Finite Group Databasebeta

← C89×#61434 #36439282

Subgroups — C89×#61434 #36439282

Order 22784.

1906 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3483C211yesmaximal, characteristic, complemented
2C89×#3482C211yesmaximal, characteristic, complemented
3C89×#3392C211yesmaximal, characteristic, complemented
4Q8×SD16×C89C211yesmaximal, characteristic, complemented
5C89×#3363C211yesmaximal, characteristic, complemented
6C89×#3358C211yesmaximal, characteristic, complemented
7C89×#3340C213yesmaximal, complemented
8C89×#3304C213yesmaximal, complemented
9C89×#3303C213yesmaximal, complemented
10C89×#3284C213yesmaximal, complemented
11C89×#3167C213yesmaximal, complemented
12C89×#3107C213yesmaximal, complemented
13C89×#3106C213yesmaximal, complemented
14C8×#41×C89C211yesmaximal, characteristic, complemented
15C4×#135×C89C213yesmaximal, complemented
16C2×#135×C89C2213yes—
17Q82×C89C2211yescharacteristic
18#557×C89C2211yescharacteristic
19#556×C89C2213yes—
20#555×C89—23no—
21#555×C89C2213yes—
22#552×C89—23no—
23#550×C89—21no—
24#550×C89C2213yes—
25#550×C89C2211yescharacteristic
26D4×Q8×C89C2211yescharacteristic
27#548×C89—21no—
28#547×C89C2213yes—
29#533×C89C2213yes—
30#532×C89—23no—
31#532×C89C2213yes—
32#529×C89—23no—
33C4×#41×C89—21no—
34C4×#41×C89C2213yes—
35C4×#41×C89C2213yes—
36C4×#41×C89C2211yescharacteristic
37C2×C4×Q8×C89—21no—
38#498×C89C2213yes—
39#492×C89C2213yes—
40#487×C89C2213yes—
41#486×C89C2213yes—
42#485×C89C2216yes—
43#482×C89C2216yes—
44#477×C89C2213yes—
45#476×C89C2213yes—
46#475×C89C2213yes—
47#474×C89C2213yes—
48#465×C89C2216yes—
49#465×C89C2216yes—
50#462×C89C2213yes—
51#461×C89C2213yes—
52#460×C89C2213yes—
53#459×C89C2213yes—
54#452×C89C2216yes—
55#449×C89C2216yes—
56C8×Q8×C89C2211yescharacteristic
57C8×Q8×C89C2211yescharacteristic
58C4×Q16×C89C2213yes—
59C4×Q16×C89C2213yes—
60C4×SD16×C89C2213yes—
61C4×SD16×C89C2216yes—
62C4×SD16×C89C2213yes—
63C4×SD16×C89C2213yes—
64C4×SD16×C89C2211yescharacteristic
65C4×D8×C89C2213yes—
66C4×D8×C89C2213yes—
67C8×D4×C89C2213yes—
68C8×D4×C89C2213yes—
69#431×C89C2213yes—
70#427×C89C2213yes—
71#416×C89C2213yes—
72#416×C89C2213yes—
73C2×C4×C8×C89C2213yes—
74C2×#41×C89—21no—
75C2×#41×C89—23no—
76C2×#41×C89D411yescharacteristic
77C2×#41×C89C2313yes—
78C2×#41×C89C2313yes—
79C22×Q8×C89—21no—
80#135×C89—212no—
81C2×Q16×C89C2313yes—
82C2×SD16×C89C2313yes—
83C2×SD16×C89C2316yes—
84C2×SD16×C89C2311yescharacteristic
85C2×D8×C89C2313yes—
86C22×C8×C89C2313yes—
87C22×C8×C89C2313yes—
88C4⋊Q8×C89—23no—
89C4⋊Q8×C89C2313yes—
90C4⋊Q8×C89C2313yes—
91C4⋊Q8×C89C2313yes—
92#127×C89—23no—
93#127×C89C2313yes—
94#126×C89—26no—
95#125×C89—23no—
96#125×C89—23no—
97#124×C89—23no—
98#124×C89—26no—
99#124×C89—23no—
100#123×C89—26no—
101C22⋊Q8×C89—23no—
102C22⋊Q8×C89—23no—
103C22⋊Q8×C89C2316yes—
104#121×C89—23no—
105#121×C89—23no—
106#121×C89—23no—
107#121×C89—23no—
108#121×C89—26no—
109#121×C89C2316yes—
110C4×Q8×C89—21no—
111C4×Q8×C89—21no—
112C4×Q8×C89—23no—
113C4×Q8×C89—23no—
114C4×Q8×C89D411yescharacteristic
115C4×Q8×C89C2313yes—
116C4×Q8×C89C2311yescharacteristic
117C4×Q8×C89C2313yes—
118C4×Q8×C89C2313yes—
119C4×Q8×C89C2311yescharacteristic
120C4×D4×C89—23no—
121C4×D4×C89—23no—
122C4×D4×C89—23no—
123C4×D4×C89—23no—
124C4×D4×C89—23no—
125C4×D4×C89—23no—
126C4×D4×C89D413yes—
127C4×D4×C89C2311yescharacteristic
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×C89—23no—
136#117×C89D413yes—
137#117×C89C2313yes—
138#117×C89C2313yes—
139C2×C4⋊C4×C89—23no—
140C2×C42×C89—23no—
141C2×C42×C89—23no—
142C2×C42×C89C2313yes—
143#107×C89C2313yes—
144#107×C89C2313yes—
145#106×C89C2311yescharacteristic
146#106×C89C2313yes—
147#106×C89C2316yes—
148C4⋊C8×C89C2313yes—
149C4⋊C8×C89C2313yes—
150#103×C89C2311yescharacteristic
151#103×C89C2313yes—
152#103×C89C2313yes—
153#103×C89C2313yes—
154#103×C89C2316yes—
155#102×C89C2311yescharacteristic
156#102×C89C2313yes—
157#102×C89C2313yes—
158#102×C89C2313yes—
159#102×C89C2316yes—
160C22⋊C8×C89C2316yes—
161C4×C8×C89C2311yescharacteristic
162C4×C8×C89C2313yes—
163C4×C8×C89C2313yes—
164C4×C8×C89C2313yes—
165#41×C89—24no—
166#41×C89—26no—
167#41×C89—26no—
168#41×C89—43no—
169#41×C89—43no—
170#41×C89—44no—
171#41×C89—86no—
172C2×Q8×C89—23no—
173C2×Q8×C89—23no—
174C2×Q8×C89—44no—
175C2×Q8×C89C2×D411yescharacteristic
176C2×Q8×C89C2413yes—
177C2×Q8×C89C2411yescharacteristic
178C2×D4×C89—23no—
179C2×D4×C89—23no—
180C2×D4×C89—26no—
181C2×D4×C89—23no—
182C2×D4×C89—23no—
183C2×D4×C89C2×D413yes—
184C2×D4×C89C2413yes—
185C2×D4×C89C2411yescharacteristic
186C22×C4×C89—23no—
187C22×C4×C89—23no—
188C22×C4×C89—23no—
189C22×C4×C89—23no—
190C22×C4×C89—21no—
191C22×C4×C89—23no—
192C22×C4×C89C2×D413yes—
193C22×C4×C89C2×D413yes—
194C22×C4×C89C2413yes—
195Q16×C89—26no—
196SD16×C89—26no—
197SD16×C89—46no—
198SD16×C89#4114yescomplemented
199D8×C89—26no—
200C2×C8×C89—26no—
201C2×C8×C89—43no—
202C2×C8×C89C2411yescharacteristic
203C2×C8×C89C2413yes—
204C2×C8×C89C2413yes—
205C2×C8×C89C2411yescharacteristic
206C4⋊C4×C89—23no—
207C4⋊C4×C89—26no—
208C4⋊C4×C89—21no—
209C4⋊C4×C89—23no—
210C4⋊C4×C89—23no—
211C4⋊C4×C89—23no—
212C4⋊C4×C89—23no—
213C4⋊C4×C89—23no—
214C4⋊C4×C89—23no—
215C4⋊C4×C89C2×D413yes—
216C4⋊C4×C89C2×D413yes—
217C4⋊C4×C89C2413yes—
218C4⋊C4×C89C2411yescharacteristic
219C4⋊C4×C89C2413yes—
220C4⋊C4×C89C2411yescharacteristic
221C22⋊C4×C89—26no—
222C22⋊C4×C89—23no—
223C22⋊C4×C89—23no—
224C22⋊C4×C89—21no—
225C22⋊C4×C89—26no—
226C22⋊C4×C89—23no—
227C22⋊C4×C89—23no—
228C22⋊C4×C89—23no—
229C22⋊C4×C89C2×D416yes—
230C42×C89—23no—
231C42×C89—23no—
232C42×C89—23no—
233C42×C89—23no—
234C42×C89—21no—
235C42×C89—23no—
236C42×C89C2×D413yes—
237C42×C89C2×D413yes—
238C42×C89C2413yes—
239C42×C89C2411yescharacteristic
240C23×C89—21no—
241C23×C89—23no—
242C23×C89C22×D413yes—
243Q8×C89—21no—
244Q8×C89—23no—
245Q8×C89—23no—
246Q8×C89—46no—
247Q8×C89—83no—
248Q8×C89#13514yescomplemented
249Q8×C89C2×#4112yes—
250D4×C89—26no—
251D4×C89—23no—
252D4×C89—21no—
253D4×C89—23no—
254D4×C89—46no—
255D4×C89—46no—
256D4×C89—46no—
257D4×C89—83no—
258D4×C89—86no—
259D4×C89C2×#4112yes—
260C2×C4×C89—21no—
261C2×C4×C89—26no—
262C2×C4×C89—26no—
263C2×C4×C89—23no—
264C2×C4×C89—23no—
265C2×C4×C89—21no—
266C2×C4×C89—21no—
267C2×C4×C89—23no—
268C2×C4×C89—23no—
269C2×C4×C89—23no—
270C2×C4×C89—46no—
271C2×C4×C89—46no—
272C2×C4×C89—46no—
273C2×C4×C89—46no—
274C2×C4×C89—42no—
275C2×C4×C89—46no—
276C2×C4×C89C22×D413yes—
277C2×C4×C89C22×D411yescharacteristic
278C2×C4×C89C22×D413yes—
279C2×C4×C89C22×D413yes—
280C2×C4×C89C22×D411yescharacteristic
281C2×C4×C89C2511yescharacteristic
282C8×C89—23no—
283C8×C89—21no—
284C8×C89—23no—
285C8×C89C2×#4112yes—
286C22×C89—23no—
287C22×C89—22no—
288C22×C89—23no—
289C22×C89—23no—
290C22×C89—41no—
291C22×C89—41no—
292C22×C89—43no—
293C22×C89—43no—
294C22×C89—43no—
295C22×C89C23×D411yescharacteristic
296C4×C89—23no—
297C4×C89—22no—
298C4×C89—21no—
299C4×C89—23no—
300C4×C89—43no—
301C4×C89—41no—
302C4×C89—43no—
303C4×C89—41no—
304C4×C89—43no—
305C4×C89C2×#13516yes—
306C4×C89#57712yes—
307C4×C89C22×#4111yescharacteristic
308C4×C89#58511yescharacteristic
309#61434C8911yesmaximal, characteristic, direct product factor, Sylow
310C2×C89—43no—
311C2×C89—42no—
312C2×C89—83no—
313C2×C89D4×#4111yescharacteristic
314C2×C89C22×#13511yescharacteristic
315C2×C89C2×#57711yescharacteristic
316#3483C2×C8911yescharacteristic, complemented
317#3482C2×C8911yescharacteristic, complemented
318#3392C2×C8911yescharacteristic, complemented
319Q8×SD16C2×C8911yescharacteristic, complemented
320#3363C2×C8911yescharacteristic, complemented
321#3358C2×C8911yescharacteristic, complemented
322#3340C2×C8913yescomplemented
323#3304C2×C8913yescomplemented
324#3303C2×C8913yescomplemented
325#3284C2×C8913yescomplemented
326#3167C2×C8913yescomplemented
327#3107C2×C8913yescomplemented
328#3106C2×C8913yescomplemented
329C8×#41C2×C8911yescharacteristic, complemented
330C4×#135C2×C8913yescomplemented
331C89#6143411yescharacteristic, direct product factor, Sylow
332C2×#135C22×C8913yes—
333Q82C22×C8911yescharacteristic
334#557C22×C8911yescharacteristic
335#556C22×C8913yes—
336#555—23no—
337#555C22×C8913yes—
338#552—23no—
339#550—21no—
340#550C22×C8913yes—
341#550C22×C8911yescharacteristic
342D4×Q8C22×C8911yescharacteristic
343#548—21no—
344#547C22×C8913yes—
345#533C22×C8913yes—
346#532—23no—
347#532C22×C8913yes—
348#529—23no—
349C4×#41—21no—
350C4×#41C22×C8913yes—
351C4×#41C22×C8913yes—
352C4×#41C22×C8911yescharacteristic
353C2×C4×Q8—21no—
354#498C22×C8913yes—
355#492C22×C8913yes—
356#487C22×C8913yes—
357#486C22×C8913yes—
358#485C22×C8916yes—
359#482C22×C8916yes—
360#477C22×C8913yes—
361#476C22×C8913yes—
362#475C22×C8913yes—
363#474C22×C8913yes—
364#465C22×C8916yes—
365#465C22×C8916yes—
366#462C22×C8913yes—
367#461C22×C8913yes—
368#460C22×C8913yes—
369#459C22×C8913yes—
370#452C22×C8916yes—
371#449C22×C8916yes—
372C8×Q8C22×C8911yescharacteristic
373C8×Q8C22×C8911yescharacteristic
374C4×Q16C22×C8913yes—
375C4×Q16C22×C8913yes—
376C4×SD16C22×C8913yes—
377C4×SD16C22×C8916yes—
378C4×SD16C22×C8913yes—
379C4×SD16C22×C8913yes—
380C4×SD16C22×C8911yescharacteristic
381C4×D8C22×C8913yes—
382C4×D8C22×C8913yes—
383C8×D4C22×C8913yes—
384C8×D4C22×C8913yes—
385#431C22×C8913yes—
386#427C22×C8913yes—
387#416C22×C8913yes—
388#416C22×C8913yes—
389C2×C4×C8C22×C8913yes—
390C2×#41—21no—
391C2×#41—23no—
392C2×#41D4×C8911yescharacteristic
393C2×#41C23×C8913yes—
394C2×#41C23×C8913yes—
395C22×Q8—21no—
396#135—212no—
397C2×Q16C23×C8913yes—
398C2×SD16C23×C8913yes—
399C2×SD16C23×C8916yes—
400C2×SD16C23×C8911yescharacteristic
401C2×D8C23×C8913yes—
402C22×C8C23×C8913yes—
403C22×C8C23×C8913yes—
404C4⋊Q8—23no—
405C4⋊Q8C23×C8913yes—
406C4⋊Q8C23×C8913yes—
407C4⋊Q8C23×C8913yes—
408#127—23no—
409#127C23×C8913yes—
410#126—26no—
411#125—23no—
412#125—23no—
413#124—23no—
414#124—26no—
415#124—23no—
416#123—26no—
417C22⋊Q8—23no—
418C22⋊Q8—23no—
419C22⋊Q8C23×C8916yes—
420#121—23no—
421#121—23no—
422#121—23no—
423#121—23no—
424#121—26no—
425#121C23×C8916yes—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8—23no—
429C4×Q8—23no—
430C4×Q8D4×C8911yescharacteristic
431C4×Q8C23×C8911yescharacteristic
432C4×Q8C23×C8913yes—
433C4×Q8C23×C8913yes—
434C4×Q8C23×C8913yes—
435C4×Q8C23×C8911yescharacteristic
436C4×D4—23no—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—23no—
440C4×D4—23no—
441C4×D4—23no—
442C4×D4D4×C8913yes—
443C4×D4C23×C8911yescharacteristic
444C4×D4C23×C8913yes—
445C4×D4C23×C8913yes—
446C4×D4C23×C8913yes—
447C4×D4C23×C8913yes—
448#117—23no—
449#117—23no—
450#117—23no—
451#117—23no—
452#117D4×C8913yes—
453#117C23×C8913yes—
454#117C23×C8913yes—
455C2×C4⋊C4—23no—
456C2×C42—23no—
457C2×C42—23no—
458C2×C42C23×C8913yes—
459#107C23×C8913yes—
460#107C23×C8913yes—
461#106C23×C8911yescharacteristic
462#106C23×C8913yes—
463#106C23×C8916yes—
464C4⋊C8C23×C8913yes—
465C4⋊C8C23×C8913yes—
466#103C23×C8911yescharacteristic
467#103C23×C8913yes—
468#103C23×C8913yes—
469#103C23×C8913yes—
470#103C23×C8916yes—
471#102C23×C8911yescharacteristic
472#102C23×C8913yes—
473#102C23×C8913yes—
474#102C23×C8913yes—
475#102C23×C8916yes—
476C22⋊C8C23×C8916yes—
477C4×C8C23×C8911yescharacteristic
478C4×C8C23×C8913yes—
479C4×C8C23×C8913yes—
480C4×C8C23×C8913yes—
481#41—24no—
482#41—26no—
483#41—26no—
484#41—43no—
485#41—43no—
486#41—44no—
487#41—86no—
488C2×Q8—23no—
489C2×Q8—23no—
490C2×Q8—44no—
491C2×Q8C2×D4×C8911yescharacteristic
492C2×Q8C24×C8913yes—
493C2×Q8C24×C8911yescharacteristic
494C2×D4—23no—
495C2×D4—23no—
496C2×D4—26no—
497C2×D4—23no—
498C2×D4—23no—
499C2×D4C2×D4×C8913yes—
500C2×D4C24×C8913yes—
501C2×D4C24×C8911yescharacteristic
502C22×C4—21no—
503C22×C4—23no—
504C22×C4—23no—
505C22×C4—23no—
506C22×C4—23no—
507C22×C4—23no—
508C22×C4C2×D4×C8913yes—
509C22×C4C2×D4×C8913yes—
510C22×C4C24×C8913yes—
511Q16—26no—
512SD16—26no—
513SD16—46no—
514SD16#41×C8914yescomplemented
515D8—26no—
516C2×C8—26no—
517C2×C8—43no—
518C2×C8C24×C8911yescharacteristic
519C2×C8C24×C8913yes—
520C2×C8C24×C8913yes—
521C2×C8C24×C8911yescharacteristic
522C4⋊C4—23no—
523C4⋊C4—26no—
524C4⋊C4—21no—
525C4⋊C4—23no—
526C4⋊C4—23no—
527C4⋊C4—23no—
528C4⋊C4—23no—
529C4⋊C4—23no—
530C4⋊C4—23no—
531C4⋊C4C2×D4×C8913yes—
532C4⋊C4C2×D4×C8913yes—
533C4⋊C4C24×C8913yes—
534C4⋊C4C24×C8911yescharacteristic
535C4⋊C4C24×C8913yes—
536C4⋊C4C24×C8911yescharacteristic
537C22⋊C4—26no—
538C22⋊C4—23no—
539C22⋊C4—23no—
540C22⋊C4—21no—
541C22⋊C4—26no—
542C22⋊C4—23no—
543C22⋊C4—23no—
544C22⋊C4—23no—
545C22⋊C4C2×D4×C8916yes—
546C42—21no—
547C42—23no—
548C42—23no—
549C42—23no—
550C42—23no—
551C42—23no—
552C42C2×D4×C8913yes—
553C42C2×D4×C8913yes—
554C42C24×C8911yescharacteristic
555C42C24×C8913yes—
556C23—21no—
557C23—23no—
558C23C22×D4×C8913yes—
559Q8—21no—
560Q8—23no—
561Q8—23no—
562Q8—46no—
563Q8—83no—
564Q8#135×C8914yescomplemented
565Q8C2×#41×C8912yes—
566D4—26no—
567D4—23no—
568D4—21no—
569D4—23no—
570D4—46no—
571D4—46no—
572D4—46no—
573D4—83no—
574D4—86no—
575D4C2×#41×C8912yes—
576C2×C4—21no—
577C2×C4—26no—
578C2×C4—26no—
579C2×C4—23no—
580C2×C4—23no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—23no—
584C2×C4—23no—
585C2×C4—23no—
586C2×C4—46no—
587C2×C4—46no—
588C2×C4—46no—
589C2×C4—46no—
590C2×C4—42no—
591C2×C4—46no—
592C2×C4C22×D4×C8913yes—
593C2×C4C22×D4×C8911yescharacteristic
594C2×C4C22×D4×C8913yes—
595C2×C4C22×D4×C8913yes—
596C2×C4C22×D4×C8911yescharacteristic
597C2×C4C25×C8911yescharacteristic
598C8—23no—
599C8—21no—
600C8—23no—
601C8C2×#41×C8912yes—
602C22—23no—
603C22—23no—
604C22—23no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—43no—
609C22—43no—
610C22—43no—
611C22C23×D4×C8911yescharacteristic
612C4—21no—
613C4—23no—
614C4—23no—
615C4—22no—
616C4—41no—
617C4—43no—
618C4—43no—
619C4—41no—
620C4—43no—
621C4C2×#135×C8916yes—
622C4#577×C8912yes—
623C4C22×#41×C8911yescharacteristic
624C4#585×C8911yescharacteristic
625C2—43no—
626C2—42no—
627C2—83no—
628C2D4×#41×C8911yescharacteristic
629C2C22×#135×C8911yescharacteristic
630C2C2×#577×C8911yescharacteristic
subgroupG/H
centerC2×C4×C89C22×D4
commutator, Frattini, cosocleC2×C4C25×C89
socleC22×C89C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C89 ◁ C4×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3483C2semidirect
2. C89×#3482C2semidirect
3. C89×#3392C2semidirect
4. Q8×SD16×C89C2semidirect
5. C89×#3363C2semidirect
6. C89×#3358C2semidirect
7. C89×#3340C2semidirect
8. C89×#3304C2semidirect
9. C89×#3303C2semidirect
10. C89×#3284C2semidirect
11. C89×#3167C2semidirect
12. C89×#3107C2semidirect
13. C89×#3106C2semidirect
14. C8×#41×C89C2semidirect
15. C4×#135×C89C2semidirect
198. SD16×C89#41semidirect
248. Q8×C89#135semidirect
309. #61434C89direct (N × H)
316. #3483C2×C89semidirect
317. #3482C2×C89semidirect
318. #3392C2×C89semidirect
319. Q8×SD16C2×C89semidirect
320. #3363C2×C89semidirect
321. #3358C2×C89semidirect
322. #3340C2×C89semidirect
323. #3304C2×C89semidirect
324. #3303C2×C89semidirect
325. #3284C2×C89semidirect
326. #3167C2×C89semidirect
327. #3107C2×C89semidirect
328. #3106C2×C89semidirect
329. C8×#41C2×C89semidirect
330. C4×#135C2×C89semidirect
331. C89#61434direct (N × H)
514. SD16#41×C89semidirect
564. Q8#135×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#34832
C89×#34822
C89×#33922
Q8×SD16×C892
C89×#33632
C89×#33582
C89×#33402
C89×#33042
C89×#33032
C89×#32842
C89×#31672
C89×#31072
C89×#31062
C8×#41×C892
C4×#135×C892
#6143489