Finite Group Databasebeta

← #36526009

Subgroups — #36526009

Order 22784.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029270C211yesmaximal, characteristic
2M16×Dic178C211yesmaximal, characteristic, complemented
3C4×#15668045C211yesmaximal, characteristic
4#26019156C211yesmaximal, characteristic, complemented
5#26019144C211yesmaximal, characteristic, complemented
6#26019132C211yesmaximal, characteristic, complemented
7#26019125C211yesmaximal, characteristic, complemented
8#26019104C211yesmaximal, characteristic, complemented
9#26019097C211yesmaximal, characteristic, complemented
10#26019072C211yesmaximal, characteristic, complemented
11C8×Dic356C211yesmaximal, characteristic, complemented
12#26019019C211yesmaximal, characteristic, complemented
13#26018700C211yesmaximal, characteristic, complemented
14#26018692C211yesmaximal, characteristic
15C89×#1455C211yesmaximal, characteristic, complemented
16C2×#15668045C411yescharacteristic
17C4×#13083574C2211yescharacteristic
18C2×C4×Dic178C2211yescharacteristic
19#19151728C2211yescharacteristic
20#19151728C2211yescharacteristic
21#19151726—21no—
22#19151726—21no—
23C8×Dic178C2211yescharacteristic
24C8×Dic178C2211yescharacteristic
25C8×Dic178C2211yescharacteristic
26C8×Dic178C2211yescharacteristic
27#19151706C411yescharacteristic
28#19151706C411yescharacteristic
29#19151700C411yescharacteristic
30#19151700C411yescharacteristic
31C4×Dic356C2211yescharacteristic
32C4×Dic356C2211yescharacteristic
33C4×#13083563C2211yescharacteristic
34C4×#13083563C2211yescharacteristic
35#19151677C2211yescharacteristic
36#19151677C2211yescharacteristic
37#19151674—21no—
38#19151674—21no—
39#19151655C2211yescharacteristic
40#19151653—21no—
41#19151640C411yescharacteristic
42#19151622C2211yescharacteristic
43#19151621—21no—
44#19151608C411yescharacteristic
45#19151600C411yescharacteristic
46#19151550—21no—
47M16×Dic89—21no—
48#19151351C2211yescharacteristic
49#19151351C2211yescharacteristic
50#19151349C2211yescharacteristic
51#19151349C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151330C2211yescharacteristic
55#19151330C2211yescharacteristic
56#19151330C2211yescharacteristic
57#19151330C2211yescharacteristic
58#19151327C2211yescharacteristic
59#19151327C2211yescharacteristic
60C4×M16×C89C2211yescharacteristic
61C4×M16×C89C2211yescharacteristic
62C4×M16×C89C2211yescharacteristic
63#322×C89C2211yescharacteristic
64#322×C89C2211yescharacteristic
65#322×C89C2211yescharacteristic
66C82×C89C2211yescharacteristic
67C2×#13083574C2×C411yescharacteristic
68C22×Dic178C2×C411yescharacteristic
69#15668045—24no—
70C2×Dic356C2×C411yescharacteristic
71C2×Dic356C2×C411yescharacteristic
72C2×#13083563C2×C411yescharacteristic
73C2×#13083563C2×C411yescharacteristic
74C2×#13083558—21no—
75C2×#13083557—21no—
76C2×#13083557—21no—
77C4×Dic178—21no—
78C4×Dic178—21no—
79C4×Dic178C2×C411yescharacteristic
80C4×Dic178C2×C411yescharacteristic
81C4×Dic178C2×C411yescharacteristic
82C4×Dic178C2311yescharacteristic
83C4×Dic178C2311yescharacteristic
84C4×Dic178C2311yescharacteristic
85C4×D356C2×C411yescharacteristic
86C4×D356C2311yescharacteristic
87C4×C22⋊D89—21no—
88C4×C22⋊D89—21no—
89#15667971C2×C411yescharacteristic
90#15667969—21no—
91C2×C4⋊Dic89C2×C411yescharacteristic
92C2×#13083553—21no—
93C2×C4×Dic89—21no—
94C42×D89—21no—
95#15667946C2×C411yescharacteristic
96#15667946C2×C411yescharacteristic
97#15667944C2×C411yescharacteristic
98#15667944C2×C411yescharacteristic
99C4⋊#10623494—21no—
100C4⋊#10623494C2311yescharacteristic
101C4⋊#10623494C2311yescharacteristic
102C4⋊#10623494C2311yescharacteristic
103C4⋊#10623494C2311yescharacteristic
104#15667941—21no—
105#15667941—21no—
106#15667941—21no—
107#15667941—21no—
108#15667941—21no—
109#15667941—21no—
110#15667937C2×C411yescharacteristic
111#15667937C2×C411yescharacteristic
112#15667937C2×C411yescharacteristic
113#15667937C2×C411yescharacteristic
114#15667937C2×C411yescharacteristic
115#15667937C2×C411yescharacteristic
116#15667934C2×C411yescharacteristic
117#15667934C2×C411yescharacteristic
118C22⋊#10623494—21no—
119#15667924—21no—
120#15667924—21no—
121#15667923—21no—
122#15667922—21no—
123#15667922—21no—
124C8×Dic89—21no—
125C8×Dic89—21no—
126C8×Dic89—21no—
127C8×Dic89—21no—
128C2×M16×C89—21no—
129C2×M16×C89C2×C411yescharacteristic
130C2×M16×C89C2×C411yescharacteristic
131C2×M16×C89D411yescharacteristic
132C2×M16×C89D411yescharacteristic
133C2×C42×C89C2311yescharacteristic
134#97×C89—21no—
135#97×C89C2×C411yescharacteristic
136#97×C89C2×C411yescharacteristic
137#97×C89D411yescharacteristic
138#97×C89D411yescharacteristic
139C4×C8×C89C2311yescharacteristic
140C4×C8×C89C2311yescharacteristic
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145#13083574—22no—
146#13083574—41no—
147C2×Dic178—22no—
148C2×Dic178—21no—
149C2×Dic178—41no—
150C2×Dic178C22×C411yescharacteristic
151C2×Dic178C22×C411yescharacteristic
152C2×Dic178C22×C411yescharacteristic
153C2×D356C22×C411yescharacteristic
154C2×C22⋊D89—21no—
155C22×Dic89—21no—
156C22×Dic89—21no—
157C2×C4×D89—21no—
158C2×C4×D89—21no—
159C2×C4×D89—21no—
160Dic356—22no—
161Dic356M1614yescomplemented
162#13083563—22no—
163#13083563—22no—
164#13083558—42no—
165#13083557—42no—
166#13083557—42no—
167C2×#10623494—21no—
168C2×#10623494—21no—
169C2×#10623494—21no—
170C2×#10623494—21no—
171C4⋊Dic89—21no—
172C4⋊Dic89C22×C411yescharacteristic
173C4⋊Dic89C22×C411yescharacteristic
174C4⋊Dic89C22×C411yescharacteristic
175C4⋊Dic89C22×C411yescharacteristic
176#13083553—21no—
177#13083553—21no—
178#13083553—21no—
179#13083553—21no—
180#13083553—21no—
181#13083553—21no—
182C22⋊Dic89—21no—
183#13083551—21no—
184#13083551—21no—
185C4×Dic89—21no—
186C4×Dic89—21no—
187C4×Dic89—21no—
188C4×Dic89—21no—
189C4×Dic89—21no—
190C4×Dic89—21no—
191C4×Dic89—21no—
192C22×C4×C89C22×C411yescharacteristic
193C22×C4×C89C2×D411yescharacteristic
194C22×C4×C89#4111yescharacteristic
195M16×C89—22no—
196M16×C89—22no—
197M16×C89—41no—
198M16×C89—41no—
199M16×C89—42no—
200C2×C8×C89—21no—
201C2×C8×C89—21no—
202C2×C8×C89C22×C411yescharacteristic
203C2×C8×C89C22×C411yescharacteristic
204C2×C8×C89C22×C411yescharacteristic
205C2×C8×C89C22×C411yescharacteristic
206C2×C8×C89C2×D411yescharacteristic
207C2×C8×C89C2×D411yescharacteristic
208C2×C8×C89C2×D411yescharacteristic
209C2×C8×C89C2×D411yescharacteristic
210C42×C89C22×C411yescharacteristic
211C42×C89C2×D411yescharacteristic
212C42×C89#4111yescharacteristic
213C42×C89C2411yescharacteristic
214C23×C89C4×D411yescharacteristic
215C22×D89—21no—
216C2×C4×C89—21no—
217C2×C4×C89—21no—
218C2×C4×C89—21no—
219C2×C4×C89—21no—
220C2×C4×C89—41no—
221C2×C4×C89C4×D411yescharacteristic
222C2×C4×C89C4×D411yescharacteristic
223C2×C4×C89C4×D411yescharacteristic
224C2×C4×C89C23×C411yescharacteristic
225C2×C4×C89C22×D411yescharacteristic
226C2×C4×C89C2×#4111yescharacteristic
227C22⋊D89—42no—
228C2×Dic89—21no—
229C2×Dic89—21no—
230C2×Dic89—21no—
231C2×Dic89—21no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—42no—
236C2×Dic89—81no—
237D356—21no—
238D356—21no—
239C4×D89—42no—
240C4×D89—41no—
241C4×D89—41no—
242C4×D89—81no—
243Dic178—21no—
244Dic178—21no—
245Dic178—21no—
246Dic178—21no—
247Dic178—42no—
248Dic178C2×M1612yes—
249Dic178C2×M1612yes—
250C8×C89—22no—
251C8×C89—21no—
252C8×C89—21no—
253C8×C89—21no—
254C8×C89—21no—
255C8×C89—21no—
256C8×C89—21no—
257C8×C89—41no—
258C8×C89C2×M1612yes—
259C8×C89#13712yescomplemented
260#10623494—41no—
261#10623494—41no—
262#10623494—41no—
263#10623494—41no—
264C22×C89—21no—
265C22×C89—21no—
266C22×C89—21no—
267C22×C89C2×C4×D411yescharacteristic
268C2×D89—41no—
269C2×D89—41no—
270C2×D89—41no—
271C4×C89—21no—
272C4×C89—21no—
273C4×C89—21no—
274C4×C89—21no—
275C4×C89C22×M1611yescharacteristic
276C4×C89#56811yescharacteristic
277C4×C89C2×#13711yescharacteristic
278C4×C89#57511yescharacteristic
279Dic89—22no—
280Dic89—22no—
281Dic89—41no—
282Dic89—41no—
283Dic89—41no—
284Dic89—41no—
285Dic89—41no—
286#18049—891nomaximal, Sylow
287C2×C89—41no—
288C2×C89D4×M1611yescharacteristic
289C2×C89C4×#13711yescharacteristic
290C2×C89#295611yescharacteristic
291D89—81no—
292#2971—891no—
293Q8×M16—891no—
294C4×#137—891no—
295#1592—891no—
296#1590—891no—
297#1588—891no—
298#1587—891no—
299#1584—891no—
300#1583—891no—
301#1579—891no—
302C8×Q16—891no—
303#1569—891no—
304#1491—891no—
305#1489—891no—
306#1455D8911yescharacteristic, complemented
307C89#1804911yescharacteristic, complemented, Sylow
308C2×#137—891no—
309C4×#41—891no—
310C2×C4×Q8—891no—
311#446—891no—
312#446—891no—
313#446—1781no—
314#446—1781no—
315C8×Q8—891no—
316C8×Q8—891no—
317C8×Q8—891no—
318C8×Q8—891no—
319#441—891no—
320#441—891no—
321#440—891no—
322#440—891no—
323C4×Q16—891no—
324C4×Q16—891no—
325C4×SD16—891no—
326C4×SD16—891no—
327#436—891no—
328#436—891no—
329#436—1781no—
330#436—1781no—
331#432—891no—
332#432—1781no—
333#428—891no—
334#423—891no—
335#423—1781no—
336#419—891no—
337#417—891no—
338C4×M16—1781no—
339C4×M16—1781no—
340C4×M16C2×D8911yescharacteristic
341C4×M16C2×D8911yescharacteristic
342C4×M16C2×D8911yescharacteristic
343#335—891no—
344#335—891no—
345#334—891no—
346#334—891no—
347Q8⋊C8—891no—
348Q8⋊C8—891no—
349Q8⋊C8—891no—
350Q8⋊C8—891no—
351Q8⋊C8—891no—
352Q8⋊C8—891no—
353D4⋊C8—891no—
354D4⋊C8—891no—
355#322C2×D8911yescharacteristic
356#322C2×D8911yescharacteristic
357#322C2×D8911yescharacteristic
358C82C2×D8911yescharacteristic
359C2×#41—891no—
360C22×Q8—891no—
361#137—1784no—
362C2×Q16—891no—
363C2×Q16—891no—
364C2×SD16—891no—
365C2×SD16—891no—
366C2×M16—21no—
367C2×M16—1781no—
368C2×M16—1781no—
369C2×M16—1781no—
370C2×M16C4×D8911yescharacteristic
371C2×M16C4×D8911yescharacteristic
372C2×M16D35611yescharacteristic
373C2×M16D35611yescharacteristic
374C4×Q8—891no—
375C4×Q8—891no—
376C4×Q8—891no—
377C4×Q8—891no—
378C4×Q8—891no—
379C4×Q8—891no—
380C4×Q8—1781no—
381C4×Q8—1781no—
382C4×D4—891no—
383C4×D4—891no—
384C4×D4—1781no—
385C4×D4—1781no—
386#117—891no—
387#117—1781no—
388C2×C4⋊C4—891no—
389C2×C4⋊C4—1781no—
390C2×C42—1781no—
391C2×C42—1781no—
392C2×C42C22×D8911yescharacteristic
393#107—891no—
394#107—891no—
395#106—891no—
396#106—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—891no—
400C4⋊C8—891no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8—1781no—
404C4⋊C8—1781no—
405C4⋊C8—1781no—
406C4⋊C8—1781no—
407C4⋊C8—1781no—
408#103—891no—
409#103—891no—
410#103—891no—
411#103—891no—
412#103—891no—
413#103—891no—
414#102—891no—
415#102—891no—
416C22⋊C8—1781no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419#97—21no—
420#97—1781no—
421#97—1781no—
422#97—1781no—
423#97C4×D8911yescharacteristic
424#97C4×D8911yescharacteristic
425#97D35611yescharacteristic
426#97D35611yescharacteristic
427C4×C8—1781no—
428C4×C8—1781no—
429C4×C8—1781no—
430C4×C8—1781no—
431C4×C8C22×D8911yescharacteristic
432C4×C8C22×D8911yescharacteristic
433C4×C8C22×D8911yescharacteristic
434C4×C8C22×D8911yescharacteristic
435C4×C8C22×D8911yescharacteristic
436C4×C8C22×D8911yescharacteristic
437#41—3561no—
438#41—1782no—
439C2×Q8—891no—
440C2×Q8—891no—
441C2×Q8—891no—
442C2×Q8—3561no—
443C2×Q8—1782no—
444C2×Q8—1781no—
445C2×D4—891no—
446C2×D4—1781no—
447C22×C4—1781no—
448C22×C4—1781no—
449C22×C4—1781no—
450C22×C4—1781no—
451C22×C4—1781no—
452C22×C4C2×C4×D8911yescharacteristic
453C22×C4C2×D35611yescharacteristic
454C22×C4#1308357411yescharacteristic
455Q16—894no—
456Q16—1782no—
457SD16—1782no—
458SD16—1782no—
459M16—22no—
460M16—22no—
461M16—41no—
462M16—41no—
463M16—42no—
464M16—3562no—
465M16—3562no—
466M16—3562no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—1781no—
470C2×C8—1781no—
471C2×C8—1781no—
472C2×C8—1781no—
473C2×C8C2×C4×D8911yescharacteristic
474C2×C8C2×C4×D8911yescharacteristic
475C2×C8C2×C4×D8911yescharacteristic
476C2×C8C2×C4×D8911yescharacteristic
477C2×C8C2×D35611yescharacteristic
478C2×C8C2×D35611yescharacteristic
479C2×C8C2×D35611yescharacteristic
480C2×C8C2×D35611yescharacteristic
481C4⋊C4—891no—
482C4⋊C4—891no—
483C4⋊C4—891no—
484C4⋊C4—891no—
485C4⋊C4—1781no—
486C4⋊C4—1781no—
487C4⋊C4—1781no—
488C4⋊C4—1781no—
489C4⋊C4—1781no—
490C4⋊C4—1781no—
491C4⋊C4—1781no—
492C22⋊C4—1781no—
493C22⋊C4—1781no—
494C22⋊C4—1781no—
495C42—1781no—
496C42—1781no—
497C42—1781no—
498C42—1781no—
499C42—1781no—
500C42—1781no—
501C42—1781no—
502C42C2×C4×D8911yescharacteristic
503C42C2×D35611yescharacteristic
504C42#1308357411yescharacteristic
505C42C23×D8911yescharacteristic
506C23—1781no—
507C23C4×D35611yescharacteristic
508Q8—892no—
509Q8—892no—
510Q8—3562no—
511Q8—1781no—
512Q8—1781no—
513Q8—1781no—
514Q8—1781no—
515D4—3562no—
516D4—1781no—
517D4—1781no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—3562no—
524C2×C4—3562no—
525C2×C4—3561no—
526C2×C4—3561no—
527C2×C4—1781no—
528C2×C4—1781no—
529C2×C4—1781no—
530C2×C4—1781no—
531C2×C4—1781no—
532C2×C4—1781no—
533C2×C4—1781no—
534C2×C4—7121no—
535C2×C4—7121no—
536C2×C4C4×D35611yescharacteristic
537C2×C4C4×D35611yescharacteristic
538C2×C4C4×D35611yescharacteristic
539C2×C4C22×C4×D8911yescharacteristic
540C2×C4C22×D35611yescharacteristic
541C2×C4C2×#1308357411yescharacteristic
542C8—22no—
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—3561no—
551C8—3561no—
552C8—3561no—
553C8—3561no—
554C8M16×D8912yes—
555C8#1566804512yescomplemented
556C22—21no—
557C22—21no—
558C22—21no—
559C22—3561no—
560C22—3561no—
561C22—3561no—
562C22C2×C4×D35611yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—3561no—
568C4—3561no—
569C4—3561no—
570C4—3561no—
571C4—3561no—
572C4—1782no—
573C4—1782no—
574C4C2×M16×D8911yescharacteristic
575C4#1915236411yescharacteristic
576C4C2×#1566804511yescharacteristic
577C4#1915239811yescharacteristic
578C2—41no—
579C2—7121no—
580C2M16×D35611yescharacteristic
581C2C4×#1566804511yescharacteristic
582C2#2602916311yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1455C2
socleC22×C89C2×C4×D4
prime radicalC2×#13083574C2×C4
cosocleC42×C89C24
hypercenter#1455D89
hypocenterC89#18049
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1455

Complements

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

NH ≅ G/Ntype
2. M16×Dic178C2semidirect
4. #26019156C2semidirect
5. #26019144C2semidirect
6. #26019132C2semidirect
7. #26019125C2semidirect
8. #26019104C2semidirect
9. #26019097C2semidirect
10. #26019072C2semidirect
11. C8×Dic356C2semidirect
12. #26019019C2semidirect
13. #26018700C2semidirect
15. C89×#1455C2semidirect
161. Dic356M16semidirect
259. C8×C89#137semidirect
306. #1455D89semidirect
307. C89#18049semidirect
555. C8#15668045semidirect

Maximal subgroups

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

subgroupindex
#260292702
M16×Dic1782
C4×#156680452
#260191562
#260191442
#260191322
#260191252
#260191042
#260190972
#260190722
C8×Dic3562
#260190192
#260187002
#260186922
C89×#14552
#1804989