Finite Group Databasebeta

← C89×#61469 #36439317

Subgroups — C89×#61469 #36439317

Order 22784.

1866 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3483C211yesmaximal, characteristic
2Q8×#41×C89C211yesmaximal, characteristic, complemented
3C89×#3392C211yesmaximal, characteristic, complemented
4Q8×Q16×C89C211yesmaximal, characteristic, complemented
5C89×#3363C211yesmaximal, characteristic, complemented
6C89×#3362C211yesmaximal, characteristic, complemented
7C89×#3357C213yesmaximal, complemented
8C89×#3333C213yesmaximal, complemented
9C89×#3329C213yesmaximal, complemented
10C89×#3278C213yesmaximal, complemented
11C89×#3182C213yesmaximal, complemented
12C89×#3116C213yesmaximal, complemented
13C89×#3112C213yesmaximal, complemented
14C89×#2971C211yesmaximal, characteristic, complemented
15C4×#137×C89C213yesmaximal, complemented
16C2×#137×C89C2213yes—
17Q82×C89C2211yescharacteristic
18Q82×C89C2211yescharacteristic
19#557×C89—21no—
20#557×C89C2211yescharacteristic
21#555×C89—23no—
22#555×C89C2213yes—
23#554×C89—23no—
24#554×C89C2213yes—
25#550×C89—21no—
26#550×C89C2213yes—
27#550×C89C2211yescharacteristic
28D4×Q8×C89C2213yes—
29#533×C89—23no—
30#533×C89C2213yes—
31#532×C89—23no—
32#532×C89C2213yes—
33C4×#41×C89—21no—
34C4×#41×C89—21no—
35C4×#41×C89C2213yes—
36C4×#41×C89C2211yescharacteristic
37C2×C4×Q8×C89C2213yes—
38#500×C89C2213yes—
39#492×C89C2213yes—
40#487×C89C2213yes—
41#486×C89C2213yes—
42#485×C89C2216yes—
43#480×C89C2216yes—
44#477×C89C2213yes—
45#477×C89C2213yes—
46#477×C89C2213yes—
47#476×C89C2213yes—
48#468×C89C2216yes—
49#468×C89C2216yes—
50#462×C89C2213yes—
51#461×C89C2213yes—
52#460×C89C2213yes—
53#460×C89C2213yes—
54#452×C89C2216yes—
55#448×C89C2216yes—
56C8×Q8×C89C2211yescharacteristic
57C8×Q8×C89C2211yescharacteristic
58#441×C89C2213yes—
59#441×C89C2211yescharacteristic
60#440×C89C2213yes—
61#440×C89C2216yes—
62#440×C89C2213yes—
63C4×Q16×C89C2213yes—
64C4×Q16×C89C2213yes—
65C4×SD16×C89C2213yes—
66C4×SD16×C89C2213yes—
67#436×C89C2213yes—
68#436×C89C2213yes—
69#432×C89C2213yes—
70#428×C89C2213yes—
71#419×C89C2213yes—
72#417×C89C2213yes—
73C4×M16×C89C2213yes—
74C2×#41×C89—21no—
75C2×#41×C89—23no—
76C2×#41×C89D411yescharacteristic
77C2×#41×C89C2313yes—
78C22×Q8×C89C2313yes—
79#137×C89—212no—
80C2×Q16×C89C2313yes—
81C2×Q16×C89C2311yescharacteristic
82C2×SD16×C89C2313yes—
83C2×SD16×C89C2313yes—
84C2×SD16×C89C2316yes—
85C2×M16×C89C2313yes—
86C2×M16×C89C2313yes—
87C4⋊Q8×C89—23no—
88C4⋊Q8×C89—23no—
89C4⋊Q8×C89—23no—
90C4⋊Q8×C89C2313yes—
91C4⋊Q8×C89C2313yes—
92C4⋊Q8×C89C2313yes—
93#127×C89—23no—
94#127×C89C2313yes—
95#125×C89—23no—
96#125×C89—23no—
97#125×C89—23no—
98#124×C89—26no—
99#124×C89—23no—
100C22⋊Q8×C89—23no—
101C22⋊Q8×C89—26no—
102C22⋊Q8×C89—26no—
103C22⋊Q8×C89C2316yes—
104#121×C89—23no—
105#121×C89—23no—
106#121×C89—23no—
107#121×C89—26no—
108#121×C89C2316yes—
109C4×Q8×C89—21no—
110C4×Q8×C89—21no—
111C4×Q8×C89—21no—
112C4×Q8×C89—23no—
113C4×Q8×C89—23no—
114C4×Q8×C89—23no—
115C4×Q8×C89D411yescharacteristic
116C4×Q8×C89C2313yes—
117C4×Q8×C89C2313yes—
118C4×Q8×C89C2313yes—
119C4×Q8×C89C2311yescharacteristic
120C4×Q8×C89C2313yes—
121C4×Q8×C89C2311yescharacteristic
122C4×Q8×C89C2311yescharacteristic
123C4×D4×C89—23no—
124C4×D4×C89—23no—
125C4×D4×C89—23no—
126C4×D4×C89—23no—
127C4×D4×C89—23no—
128C4×D4×C89—23no—
129C4×D4×C89D413yes—
130C4×D4×C89C2313yes—
131C4×D4×C89C2313yes—
132C4×D4×C89C2313yes—
133#117×C89—23no—
134#117×C89—23no—
135#117×C89—23no—
136#117×C89D413yes—
137#117×C89C2313yes—
138C2×C4⋊C4×C89—23no—
139C2×C4⋊C4×C89C2313yes—
140C2×C42×C89—23no—
141C2×C42×C89—23no—
142C2×C42×C89C2313yes—
143#107×C89C2313yes—
144#107×C89C2313yes—
145#107×C89C2316yes—
146#106×C89C2311yescharacteristic
147#106×C89C2313yes—
148C4⋊C8×C89C2313yes—
149C4⋊C8×C89C2313yes—
150#103×C89C2311yescharacteristic
151#103×C89C2313yes—
152#103×C89C2311yescharacteristic
153#103×C89C2313yes—
154#103×C89C2313yes—
155#103×C89C2313yes—
156#103×C89C2316yes—
157#102×C89C2313yes—
158#102×C89C2313yes—
159#102×C89C2316yes—
160C22⋊C8×C89C2316yes—
161#97×C89C2311yescharacteristic
162#97×C89C2313yes—
163C4×C8×C89C2313yes—
164C4×C8×C89C2313yes—
165#41×C89—24no—
166#41×C89—26no—
167#41×C89—43no—
168#41×C89—44no—
169#41×C89—86no—
170C2×Q8×C89—23no—
171C2×Q8×C89—23no—
172C2×Q8×C89—23no—
173C2×Q8×C89—26no—
174C2×Q8×C89—43no—
175C2×Q8×C89C2×D411yescharacteristic
176C2×Q8×C89C2413yes—
177C2×Q8×C89C2411yescharacteristic
178C2×Q8×C89C2411yescharacteristic
179C2×D4×C89—23no—
180C2×D4×C89—26no—
181C2×D4×C89—23no—
182C2×D4×C89—23no—
183C2×D4×C89C2×D413yes—
184C2×D4×C89C2413yes—
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×C89C2413yes—
193Q16×C89—26no—
194Q16×C89#4114yescomplemented
195SD16×C89—26no—
196SD16×C89—26no—
197SD16×C89—46no—
198M16×C89—26no—
199M16×C89—43no—
200C2×C8×C89C2411yescharacteristic
201C2×C8×C89C2413yes—
202C2×C8×C89C2413yes—
203C2×C8×C89C2411yescharacteristic
204C4⋊C4×C89—21no—
205C4⋊C4×C89—23no—
206C4⋊C4×C89—23no—
207C4⋊C4×C89—23no—
208C4⋊C4×C89—23no—
209C4⋊C4×C89—21no—
210C4⋊C4×C89—23no—
211C4⋊C4×C89—23no—
212C4⋊C4×C89—23no—
213C4⋊C4×C89—23no—
214C4⋊C4×C89—23no—
215C4⋊C4×C89—26no—
216C4⋊C4×C89C2×D413yes—
217C4⋊C4×C89C2×D413yes—
218C4⋊C4×C89C2411yescharacteristic
219C4⋊C4×C89C2413yes—
220C4⋊C4×C89C2411yescharacteristic
221C4⋊C4×C89C2413yes—
222C22⋊C4×C89—26no—
223C22⋊C4×C89—23no—
224C22⋊C4×C89—26no—
225C22⋊C4×C89—23no—
226C22⋊C4×C89—23no—
227C22⋊C4×C89C2×D416yes—
228C42×C89—21no—
229C42×C89—21no—
230C42×C89—23no—
231C42×C89—23no—
232C42×C89—23no—
233C42×C89—23no—
234C42×C89—23no—
235C42×C89C2×D413yes—
236C42×C89C2×D413yes—
237C42×C89C2413yes—
238C42×C89C2411yescharacteristic
239C23×C89—23no—
240C23×C89C22×D413yes—
241Q8×C89—21no—
242Q8×C89—21no—
243Q8×C89—23no—
244Q8×C89—23no—
245Q8×C89—46no—
246Q8×C89—83no—
247Q8×C89—83no—
248Q8×C89#13714yescomplemented
249Q8×C89C2×#4112yes—
250Q8×C89C2×#4112yes—
251D4×C89—26no—
252D4×C89—23no—
253D4×C89—23no—
254D4×C89—46no—
255D4×C89—46no—
256D4×C89—83no—
257D4×C89—86no—
258C2×C4×C89—21no—
259C2×C4×C89—21no—
260C2×C4×C89—26no—
261C2×C4×C89—23no—
262C2×C4×C89—23no—
263C2×C4×C89—21no—
264C2×C4×C89—21no—
265C2×C4×C89—23no—
266C2×C4×C89—23no—
267C2×C4×C89—23no—
268C2×C4×C89—43no—
269C2×C4×C89—46no—
270C2×C4×C89—46no—
271C2×C4×C89—46no—
272C2×C4×C89—46no—
273C2×C4×C89—83no—
274C2×C4×C89C22×D411yescharacteristic
275C2×C4×C89C22×D411yescharacteristic
276C2×C4×C89C22×D413yes—
277C2×C4×C89C22×D413yes—
278C2×C4×C89C22×D413yes—
279C2×C4×C89C2511yescharacteristic
280C8×C89—23no—
281C8×C89—21no—
282C8×C89—23no—
283C8×C89C2×#4112yes—
284C22×C89—23no—
285C22×C89—23no—
286C22×C89—23no—
287C22×C89—43no—
288C22×C89—43no—
289C22×C89—43no—
290C22×C89C23×D411yescharacteristic
291C4×C89—21no—
292C4×C89—23no—
293C4×C89—23no—
294C4×C89—22no—
295C4×C89—22no—
296C4×C89—21no—
297C4×C89—43no—
298C4×C89—41no—
299C4×C89—43no—
300C4×C89—43no—
301C4×C89—41no—
302C4×C89C2×#13716yes—
303C4×C89C22×#4111yescharacteristic
304C4×C89#58511yescharacteristic
305#61469C8911yesmaximal, characteristic, direct product factor, Sylow
306C2×C89—43no—
307C2×C89—83no—
308C2×C89D4×#4111yescharacteristic
309C2×C89C22×#13711yescharacteristic
310C2×C89#358511yescharacteristic
311#3483C2×C8911yescharacteristic
312Q8×#41C2×C8911yescharacteristic, complemented
313#3392C2×C8911yescharacteristic, complemented
314Q8×Q16C2×C8911yescharacteristic, complemented
315#3363C2×C8911yescharacteristic, complemented
316#3362C2×C8911yescharacteristic, complemented
317#3357C2×C8913yescomplemented
318#3333C2×C8913yescomplemented
319#3329C2×C8913yescomplemented
320#3278C2×C8913yescomplemented
321#3182C2×C8913yescomplemented
322#3116C2×C8913yescomplemented
323#3112C2×C8913yescomplemented
324#2971C2×C8911yescharacteristic, complemented
325C4×#137C2×C8913yescomplemented
326C89#6146911yescharacteristic, direct product factor, Sylow
327C2×#137C22×C8913yes—
328Q82C22×C8911yescharacteristic
329Q82C22×C8911yescharacteristic
330#557—21no—
331#557C22×C8911yescharacteristic
332#555—23no—
333#555C22×C8913yes—
334#554—23no—
335#554C22×C8913yes—
336#550—21no—
337#550C22×C8913yes—
338#550C22×C8911yescharacteristic
339D4×Q8C22×C8913yes—
340#533—23no—
341#533C22×C8913yes—
342#532—23no—
343#532C22×C8913yes—
344C4×#41—21no—
345C4×#41—21no—
346C4×#41C22×C8913yes—
347C4×#41C22×C8911yescharacteristic
348C2×C4×Q8C22×C8913yes—
349#500C22×C8913yes—
350#492C22×C8913yes—
351#487C22×C8913yes—
352#486C22×C8913yes—
353#485C22×C8916yes—
354#480C22×C8916yes—
355#477C22×C8913yes—
356#477C22×C8913yes—
357#477C22×C8913yes—
358#476C22×C8913yes—
359#468C22×C8916yes—
360#468C22×C8916yes—
361#462C22×C8913yes—
362#461C22×C8913yes—
363#460C22×C8913yes—
364#460C22×C8913yes—
365#452C22×C8916yes—
366#448C22×C8916yes—
367C8×Q8C22×C8911yescharacteristic
368C8×Q8C22×C8911yescharacteristic
369#441C22×C8913yes—
370#441C22×C8911yescharacteristic
371#440C22×C8913yes—
372#440C22×C8916yes—
373#440C22×C8913yes—
374C4×Q16C22×C8913yes—
375C4×Q16C22×C8913yes—
376C4×SD16C22×C8913yes—
377C4×SD16C22×C8913yes—
378#436C22×C8913yes—
379#436C22×C8913yes—
380#432C22×C8913yes—
381#428C22×C8913yes—
382#419C22×C8913yes—
383#417C22×C8913yes—
384C4×M16C22×C8913yes—
385C2×#41—21no—
386C2×#41—23no—
387C2×#41D4×C8911yescharacteristic
388C2×#41C23×C8913yes—
389C22×Q8C23×C8913yes—
390#137—212no—
391C2×Q16C23×C8913yes—
392C2×Q16C23×C8911yescharacteristic
393C2×SD16C23×C8913yes—
394C2×SD16C23×C8913yes—
395C2×SD16C23×C8916yes—
396C2×M16C23×C8913yes—
397C2×M16C23×C8913yes—
398C4⋊Q8—23no—
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8C23×C8913yes—
402C4⋊Q8C23×C8913yes—
403C4⋊Q8C23×C8913yes—
404#127—23no—
405#127C23×C8913yes—
406#125—23no—
407#125—23no—
408#125—23no—
409#124—26no—
410#124—23no—
411C22⋊Q8—23no—
412C22⋊Q8—26no—
413C22⋊Q8—26no—
414C22⋊Q8C23×C8916yes—
415#121—23no—
416#121—23no—
417#121—23no—
418#121—26no—
419#121C23×C8916yes—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—23no—
426C4×Q8D4×C8911yescharacteristic
427C4×Q8C23×C8911yescharacteristic
428C4×Q8C23×C8913yes—
429C4×Q8C23×C8913yes—
430C4×Q8C23×C8913yes—
431C4×Q8C23×C8911yescharacteristic
432C4×Q8C23×C8913yes—
433C4×Q8C23×C8911yescharacteristic
434C4×D4—23no—
435C4×D4—23no—
436C4×D4—23no—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—23no—
440C4×D4D4×C8913yes—
441C4×D4C23×C8913yes—
442C4×D4C23×C8913yes—
443C4×D4C23×C8913yes—
444#117—23no—
445#117—23no—
446#117—23no—
447#117D4×C8913yes—
448#117C23×C8913yes—
449C2×C4⋊C4—23no—
450C2×C4⋊C4C23×C8913yes—
451C2×C42—23no—
452C2×C42—23no—
453C2×C42C23×C8913yes—
454#107C23×C8913yes—
455#107C23×C8913yes—
456#107C23×C8916yes—
457#106C23×C8911yescharacteristic
458#106C23×C8913yes—
459C4⋊C8C23×C8913yes—
460C4⋊C8C23×C8913yes—
461#103C23×C8911yescharacteristic
462#103C23×C8913yes—
463#103C23×C8911yescharacteristic
464#103C23×C8913yes—
465#103C23×C8913yes—
466#103C23×C8913yes—
467#103C23×C8916yes—
468#102C23×C8913yes—
469#102C23×C8913yes—
470#102C23×C8916yes—
471C22⋊C8C23×C8916yes—
472#97C23×C8911yescharacteristic
473#97C23×C8913yes—
474C4×C8C23×C8913yes—
475C4×C8C23×C8913yes—
476#41—24no—
477#41—26no—
478#41—43no—
479#41—44no—
480#41—86no—
481C2×Q8—23no—
482C2×Q8—23no—
483C2×Q8—23no—
484C2×Q8—26no—
485C2×Q8—43no—
486C2×Q8C2×D4×C8911yescharacteristic
487C2×Q8C24×C8913yes—
488C2×Q8C24×C8911yescharacteristic
489C2×Q8C24×C8911yescharacteristic
490C2×D4—23no—
491C2×D4—26no—
492C2×D4—23no—
493C2×D4—23no—
494C2×D4C2×D4×C8913yes—
495C2×D4C24×C8913yes—
496C22×C4—23no—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—23no—
501C22×C4C2×D4×C8913yes—
502C22×C4C2×D4×C8913yes—
503C22×C4C24×C8913yes—
504Q16—26no—
505Q16#41×C8914yescomplemented
506SD16—26no—
507SD16—26no—
508SD16—46no—
509M16—26no—
510M16—43no—
511C2×C8C24×C8911yescharacteristic
512C2×C8C24×C8913yes—
513C2×C8C24×C8913yes—
514C2×C8C24×C8911yescharacteristic
515C4⋊C4—21no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—21no—
521C4⋊C4—23no—
522C4⋊C4—23no—
523C4⋊C4—23no—
524C4⋊C4—23no—
525C4⋊C4—23no—
526C4⋊C4—26no—
527C4⋊C4C2×D4×C8913yes—
528C4⋊C4C2×D4×C8913yes—
529C4⋊C4C24×C8911yescharacteristic
530C4⋊C4C24×C8913yes—
531C4⋊C4C24×C8911yescharacteristic
532C4⋊C4C24×C8913yes—
533C22⋊C4—26no—
534C22⋊C4—23no—
535C22⋊C4—26no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4C2×D4×C8916yes—
539C42—21no—
540C42—21no—
541C42—23no—
542C42—23no—
543C42—23no—
544C42—23no—
545C42—23no—
546C42C2×D4×C8913yes—
547C42C2×D4×C8913yes—
548C42C24×C8913yes—
549C42C24×C8911yescharacteristic
550C23—23no—
551C23C22×D4×C8913yes—
552Q8—21no—
553Q8—21no—
554Q8—23no—
555Q8—23no—
556Q8—46no—
557Q8—83no—
558Q8—83no—
559Q8#137×C8914yescomplemented
560Q8C2×#41×C8912yes—
561Q8C2×#41×C8912yes—
562D4—26no—
563D4—23no—
564D4—23no—
565D4—46no—
566D4—46no—
567D4—83no—
568D4—86no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—26no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—23no—
577C2×C4—23no—
578C2×C4—23no—
579C2×C4—43no—
580C2×C4—46no—
581C2×C4—46no—
582C2×C4—46no—
583C2×C4—46no—
584C2×C4—83no—
585C2×C4C22×D4×C8911yescharacteristic
586C2×C4C22×D4×C8911yescharacteristic
587C2×C4C22×D4×C8913yes—
588C2×C4C22×D4×C8913yes—
589C2×C4C22×D4×C8913yes—
590C2×C4C25×C8911yescharacteristic
591C8—23no—
592C8—21no—
593C8—23no—
594C8C2×#41×C8912yes—
595C22—23no—
596C22—23no—
597C22—23no—
598C22—43no—
599C22—43no—
600C22—43no—
601C22C23×D4×C8911yescharacteristic
602C4—21no—
603C4—21no—
604C4—23no—
605C4—23no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—43no—
610C4—41no—
611C4—43no—
612C4—43no—
613C4C2×#137×C8916yes—
614C4C22×#41×C8911yescharacteristic
615C4#585×C8911yescharacteristic
616C2—43no—
617C2—83no—
618C2D4×#41×C8911yescharacteristic
619C2C22×#137×C8911yescharacteristic
620C2C89×#358511yescharacteristic
subgroupG/H
center, socleC22×C89C23×D4
commutator, Frattini, cosocleC2×C4C25×C89
prime radicalC89×#3483C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×#41×C89C2semidirect
3. C89×#3392C2semidirect
4. Q8×Q16×C89C2semidirect
5. C89×#3363C2semidirect
6. C89×#3362C2semidirect
7. C89×#3357C2semidirect
8. C89×#3333C2semidirect
9. C89×#3329C2semidirect
10. C89×#3278C2semidirect
11. C89×#3182C2semidirect
12. C89×#3116C2semidirect
13. C89×#3112C2semidirect
14. C89×#2971C2semidirect
15. C4×#137×C89C2semidirect
194. Q16×C89#41semidirect
248. Q8×C89#137semidirect
305. #61469C89direct (N × H)
312. Q8×#41C2×C89semidirect
313. #3392C2×C89semidirect
314. Q8×Q16C2×C89semidirect
315. #3363C2×C89semidirect
316. #3362C2×C89semidirect
317. #3357C2×C89semidirect
318. #3333C2×C89semidirect
319. #3329C2×C89semidirect
320. #3278C2×C89semidirect
321. #3182C2×C89semidirect
322. #3116C2×C89semidirect
323. #3112C2×C89semidirect
324. #2971C2×C89semidirect
325. C4×#137C2×C89semidirect
326. C89#61469direct (N × H)
505. Q16#41×C89semidirect
559. Q8#137×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#34832
Q8×#41×C892
C89×#33922
Q8×Q16×C892
C89×#33632
C89×#33622
C89×#33572
C89×#33332
C89×#33292
C89×#32782
C89×#31822
C89×#31162
C89×#31122
C89×#29712
C4×#137×C892
#6146989