Finite Group Databasebeta

← C89×#29382 #36407230

Subgroups — C89×#29382 #36407230

Order 22784.

748 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×#2326C211yesmaximal, characteristic
2C89×#2320C211yesmaximal, characteristic, complemented
3C89×#2099C211yesmaximal, characteristic, complemented
4C89×#2095C211yesmaximal, characteristic, complemented
5C89×#2042C211yesmaximal, characteristic
6C89×#2015C211yesmaximal, characteristic
7C89×#1928C211yesmaximal, characteristic, complemented
8C89×#1925C211yesmaximal, characteristic
9C89×#1880C211yesmaximal, characteristic
10C89×#1878C211yesmaximal, characteristic, complemented
11C89×#1878C211yesmaximal, characteristic, complemented
12C89×#1877C211yesmaximal, characteristic
13C89×#1863C211yesmaximal, characteristic, complemented
14C89×#1862C211yesmaximal, characteristic
15C89×#1840C211yesmaximal, characteristic, complemented
16C22×Q16×C89C2211yescharacteristic
17C22×SD16×C89C2211yescharacteristic
18C2×#124×C89C2211yescharacteristic
19C2×C4×Q8×C89C2211yescharacteristic
20C2×C4×Q8×C89C2211yescharacteristic
21C2×C4×D4×C89C2211yescharacteristic
22C2×#107×C89C2211yescharacteristic
23C2×#107×C89C2211yescharacteristic
24C2×#103×C89C2211yescharacteristic
25C2×#103×C89C2211yescharacteristic
26C2×#103×C89C2211yescharacteristic
27C2×#103×C89C2211yescharacteristic
28C2×#103×C89C2211yescharacteristic
29C2×#102×C89C2211yescharacteristic
30C2×#102×C89C2211yescharacteristic
31C2×#102×C89C2211yescharacteristic
32#391×C89C2211yescharacteristic
33#391×C89C2211yescharacteristic
34#391×C89C2211yescharacteristic
35#390×C89C2211yescharacteristic
36#389×C89C2211yescharacteristic
37#389×C89C2211yescharacteristic
38#388×C89—21no—
39#388×C89—21no—
40#387×C89—21no—
41#387×C89—21no—
42#385×C89—21no—
43#385×C89—21no—
44#384×C89C2211yescharacteristic
45#383×C89—21no—
46#382×C89—21no—
47C4×C4⋊C4×C89C2211yescharacteristic
48#376×C89C2211yescharacteristic
49#340×C89C2211yescharacteristic
50#340×C89C2211yescharacteristic
51#340×C89C2211yescharacteristic
52#340×C89C2211yescharacteristic
53#336×C89C2211yescharacteristic
54#336×C89C2211yescharacteristic
55#336×C89C2211yescharacteristic
56#336×C89C2211yescharacteristic
57#336×C89C2211yescharacteristic
58#336×C89C2211yescharacteristic
59C22×Q8×C89C2311yescharacteristic
60C22×Q8×C89C2311yescharacteristic
61C22×D4×C89C2311yescharacteristic
62C23×C4×C89—21no—
63C2×Q16×C89—22no—
64C2×Q16×C89—22no—
65C2×Q16×C89—41no—
66C2×SD16×C89—22no—
67C2×SD16×C89—22no—
68C2×SD16×C89—41no—
69C22×C8×C89C2311yescharacteristic
70C22×C8×C89C2311yescharacteristic
71C22×C8×C89C2311yescharacteristic
72C22×C8×C89C2311yescharacteristic
73#124×C89—22no—
74#124×C89—22no—
75C4×Q8×C89—22no—
76C4×Q8×C89—22no—
77C4×Q8×C89—41no—
78C4×Q8×C89—41no—
79C4×D4×C89—41no—
80C4×D4×C89—41no—
81C2×C4⋊C4×C89—21no—
82C2×C4⋊C4×C89—21no—
83C2×C4⋊C4×C89—21no—
84C2×C4⋊C4×C89—21no—
85C2×C4⋊C4×C89—21no—
86C2×C4⋊C4×C89—21no—
87C2×C4⋊C4×C89C2311yescharacteristic
88C2×C4⋊C4×C89C2311yescharacteristic
89C2×C4⋊C4×C89C2311yescharacteristic
90C2×C4⋊C4×C89C2311yescharacteristic
91C2×C4⋊C4×C89C2311yescharacteristic
92C2×C22⋊C4×C89—21no—
93C2×C22⋊C4×C89—21no—
94C2×C22⋊C4×C89—21no—
95C2×C22⋊C4×C89—21no—
96C2×C22⋊C4×C89—21no—
97C2×C42×C89—21no—
98C2×C42×C89—21no—
99C2×C42×C89—21no—
100C2×C42×C89C2311yescharacteristic
101C2×C42×C89C2311yescharacteristic
102C2×C42×C89C2311yescharacteristic
103#107×C89—41no—
104#107×C89—41no—
105#103×C89—22no—
106#103×C89—22no—
107#103×C89—41no—
108#103×C89—41no—
109#103×C89—41no—
110#102×C89—22no—
111#102×C89—22no—
112#102×C89—41no—
113#95×C89—21no—
114#95×C89—21no—
115#95×C89—21no—
116#95×C89—21no—
117#95×C89—21no—
118#95×C89—21no—
119#95×C89—21no—
120#95×C89—21no—
121#95×C89—21no—
122#95×C89D411yescharacteristic
123#95×C89D411yescharacteristic
124#95×C89D411yescharacteristic
125#95×C89D411yescharacteristic
126C24×C89—21no—
127C2×Q8×C89—21no—
128C2×Q8×C89—21no—
129C2×Q8×C89—21no—
130C2×Q8×C89—21no—
131C2×Q8×C89—21no—
132C2×Q8×C89—21no—
133C2×Q8×C89—21no—
134C2×Q8×C89—21no—
135C2×Q8×C89—21no—
136C2×Q8×C89—21no—
137C2×Q8×C89#4112yes—
138C2×Q8×C89#4112yes—
139C2×D4×C89—21no—
140C2×D4×C89—21no—
141C2×D4×C89—21no—
142C2×D4×C89—21no—
143C2×D4×C89—21no—
144C2×D4×C89#4112yes—
145C22×C4×C89—21no—
146C22×C4×C89—21no—
147C22×C4×C89—21no—
148C22×C4×C89—21no—
149C22×C4×C89—21no—
150C22×C4×C89—21no—
151C22×C4×C89—21no—
152C22×C4×C89—42no—
153C22×C4×C89—81no—
154C22×C4×C89—81no—
155C22×C4×C89C2×D411yescharacteristic
156C22×C4×C89C2×D411yescharacteristic
157C22×C4×C89C2×D411yescharacteristic
158C22×C4×C89C2×D411yescharacteristic
159C22×C4×C89C2×D411yescharacteristic
160C22×C4×C89C2×D411yescharacteristic
161C22×C4×C89C2411yescharacteristic
162Q16×C89—44no—
163SD16×C89—44no—
164C2×C8×C89—21no—
165C2×C8×C89—21no—
166C2×C8×C89—21no—
167C2×C8×C89—21no—
168C2×C8×C89—21no—
169C2×C8×C89—21no—
170C2×C8×C89—21no—
171C2×C8×C89—21no—
172C2×C8×C89—21no—
173C2×C8×C89—21no—
174C2×C8×C89—21no—
175C2×C8×C89—21no—
176C4⋊C4×C89—24no—
177C4⋊C4×C89—21no—
178C4⋊C4×C89—21no—
179C4⋊C4×C89—21no—
180C4⋊C4×C89—21no—
181C4⋊C4×C89—21no—
182C4⋊C4×C89—21no—
183C4⋊C4×C89—21no—
184C4⋊C4×C89—21no—
185C4⋊C4×C89—21no—
186C4⋊C4×C89—42no—
187C4⋊C4×C89—81no—
188C4⋊C4×C89—81no—
189C4⋊C4×C89—81no—
190C4⋊C4×C89—81no—
191C4⋊C4×C89#4112yes—
192C22⋊C4×C89—24no—
193C22⋊C4×C89—24no—
194C22⋊C4×C89—42no—
195C22⋊C4×C89—81no—
196C22⋊C4×C89—81no—
197C42×C89—24no—
198C42×C89—21no—
199C42×C89—21no—
200C42×C89—21no—
201C42×C89—21no—
202C42×C89—21no—
203C42×C89—21no—
204C42×C89—42no—
205C42×C89—81no—
206C23×C89—22no—
207C23×C89—41no—
208C23×C89—41no—
209C23×C89—41no—
210C23×C89—41no—
211C23×C89—41no—
212C23×C89—41no—
213C23×C89C22×D411yescharacteristic
214Q8×C89—22no—
215Q8×C89—22no—
216Q8×C89—22no—
217Q8×C89—22no—
218Q8×C89—41no—
219Q8×C89—41no—
220Q8×C89—41no—
221Q8×C89—41no—
222D4×C89—22no—
223D4×C89—22no—
224D4×C89—41no—
225D4×C89—41no—
226C2×C4×C89—21no—
227C2×C4×C89—21no—
228C2×C4×C89—21no—
229C2×C4×C89—21no—
230C2×C4×C89—22no—
231C2×C4×C89—22no—
232C2×C4×C89—22no—
233C2×C4×C89—21no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—21no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—21no—
240C2×C4×C89—21no—
241C2×C4×C89—21no—
242C2×C4×C89—21no—
243C2×C4×C89—21no—
244C2×C4×C89—21no—
245C2×C4×C89—21no—
246C2×C4×C89—21no—
247C2×C4×C89—41no—
248C2×C4×C89—41no—
249C2×C4×C89—41no—
250C2×C4×C89—41no—
251C2×C4×C89—41no—
252C2×C4×C89—41no—
253C2×C4×C89—41no—
254C2×C4×C89—41no—
255C2×C4×C89—41no—
256C2×C4×C89—41no—
257C2×C4×C89—41no—
258C2×C4×C89—41no—
259C2×C4×C89—41no—
260C2×C4×C89—41no—
261C2×C4×C89—41no—
262C2×C4×C89—41no—
263C2×C4×C89—41no—
264C2×C4×C89—41no—
265C2×C4×C89—84no—
266C2×C4×C89C2×#4111yescharacteristic
267C2×C4×C89C2×#4111yescharacteristic
268C2×C4×C89C2×#4111yescharacteristic
269C2×C4×C89C2×#4111yescharacteristic
270C2×C4×C89H(5,𝔽2)11yescharacteristic
271C2×C4×C89H(5,𝔽2)11yescharacteristic
272C8×C89—41no—
273C8×C89—41no—
274C8×C89—41no—
275C8×C89—41no—
276C22×C89—42no—
277C22×C89—42no—
278C22×C89—42no—
279C22×C89—81no—
280C22×C89—81no—
281C22×C89—81no—
282C22×C89—81no—
283C22×C89#52511yescharacteristic
284C22×C89#52511yescharacteristic
285C22×C89#53511yescharacteristic
286C22×C89#57611yescharacteristic
287C22×C89#57711yescharacteristic
288C22×C89#57811yescharacteristic
289C22×C89#57811yescharacteristic
290C4×C89—42no—
291C4×C89—42no—
292C4×C89—41no—
293C4×C89—41no—
294C4×C89—41no—
295C4×C89—41no—
296C4×C89—42no—
297C4×C89—41no—
298C4×C89—41no—
299C4×C89—81no—
300C4×C89—81no—
301C4×C89—81no—
302C4×C89—81no—
303C4×C89#55111yescharacteristic
304C4×C89#55111yescharacteristic
305C4×C89#55311yescharacteristic
306C4×C89#55311yescharacteristic
307#29382C8911yesmaximal, characteristic, direct product factor, Sylow
308C2×C89—82no—
309C2×C89#259811yescharacteristic
310C2×C89#312011yescharacteristic
311C2×C89#312111yescharacteristic
312C2×C89#312411yescharacteristic
313C2×C89#312411yescharacteristic
314C2×C89#322311yescharacteristic
315C2×C89#322411yescharacteristic
316#2326C2×C8911yescharacteristic
317#2320C2×C8911yescharacteristic, complemented
318#2099C2×C8911yescharacteristic, complemented
319#2095C2×C8911yescharacteristic, complemented
320#2042C2×C8911yescharacteristic
321#2015C2×C8911yescharacteristic
322#1928C2×C8911yescharacteristic, complemented
323#1925C2×C8911yescharacteristic
324#1880C2×C8911yescharacteristic
325#1878C2×C8911yescharacteristic, complemented
326#1878C2×C8911yescharacteristic, complemented
327#1877C2×C8911yescharacteristic
328#1863C2×C8911yescharacteristic, complemented
329#1862C2×C8911yescharacteristic
330#1840C2×C8911yescharacteristic, complemented
331C89#2938211yescharacteristic, direct product factor, Sylow
332C22×Q16C22×C8911yescharacteristic
333C22×SD16C22×C8911yescharacteristic
334C2×#124C22×C8911yescharacteristic
335C2×C4×Q8C22×C8911yescharacteristic
336C2×C4×Q8C22×C8911yescharacteristic
337C2×C4×D4C22×C8911yescharacteristic
338C2×#107C22×C8911yescharacteristic
339C2×#107C22×C8911yescharacteristic
340C2×#103C22×C8911yescharacteristic
341C2×#103C22×C8911yescharacteristic
342C2×#103C22×C8911yescharacteristic
343C2×#103C22×C8911yescharacteristic
344C2×#103C22×C8911yescharacteristic
345C2×#102C22×C8911yescharacteristic
346C2×#102C22×C8911yescharacteristic
347C2×#102C22×C8911yescharacteristic
348#391C22×C8911yescharacteristic
349#391C22×C8911yescharacteristic
350#391C22×C8911yescharacteristic
351#390C22×C8911yescharacteristic
352#389C22×C8911yescharacteristic
353#389C22×C8911yescharacteristic
354#388—21no—
355#388—21no—
356#387—21no—
357#387—21no—
358#385—21no—
359#385—21no—
360#384C22×C8911yescharacteristic
361#383—21no—
362#382—21no—
363C4×C4⋊C4C22×C8911yescharacteristic
364#376C22×C8911yescharacteristic
365#340C22×C8911yescharacteristic
366#340C22×C8911yescharacteristic
367#340C22×C8911yescharacteristic
368#340C22×C8911yescharacteristic
369#336C22×C8911yescharacteristic
370#336C22×C8911yescharacteristic
371#336C22×C8911yescharacteristic
372#336C22×C8911yescharacteristic
373#336C22×C8911yescharacteristic
374#336C22×C8911yescharacteristic
375C22×Q8C23×C8911yescharacteristic
376C22×Q8C23×C8911yescharacteristic
377C22×D4C23×C8911yescharacteristic
378C23×C4—21no—
379C2×Q16—22no—
380C2×Q16—22no—
381C2×Q16—41no—
382C2×SD16—22no—
383C2×SD16—22no—
384C2×SD16—41no—
385C22×C8C23×C8911yescharacteristic
386C22×C8C23×C8911yescharacteristic
387C22×C8C23×C8911yescharacteristic
388C22×C8C23×C8911yescharacteristic
389#124—22no—
390#124—22no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—41no—
394C4×Q8—41no—
395C4×D4—41no—
396C4×D4—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4C23×C8911yescharacteristic
404C2×C4⋊C4C23×C8911yescharacteristic
405C2×C4⋊C4C23×C8911yescharacteristic
406C2×C4⋊C4C23×C8911yescharacteristic
407C2×C4⋊C4C23×C8911yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42C23×C8911yescharacteristic
417C2×C42C23×C8911yescharacteristic
418C2×C42C23×C8911yescharacteristic
419#107—41no—
420#107—41no—
421#103—22no—
422#103—22no—
423#103—41no—
424#103—41no—
425#103—41no—
426#102—22no—
427#102—22no—
428#102—41no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95D4×C8911yescharacteristic
439#95D4×C8911yescharacteristic
440#95D4×C8911yescharacteristic
441#95D4×C8911yescharacteristic
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8#41×C8912yes—
454C2×Q8#41×C8912yes—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4#41×C8912yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4C2×D4×C8911yescharacteristic
472C22×C4C2×D4×C8911yescharacteristic
473C22×C4C2×D4×C8911yescharacteristic
474C22×C4C2×D4×C8911yescharacteristic
475C22×C4C2×D4×C8911yescharacteristic
476C22×C4C2×D4×C8911yescharacteristic
477C22×C4C24×C8911yescharacteristic
478Q16—44no—
479SD16—44no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4#41×C8912yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C42—24no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—81no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C22×D4×C8911yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4C2×#41×C8911yescharacteristic
583C2×C4C2×#41×C8911yescharacteristic
584C2×C4C2×#41×C8911yescharacteristic
585C2×C4C2×#41×C8911yescharacteristic
586C2×C4H(5,𝔽2)×C8911yescharacteristic
587C2×C4H(5,𝔽2)×C8911yescharacteristic
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#525×C8911yescharacteristic
600C22#525×C8911yescharacteristic
601C22#535×C8911yescharacteristic
602C22#576×C8911yescharacteristic
603C22#577×C8911yescharacteristic
604C22#578×C8911yescharacteristic
605C22#578×C8911yescharacteristic
606C4—42no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4#551×C8911yescharacteristic
620C4#551×C8911yescharacteristic
621C4#553×C8911yescharacteristic
622C4#553×C8911yescharacteristic
623C2—82no—
624C2C89×#259811yescharacteristic
625C2C89×#312011yescharacteristic
626C2C89×#312111yescharacteristic
627C2C89×#312411yescharacteristic
628C2C89×#312411yescharacteristic
629C2C89×#322311yescharacteristic
630C2C89×#322411yescharacteristic
subgroupG/H
center, socleC23×C89C22×D4
commutator, Frattini, cosocleC22×C4C24×C89
prime radicalC22×D4×C89C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C89 ◁ #376×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2320C2semidirect
3. C89×#2099C2semidirect
4. C89×#2095C2semidirect
7. C89×#1928C2semidirect
10. C89×#1878C2semidirect
11. C89×#1878C2semidirect
13. C89×#1863C2semidirect
15. C89×#1840C2semidirect
307. #29382C89direct (N × H)
317. #2320C2×C89semidirect
318. #2099C2×C89semidirect
319. #2095C2×C89semidirect
322. #1928C2×C89semidirect
325. #1878C2×C89semidirect
326. #1878C2×C89semidirect
328. #1863C2×C89semidirect
330. #1840C2×C89semidirect
331. C89#29382direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#23262
C89×#23202
C89×#20992
C89×#20952
C89×#20422
C89×#20152
C89×#19282
C89×#19252
C89×#18802
C89×#18782
C89×#18772
C89×#18632
C89×#18622
C89×#18402
#2938289