Finite Group Databasebeta

← C89×#24724 #36402572

Subgroups — C89×#24724 #36402572

Order 22784.

690 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3080C211yesmaximal, characteristic, complemented
2C2×#478×C89C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)×C89C211yesmaximal, characteristic
4C89×#2925C211yesmaximal, characteristic, complemented
5C89×#2079C211yesmaximal, characteristic, complemented
6C89×#2078C211yesmaximal, characteristic, complemented
7C89×#2051C211yesmaximal, characteristic
8C89×#2051C211yesmaximal, characteristic
9C89×#1998C211yesmaximal, characteristic, complemented
10C89×#1952C211yesmaximal, characteristic, complemented
11C89×#1951C211yesmaximal, characteristic, complemented
12C89×#1950C211yesmaximal, characteristic, complemented
13C89×#1931C211yesmaximal, characteristic, complemented
14C89×#1931C211yesmaximal, characteristic, complemented
15C89×#1800C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)×C89C2211yescharacteristic
17C2×#125×C89C2211yescharacteristic
18C4×#41×C89C2211yescharacteristic
19C2×C4×D4×C89C2211yescharacteristic
20#479×C89—21no—
21#479×C89—21no—
22#478×C89—21no—
23#478×C89—21no—
24#478×C89—21no—
25#478×C89—21no—
26#478×C89—21no—
27#478×C89—21no—
28#442×C89—21no—
29#440×C89—21no—
30C4×SD16×C89—21no—
31C4×D8×C89—21no—
32#433×C89—21no—
33#433×C89—21no—
34#432×C89—21no—
35#431×C89—21no—
36C2×#108×C89C2211yescharacteristic
37C2×#108×C89C2211yescharacteristic
38#428×C89C2211yescharacteristic
39#428×C89C2211yescharacteristic
40#428×C89C2211yescharacteristic
41C2×#107×C89C2211yescharacteristic
42C2×#106×C89C2211yescharacteristic
43#423×C89C2211yescharacteristic
44#423×C89C2211yescharacteristic
45C2×C4⋊C8×C89C2211yescharacteristic
46#418×C89C2211yescharacteristic
47#417×C89C2211yescharacteristic
48#417×C89C2211yescharacteristic
49#417×C89C2211yescharacteristic
50C2×#102×C89C2211yescharacteristic
51C2×#102×C89C2211yescharacteristic
52C2×#100×C89C2211yescharacteristic
53C2×#100×C89C2211yescharacteristic
54#408×C89C2211yescharacteristic
55#408×C89C2211yescharacteristic
56#405×C89C2211yescharacteristic
57#405×C89C2211yescharacteristic
58C4×M16×C89C2211yescharacteristic
59#343×C89C2211yescharacteristic
60#343×C89C2211yescharacteristic
61#343×C89C2211yescharacteristic
62#343×C89C2211yescharacteristic
63#343×C89C2211yescharacteristic
64#343×C89C2211yescharacteristic
65#341×C89C2211yescharacteristic
66#341×C89C2211yescharacteristic
67C2×#41×C89C2311yescharacteristic
68C22×D4×C89C2311yescharacteristic
69C23×C4×C89—21no—
70AGL(1,ℤ/8)×C89—22no—
71AGL(1,ℤ/8)×C89—22no—
72C2×SD16×C89—21no—
73C2×D8×C89—21no—
74C2×M16×C89C2311yescharacteristic
75C2×M16×C89C2311yescharacteristic
76C2×M16×C89C2311yescharacteristic
77C2×M16×C89C2311yescharacteristic
78C2×M16×C89C2311yescharacteristic
79C2×M16×C89C2311yescharacteristic
80C22×C8×C89C2311yescharacteristic
81C22×C8×C89C2311yescharacteristic
82#125×C89—21no—
83#125×C89—21no—
84#125×C89—21no—
85#125×C89—21no—
86C4×Q8×C89D411yescharacteristic
87C4×Q8×C89D411yescharacteristic
88C4×D4×C89—21no—
89C4×D4×C89—21no—
90C4×D4×C89—21no—
91C4×D4×C89—21no—
92C4×D4×C89—21no—
93C4×D4×C89—21no—
94C4×D4×C89D411yescharacteristic
95C4×D4×C89D411yescharacteristic
96#117×C89—21no—
97#117×C89C2311yescharacteristic
98C2×C4⋊C4×C89—21no—
99C2×C4⋊C4×C89—21no—
100C2×C4⋊C4×C89C2311yescharacteristic
101C2×C4⋊C4×C89C2311yescharacteristic
102C2×C4⋊C4×C89C2311yescharacteristic
103C2×C22⋊C4×C89—21no—
104C2×C42×C89—21no—
105C2×C42×C89C2311yescharacteristic
106#108×C89—22no—
107#108×C89—22no—
108#107×C89—21no—
109#107×C89—21no—
110#107×C89—21no—
111#107×C89—21no—
112#107×C89—21no—
113#106×C89—21no—
114#106×C89—21no—
115#106×C89—21no—
116#106×C89—21no—
117#106×C89—21no—
118C4⋊C8×C89—21no—
119C4⋊C8×C89—21no—
120C4⋊C8×C89—21no—
121C4⋊C8×C89—21no—
122C4⋊C8×C89D411yescharacteristic
123C4⋊C8×C89D411yescharacteristic
124C4⋊C8×C89D411yescharacteristic
125C4⋊C8×C89D411yescharacteristic
126#103×C89—21no—
127#103×C89—21no—
128#103×C89—21no—
129#102×C89—21no—
130#102×C89—21no—
131#102×C89—21no—
132#102×C89—21no—
133#102×C89—21no—
134#102×C89—21no—
135#102×C89—21no—
136#102×C89—21no—
137#102×C89—21no—
138#100×C89—22no—
139#100×C89—22no—
140C22⋊C8×C89—21no—
141C22⋊C8×C89—21no—
142C22⋊C8×C89—21no—
143C22⋊C8×C89—21no—
144#97×C89—21no—
145#97×C89—21no—
146#97×C89—21no—
147C4×C8×C89—21no—
148C4×C8×C89—21no—
149C4×C8×C89—21no—
150C24×C89—21no—
151#41×C89—21no—
152#41×C89—21no—
153#41×C89—21no—
154#41×C89—21no—
155C2×Q8×C89C2×D411yescharacteristic
156C2×D4×C89—21no—
157C2×D4×C89—21no—
158C2×D4×C89—21no—
159C2×D4×C89—21no—
160C2×D4×C89—21no—
161C2×D4×C89—21no—
162C2×D4×C89C2×D411yescharacteristic
163C2×D4×C89#4112yes—
164C22×C4×C89—21no—
165C22×C4×C89—21no—
166C22×C4×C89—21no—
167C22×C4×C89—21no—
168C22×C4×C89—21no—
169C22×C4×C89—21no—
170C22×C4×C89—41no—
171C22×C4×C89—41no—
172C22×C4×C89—82no—
173C22×C4×C89#4111yescharacteristic
174C22×C4×C89#4111yescharacteristic
175C22×C4×C89C2411yescharacteristic
176SD16×C89—24no—
177D8×C89—24no—
178M16×C89—21no—
179M16×C89—21no—
180M16×C89—21no—
181M16×C89—21no—
182M16×C89—21no—
183M16×C89—21no—
184M16×C89—21no—
185M16×C89—21no—
186M16×C89—21no—
187M16×C89—21no—
188M16×C89—21no—
189M16×C89#4112yes—
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C2×C8×C89—21no—
193C2×C8×C89—21no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C2×C8×C89—21no—
197C2×C8×C89—21no—
198C2×C8×C89—21no—
199C2×C8×C89—21no—
200C2×C8×C89C2×D411yescharacteristic
201C2×C8×C89C2×D411yescharacteristic
202C2×C8×C89C2×D411yescharacteristic
203C2×C8×C89C2×D411yescharacteristic
204C4⋊C4×C89—21no—
205C4⋊C4×C89—21no—
206C4⋊C4×C89—21no—
207C4⋊C4×C89—21no—
208C4⋊C4×C89—21no—
209C4⋊C4×C89—21no—
210C4⋊C4×C89—21no—
211C4⋊C4×C89—21no—
212C4⋊C4×C89—41no—
213C4⋊C4×C89—41no—
214C4⋊C4×C89—41no—
215C4⋊C4×C89—41no—
216C4⋊C4×C89C2×D411yescharacteristic
217C4⋊C4×C89C2×D411yescharacteristic
218C22⋊C4×C89—21no—
219C22⋊C4×C89—21no—
220C22⋊C4×C89—21no—
221C22⋊C4×C89—41no—
222C22⋊C4×C89—41no—
223C42×C89—21no—
224C42×C89—21no—
225C42×C89—21no—
226C42×C89C2×D411yescharacteristic
227C42×C89C2×D411yescharacteristic
228C42×C89C2×D411yescharacteristic
229C42×C89C2×D411yescharacteristic
230C23×C89—21no—
231C23×C89—22no—
232C23×C89—41no—
233C23×C89—41no—
234C23×C89—81no—
235C23×C89—81no—
236C23×C89C2×#4111yescharacteristic
237Q8×C89—21no—
238Q8×C89—21no—
239D4×C89—21no—
240D4×C89—22no—
241D4×C89—21no—
242D4×C89—22no—
243D4×C89—41no—
244D4×C89—41no—
245D4×C89—41no—
246D4×C89—41no—
247C2×C4×C89—21no—
248C2×C4×C89—21no—
249C2×C4×C89—21no—
250C2×C4×C89—41no—
251C2×C4×C89—41no—
252C2×C4×C89—42no—
253C2×C4×C89—41no—
254C2×C4×C89—41no—
255C2×C4×C89—41no—
256C2×C4×C89—41no—
257C2×C4×C89—41no—
258C2×C4×C89—81no—
259C2×C4×C89—82no—
260C2×C4×C89—82no—
261C2×C4×C89—81no—
262C2×C4×C89—81no—
263C2×C4×C89—81no—
264C2×C4×C89#12111yescharacteristic
265C2×C4×C89#12111yescharacteristic
266C2×C4×C89#12111yescharacteristic
267C2×C4×C89#12111yescharacteristic
268C2×C4×C89C22×D411yescharacteristic
269C2×C4×C89C22×D411yescharacteristic
270C2×C4×C89C2×#4111yescharacteristic
271C2×C4×C89C2×#4111yescharacteristic
272C2×C4×C892−1+411yescharacteristic
273C2×C4×C892−1+411yescharacteristic
274C8×C89—22no—
275C8×C89—21no—
276C8×C89—21no—
277C8×C89—21no—
278C8×C89—21no—
279C8×C89—41no—
280C8×C89—41no—
281C8×C89—41no—
282C8×C89—41no—
283C8×C89—41no—
284C22×C89—21no—
285C22×C89—21no—
286C22×C89—41no—
287C22×C89—42no—
288C22×C89—41no—
289C22×C89—41no—
290C22×C89—82no—
291C22×C89—82no—
292C22×C89—81no—
293C22×C89—81no—
294C22×C89C2×#12111yescharacteristic
295C22×C89#55211yescharacteristic
296C22×C89#55611yescharacteristic
297C4×C89—21no—
298C4×C89—21no—
299C4×C89—21no—
300C4×C89—21no—
301C4×C89—41no—
302C4×C89—41no—
303C4×C89—41no—
304C4×C89—81no—
305C4×C89—81no—
306C4×C89—81no—
307C4×C89#52511yescharacteristic
308C4×C89#53611yescharacteristic
309C4×C89#54711yescharacteristic
310C4×C89#54811yescharacteristic
311#24724C8911yesmaximal, characteristic, direct product factor, Sylow
312C2×C89—21no—
313C2×C89—21no—
314C2×C89—81no—
315C2×C89—82no—
316C2×C89#241511yescharacteristic
317C2×C89#306711yescharacteristic
318C2×C89#306811yescharacteristic
319#3080C2×C8911yescharacteristic, complemented
320C2×#478C2×C8911yescharacteristic, complemented
321C4×AGL(1,ℤ/8)C2×C8911yescharacteristic
322#2925C2×C8911yescharacteristic, complemented
323#2079C2×C8911yescharacteristic, complemented
324#2078C2×C8911yescharacteristic, complemented
325#2051C2×C8911yescharacteristic
326#2051C2×C8911yescharacteristic
327#1998C2×C8911yescharacteristic, complemented
328#1952C2×C8911yescharacteristic, complemented
329#1951C2×C8911yescharacteristic, complemented
330#1950C2×C8911yescharacteristic, complemented
331#1931C2×C8911yescharacteristic, complemented
332#1931C2×C8911yescharacteristic, complemented
333#1800C2×C8911yescharacteristic, complemented
334C89#2472411yescharacteristic, direct product factor, Sylow
335C2×AGL(1,ℤ/8)C22×C8911yescharacteristic
336C2×#125C22×C8911yescharacteristic
337C4×#41C22×C8911yescharacteristic
338C2×C4×D4C22×C8911yescharacteristic
339#479—21no—
340#479—21no—
341#478—21no—
342#478—21no—
343#478—21no—
344#478—21no—
345#478—21no—
346#478—21no—
347#442—21no—
348#440—21no—
349C4×SD16—21no—
350C4×D8—21no—
351#433—21no—
352#433—21no—
353#432—21no—
354#431—21no—
355C2×#108C22×C8911yescharacteristic
356C2×#108C22×C8911yescharacteristic
357#428C22×C8911yescharacteristic
358#428C22×C8911yescharacteristic
359#428C22×C8911yescharacteristic
360C2×#107C22×C8911yescharacteristic
361C2×#106C22×C8911yescharacteristic
362#423C22×C8911yescharacteristic
363#423C22×C8911yescharacteristic
364C2×C4⋊C8C22×C8911yescharacteristic
365#418C22×C8911yescharacteristic
366#417C22×C8911yescharacteristic
367#417C22×C8911yescharacteristic
368#417C22×C8911yescharacteristic
369C2×#102C22×C8911yescharacteristic
370C2×#102C22×C8911yescharacteristic
371C2×#100C22×C8911yescharacteristic
372C2×#100C22×C8911yescharacteristic
373#408C22×C8911yescharacteristic
374#408C22×C8911yescharacteristic
375#405C22×C8911yescharacteristic
376#405C22×C8911yescharacteristic
377C4×M16C22×C8911yescharacteristic
378#343C22×C8911yescharacteristic
379#343C22×C8911yescharacteristic
380#343C22×C8911yescharacteristic
381#343C22×C8911yescharacteristic
382#343C22×C8911yescharacteristic
383#343C22×C8911yescharacteristic
384#341C22×C8911yescharacteristic
385#341C22×C8911yescharacteristic
386C2×#41C23×C8911yescharacteristic
387C22×D4C23×C8911yescharacteristic
388C23×C4—21no—
389AGL(1,ℤ/8)—22no—
390AGL(1,ℤ/8)—22no—
391C2×SD16—21no—
392C2×D8—21no—
393C2×M16C23×C8911yescharacteristic
394C2×M16C23×C8911yescharacteristic
395C2×M16C23×C8911yescharacteristic
396C2×M16C23×C8911yescharacteristic
397C2×M16C23×C8911yescharacteristic
398C2×M16C23×C8911yescharacteristic
399C22×C8C23×C8911yescharacteristic
400C22×C8C23×C8911yescharacteristic
401#125—21no—
402#125—21no—
403#125—21no—
404#125—21no—
405C4×Q8D4×C8911yescharacteristic
406C4×Q8D4×C8911yescharacteristic
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4D4×C8911yescharacteristic
414C4×D4D4×C8911yescharacteristic
415#117—21no—
416#117C23×C8911yescharacteristic
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4C23×C8911yescharacteristic
420C2×C4⋊C4C23×C8911yescharacteristic
421C2×C4⋊C4C23×C8911yescharacteristic
422C2×C22⋊C4—21no—
423C2×C42—21no—
424C2×C42C23×C8911yescharacteristic
425#108—22no—
426#108—22no—
427#107—21no—
428#107—21no—
429#107—21no—
430#107—21no—
431#107—21no—
432#106—21no—
433#106—21no—
434#106—21no—
435#106—21no—
436#106—21no—
437C4⋊C8—21no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8D4×C8911yescharacteristic
442C4⋊C8D4×C8911yescharacteristic
443C4⋊C8D4×C8911yescharacteristic
444C4⋊C8D4×C8911yescharacteristic
445#103—21no—
446#103—21no—
447#103—21no—
448#102—21no—
449#102—21no—
450#102—21no—
451#102—21no—
452#102—21no—
453#102—21no—
454#102—21no—
455#102—21no—
456#102—21no—
457#100—22no—
458#100—22no—
459C22⋊C8—21no—
460C22⋊C8—21no—
461C22⋊C8—21no—
462C22⋊C8—21no—
463#97—21no—
464#97—21no—
465#97—21no—
466C4×C8—21no—
467C4×C8—21no—
468C4×C8—21no—
469C24—21no—
470#41—21no—
471#41—21no—
472#41—21no—
473#41—21no—
474C2×Q8C2×D4×C8911yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4C2×D4×C8911yescharacteristic
482C2×D4#41×C8912yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—82no—
492C22×C4#41×C8911yescharacteristic
493C22×C4#41×C8911yescharacteristic
494C22×C4C24×C8911yescharacteristic
495SD16—24no—
496D8—24no—
497M16—21no—
498M16—21no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—21no—
503M16—21no—
504M16—21no—
505M16—21no—
506M16—21no—
507M16—21no—
508M16#41×C8912yes—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8C2×D4×C8911yescharacteristic
520C2×C8C2×D4×C8911yescharacteristic
521C2×C8C2×D4×C8911yescharacteristic
522C2×C8C2×D4×C8911yescharacteristic
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4C2×D4×C8911yescharacteristic
536C4⋊C4C2×D4×C8911yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42C2×D4×C8911yescharacteristic
546C42C2×D4×C8911yescharacteristic
547C42C2×D4×C8911yescharacteristic
548C42C2×D4×C8911yescharacteristic
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—81no—
554C23—81no—
555C23C2×#41×C8911yescharacteristic
556Q8—21no—
557Q8—21no—
558D4—21no—
559D4—22no—
560D4—21no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4#121×C8911yescharacteristic
584C2×C4#121×C8911yescharacteristic
585C2×C4#121×C8911yescharacteristic
586C2×C4#121×C8911yescharacteristic
587C2×C4C22×D4×C8911yescharacteristic
588C2×C4C22×D4×C8911yescharacteristic
589C2×C4C2×#41×C8911yescharacteristic
590C2×C4C2×#41×C8911yescharacteristic
591C2×C42−1+4×C8911yescharacteristic
592C2×C42−1+4×C8911yescharacteristic
593C8—22no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—41no—
599C8—41no—
600C8—41no—
601C8—41no—
602C8—41no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22C2×#121×C8911yescharacteristic
614C22#552×C8911yescharacteristic
615C22#556×C8911yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4#525×C8911yescharacteristic
627C4#536×C8911yescharacteristic
628C4#547×C8911yescharacteristic
629C4#548×C8911yescharacteristic
630C2—21no—
631C2—21no—
632C2—81no—
633C2—82no—
634C2C89×#241511yescharacteristic
635C2C89×#306711yescharacteristic
636C2C89×#306811yescharacteristic
subgroupG/H
center, socleC22×C89C2×#121
commutator, Frattini, cosocleC22×C4C24×C89
prime radicalC2×AGL(1,ℤ/8)×C89C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3080C2semidirect
2. C2×#478×C89C2semidirect
4. C89×#2925C2semidirect
5. C89×#2079C2semidirect
6. C89×#2078C2semidirect
9. C89×#1998C2semidirect
10. C89×#1952C2semidirect
11. C89×#1951C2semidirect
12. C89×#1950C2semidirect
13. C89×#1931C2semidirect
14. C89×#1931C2semidirect
15. C89×#1800C2semidirect
311. #24724C89direct (N × H)
319. #3080C2×C89semidirect
320. C2×#478C2×C89semidirect
322. #2925C2×C89semidirect
323. #2079C2×C89semidirect
324. #2078C2×C89semidirect
327. #1998C2×C89semidirect
328. #1952C2×C89semidirect
329. #1951C2×C89semidirect
330. #1950C2×C89semidirect
331. #1931C2×C89semidirect
332. #1931C2×C89semidirect
333. #1800C2×C89semidirect
334. C89#24724direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#30802
C2×#478×C892
C4×AGL(1,ℤ/8)×C892
C89×#29252
C89×#20792
C89×#20782
C89×#20512
C89×#19982
C89×#19522
C89×#19512
C89×#19502
C89×#19312
C89×#18002
#2472489