Finite Group Databasebeta

← C89×#32616 #36410464

Subgroups — C89×#32616 #36410464

Order 22784.

700 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2592C211yesmaximal, characteristic
2C89×#2582C211yesmaximal, characteristic, complemented
3C89×#2094C211yesmaximal, characteristic
4C89×#2072C211yesmaximal, characteristic, complemented
5C89×#2066C211yesmaximal, characteristic, complemented
6C89×#2055C211yesmaximal, characteristic
7C89×#2042C211yesmaximal, characteristic
8C89×#2015C211yesmaximal, characteristic
9C89×#1985C211yesmaximal, characteristic, complemented
10C89×#1971C211yesmaximal, characteristic, complemented
11C89×#1936C211yesmaximal, characteristic
12C89×#1927C211yesmaximal, characteristic, complemented
13C89×#1919C211yesmaximal, characteristic, complemented
14C89×#1880C211yesmaximal, characteristic
15C89×#1878C211yesmaximal, characteristic, complemented
16C22×Q16×C89C2211yescharacteristic
17C22×SD16×C89C2211yescharacteristic
18C2×#124×C89C2211yescharacteristic
19C2×C4×Q8×C89C2211yescharacteristic
20C2×C4×Q8×C89C2211yescharacteristic
21C2×#106×C89C2211yescharacteristic
22C2×#106×C89C2211yescharacteristic
23C2×C4⋊C8×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
32C2×#97×C89C2211yescharacteristic
33#398×C89—21no—
34#398×C89—21no—
35#397×C89—21no—
36#393×C89—21no—
37#391×C89—21no—
38#391×C89C2211yescharacteristic
39#391×C89C2211yescharacteristic
40#391×C89C2211yescharacteristic
41#391×C89C2211yescharacteristic
42#390×C89C2211yescharacteristic
43#390×C89C2211yescharacteristic
44#389×C89—21no—
45#389×C89C2211yescharacteristic
46#389×C89C2211yescharacteristic
47#388×C89—21no—
48#388×C89—21no—
49#384×C89C2211yescharacteristic
50C4×C4⋊C4×C89C2211yescharacteristic
51#340×C89C2211yescharacteristic
52#340×C89C2211yescharacteristic
53#340×C89C2211yescharacteristic
54#340×C89C2211yescharacteristic
55#336×C89C2211yescharacteristic
56#336×C89C2211yescharacteristic
57#336×C89C2211yescharacteristic
58#336×C89C2211yescharacteristic
59C22×Q8×C89D411yescharacteristic
60C22×Q8×C89C2311yescharacteristic
61C22×Q8×C89C2311yescharacteristic
62C22×D4×C89C2311yescharacteristic
63C2×Q16×C89—22no—
64C2×Q16×C89—22no—
65C2×Q16×C89—41no—
66C2×SD16×C89—22no—
67C2×SD16×C89—41no—
68C2×SD16×C89—41no—
69C22×C8×C89C2311yescharacteristic
70C22×C8×C89C2311yescharacteristic
71C22×C8×C89C2311yescharacteristic
72C22×C8×C89C2311yescharacteristic
73#124×C89—41no—
74#124×C89—41no—
75C4×Q8×C89—41no—
76C4×Q8×C89—41no—
77C4×Q8×C89—41no—
78C4×Q8×C89—41no—
79C2×C4⋊C4×C89—21no—
80C2×C4⋊C4×C89—21no—
81C2×C4⋊C4×C89—21no—
82C2×C4⋊C4×C89—21no—
83C2×C4⋊C4×C89—21no—
84C2×C4⋊C4×C89D411yescharacteristic
85C2×C4⋊C4×C89D411yescharacteristic
86C2×C4⋊C4×C89D411yescharacteristic
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×C22⋊C4×C89—21no—
98C2×C42×C89—21no—
99C2×C42×C89—21no—
100C2×C42×C89C2311yescharacteristic
101C2×C42×C89C2311yescharacteristic
102C2×C42×C89C2311yescharacteristic
103#106×C89—22no—
104#106×C89—41no—
105C4⋊C8×C89—41no—
106#103×C89—22no—
107#103×C89—41no—
108#103×C89—41no—
109#103×C89—41no—
110#103×C89—41no—
111#102×C89—22no—
112#102×C89—41no—
113#102×C89—41no—
114#97×C89—41no—
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×C89—21no—
123#95×C89—21no—
124#95×C89—21no—
125#95×C89—21no—
126C24×C89—21no—
127C2×Q8×C89—22no—
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—21no—
138C2×Q8×C89—21no—
139C2×Q8×C89—41no—
140C2×Q8×C89—41no—
141C2×Q8×C89#4112yes—
142C2×D4×C89—21no—
143C2×D4×C89—21no—
144C2×D4×C89—21no—
145C2×D4×C89—21no—
146C2×D4×C89—21no—
147C2×D4×C89—21no—
148C22×C4×C89—21no—
149C22×C4×C89—21no—
150C22×C4×C89—21no—
151C22×C4×C89—21no—
152C22×C4×C89—21no—
153C22×C4×C89—21no—
154C22×C4×C89—21no—
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#4112yes—
176C4⋊C4×C89—21no—
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—41no—
187C4⋊C4×C89—41no—
188C4⋊C4×C89—41no—
189C4⋊C4×C89—42no—
190C4⋊C4×C89—42no—
191C4⋊C4×C89—81no—
192C4⋊C4×C89—81no—
193C4⋊C4×C89—81no—
194C22⋊C4×C89—42no—
195C22⋊C4×C89—42no—
196C22⋊C4×C89—42no—
197C22⋊C4×C89—42no—
198C22⋊C4×C89—81no—
199C22⋊C4×C89—81no—
200C42×C89—21no—
201C42×C89—21no—
202C42×C89—21no—
203C42×C89—21no—
204C42×C89—21no—
205C42×C89—21no—
206C42×C89—42no—
207C42×C89—81no—
208C23×C89—41no—
209C23×C89—41no—
210C23×C89—41no—
211C23×C89—41no—
212C23×C89—41no—
213C23×C89—41no—
214C23×C89—41no—
215C23×C89C22×D411yescharacteristic
216Q8×C89—22no—
217Q8×C89—22no—
218Q8×C89—44no—
219Q8×C89—41no—
220Q8×C89—41no—
221Q8×C89—41no—
222Q8×C89—41no—
223Q8×C89—41no—
224Q8×C89—41no—
225D4×C89—41no—
226D4×C89—41no—
227D4×C89—41no—
228D4×C89—41no—
229C2×C4×C89—21no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—21no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—22no—
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—41no—
266C2×C4×C89—41no—
267C2×C4×C89#13712yes—
268C2×C4×C89C2×#4111yescharacteristic
269C2×C4×C89C2×#4111yescharacteristic
270C2×C4×C89H(5,𝔽2)11yescharacteristic
271C2×C4×C89H(5,𝔽2)11yescharacteristic
272C2×C4×C89H(5,𝔽2)11yescharacteristic
273C2×C4×C892−1+411yescharacteristic
274C8×C89—22no—
275C8×C89—41no—
276C8×C89—41no—
277C8×C89—41no—
278C22×C89—81no—
279C22×C89—81no—
280C22×C89—81no—
281C22×C89—81no—
282C22×C89—81no—
283C22×C89—81no—
284C22×C89—81no—
285C22×C89#53411yescharacteristic
286C22×C89#54811yescharacteristic
287C22×C89#55011yescharacteristic
288C22×C89C2×#13711yescharacteristic
289C22×C89#57511yescharacteristic
290C22×C89#57711yescharacteristic
291C22×C89#57711yescharacteristic
292C4×C89—22no—
293C4×C89—41no—
294C4×C89—41no—
295C4×C89—41no—
296C4×C89—41no—
297C4×C89—41no—
298C4×C89—42no—
299C4×C89—81no—
300C4×C89—81no—
301C4×C89—81no—
302C4×C89—81no—
303C4×C89—81no—
304C4×C89—81no—
305C4×C89#53811yescharacteristic
306C4×C89#54211yescharacteristic
307C4×C89#55111yescharacteristic
308C4×C89#56211yescharacteristic
309#32616C8911yesmaximal, characteristic, direct product factor, Sylow
310C2×C89—161no—
311C2×C89#273011yescharacteristic
312C2×C89#319611yescharacteristic
313C2×C89#320411yescharacteristic
314C2×C89#336911yescharacteristic
315C2×C89#337111yescharacteristic
316C2×C89#340011yescharacteristic
317C2×C89#340111yescharacteristic
318#2592C2×C8911yescharacteristic
319#2582C2×C8911yescharacteristic, complemented
320#2094C2×C8911yescharacteristic
321#2072C2×C8911yescharacteristic, complemented
322#2066C2×C8911yescharacteristic, complemented
323#2055C2×C8911yescharacteristic
324#2042C2×C8911yescharacteristic
325#2015C2×C8911yescharacteristic
326#1985C2×C8911yescharacteristic, complemented
327#1971C2×C8911yescharacteristic, complemented
328#1936C2×C8911yescharacteristic
329#1927C2×C8911yescharacteristic, complemented
330#1919C2×C8911yescharacteristic, complemented
331#1880C2×C8911yescharacteristic
332#1878C2×C8911yescharacteristic, complemented
333C89#3261611yescharacteristic, direct product factor, Sylow
334C22×Q16C22×C8911yescharacteristic
335C22×SD16C22×C8911yescharacteristic
336C2×#124C22×C8911yescharacteristic
337C2×C4×Q8C22×C8911yescharacteristic
338C2×C4×Q8C22×C8911yescharacteristic
339C2×#106C22×C8911yescharacteristic
340C2×#106C22×C8911yescharacteristic
341C2×C4⋊C8C22×C8911yescharacteristic
342C2×#103C22×C8911yescharacteristic
343C2×#103C22×C8911yescharacteristic
344C2×#103C22×C8911yescharacteristic
345C2×#103C22×C8911yescharacteristic
346C2×#103C22×C8911yescharacteristic
347C2×#102C22×C8911yescharacteristic
348C2×#102C22×C8911yescharacteristic
349C2×#102C22×C8911yescharacteristic
350C2×#97C22×C8911yescharacteristic
351#398—21no—
352#398—21no—
353#397—21no—
354#393—21no—
355#391—21no—
356#391C22×C8911yescharacteristic
357#391C22×C8911yescharacteristic
358#391C22×C8911yescharacteristic
359#391C22×C8911yescharacteristic
360#390C22×C8911yescharacteristic
361#390C22×C8911yescharacteristic
362#389—21no—
363#389C22×C8911yescharacteristic
364#389C22×C8911yescharacteristic
365#388—21no—
366#388—21no—
367#384C22×C8911yescharacteristic
368C4×C4⋊C4C22×C8911yescharacteristic
369#340C22×C8911yescharacteristic
370#340C22×C8911yescharacteristic
371#340C22×C8911yescharacteristic
372#340C22×C8911yescharacteristic
373#336C22×C8911yescharacteristic
374#336C22×C8911yescharacteristic
375#336C22×C8911yescharacteristic
376#336C22×C8911yescharacteristic
377C22×Q8D4×C8911yescharacteristic
378C22×Q8C23×C8911yescharacteristic
379C22×Q8C23×C8911yescharacteristic
380C22×D4C23×C8911yescharacteristic
381C2×Q16—22no—
382C2×Q16—22no—
383C2×Q16—41no—
384C2×SD16—22no—
385C2×SD16—41no—
386C2×SD16—41no—
387C22×C8C23×C8911yescharacteristic
388C22×C8C23×C8911yescharacteristic
389C22×C8C23×C8911yescharacteristic
390C22×C8C23×C8911yescharacteristic
391#124—41no—
392#124—41no—
393C4×Q8—41no—
394C4×Q8—41no—
395C4×Q8—41no—
396C4×Q8—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4D4×C8911yescharacteristic
403C2×C4⋊C4D4×C8911yescharacteristic
404C2×C4⋊C4D4×C8911yescharacteristic
405C2×C4⋊C4C23×C8911yescharacteristic
406C2×C4⋊C4C23×C8911yescharacteristic
407C2×C4⋊C4C23×C8911yescharacteristic
408C2×C4⋊C4C23×C8911yescharacteristic
409C2×C4⋊C4C23×C8911yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42C23×C8911yescharacteristic
419C2×C42C23×C8911yescharacteristic
420C2×C42C23×C8911yescharacteristic
421#106—22no—
422#106—41no—
423C4⋊C8—41no—
424#103—22no—
425#103—41no—
426#103—41no—
427#103—41no—
428#103—41no—
429#102—22no—
430#102—41no—
431#102—41no—
432#97—41no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444C24—21no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8#41×C8912yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4C2×D4×C8911yescharacteristic
474C22×C4C2×D4×C8911yescharacteristic
475C22×C4C2×D4×C8911yescharacteristic
476C22×C4C2×D4×C8911yescharacteristic
477C22×C4C2×D4×C8911yescharacteristic
478C22×C4C2×D4×C8911yescharacteristic
479C22×C4C24×C8911yescharacteristic
480Q16—44no—
481SD16—44no—
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—
492C2×C8—21no—
493C2×C8#41×C8912yes—
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—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—81no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×D4×C8911yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—44no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
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—21no—
564C2×C4—21no—
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—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4#137×C8912yes—
586C2×C4C2×#41×C8911yescharacteristic
587C2×C4C2×#41×C8911yescharacteristic
588C2×C4H(5,𝔽2)×C8911yescharacteristic
589C2×C4H(5,𝔽2)×C8911yescharacteristic
590C2×C4H(5,𝔽2)×C8911yescharacteristic
591C2×C42−1+4×C8911yescharacteristic
592C8—22no—
593C8—41no—
594C8—41no—
595C8—41no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#534×C8911yescharacteristic
604C22#548×C8911yescharacteristic
605C22#550×C8911yescharacteristic
606C22C2×#137×C8911yescharacteristic
607C22#575×C8911yescharacteristic
608C22#577×C8911yescharacteristic
609C22#577×C8911yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4#538×C8911yescharacteristic
624C4#542×C8911yescharacteristic
625C4#551×C8911yescharacteristic
626C4#562×C8911yescharacteristic
627C2—161no—
628C2C89×#273011yescharacteristic
629C2C89×#319611yescharacteristic
630C2C89×#320411yescharacteristic
631C2C89×#336911yescharacteristic
632C2C89×#337111yescharacteristic
633C2C89×#340011yescharacteristic
634C2C89×#340111yescharacteristic
subgroupG/H
center, socleC23×C89C22×D4
commutator, Frattini, cosocleC22×C4C24×C89
prime radicalC22×D4×C89C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C89 ◁ C2×C4×Q8×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2582C2semidirect
4. C89×#2072C2semidirect
5. C89×#2066C2semidirect
9. C89×#1985C2semidirect
10. C89×#1971C2semidirect
12. C89×#1927C2semidirect
13. C89×#1919C2semidirect
15. C89×#1878C2semidirect
309. #32616C89direct (N × H)
319. #2582C2×C89semidirect
321. #2072C2×C89semidirect
322. #2066C2×C89semidirect
326. #1985C2×C89semidirect
327. #1971C2×C89semidirect
329. #1927C2×C89semidirect
330. #1919C2×C89semidirect
332. #1878C2×C89semidirect
333. C89#32616direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#25922
C89×#25822
C89×#20942
C89×#20722
C89×#20662
C89×#20552
C89×#20422
C89×#20152
C89×#19852
C89×#19712
C89×#19362
C89×#19272
C89×#19192
C89×#18802
C89×#18782
#3261689