Finite Group Databasebeta

← #33057428

Subgroups — #33057428

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618950C211yesmaximal, characteristic, complemented
2#22618939C211yesmaximal, characteristic, complemented
3C4×#14556612C211yesmaximal, characteristic
4#22608823C211yesmaximal, characteristic, complemented
5#22608820C211yesmaximal, characteristic, complemented
6#22608781C211yesmaximal, characteristic, complemented
7#22608778C211yesmaximal, characteristic, complemented
8#22608760C211yesmaximal, characteristic, complemented
9#22608757C211yesmaximal, characteristic, complemented
10SD16×#10596831C211yesmaximal, characteristic, complemented
11#22608691C211yesmaximal, characteristic, complemented
12#22608364C211yesmaximal, characteristic, complemented
13#22608362C211yesmaximal, characteristic, complemented
14C8×#11924011C211yesmaximal, characteristic, complemented
15C8×SD16×C73C211yesmaximal, characteristic, complemented
16C2×#14556612C411yescharacteristic
17C4×#11924029C2211yescharacteristic
18C4×#11924027C2211yescharacteristic
19#17988717C2211yescharacteristic
20#17988714C2211yescharacteristic
21Q8×#10596831C2211yescharacteristic
22C4×#11924018C2211yescharacteristic
23#17988679C411yescharacteristic
24C4×#11924017C2211yescharacteristic
25SD16×Dic73C411yescharacteristic
26#17988674C411yescharacteristic
27C4×#11924013C2211yescharacteristic
28#17988667C411yescharacteristic
29#17988666—21no—
30#17988666C2211yescharacteristic
31#17988664—21no—
32#17988664C2211yescharacteristic
33#17988658—21no—
34#17988655—21no—
35D4×#10596831C2211yescharacteristic
36#17988643—21no—
37#17988643C2211yescharacteristic
38#17988643C2211yescharacteristic
39#17988641—21no—
40#17988638C2211yescharacteristic
41#17988622C411yescharacteristic
42#17988582C411yescharacteristic
43#17988580C411yescharacteristic
44C4×#11924011—21no—
45C4×#11924011C2211yescharacteristic
46C2×#14556501—21no—
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988338C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53#17988317C2211yescharacteristic
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988308C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C2×#11924029C2×C411yescharacteristic
68C2×#11924027C2×C411yescharacteristic
69#14556612—24no—
70C2×#11924018C2×C411yescharacteristic
71C2×#11924017C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011D411yescharacteristic
76C2×#11924011D411yescharacteristic
77C22×#10596831—21no—
78C2×C8×D73C2×C411yescharacteristic
79C2×C8×D73C2×C411yescharacteristic
80Q8×Dic73C2×C411yescharacteristic
81C4×Dic146C2311yescharacteristic
82#14556556C2×C411yescharacteristic
83C4×D292—21no—
84C4×D292C2311yescharacteristic
85D4×Dic73C2×C411yescharacteristic
86#14556553—21no—
87#14556553C2×C411yescharacteristic
88C4×C22⋊D73—21no—
89#14556551—21no—
90#14556549—21no—
91#14556549C2×C411yescharacteristic
92#14556549C2×C411yescharacteristic
93#14556547—21no—
94C2×C4×Dic73—21no—
95C42×D73—21no—
96C42×D73C2311yescharacteristic
97#14556524C2×C411yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556516C2×C411yescharacteristic
107#14556515C2×C411yescharacteristic
108#14556514C2×C411yescharacteristic
109#14556513C2×C411yescharacteristic
110#14556512C2×C411yescharacteristic
111#14556511C2×C411yescharacteristic
112C22⋊#10596831—21no—
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556501—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501—21no—
122#14556501D411yescharacteristic
123#14556501D411yescharacteristic
124C4×#10596831—21no—
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73C2×C411yescharacteristic
129C8×Dic73C2×C411yescharacteristic
130C2×SD16×C73C2×C411yescharacteristic
131C22×C8×C73—21no—
132C4×Q8×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#106×C73C2×C411yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924029—22no—
146#11924029—41no—
147#11924027—22no—
148#11924027—41no—
149C2×Dic146C22×C411yescharacteristic
150C2×D292—21no—
151C2×D292C22×C411yescharacteristic
152C2×C22⋊D73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73—21no—
158C2×C4×D73C22×C411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160C2×C4×D73#4111yescharacteristic
161#11924018—22no—
162#11924017—22no—
163#11924013—22no—
164#11924011—22no—
165#11924011—22no—
166#11924011—42no—
167#11924011—42no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—42no—
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C8×D73—22no—
177C8×D73—41no—
178C4⋊Dic73—21no—
179C4⋊Dic73C22×C411yescharacteristic
180C4⋊Dic73C22×C411yescharacteristic
181#11924007—21no—
182#11924007—21no—
183C22⋊Dic73—21no—
184#11924005—21no—
185#11924005—21no—
186#11924005—21no—
187#11924005—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73C22×C411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C4×Dic73#4111yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C22×C4×C73—21no—
199SD16×C73M1614yescomplemented
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—42no—
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C4⋊C4×C73—21no—
209C4⋊C4×C73C22×C411yescharacteristic
210C4⋊C4×C73C22×C411yescharacteristic
211C22⋊C4×C73—21no—
212C42×C73—21no—
213C42×C73C2411yescharacteristic
214C23×C73—21no—
215C22×D73—21no—
216C22×D73C4×D411yescharacteristic
217Q8×C73—21no—
218Q8×C73C2×M1612yes—
219D4×C73—21no—
220D4×C73C2×M1612yes—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73C23×C411yescharacteristic
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C22⋊D73—42no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—42no—
234C2×Dic73—81no—
235C2×Dic73C4×D411yescharacteristic
236C2×Dic73C4×D411yescharacteristic
237C2×Dic73C4×D411yescharacteristic
238D292—21no—
239D292—21no—
240D292—42no—
241C4×D73—21no—
242C4×D73—21no—
243C4×D73—21no—
244C4×D73—21no—
245C4×D73—41no—
246C4×D73—42no—
247C4×D73—41no—
248C4×D73—41no—
249C4×D73—81no—
250Dic146—21no—
251Dic146—21no—
252C8×C73—21no—
253C8×C73—21no—
254C8×C73—41no—
255C8×C73—41no—
256C8×C73—41no—
257C8×C73C2×M1612yes—
258C8×C73#13512yescomplemented
259#10596831—21no—
260#10596831—21no—
261#10596831—21no—
262#10596831—41no—
263#10596831—41no—
264#10596831—41no—
265#10596831#13512yescomplemented
266C22×C73—22no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73C2×C4×D411yescharacteristic
270C2×D73—21no—
271C2×D73—21no—
272C2×D73—21no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—41no—
276C4×C73—22no—
277C4×C73—21no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73C22×M1611yescharacteristic
281C4×C73#56811yescharacteristic
282C4×C73C2×#13511yescharacteristic
283C4×C73C2×#13511yescharacteristic
284Dic73—21no—
285Dic73—21no—
286Dic73—21no—
287Dic73—41no—
288Dic73—41no—
289Dic73—41no—
290#18037—731nomaximal, Sylow
291C2×C73—42no—
292C2×C73D4×M1611yescharacteristic
293C2×C73C4×#13511yescharacteristic
294C2×C73#295711yescharacteristic
295D73—41no—
296D73—81no—
297#2971—731no—
298#2970—731no—
299C4×#135—731no—
300#1590—731no—
301#1590—731no—
302#1584—731no—
303#1584—731no—
304#1581—731no—
305#1581—731no—
306C8×SD16—731no—
307C8×SD16D7311yescharacteristic, complemented
308#1568—731no—
309#1488—731no—
310#1488—731no—
311C8×M16—731no—
312C73#1803711yescharacteristic, complemented, Sylow
313C2×#135—731no—
314C4×#41—731no—
315C4×#41—731no—
316#446—731no—
317#446—731no—
318C8×Q8—731no—
319C8×Q8C2×D7311yescharacteristic
320C4×Q16—731no—
321C4×Q16—731no—
322C4×SD16—731no—
323C4×SD16—731no—
324C4×SD16—731no—
325C4×SD16C2×D7311yescharacteristic
326C4×D8—731no—
327C4×D8—731no—
328#436—731no—
329#436—731no—
330#436—1461no—
331#436—1461no—
332#435—1461no—
333#435—1461no—
334C8×D4—731no—
335C8×D4C2×D7311yescharacteristic
336#432—731no—
337#432—731no—
338#432—1461no—
339#432—1461no—
340#431—731no—
341#427—731no—
342#416—731no—
343#416—731no—
344C4×M16—731no—
345C4×M16—1461no—
346C2×#97—1461no—
347C2×C4×C8—731no—
348#335—731no—
349#335—731no—
350#334—731no—
351#334C2×D7311yescharacteristic
352Q8⋊C8—731no—
353Q8⋊C8—731no—
354Q8⋊C8—731no—
355Q8⋊C8C2×D7311yescharacteristic
356D4⋊C8—731no—
357D4⋊C8—731no—
358D4⋊C8—731no—
359D4⋊C8C2×D7311yescharacteristic
360#322—731no—
361#322—731no—
362C82—731no—
363C82C2×D7311yescharacteristic
364C2×#41—731no—
365C2×#41—731no—
366#135—1464no—
367C2×Q16—731no—
368C2×SD16—731no—
369C2×SD16C4×D7311yescharacteristic
370C2×D8—731no—
371C2×M16—731no—
372C2×M16—731no—
373C2×M16—1461no—
374C2×M16—1461no—
375C22×C8—21no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—1461no—
379C4×Q8—731no—
380C4×Q8—731no—
381C4×Q8—731no—
382C4×Q8C22×D7311yescharacteristic
383C4×D4—731no—
384C4×D4—731no—
385C4×D4—731no—
386C4×D4—1461no—
387C4×D4—1461no—
388C4×D4—1461no—
389C4×D4—1461no—
390C4×D4C22×D7311yescharacteristic
391#117—731no—
392#117—731no—
393#117—1461no—
394#117—1461no—
395C2×C42—731no—
396C2×C42—1461no—
397C2×C42—1461no—
398#107—731no—
399#107—731no—
400#106—731no—
401#106C4×D7311yescharacteristic
402C4⋊C8—21no—
403C4⋊C8—731no—
404C4⋊C8—731no—
405C4⋊C8—1461no—
406C4⋊C8—1461no—
407C4⋊C8—1461no—
408C4⋊C8—1461no—
409C4⋊C8C22×D7311yescharacteristic
410C4⋊C8C22×D7311yescharacteristic
411#103—731no—
412#103—731no—
413#103—731no—
414#103C4×D7311yescharacteristic
415#102—731no—
416#102—731no—
417#102—731no—
418#102C4×D7311yescharacteristic
419C22⋊C8—21no—
420C22⋊C8—1461no—
421C22⋊C8—1461no—
422C22⋊C8—1461no—
423C22⋊C8—1461no—
424C22⋊C8—1461no—
425C22⋊C8—1461no—
426#97—731no—
427#97—731no—
428#97—1461no—
429#97—1461no—
430#97—1461no—
431#97—1461no—
432C4×C8—21no—
433C4×C8—731no—
434C4×C8—731no—
435C4×C8—731no—
436C4×C8—731no—
437C4×C8—731no—
438C4×C8—1461no—
439C4×C8C22×D7311yescharacteristic
440C4×C8C22×D7311yescharacteristic
441C4×C8C22×D7311yescharacteristic
442#41—2921no—
443#41—2921no—
444#41—1462no—
445#41—1462no—
446C2×Q8—731no—
447C2×Q8C2×C4×D7311yescharacteristic
448C2×D4—731no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4C2×C4×D7311yescharacteristic
452C22×C4—21no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461Q16—1462no—
462SD16—1462no—
463SD16#1192401114yescomplemented
464D8—1462no—
465M16—2922no—
466M16—2922no—
467M16—1462no—
468M16—1462no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—42no—
473C2×C8—2921no—
474C2×C8—2922no—
475C2×C8—731no—
476C2×C8—731no—
477C2×C8—731no—
478C2×C8—731no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1462no—
483C2×C8C2×C4×D7311yescharacteristic
484C2×C8C2×C4×D7311yescharacteristic
485C2×C8D4×D7311yescharacteristic
486C2×C8D4×D7311yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—731no—
489C4⋊C4—731no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4C2×C4×D7311yescharacteristic
494C4⋊C4C2×C4×D7311yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—1461no—
497C22⋊C4—1461no—
498C22⋊C4—1461no—
499C22⋊C4—1461no—
500C22⋊C4—1461no—
501C42—21no—
502C42—731no—
503C42—731no—
504C42—731no—
505C42—1461no—
506C42—1461no—
507C42—1461no—
508C42—1461no—
509C42—1461no—
510C42C23×D7311yescharacteristic
511C23—21no—
512C23—731no—
513C23—1461no—
514Q8—21no—
515Q8—1461no—
516Q8—1461no—
517Q8C2×#1192401112yes—
518D4—21no—
519D4—2922no—
520D4—2922no—
521D4—1461no—
522D4—1461no—
523D4C2×#1192401112yes—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—2922no—
530C2×C4—2921no—
531C2×C4—2922no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—5841no—
535C2×C4—5841no—
536C2×C4—731no—
537C2×C4—731no—
538C2×C4—731no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4C22×C4×D7311yescharacteristic
547C2×C4C2×D4×D7311yescharacteristic
548C2×C4#41×D7311yescharacteristic
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—2921no—
555C8—2921no—
556C8—2921no—
557C8—732no—
558C8—1461no—
559C8—1461no—
560C8—1461no—
561C8C2×#1192401112yes—
562C8#1455661212yescomplemented
563C22—22no—
564C22—41no—
565C22—41no—
566C22—2921no—
567C22—2921no—
568C22—2921no—
569C22—1461no—
570C22—1461no—
571C22—1461no—
572C22C4×D4×D7311yescharacteristic
573C4—22no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—2921no—
578C4—2921no—
579C4—2921no—
580C4—1461no—
581C4—1461no—
582C4—1461no—
583C4C22×#1192401111yescharacteristic
584C4#1798935211yescharacteristic
585C4#135×D7311yescharacteristic
586C4C2×#1455661211yescharacteristic
587C2—42no—
588C2—2921no—
589C2—5841no—
590C2D4×#1192401111yescharacteristic
591C2C4×#1455661211yescharacteristic
592C2#2261884311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×SD16×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556612C4
cosocleC42×C73C24
hypercenterC8×SD16D73
hypocenterC73#18037
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
1. #22618950C2semidirect
2. #22618939C2semidirect
4. #22608823C2semidirect
5. #22608820C2semidirect
6. #22608781C2semidirect
7. #22608778C2semidirect
8. #22608760C2semidirect
9. #22608757C2semidirect
10. SD16×#10596831C2semidirect
11. #22608691C2semidirect
12. #22608364C2semidirect
13. #22608362C2semidirect
14. C8×#11924011C2semidirect
15. C8×SD16×C73C2semidirect
199. SD16×C73M16semidirect
258. C8×C73#135semidirect
265. #10596831#135semidirect
307. C8×SD16D73semidirect
312. C73#18037semidirect
463. SD16#11924011semidirect
562. C8#14556612semidirect

Maximal subgroups

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

subgroupindex
#226189502
#226189392
C4×#145566122
#226088232
#226088202
#226087812
#226087782
#226087602
#226087572
SD16×#105968312
#226086912
#226083642
#226083622
C8×#119240112
C8×SD16×C732
#1803773