Finite Group Databasebeta

← #36525147

Subgroups — #36525147

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030835C211yesmaximal, characteristic
2#26029183C211yesmaximal, characteristic
3#26029181C211yesmaximal, characteristic
4#26019105C211yesmaximal, characteristic, complemented
5#26019104C211yesmaximal, characteristic, complemented
6#26019103C211yesmaximal, characteristic, complemented
7#26019102C211yesmaximal, characteristic, complemented
8#26019101C211yesmaximal, characteristic, complemented
9#26019100C211yesmaximal, characteristic, complemented
10#26019099C211yesmaximal, characteristic, complemented
11#26019009C211yesmaximal, characteristic
12#26018705C211yesmaximal, characteristic
13#26018705C211yesmaximal, characteristic
14#322×D89C211yesmaximal, characteristic
15C89×#1584C211yesmaximal, characteristic, complemented
16#19152152C2211yescharacteristic
17#19152152C2211yescharacteristic
18#19152002C411yescharacteristic
19#19151976C411yescharacteristic
20#19151947C411yescharacteristic
21#19151947C411yescharacteristic
22#19151840C411yescharacteristic
23#19151838C411yescharacteristic
24#19151838C411yescharacteristic
25#19151836C411yescharacteristic
26#19151729C2211yescharacteristic
27#19151728C2211yescharacteristic
28#19151727—21no—
29#19151726—21no—
30#19151725C2211yescharacteristic
31Q8×#10623494C2211yescharacteristic
32C8×Dic178—21no—
33C8×Dic178C2211yescharacteristic
34#19151722—21no—
35#19151722C2211yescharacteristic
36C4×Dic356C2211yescharacteristic
37C4×#13083564C2211yescharacteristic
38C4×#13083564C2211yescharacteristic
39#19151623—21no—
40#19151623—21no—
41#19151620C2211yescharacteristic
42#19151620C2211yescharacteristic
43#19151555—21no—
44#19151552—21no—
45C4×C8×D89C2211yescharacteristic
46C4×C8×D89C2211yescharacteristic
47C4×C8×D89C2211yescharacteristic
48#19151351C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151350C2211yescharacteristic
51#19151330C2211yescharacteristic
52#19151330C2211yescharacteristic
53Q8⋊#10623494C2211yescharacteristic
54Q8⋊#10623494C2211yescharacteristic
55#19151328C2211yescharacteristic
56#19151328C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151320C2211yescharacteristic
59#19151319C2211yescharacteristic
60#446×C89C2211yescharacteristic
61C8×Q8×C89C2211yescharacteristic
62C4×Q16×C89C2211yescharacteristic
63#335×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65Q8⋊C8×C89C2211yescharacteristic
66#322×C89C2211yescharacteristic
67C2×Dic356C2×C411yescharacteristic
68C2×#13083564C2×C411yescharacteristic
69C2×#13083564C2×C411yescharacteristic
70C2×#13083557—21no—
71C2×#13083557—21no—
72C2×C8×D89—21no—
73C2×C8×D89C2×C411yescharacteristic
74C2×C8×D89C2×C411yescharacteristic
75C2×C8×D89D411yescharacteristic
76C2×C8×D89D411yescharacteristic
77#15668011C2×C411yescharacteristic
78#15668011C2×C411yescharacteristic
79#15668010C2×C411yescharacteristic
80#15668010C2×C411yescharacteristic
81#15668002—21no—
82#15668002—21no—
83#15667992C2×C411yescharacteristic
84#15667992C2×C411yescharacteristic
85#15667990—21no—
86#15667990—21no—
87#15667988C2×C411yescharacteristic
88#15667988C2×C411yescharacteristic
89C4×Dic178—21no—
90C4×Dic178—21no—
91C4×Dic178C2311yescharacteristic
92C4×Dic178C2311yescharacteristic
93#15667969—21no—
94#15667969—21no—
95C42×D89C2311yescharacteristic
96#15667946C2×C411yescharacteristic
97#15667945C2×C411yescharacteristic
98#15667945C2×C411yescharacteristic
99C4⋊#10623494—21no—
100C4⋊#10623494—21no—
101C4⋊#10623494C2311yescharacteristic
102C4⋊#10623494C2311yescharacteristic
103#15667941—21no—
104#15667941—21no—
105#15667941—21no—
106#15667941—21no—
107#15667941—21no—
108#15667941—21no—
109#15667937C2×C411yescharacteristic
110#15667937C2×C411yescharacteristic
111#15667936C2×C411yescharacteristic
112#15667936C2×C411yescharacteristic
113#15667935C2×C411yescharacteristic
114#15667935C2×C411yescharacteristic
115#15667923—21no—
116#15667922—21no—
117#15667922—21no—
118C4×#10623494—21no—
119C4×#10623494C2311yescharacteristic
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C8×Dic89—21no—
123C8×Dic89—21no—
124C8×Dic89—21no—
125C8×Dic89C2×C411yescharacteristic
126C8×Dic89C2×C411yescharacteristic
127C8×Dic89D411yescharacteristic
128C8×Dic89D411yescharacteristic
129C2×Q16×C89C2×C411yescharacteristic
130C4×Q8×C89C2311yescharacteristic
131C4×Q8×C89C2311yescharacteristic
132#107×C89C2×C411yescharacteristic
133C4⋊C8×C89—21no—
134C4⋊C8×C89—21no—
135C4⋊C8×C89C2311yescharacteristic
136C4⋊C8×C89C2311yescharacteristic
137#103×C89C2×C411yescharacteristic
138#103×C89C2×C411yescharacteristic
139#97×C89—21no—
140C4×C8×C89—21no—
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C2×Dic178—21no—
145C2×Dic178—21no—
146C2×Dic178C22×C411yescharacteristic
147C2×Dic178C22×C411yescharacteristic
148C2×C4×D89C22×C411yescharacteristic
149C2×C4×D89C2×D411yescharacteristic
150C2×C4×D89#4111yescharacteristic
151Dic356—41no—
152#13083564—41no—
153#13083564—41no—
154#13083557—42no—
155#13083557—42no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494—21no—
159C2×#10623494C22×C411yescharacteristic
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C2×D411yescharacteristic
162C2×#10623494C2×D411yescharacteristic
163C8×D89—22no—
164C8×D89—41no—
165C8×D89—41no—
166C8×D89—42no—
167C8×D89Q1614yescomplemented
168C4⋊Dic89—21no—
169C4⋊Dic89—21no—
170C4⋊Dic89C22×C411yescharacteristic
171C4⋊Dic89C22×C411yescharacteristic
172#13083553—21no—
173#13083553—21no—
174#13083553—21no—
175#13083553—21no—
176#13083553—21no—
177#13083553—21no—
178#13083553—21no—
179#13083553—21no—
180#13083553—21no—
181#13083553—21no—
182#13083551—21no—
183#13083551—21no—
184#13083551—21no—
185#13083551—21no—
186C4×Dic89—21no—
187C4×Dic89—21no—
188C4×Dic89C22×C411yescharacteristic
189C4×Dic89C2×D411yescharacteristic
190C4×Dic89#4111yescharacteristic
191C2×Q8×C89C22×C411yescharacteristic
192C2×Q8×C89C22×C411yescharacteristic
193Q16×C89—41no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C2×C8×C89—21no—
197C2×C8×C89C22×C411yescharacteristic
198C2×C8×C89C22×C411yescharacteristic
199C2×C8×C89C2×D411yescharacteristic
200C2×C8×C89C2×D411yescharacteristic
201C4⋊C4×C89—21no—
202C4⋊C4×C89—21no—
203C4⋊C4×C89C22×C411yescharacteristic
204C4⋊C4×C89C22×C411yescharacteristic
205C42×C89—21no—
206C42×C89—21no—
207C42×C89C2411yescharacteristic
208C22×D89C4×D411yescharacteristic
209Q8×C89—21no—
210Q8×C89—21no—
211Q8×C89—21no—
212Q8×C89—21no—
213C2×C4×C89—21no—
214C2×C4×C89—21no—
215C2×C4×C89—21no—
216C2×C4×C89—21no—
217C2×C4×C89C23×C411yescharacteristic
218C2×C4×C89C22×D411yescharacteristic
219C2×C4×C89C2×#4111yescharacteristic
220C2×Dic89—21no—
221C2×Dic89—21no—
222C2×Dic89—21no—
223C2×Dic89—21no—
224C2×Dic89C4×D411yescharacteristic
225C2×Dic89C4×D411yescharacteristic
226C2×Dic89C4×D411yescharacteristic
227C4×D89—21no—
228C4×D89—21no—
229C4×D89—22no—
230C4×D89C2×Q1612yes—
231C4×D89#13512yes—
232Dic178—21no—
233Dic178—21no—
234Dic178—21no—
235Dic178—21no—
236Dic178—42no—
237Dic178—42no—
238C8×C89—21no—
239C8×C89—21no—
240C8×C89—21no—
241C8×C89—41no—
242C8×C89—41no—
243C8×C89—41no—
244C8×C89C2×Q1612yes—
245#10623494—21no—
246#10623494—21no—
247#10623494—21no—
248#10623494—41no—
249#10623494—41no—
250#10623494—41no—
251#10623494C2×Q1612yes—
252C22×C89C2×C4×D411yescharacteristic
253C2×D89—21no—
254C2×D89—21no—
255C2×D89C4×Q1612yes—
256C4×C89—21no—
257C4×C89—41no—
258C4×C89—41no—
259C4×C89—41no—
260C4×C89—41no—
261C4×C89#56811yescharacteristic
262C4×C89#56811yescharacteristic
263C4×C89C22×Q1611yescharacteristic
264C4×C89C2×#13511yescharacteristic
265Dic89—21no—
266Dic89—41no—
267Dic89—41no—
268Dic89—41no—
269Dic89—41no—
270Dic89C4×Q1612yes—
271Dic89#44412yes—
272#17977—891nomaximal, Sylow
273C2×C89#293211yescharacteristic
274C2×C89C2×C4×Q1611yescharacteristic
275C2×C89C2×#44411yescharacteristic
276D89—22no—
277#3170—891no—
278#2960—891no—
279#2960—891no—
280#1584—891no—
281#1584—891no—
282#1584—891no—
283#1584—891no—
284#1584—891no—
285#1584—891no—
286#1584—891no—
287#1584D8911yescharacteristic, complemented
288#1567—891no—
289#1493—891no—
290#1493—891no—
291C2×#322—891no—
292C89#1797711yescharacteristic, complemented, Sylow
293#533—891no—
294#533—891no—
295#500—891no—
296#494—891no—
297#487—891no—
298#487—891no—
299#467—891no—
300#467—891no—
301#467—891no—
302#467—891no—
303#446—891no—
304#446—891no—
305#446—891no—
306#446—1781no—
307#446—1781no—
308#446C2×D8911yescharacteristic
309C8×Q8—891no—
310C8×Q8—891no—
311C8×Q8—891no—
312C8×Q8—1781no—
313C8×Q8—1781no—
314C8×Q8C2×D8911yescharacteristic
315C4×Q16—891no—
316C4×Q16—891no—
317C4×Q16—891no—
318C4×Q16C2×D8911yescharacteristic
319#424—1781no—
320#424—1781no—
321#423—891no—
322#423—891no—
323#405—1781no—
324#405—1781no—
325C2×C4×C8—891no—
326C2×C4×C8—891no—
327C2×C4×C8—891no—
328#335—891no—
329#335—891no—
330#335—891no—
331#335C2×D8911yescharacteristic
332Q8⋊C8—891no—
333Q8⋊C8—891no—
334Q8⋊C8—891no—
335Q8⋊C8—891no—
336Q8⋊C8—891no—
337Q8⋊C8—891no—
338Q8⋊C8C2×D8911yescharacteristic
339Q8⋊C8C2×D8911yescharacteristic
340#322—891no—
341#322—891no—
342#322—891no—
343#322C2×D8911yescharacteristic
344C2×Q16—891no—
345C2×Q16—891no—
346C2×Q16—891no—
347C2×Q16C4×D8911yescharacteristic
348C2×M16—1781no—
349C2×M16—1781no—
350C22×C8—891no—
351C22×C8—891no—
352C22×C8—891no—
353C22×C8—891no—
354C22×C8—1781no—
355C4⋊Q8—891no—
356C4⋊Q8—891no—
357C4⋊Q8—891no—
358C4⋊Q8—891no—
359#125—1781no—
360#125—1781no—
361C22⋊Q8—891no—
362C22⋊Q8—891no—
363C22⋊Q8—891no—
364C22⋊Q8—891no—
365C22⋊Q8—1781no—
366C22⋊Q8—1781no—
367C4×Q8—891no—
368C4×Q8—891no—
369C4×Q8—1781no—
370C4×Q8—1781no—
371C4×Q8C22×D8911yescharacteristic
372C4×Q8C22×D8911yescharacteristic
373#117—1781no—
374#117—1781no—
375C2×C42—891no—
376#107—891no—
377#107—891no—
378#107—891no—
379#107C4×D8911yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—891no—
383C4⋊C8—891no—
384C4⋊C8—1781no—
385C4⋊C8—1781no—
386C4⋊C8—1781no—
387C4⋊C8—1781no—
388C4⋊C8—1781no—
389C4⋊C8—1781no—
390C4⋊C8—1781no—
391C4⋊C8—1781no—
392C4⋊C8C22×D8911yescharacteristic
393C4⋊C8C22×D8911yescharacteristic
394#103—891no—
395#103—891no—
396#103—891no—
397#103—891no—
398#103—891no—
399#103—891no—
400#103C4×D8911yescharacteristic
401#103C4×D8911yescharacteristic
402#97—21no—
403#97—1781no—
404#97—1781no—
405#97—1781no—
406C4×C8—21no—
407C4×C8—891no—
408C4×C8—891no—
409C4×C8—891no—
410C4×C8—891no—
411C4×C8—891no—
412C4×C8—891no—
413C4×C8—891no—
414C4×C8—1781no—
415C4×C8—1781no—
416C4×C8—1781no—
417C4×C8—1781no—
418C4×C8C22×D8911yescharacteristic
419C4×C8C22×D8911yescharacteristic
420C4×C8C22×D8911yescharacteristic
421C2×Q8—891no—
422C2×Q8—891no—
423C2×Q8—1781no—
424C2×Q8—1781no—
425C2×Q8C2×C4×D8911yescharacteristic
426C2×Q8C2×C4×D8911yescharacteristic
427C22×C4—891no—
428C22×C4—891no—
429C22×C4—891no—
430Q16—41no—
431Q16—3561no—
432Q16—3561no—
433Q16—3561no—
434M16—3562no—
435M16—3562no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—891no—
440C2×C8—894no—
441C2×C8—891no—
442C2×C8—891no—
443C2×C8—891no—
444C2×C8—3561no—
445C2×C8—3561no—
446C2×C8—3562no—
447C2×C8—1782no—
448C2×C8—1781no—
449C2×C8—1781no—
450C2×C8—1781no—
451C2×C8C2×C4×D8911yescharacteristic
452C2×C8C2×C4×D8911yescharacteristic
453C2×C8D4×D8911yescharacteristic
454C2×C8D4×D8911yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—891no—
458C4⋊C4—891no—
459C4⋊C4—1781no—
460C4⋊C4—1781no—
461C4⋊C4—1781no—
462C4⋊C4—1781no—
463C4⋊C4—1781no—
464C4⋊C4—1781no—
465C4⋊C4—1781no—
466C4⋊C4—1781no—
467C4⋊C4—1781no—
468C4⋊C4—1781no—
469C4⋊C4—1781no—
470C4⋊C4—1781no—
471C4⋊C4C2×C4×D8911yescharacteristic
472C4⋊C4C2×C4×D8911yescharacteristic
473C22⋊C4—1781no—
474C22⋊C4—1781no—
475C22⋊C4—1781no—
476C22⋊C4—1781no—
477C42—21no—
478C42—21no—
479C42—891no—
480C42—891no—
481C42—891no—
482C42—1781no—
483C42—1781no—
484C42C23×D8911yescharacteristic
485C23—891no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—3562no—
491Q8—3562no—
492Q8—1781no—
493Q8—1781no—
494Q8—1781no—
495Q8—1781no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—891no—
501C2×C4—891no—
502C2×C4—892no—
503C2×C4—892no—
504C2×C4—891no—
505C2×C4—1782no—
506C2×C4—1781no—
507C2×C4—1781no—
508C2×C4—1781no—
509C2×C4—1781no—
510C2×C4—1781no—
511C2×C4—1781no—
512C2×C4C22×C4×D8911yescharacteristic
513C2×C4C2×D4×D8911yescharacteristic
514C2×C4#41×D8911yescharacteristic
515C8—21no—
516C8—21no—
517C8—21no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—892no—
522C8—3561no—
523C8—3561no—
524C8—3561no—
525C8—1781no—
526C8—1781no—
527C8—1781no—
528C8Q16×D8912yes—
529C22—892no—
530C22—1781no—
531C22—1781no—
532C22C4×D4×D8911yescharacteristic
533C4—21no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—892no—
539C4—892no—
540C4—3561no—
541C4—3561no—
542C4—3561no—
543C4—3561no—
544C4—1781no—
545C4#1915236311yescharacteristic
546C4#1915236411yescharacteristic
547C4C2×Q16×D8911yescharacteristic
548C4#135×D8911yescharacteristic
549C2—1782no—
550C2#2602896611yescharacteristic
551C2C4×Q16×D8911yescharacteristic
552C2#444×D8911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1584C2
socleC22×C89C2×C4×D4
prime radicalC22×D89C4×D4
cosocleC42×C89C24
hypercenter#1584D89
hypocenterC89#17977
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #26019105C2semidirect
5. #26019104C2semidirect
6. #26019103C2semidirect
7. #26019102C2semidirect
8. #26019101C2semidirect
9. #26019100C2semidirect
10. #26019099C2semidirect
15. C89×#1584C2semidirect
167. C8×D89Q16semidirect
287. #1584D89semidirect
292. C89#17977semidirect

Maximal subgroups

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

subgroupindex
#260308352
#260291832
#260291812
#260191052
#260191042
#260191032
#260191022
#260191012
#260191002
#260190992
#260190092
#260187052
#322×D892
C89×#15842
#1797789