Finite Group Databasebeta

← C89×#61303 #36439151

Subgroups — C89×#61303 #36439151

Order 22784.

1840 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3480C211yesmaximal, characteristic, complemented
2Q8×#41×C89C211yesmaximal, characteristic, complemented
3C89×#3365C213yesmaximal, complemented
4C89×#3364C213yesmaximal, complemented
5C89×#3363C211yesmaximal, characteristic, complemented
6C89×#3362C211yesmaximal, characteristic, complemented
7C89×#3361C213yesmaximal, complemented
8C89×#3360C213yesmaximal, complemented
9C89×#3359C211yesmaximal, characteristic, complemented
10C89×#3358C211yesmaximal, characteristic, complemented
11C89×#3131C213yesmaximal, complemented
12C89×#3076C213yesmaximal, complemented
13C89×#3035C213yesmaximal, complemented
14C2×C8×Q8×C89C211yesmaximal, characteristic, complemented
15C4×#135×C89C213yesmaximal, complemented
16C2×#135×C89C2213yes—
17#559×C89C2213yes—
18Q82×C89C2211yescharacteristic
19#557×C89C2211yescharacteristic
20#557×C89C2213yes—
21#556×C89C2213yes—
22#554×C89—23no—
23#554×C89C2213yes—
24#553×C89—23no—
25#550×C89C2211yescharacteristic
26D4×Q8×C89—21no—
27D4×Q8×C89C2211yescharacteristic
28#548×C89—21no—
29#533×C89—23no—
30#533×C89C2213yes—
31#532×C89C2213yes—
32#529×C89—23no—
33C4×#41×C89—21no—
34C4×#41×C89C2213yes—
35C4×#41×C89C2213yes—
36C2×C4×Q8×C89—21no—
37C2×C4×Q8×C89C2211yescharacteristic
38#488×C89C2216yes—
39#488×C89C2216yes—
40#487×C89C2213yes—
41#487×C89C2213yes—
42#486×C89C2213yes—
43#486×C89C2213yes—
44#479×C89C2216yes—
45#478×C89C2216yes—
46#477×C89C2213yes—
47#476×C89C2213yes—
48#475×C89C2213yes—
49#474×C89C2213yes—
50#464×C89C2216yes—
51#463×C89C2216yes—
52#462×C89C2213yes—
53#461×C89C2213yes—
54#460×C89C2213yes—
55#459×C89C2213yes—
56C8×Q8×C89C2211yescharacteristic
57C8×Q8×C89C2211yescharacteristic
58C8×Q8×C89C2213yes—
59C8×Q8×C89C2213yes—
60C4×Q16×C89C2213yes—
61C4×Q16×C89C2213yes—
62C4×SD16×C89C2213yes—
63C4×SD16×C89C2213yes—
64C4×SD16×C89C2213yes—
65C4×SD16×C89C2213yes—
66C4×D8×C89C2213yes—
67C4×D8×C89C2213yes—
68#427×C89C2213yes—
69C2×C4⋊C8×C89C2213yes—
70#416×C89C2216yes—
71#416×C89C2213yes—
72#416×C89C2211yescharacteristic
73C2×C4×C8×C89C2213yes—
74C2×#41×C89—21no—
75C2×#41×C89C2313yes—
76C2×#41×C89C2311yescharacteristic
77C22×Q8×C89—21no—
78C22×Q8×C89D411yescharacteristic
79#135×C89—212no—
80C2×Q16×C89C2313yes—
81C2×SD16×C89C2313yes—
82C2×SD16×C89C2313yes—
83C2×D8×C89C2313yes—
84C22×C8×C89C2311yescharacteristic
85C22×C8×C89C2313yes—
86C4⋊Q8×C89—23no—
87C4⋊Q8×C89—23no—
88C4⋊Q8×C89C2313yes—
89C4⋊Q8×C89C2313yes—
90C4⋊Q8×C89C2313yes—
91#127×C89C2313yes—
92#126×C89—26no—
93#125×C89—23no—
94#125×C89—26no—
95#125×C89C2316yes—
96#124×C89—23no—
97#124×C89C2316yes—
98#123×C89—26no—
99C22⋊Q8×C89—23no—
100C22⋊Q8×C89—23no—
101C22⋊Q8×C89—23no—
102C22⋊Q8×C89—23no—
103C22⋊Q8×C89—23no—
104C22⋊Q8×C89—26no—
105#121×C89—23no—
106#121×C89—23no—
107#121×C89—23no—
108C4×Q8×C89—21no—
109C4×Q8×C89—21no—
110C4×Q8×C89—23no—
111C4×Q8×C89—23no—
112C4×Q8×C89—23no—
113C4×Q8×C89D411yescharacteristic
114C4×Q8×C89D413yes—
115C4×Q8×C89C2313yes—
116C4×Q8×C89C2313yes—
117C4×Q8×C89C2313yes—
118C4×Q8×C89C2313yes—
119C4×Q8×C89C2311yescharacteristic
120C4×Q8×C89C2313yes—
121C4×D4×C89—23no—
122C4×D4×C89—23no—
123C4×D4×C89—23no—
124C4×D4×C89—23no—
125C4×D4×C89—23no—
126C4×D4×C89—23no—
127C4×D4×C89—23no—
128C4×D4×C89C2313yes—
129C4×D4×C89C2313yes—
130C4×D4×C89C2313yes—
131C4×D4×C89C2313yes—
132#117×C89—23no—
133#117×C89—23no—
134#117×C89—23no—
135#117×C89C2313yes—
136#117×C89C2311yescharacteristic
137C2×C4⋊C4×C89—23no—
138C2×C4⋊C4×C89D413yes—
139C2×C42×C89—23no—
140C2×C42×C89—23no—
141C2×C42×C89C2313yes—
142#107×C89C2313yes—
143#107×C89C2313yes—
144#106×C89C2313yes—
145#106×C89C2313yes—
146C4⋊C8×C89C2313yes—
147C4⋊C8×C89C2313yes—
148C4⋊C8×C89C2316yes—
149#103×C89C2311yescharacteristic
150#103×C89C2311yescharacteristic
151#103×C89C2313yes—
152#103×C89C2313yes—
153#103×C89C2316yes—
154#103×C89C2316yes—
155#102×C89C2311yescharacteristic
156#102×C89C2311yescharacteristic
157#102×C89C2313yes—
158#102×C89C2313yes—
159#102×C89C2316yes—
160#102×C89C2316yes—
161C4×C8×C89C2313yes—
162C4×C8×C89C2313yes—
163C4×C8×C89C2313yes—
164C4×C8×C89C2313yes—
165#41×C89—22no—
166#41×C89—26no—
167#41×C89—26no—
168#41×C89—44no—
169#41×C89#4114yescomplemented
170C2×Q8×C89—24no—
171C2×Q8×C89—23no—
172C2×Q8×C89—23no—
173C2×Q8×C89—44no—
174C2×Q8×C89C2×D413yes—
175C2×Q8×C89C2×D411yescharacteristic
176C2×Q8×C89C2413yes—
177C2×Q8×C89C2411yescharacteristic
178C2×D4×C89—21no—
179C2×D4×C89—23no—
180C2×D4×C89—23no—
181C2×D4×C89—23no—
182C2×D4×C89C2413yes—
183C2×D4×C89C2411yescharacteristic
184C22×C4×C89—21no—
185C22×C4×C89—23no—
186C22×C4×C89—23no—
187C22×C4×C89—23no—
188C22×C4×C89—23no—
189C22×C4×C89—23no—
190C22×C4×C89C2×D413yes—
191C22×C4×C89C2×D413yes—
192C22×C4×C89C2411yescharacteristic
193Q16×C89—26no—
194SD16×C89—26no—
195SD16×C89—26no—
196D8×C89—26no—
197C2×C8×C89—26no—
198C2×C8×C89—22no—
199C2×C8×C89C2411yescharacteristic
200C2×C8×C89C2411yescharacteristic
201C2×C8×C89C2413yes—
202C2×C8×C89C2413yes—
203C4⋊C4×C89—26no—
204C4⋊C4×C89—23no—
205C4⋊C4×C89—23no—
206C4⋊C4×C89—23no—
207C4⋊C4×C89—23no—
208C4⋊C4×C89—23no—
209C4⋊C4×C89—23no—
210C4⋊C4×C89—23no—
211C4⋊C4×C89—23no—
212C4⋊C4×C89—26no—
213C4⋊C4×C89C2×D413yes—
214C4⋊C4×C89C2×D413yes—
215C4⋊C4×C89C2×D416yes—
216C4⋊C4×C89C2413yes—
217C4⋊C4×C89C2413yes—
218C4⋊C4×C89C2411yescharacteristic
219C4⋊C4×C89C2411yescharacteristic
220C22⋊C4×C89—21no—
221C22⋊C4×C89—23no—
222C22⋊C4×C89—26no—
223C22⋊C4×C89—23no—
224C22⋊C4×C89—23no—
225C22⋊C4×C89—23no—
226C22⋊C4×C89—23no—
227C22⋊C4×C89—26no—
228C42×C89—23no—
229C42×C89—23no—
230C42×C89—23no—
231C42×C89—23no—
232C42×C89—21no—
233C42×C89—23no—
234C42×C89C2×D413yes—
235C42×C89C2×D413yes—
236C42×C89C2413yes—
237C42×C89C2413yes—
238C23×C89—21no—
239C23×C89—23no—
240C23×C89C22×D411yescharacteristic
241Q8×C89—26no—
242Q8×C89—21no—
243Q8×C89—23no—
244Q8×C89—23no—
245Q8×C89—46no—
246Q8×C89—83no—
247Q8×C89#13514yescomplemented
248Q8×C89C2×#4112yes—
249D4×C89—24no—
250D4×C89—23no—
251D4×C89—21no—
252D4×C89—23no—
253D4×C89—46no—
254D4×C89—46no—
255D4×C89—83no—
256D4×C89C2×#4112yes—
257C2×C4×C89—21no—
258C2×C4×C89—24no—
259C2×C4×C89—26no—
260C2×C4×C89—21no—
261C2×C4×C89—21no—
262C2×C4×C89—23no—
263C2×C4×C89—23no—
264C2×C4×C89—23no—
265C2×C4×C89—43no—
266C2×C4×C89—46no—
267C2×C4×C89—46no—
268C2×C4×C89—46no—
269C2×C4×C89—83no—
270C2×C4×C89—83no—
271C2×C4×C89C22×D413yes—
272C2×C4×C89C22×D413yes—
273C2×C4×C89C22×D413yes—
274C2×C4×C89C22×D413yes—
275C2×C4×C89C22×D411yescharacteristic
276C2×C4×C89C2×#4112yes—
277C2×C4×C892−1+412yes—
278C2×C4×C89C2511yescharacteristic
279C8×C89—23no—
280C8×C89—23no—
281C8×C89—21no—
282C8×C89—21no—
283C22×C89—21no—
284C22×C89—21no—
285C22×C89—22no—
286C22×C89—41no—
287C22×C89—41no—
288C22×C89—43no—
289C22×C89—43no—
290C22×C89—43no—
291C22×C89#54812yes—
292C22×C89C23×D411yescharacteristic
293C4×C89—23no—
294C4×C89—23no—
295C4×C89—22no—
296C4×C89—23no—
297C4×C89—41no—
298C4×C89—43no—
299C4×C89—43no—
300C4×C89—41no—
301C4×C89—43no—
302C4×C89#54812yes—
303C4×C89C2×#13516yes—
304C4×C89C22×#4111yescharacteristic
305C4×C89C2×2−1+411yescharacteristic
306#61303C8911yesmaximal, characteristic, direct product factor, Sylow
307C2×C89—22no—
308C2×C89—42no—
309C2×C89—83no—
310C2×C89C2×#54811yescharacteristic
311C2×C89C22×#13511yescharacteristic
312C2×C89#358411yescharacteristic
313#3480C2×C8911yescharacteristic, complemented
314Q8×#41C2×C8911yescharacteristic, complemented
315#3365C2×C8913yescomplemented
316#3364C2×C8913yescomplemented
317#3363C2×C8911yescharacteristic, complemented
318#3362C2×C8911yescharacteristic, complemented
319#3361C2×C8913yescomplemented
320#3360C2×C8913yescomplemented
321#3359C2×C8911yescharacteristic, complemented
322#3358C2×C8911yescharacteristic, complemented
323#3131C2×C8913yescomplemented
324#3076C2×C8913yescomplemented
325#3035C2×C8913yescomplemented
326C2×C8×Q8C2×C8911yescharacteristic, complemented
327C4×#135C2×C8913yescomplemented
328C89#6130311yescharacteristic, direct product factor, Sylow
329C2×#135C22×C8913yes—
330#559C22×C8913yes—
331Q82C22×C8911yescharacteristic
332#557C22×C8911yescharacteristic
333#557C22×C8913yes—
334#556C22×C8913yes—
335#554—23no—
336#554C22×C8913yes—
337#553—23no—
338#550C22×C8911yescharacteristic
339D4×Q8—21no—
340D4×Q8C22×C8911yescharacteristic
341#548—21no—
342#533—23no—
343#533C22×C8913yes—
344#532C22×C8913yes—
345#529—23no—
346C4×#41—21no—
347C4×#41C22×C8913yes—
348C4×#41C22×C8913yes—
349C2×C4×Q8—21no—
350C2×C4×Q8C22×C8911yescharacteristic
351#488C22×C8916yes—
352#488C22×C8916yes—
353#487C22×C8913yes—
354#487C22×C8913yes—
355#486C22×C8913yes—
356#486C22×C8913yes—
357#479C22×C8916yes—
358#478C22×C8916yes—
359#477C22×C8913yes—
360#476C22×C8913yes—
361#475C22×C8913yes—
362#474C22×C8913yes—
363#464C22×C8916yes—
364#463C22×C8916yes—
365#462C22×C8913yes—
366#461C22×C8913yes—
367#460C22×C8913yes—
368#459C22×C8913yes—
369C8×Q8C22×C8911yescharacteristic
370C8×Q8C22×C8911yescharacteristic
371C8×Q8C22×C8913yes—
372C8×Q8C22×C8913yes—
373C4×Q16C22×C8913yes—
374C4×Q16C22×C8913yes—
375C4×SD16C22×C8913yes—
376C4×SD16C22×C8913yes—
377C4×SD16C22×C8913yes—
378C4×SD16C22×C8913yes—
379C4×D8C22×C8913yes—
380C4×D8C22×C8913yes—
381#427C22×C8913yes—
382C2×C4⋊C8C22×C8913yes—
383#416C22×C8916yes—
384#416C22×C8913yes—
385#416C22×C8911yescharacteristic
386C2×C4×C8C22×C8913yes—
387C2×#41—21no—
388C2×#41C23×C8913yes—
389C2×#41C23×C8911yescharacteristic
390C22×Q8—21no—
391C22×Q8D4×C8911yescharacteristic
392#135—212no—
393C2×Q16C23×C8913yes—
394C2×SD16C23×C8913yes—
395C2×SD16C23×C8913yes—
396C2×D8C23×C8913yes—
397C22×C8C23×C8913yes—
398C22×C8C23×C8911yescharacteristic
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8C23×C8913yes—
402C4⋊Q8C23×C8913yes—
403C4⋊Q8C23×C8913yes—
404#127C23×C8913yes—
405#126—26no—
406#125—23no—
407#125—26no—
408#125C23×C8916yes—
409#124—23no—
410#124C23×C8916yes—
411#123—26no—
412C22⋊Q8—23no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—23no—
416C22⋊Q8—23no—
417C22⋊Q8—26no—
418#121—23no—
419#121—23no—
420#121—23no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—23no—
426C4×Q8D4×C8911yescharacteristic
427C4×Q8D4×C8913yes—
428C4×Q8C23×C8913yes—
429C4×Q8C23×C8913yes—
430C4×Q8C23×C8913yes—
431C4×Q8C23×C8913yes—
432C4×Q8C23×C8911yescharacteristic
433C4×Q8C23×C8913yes—
434C4×D4—23no—
435C4×D4—23no—
436C4×D4—23no—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—23no—
440C4×D4—23no—
441C4×D4C23×C8913yes—
442C4×D4C23×C8913yes—
443C4×D4C23×C8913yes—
444C4×D4C23×C8913yes—
445#117—23no—
446#117—23no—
447#117—23no—
448#117C23×C8911yescharacteristic
449#117C23×C8913yes—
450C2×C4⋊C4—23no—
451C2×C4⋊C4D4×C8913yes—
452C2×C42—23no—
453C2×C42—23no—
454C2×C42C23×C8913yes—
455#107C23×C8913yes—
456#107C23×C8913yes—
457#106C23×C8913yes—
458#106C23×C8913yes—
459C4⋊C8C23×C8913yes—
460C4⋊C8C23×C8913yes—
461C4⋊C8C23×C8916yes—
462#103C23×C8911yescharacteristic
463#103C23×C8911yescharacteristic
464#103C23×C8913yes—
465#103C23×C8913yes—
466#103C23×C8916yes—
467#103C23×C8916yes—
468#102C23×C8911yescharacteristic
469#102C23×C8911yescharacteristic
470#102C23×C8913yes—
471#102C23×C8913yes—
472#102C23×C8916yes—
473#102C23×C8916yes—
474C4×C8C23×C8913yes—
475C4×C8C23×C8913yes—
476C4×C8C23×C8913yes—
477C4×C8C23×C8913yes—
478#41—22no—
479#41—26no—
480#41—26no—
481#41—44no—
482#41#41×C8914yescomplemented
483C2×Q8—24no—
484C2×Q8—23no—
485C2×Q8—23no—
486C2×Q8—44no—
487C2×Q8C2×D4×C8913yes—
488C2×Q8C2×D4×C8911yescharacteristic
489C2×Q8C24×C8913yes—
490C2×Q8C24×C8911yescharacteristic
491C2×D4—21no—
492C2×D4—23no—
493C2×D4—23no—
494C2×D4—23no—
495C2×D4C24×C8913yes—
496C2×D4C24×C8911yescharacteristic
497C22×C4—21no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—23no—
501C22×C4—23no—
502C22×C4—23no—
503C22×C4C2×D4×C8913yes—
504C22×C4C2×D4×C8913yes—
505C22×C4C24×C8911yescharacteristic
506Q16—26no—
507SD16—26no—
508SD16—26no—
509D8—26no—
510C2×C8—26no—
511C2×C8—22no—
512C2×C8C24×C8911yescharacteristic
513C2×C8C24×C8911yescharacteristic
514C2×C8C24×C8913yes—
515C2×C8C24×C8913yes—
516C4⋊C4—26no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4—23no—
522C4⋊C4—23no—
523C4⋊C4—23no—
524C4⋊C4—23no—
525C4⋊C4—26no—
526C4⋊C4C2×D4×C8913yes—
527C4⋊C4C2×D4×C8913yes—
528C4⋊C4C2×D4×C8916yes—
529C4⋊C4C24×C8913yes—
530C4⋊C4C24×C8913yes—
531C4⋊C4C24×C8911yescharacteristic
532C4⋊C4C24×C8911yescharacteristic
533C22⋊C4—21no—
534C22⋊C4—23no—
535C22⋊C4—26no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—23no—
540C22⋊C4—26no—
541C42—21no—
542C42—23no—
543C42—23no—
544C42—23no—
545C42—23no—
546C42—23no—
547C42C2×D4×C8913yes—
548C42C2×D4×C8913yes—
549C42C24×C8913yes—
550C42C24×C8913yes—
551C23—21no—
552C23—23no—
553C23C22×D4×C8911yescharacteristic
554Q8—26no—
555Q8—21no—
556Q8—23no—
557Q8—23no—
558Q8—46no—
559Q8—83no—
560Q8#135×C8914yescomplemented
561Q8C2×#41×C8912yes—
562D4—24no—
563D4—23no—
564D4—21no—
565D4—23no—
566D4—46no—
567D4—46no—
568D4—83no—
569D4C2×#41×C8912yes—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—26no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—23no—
576C2×C4—23no—
577C2×C4—23no—
578C2×C4—43no—
579C2×C4—46no—
580C2×C4—46no—
581C2×C4—46no—
582C2×C4—83no—
583C2×C4—83no—
584C2×C4C22×D4×C8913yes—
585C2×C4C22×D4×C8913yes—
586C2×C4C22×D4×C8913yes—
587C2×C4C22×D4×C8911yescharacteristic
588C2×C4C22×D4×C8913yes—
589C2×C4C2×#41×C8912yes—
590C2×C42−1+4×C8912yes—
591C2×C4C25×C8911yescharacteristic
592C8—23no—
593C8—23no—
594C8—21no—
595C8—21no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—43no—
602C22—43no—
603C22—43no—
604C22#548×C8912yes—
605C22C23×D4×C8911yescharacteristic
606C4—23no—
607C4—23no—
608C4—23no—
609C4—22no—
610C4—41no—
611C4—43no—
612C4—41no—
613C4—43no—
614C4—43no—
615C4#548×C8912yes—
616C4C2×#135×C8916yes—
617C4C22×#41×C8911yescharacteristic
618C4C2×2−1+4×C8911yescharacteristic
619C2—22no—
620C2—42no—
621C2—83no—
622C2C2×#548×C8911yescharacteristic
623C2C22×#135×C8911yescharacteristic
624C2C89×#358411yescharacteristic
subgroupG/H
centerC2×C4×C89C22×D4
commutator, Frattini, cosocleC2×C4C25×C89
socleC22×C89C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C89 ◁ C2×C4×Q8×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3480C2semidirect
2. Q8×#41×C89C2semidirect
3. C89×#3365C2semidirect
4. C89×#3364C2semidirect
5. C89×#3363C2semidirect
6. C89×#3362C2semidirect
7. C89×#3361C2semidirect
8. C89×#3360C2semidirect
9. C89×#3359C2semidirect
10. C89×#3358C2semidirect
11. C89×#3131C2semidirect
12. C89×#3076C2semidirect
13. C89×#3035C2semidirect
14. C2×C8×Q8×C89C2semidirect
15. C4×#135×C89C2semidirect
169. #41×C89#41semidirect
247. Q8×C89#135semidirect
306. #61303C89direct (N × H)
313. #3480C2×C89semidirect
314. Q8×#41C2×C89semidirect
315. #3365C2×C89semidirect
316. #3364C2×C89semidirect
317. #3363C2×C89semidirect
318. #3362C2×C89semidirect
319. #3361C2×C89semidirect
320. #3360C2×C89semidirect
321. #3359C2×C89semidirect
322. #3358C2×C89semidirect
323. #3131C2×C89semidirect
324. #3076C2×C89semidirect
325. #3035C2×C89semidirect
326. C2×C8×Q8C2×C89semidirect
327. C4×#135C2×C89semidirect
328. C89#61303direct (N × H)
482. #41#41×C89semidirect
560. Q8#135×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#34802
Q8×#41×C892
C89×#33652
C89×#33642
C89×#33632
C89×#33622
C89×#33612
C89×#33602
C89×#33592
C89×#33582
C89×#31312
C89×#30762
C89×#30352
C2×C8×Q8×C892
C4×#135×C892
#6130389