤ/8)×Dic73C211yesmaximal, characteristic
16C22×#11924037—21no—
17#17989638C2211yescharacteristic, complemented
18#17989638C2211yescharacteristic, complemented
19#17989622C2211yescharacteristic
20#17989622C2211yescharacteristic
21#17989618—21no—
22#17989618—21no—
23#17989618—21no—
24#17989618—21no—
25D4×#10596833C2211yescharacteristic
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic, complemented
28D4×#10596833C2211yescharacteristic, complemented
29#17989608C2211yescharacteristic
30#17989607—21no—
31#17989590C2211yescharacteristic
32#17989557—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
42D4⋊#10596833C2211yescharacteristic
43D4⋊#10596833C2211yescharacteristic
44D4⋊#10596833C2211yescharacteristic
45D4⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic, complemented
47D4⋊#10596833C2211yescharacteristic, complemented
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50#17989428C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52#41×Dic73C2211yescharacteristic
53C2×D4×Dic73C2211yescharacteristic
54#17988699C411yescharacteristic
55#17988699C411yescharacteristic
56#17988688C411yescharacteristic
57#17988688C411yescharacteristic
58SD16×Dic73C2211yescharacteristic
59SD16×Dic73C2211yescharacteristic
60D8×Dic73C2211yescharacteristic
61D8×Dic73C2211yescharacteristic
62#17988629C411yescharacteristic
63#17988592C411yescharacteristic
64#17988588C411yescharacteristic
65M16×Dic73C2211yescharacteristic
66C2×AGL(1,ℤ/8)×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C2×#11924037—21no—
70C2×#11924037—21no—
71C2×#11924037—44no—
72C2×#11924037D411yescharacteristic
73C2×#11924037D411yescharacteristic
74C22×#10596833—21no—
75C22×#10596833—21no—
76C22×#10596833—21no—
77C22×#10596833—21no—
78#14556658—21no—
79#14556658—21no—
80#14556658—21no—
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C4⋊#10596833C2311yescharacteristic
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833—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
102C23×Dic73—21no—
103C2×#11924012C2×C411yescharacteristic
104Q8×Dic73C2×C411yescharacteristic
105Q8×Dic73C2311yescharacteristic
106D4×Dic73—21no—
107D4×Dic73—21no—
108D4×Dic73—21no—
109D4×Dic73—21no—
110D4×Dic73C2×C411yescharacteristic
111D4×Dic73C2×C411yescharacteristic
112D4×Dic73C2×C411yescharacteristic
113D4×Dic73C2311yescharacteristic
114D4×Dic73C2311yescharacteristic
115D4×Dic73C2311yescharacteristic
116#14556550—21no—
117#14556550C2×C411yescharacteristic
118C2×C4⋊Dic73C2×C411yescharacteristic
119C2×C22⋊Dic73—21no—
120C2×C4×Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556525C2×C411yescharacteristic
124#14556523C2×C411yescharacteristic
125#14556523C2×C411yescharacteristic
126#14556515C2×C411yescharacteristic
127#14556515C2×C411yescharacteristic
128#14556512C2×C411yescharacteristic
129#14556512C2×C411yescharacteristic
130#14556512C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134#14556501C2×C411yescharacteristic
135#14556501C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C2×#41×C73C2×C411yescharacteristic
139C22×D4×C73C2×C411yescharacteristic
140AGL(1,ℤ/8)×C73—24no—
141C2×SD16×C73C2×C411yescharacteristic
142C2×SD16×C73C2×C411yescharacteristic
143C2×D8×C73C2×C411yescharacteristic
144C2×D8×C73C2×C411yescharacteristic
145C2×M16×C73C2×C411yescharacteristic
146#11924037—22no—
147#11924037—41no—
148#11924037—42no—
149#11924037—42no—
150#11924037—42no—
151#11924037—42no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—21no—
158C2×#10596833—42no—
159C2×#10596833—42no—
160C2×#10596833—42no—
161C2×#10596833—42no—
162C2×#10596833C2×D411yescharacteristic
163C2×#10596833C2×D411yescharacteristic
164C2×#10596833C2×D411yescharacteristic
165C2×#10596833C2×D411yescharacteristic
166C22×Dic73—21no—
167C22×Dic73—21no—
168C22×Dic73—21no—
169C22×Dic73—21no—
170C22×Dic73—21no—
171C22×Dic73—21no—
172C22×Dic73—21no—
173C22×Dic73—42no—
174C22×Dic73—42no—
175C22×Dic73C2×D411yescharacteristic
176C22×Dic73#4111yescharacteristic
177#11924012—41no—
178C2×#10596831C22×C411yescharacteristic
179C2×#10596831C22×C411yescharacteristic
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—21no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186C4⋊Dic73C22×C411yescharacteristic
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C22⋊Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73C22×C411yescharacteristic
198C4×Dic73C2×D411yescharacteristic
199C4×Dic73#4111yescharacteristic
200C4×Dic73C2411yescharacteristic
201C24×C73—21no—
202#41×C73—22no—
203#41×C73—41no—
204C2×Q8×C73C22×C411yescharacteristic
205C2×D4×C73—22no—
206C2×D4×C73—21no—
207C2×D4×C73—21no—
208C2×D4×C73—41no—
209C2×D4×C73C22×C411yescharacteristic
210C2×D4×C73C22×C411yescharacteristic
211C2×D4×C73C22×C411yescharacteristic
212C22×C4×C73—21no—
213C22×C4×C73C22×C411yescharacteristic
214SD16×C73—22no—
215SD16×C73—22no—
216D8×C73—22no—
217D8×C73M1614yescomplemented
218M16×C73—22no—
219C2×C8×C73C22×C411yescharacteristic
220C2×C8×C73C22×C411yescharacteristic
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—42no—
225C23×C73—42no—
226C23×C73—81no—
227C23×C73C4×D411yescharacteristic
228Q8×C73—21no—
229Q8×C73—21no—
230D4×C73—21no—
231D4×C73—21no—
232D4×C73—21no—
233D4×C73—21no—
234D4×C73—42no—
235D4×C73—42no—
236D4×C73C2×M1612yes—
237D4×C73C2×M1612yescomplemented
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—42no—
242C2×C4×C73C4×D411yescharacteristic
243C2×C4×C73C23×C411yescharacteristic
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—41no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—81no—
261C2×Dic73C4×D411yescharacteristic
262C2×Dic73C4×D411yescharacteristic
263C2×Dic73C22×D411yescharacteristic
264C2×Dic73C2×#4111yescharacteristic
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×C73C2×M1612yescomplemented
277#10596831—21no—
278#10596831—21no—
279C22×C73—21no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—21no—
283C22×C73—21no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73—84no—
292C22×C73C2×C4×D411yescharacteristic
293C4×C73—21no—
294C4×C73—41no—
295C4×C73C22×M1611yescharacteristic
296C4×C73#56811yescharacteristic
297Dic73—21no—
298Dic73—21no—
299Dic73—21no—
300Dic73—41no—
301Dic73—41no—
302Dic73—41no—
303Dic73—41no—
304Dic73C2×AGL(1,ℤ/8)11yescharacteristic
305Dic73#57511yescharacteristic
306#18046—731nomaximal, Sylow
307C2×C73—41no—
308C2×C73—42no—
309C2×C73—42no—
310C2×C73—81no—
311C2×C73D4×M1611yescharacteristic
312C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
313C2×C73#295611yescharacteristic
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)#6440811yescharacteristic
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—731no—
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×D4C2×#6440811yescharacteristic
383C23×C4—1461no—
384AGL(1,ℤ/8)—24no—
385C2×SD16C2×#6440811yescharacteristic
386C2×SD16C2×#6440811yescharacteristic
387C2×D8C2×#6440811yescharacteristic
388C2×D8C2×#6440811yescharacteristic
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⋊C4—731no—
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#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—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—21no—
461#41—22no—
462#41—41no—
463C2×Q8C22×#6440811yescharacteristic
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—41no—
468C2×D4C22×#6440811yescharacteristic
469C2×D4C22×#6440811yescharacteristic
470C2×D4C22×#6440811yescharacteristic
471C22×C4—21no—
472C22×C4—2922no—
473C22×C4—2922no—
474C22×C4—731no—
475C22×C4—731no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4C22×#6440811yescharacteristic
484SD16—22no—
485SD16—22no—
486D8—22no—
487D8#1192403714yescomplemented
488M16—22no—
489M16—2922no—
490M16—2922no—
491M16—2922no—
492M16—2922no—
493M16—2921no—
494M16—2921no—
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—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—731no—
529C42—731no—
530C42—731no—
531C42—731no—
532C42—1461no—
533C42—1461no—
534C42—1461no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—42no—
539C23—42no—
540C23—81no—
541C23D4×#6440811yescharacteristic
542Q8—21no—
543Q8—21no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—42no—
549D4—42no—
550D4C2×#1192403712yes—
551D4C2×#1192403712yescomplemented
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—42no—
556C2×C4—2922no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2921no—
566C2×C4—5841no—
567C2×C4—731no—
568C2×C4—731no—
569C2×C4—731no—
570C2×C4—731no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4D4×#6440811yescharacteristic
578C2×C4C23×#6440811yescharacteristic
579C8—21no—
580C8—2921no—
581C8—2921no—
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—732no—
586C8—1461no—
587C8—1461no—
588C8—1461no—
589C8—1462no—
590C8—1461no—
591C8—1461no—
592C8C2×#1192403712yescomplemented
593C22—21no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—84no—
606C22C2×D4×#6440811yescharacteristic
607C4—21no—
608C4—41no—
609C4—2921no—
610C4—2921no—
611C4—2921no—
612C4—2921no—
613C4—731no—
614C4—731no—
615C4—1461no—
616C4—1461no—
617C4—1461no—
618C4C22×#1192403711yescharacteristic
619C4#1798966511yescharacteristic
620C2—41no—
621C2—42no—
622C2—42no—
623C2—81no—
624C2D4×#1192403711yescharacteristic
625C2AGL(1,ℤ/8)×#6440811yescharacteristic
626C2#2262670511yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting, prime radicalC2×AGL(1,ℤ/8)×C73C4
socleC22×C73C2×C4×D4
cosocleC4×Dic73C24
hypercenterC2×AGL(1,ℤ/8)#64408
hypocenterC73#18046
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22626768C2semidirect
2. D4×#11924037C2semidirect
3. #22625574C2semidirect
4. #22625567C2semidirect
5. #22625552C2semidirect
6. #22625548C2semidirect
7. #22625536C2semidirect
8. #22625532C2semidirect
9. #22625527C2semidirect
10. D8×#10596833C2semidirect
11. #22625495C2semidirect
12. #22625270C2semidirect
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
217. D8×C73M16semidirect
237. D4×C73C2×M16semidirect
274. #10596833AGL(1,ℤ/8)semidirect
276. C8×C73C2×M16semidirect
329. C73#18046semidirect
487. D8#11924037semidirect
551. D4C2×#11924037semidirect
592. C8C2×#11924037semidirect

Maximal subgroups

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

subgroupindex
#226267682
D4×#119240372
#226255742
#226255672
#226255522
#226255482
#226255362
#226255322
#226255272
D8×#105968332
#226254952
#226252702
#226252662
#226251542
AGL(1,ℤ/8)×Dic732
#1804673