Finite Group Databasebeta

← #36715541

Subgroups — #36715541

Order 22784.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152074C211yesmaximal, characteristic
2#26023213C213yesmaximal, complemented
3#26023205C213yesmaximal, complemented
4#26023170C211yesmaximal, characteristic, complemented
5Q8×#13083553C211yesmaximal, characteristic, complemented
6#26023141C213yesmaximal
7#26023133C213yesmaximal
8Q8×#13083551C211yesmaximal, characteristic
9Q8×#13083551C211yesmaximal, characteristic
10#26022898C213yesmaximal, complemented
11#26022857C213yesmaximal
12#26022749C213yesmaximal
13#26022729C213yesmaximal, complemented
14#26022617C211yesmaximal, characteristic, complemented
15C42×Q8×C89C211yesmaximal, characteristic, complemented
16C22×Q8×D89C2211yescharacteristic
17#19152074C4116yescomplemented
18C2×Q8×Dic89C2211yescharacteristic
19C2×Q8×Dic89C2211yescharacteristic
20C2×Q8×Dic89C2211yescharacteristic
21C2×C4×Dic178C2213yes—
22C2×#15667977C2213yes—
23C2×#15667970C2213yes—
24C2×#15667969C2213yes—
25#19151504C2213yes—
26#19151504C2213yes—
27#19151503C2213yes—
28#19151503C2213yes—
29#19151496C2213yes—
30#19151496C2213yes—
31#19151494C2213yes—
32#19151494C2213yes—
33#19151489C2211yescharacteristic
34#19151489C2213yes—
35#19151489C2213yes—
36#19151489C2213yes—
37#19151487C2216yes—
38#19151482C2213yes—
39#19151482C2213yes—
40#19151482C2213yes—
41#19151482C2216yes—
42#19151482C2213yes—
43#19151480C2216yes—
44#19151467C2213yes—
45#19151465C2216yes—
46#19151463C2213yes—
47#19151462C2216yes—
48#19151455C2213yes—
49#19151455C2213yes—
50C4⋊C4×Dic89C2213yes—
51C4⋊C4×Dic89C2213yes—
52C4×#13083553C2213yes—
53C4×#13083553C2216yes—
54C4×#13083553C2213yes—
55C4×C4⋊Dic89C2213yes—
56C4×#13083551C2213yes—
57C4×#13083551C2213yes—
58#19151444C2216yes—
59#19151444C2213yes—
60C2×C4×Q8×C89C2211yescharacteristic
61C2×C4×Q8×C89C2211yescharacteristic
62C2×C4×Q8×C89C2211yescharacteristic
63C4×C4⋊C4×C89C2213yes—
64C4×C4⋊C4×C89C2213yes—
65C4×C4⋊C4×C89C2213yes—
66C43×C89C2213yes—
67C22×Dic178C2313yes—
68C2×Q8×D89—24no—
69C2×Q8×D89—24no—
70C2×Q8×D89C2×C418yes—
71C22×C4×D89C2313yes—
72Q8×Dic89—24no—
73Q8×Dic89—24no—
74Q8×Dic89C2×C418yes—
75C4×Dic178C2×C4124yes—
76#15667977C2×C4124yes—
77#15667970C2×C4124yes—
78#15667969C2×C4124yes—
79C2×C4⋊Dic89C2313yes—
80C2×C4⋊Dic89C2313yes—
81C2×C4⋊Dic89C2313yes—
82C2×#13083553C2311yescharacteristic
83C2×#13083553C2313yes—
84C2×#13083553C2313yes—
85C2×#13083553C2316yes—
86C2×#13083553C2313yes—
87C2×#13083553C2313yes—
88C2×#13083553C2313yes—
89C2×#13083551C2311yescharacteristic
90C2×#13083551C2311yescharacteristic
91C2×#13083551C2313yes—
92C2×#13083551C2313yes—
93C2×#13083551C2313yes—
94C2×#13083551C2311yescharacteristic
95C2×C4×Dic89C2313yes—
96C2×C4×Dic89C2313yes—
97C2×C4×Dic89C2316yes—
98C2×C4×Dic89C2316yes—
99C2×C4×Dic89C2313yes—
100#15667919C2311yescharacteristic
101#15667919C2313yes—
102#15667919C2313yes—
103#15667919C2313yes—
104#15667919C2313yes—
105#15667919C2313yes—
106#15667919C2316yes—
107#15667919C2316yes—
108#15667919C2313yes—
109#15667919C2313yes—
110#15667919C2313yes—
111#15667919C2311yescharacteristic
112#15667919C2316yes—
113#15667919C2313yes—
114#15667919C2313yes—
115C22×Q8×C89C2311yescharacteristic
116C4×Q8×C89—24no—
117C4×Q8×C89—24no—
118C4×Q8×C89C2×C418yes—
119C2×C4⋊C4×C89C2313yes—
120C2×C4⋊C4×C89C2313yes—
121C2×C4⋊C4×C89C2313yes—
122C2×C42×C89C2311yescharacteristic
123C2×C42×C89C2313yes—
124C2×C42×C89C2313yes—
125C2×C42×C89C2313yes—
126C2×C42×C89C2313yes—
127C2×C42×C89C2313yes—
128C2×C42×C89C2313yes—
129#95×C89C2316yes—
130C23×D89C2411yescharacteristic
131C2×Dic178—26no—
132C2×Dic178—26no—
133C2×Dic178C22×C4112yes—
134Q8×D89—232no—
135C22×Dic89C2411yescharacteristic
136C22×Dic89C2411yescharacteristic
137C22×Dic89C2413yes—
138C22×Dic89C2413yes—
139C22×Dic89C2413yes—
140C22×Dic89C2411yescharacteristic
141C22×Dic89C2413yes—
142C2×C4×D89—26no—
143C2×C4×D89—26no—
144C2×C4×D89C22×C4112yes—
145C4⋊Dic89—26no—
146C4⋊Dic89—26no—
147C4⋊Dic89C22×C4112yes—
148#13083553—22no—
149#13083553—26no—
150#13083553—43no—
151#13083553—46no—
152#13083553—43no—
153#13083553C22×C4112yes—
154#13083553C22×C4112yes—
155#13083551—22no—
156#13083551—22no—
157#13083551—43no—
158#13083551—43no—
159#13083551C22×C414yes—
160#13083551C22×C4112yes—
161C4×Dic89—26no—
162C4×Dic89—26no—
163C4×Dic89—46no—
164C4×Dic89C22×C4112yes—
165C4×Dic89C22×C4124yes—
166C2×Q8×C89C22×C414yes—
167C2×Q8×C89#4114yes—
168C2×Q8×C89#4114yes—
169C22×C4×C89C2413yes—
170C22×C4×C89C2413yes—
171C22×C4×C89C2413yes—
172C22×C4×C89C2411yescharacteristic
173C22×C4×C89C2411yescharacteristic
174C22×C4×C89C2411yescharacteristic
175C22×C4×C89C2413yes—
176C4⋊C4×C89—26no—
177C4⋊C4×C89—26no—
178C4⋊C4×C89C22×C4112yes—
179C42×C89—26no—
180C42×C89—26no—
181C42×C89—22no—
182C42×C89—26no—
183C42×C89—43no—
184C42×C89—43no—
185C42×C89C22×C4112yes—
186C23×C89C2511yescharacteristic
187C22×D89—21no—
188C22×D89—21no—
189C22×D89—21no—
190C22×D89—21no—
191C22×D89—21no—
192C22×D89—21no—
193C22×D89C23×C412yes—
194Q8×C89#117116yes—
195C2×C4×C89—21no—
196C2×C4×C89—21no—
197C2×C4×C89—23no—
198C2×C4×C89—21no—
199C2×C4×C89—21no—
200C2×C4×C89—21no—
201C2×C4×C89—21no—
202C2×C4×C89—23no—
203C2×C4×C89—23no—
204C2×C4×C89—23no—
205C2×C4×C89—23no—
206C2×C4×C89—23no—
207C2×C4×C89—23no—
208C2×C4×C89—23no—
209C2×C4×C89C23×C412yes—
210C2×C4×C89C23×C416yes—
211C2×C4×C89C23×C416yes—
212C2×C4×C89C2×#4116yes—
213C2×C4×C89C2×#4116yes—
214C2×C4×C892−1+412yes—
215C2×C4×C892−1+412yes—
216C2×Dic89—23no—
217C2×Dic89—23no—
218C2×Dic89—21no—
219C2×Dic89—23no—
220C2×Dic89—21no—
221C2×Dic89—23no—
222C2×Dic89—21no—
223C2×Dic89—21no—
224C2×Dic89—23no—
225C2×Dic89—23no—
226C2×Dic89—21no—
227C2×Dic89—21no—
228C2×Dic89—21no—
229C2×Dic89—23no—
230C2×Dic89—23no—
231C2×Dic89—23no—
232C2×Dic89—23no—
233C2×Dic89—21no—
234C2×Dic89C23×C412yes—
235C2×Dic89C23×C416yes—
236C2×Dic89C23×C416yes—
237C4×D89—224no—
238Dic178—224no—
239C22×C89C24×C411yescharacteristic
240C22×C89C22×#4111yescharacteristic
241C22×C89C22×#4111yescharacteristic
242C22×C89C2×2−1+411yescharacteristic
243C22×C89C2×2−1+411yescharacteristic
244C22×C89#58511yescharacteristic
245C22×C89#58511yescharacteristic
246C2×D89—22no—
247C2×D89—22no—
248C2×D89—22no—
249C2×D89—41no—
250C2×D89—41no—
251C2×D89—41no—
252C2×D89—41no—
253C4×C89—26no—
254C4×C89—22no—
255C4×C89—22no—
256C4×C89—22no—
257C4×C89—43no—
258C4×C89—43no—
259C4×C89C2×#117112yes—
260Dic89—26no—
261Dic89—22no—
262Dic89—26no—
263Dic89—43no—
264Dic89—41no—
265Dic89—41no—
266Dic89—43no—
267#34262—891nomaximal, Sylow
268C2×C89C22×#11711yescharacteristic
269C2×C89C4×2−1+411yescharacteristic
270C2×C89C4×2−1+411yescharacteristic
271C2×C89#351611yescharacteristic
272C2×C89#351811yescharacteristic
273C2×C89#352311yescharacteristic
274C2×C89#352411yescharacteristic
275D89—42no—
276C2×#519—891no—
277#2356—893no—
278#2355—893no—
279#2351—891no—
280Q8×C4⋊C4—891no—
281#2347—893no—
282#2346—893no—
283Q8×C22⋊C4—891no—
284Q8×C22⋊C4—891no—
285#2320—893no—
286#2314—893no—
287#2298—893no—
288#2295—893no—
289#2276—891no—
290C42×Q8D8911yescharacteristic, complemented
291C89#3426211yescharacteristic, complemented, Sylow
292C23×Q8—891no—
293#519—8916no—
294C2×C4×Q8—891no—
295C2×C4×Q8—891no—
296C2×C4×Q8—891no—
297C2×C4×Q8—893no—
298C2×C4×Q8—893no—
299C2×C4×Q8C2×D8911yescharacteristic
300C2×C4×Q8C2×D8911yescharacteristic
301C2×C4×Q8C2×D8911yescharacteristic
302C2×#117—893no—
303C2×#117—893no—
304#391—893no—
305#391—893no—
306#391—893no—
307#391—893no—
308#389—893no—
309#389—893no—
310#389—893no—
311#389—893no—
312#388—893no—
313#388—893no—
314#388—893no—
315#388—891no—
316#388—896no—
317#387—896no—
318#387—893no—
319#387—893no—
320#387—893no—
321#387—893no—
322#387—896no—
323#384—893no—
324#383—896no—
325#382—893no—
326#382—896no—
327#380—893no—
328#380—893no—
329C4×C4⋊C4—893no—
330C4×C4⋊C4—893no—
331C4×C4⋊C4—893no—
332C4×C4⋊C4—896no—
333C4×C4⋊C4—893no—
334C4×C4⋊C4—893no—
335C4×C4⋊C4C2×D8913yes—
336C4×C4⋊C4C2×D8913yes—
337C4×C4⋊C4C2×D8913yes—
338C4×C22⋊C4—893no—
339C4×C22⋊C4—893no—
340#376—893no—
341#376—896no—
342C43C2×D8913yes—
343C22×Q8—898no—
344C22×Q8—893no—
345C22×Q8—1784no—
346C22×Q8—1784no—
347C22×Q8C22×D8911yescharacteristic
348C23×C4—893no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—898no—
352C4×Q8—8924no—
353C4×Q8—8924no—
354C4×Q8—1784no—
355C4×Q8—1784no—
356C4×Q8C4×D8918yes—
357#117—8924no—
358#117—8924no—
359C2×C4⋊C4—893no—
360C2×C4⋊C4—893no—
361C2×C4⋊C4—893no—
362C2×C4⋊C4—893no—
363C2×C4⋊C4—896no—
364C2×C4⋊C4—893no—
365C2×C4⋊C4—893no—
366C2×C4⋊C4—893no—
367C2×C4⋊C4—891no—
368C2×C4⋊C4—893no—
369C2×C4⋊C4C22×D8913yes—
370C2×C4⋊C4C22×D8913yes—
371C2×C4⋊C4C22×D8913yes—
372C2×C22⋊C4—891no—
373C2×C22⋊C4—893no—
374C2×C22⋊C4—893no—
375C2×C22⋊C4—893no—
376C2×C22⋊C4—891no—
377C2×C22⋊C4—891no—
378C2×C42—896no—
379C2×C42—893no—
380C2×C42—896no—
381C2×C42—893no—
382C2×C42—893no—
383C2×C42C22×D8913yes—
384C2×C42C22×D8913yes—
385C2×C42C22×D8911yescharacteristic
386C2×C42C22×D8913yes—
387C2×C42C22×D8913yes—
388C2×C42C22×D8913yes—
389C2×C42C22×D8913yes—
390#95—891no—
391#95—893no—
392#95—893no—
393#95—893no—
394#95—893no—
395#95—893no—
396#95—896no—
397#95—896no—
398#95—893no—
399#95—893no—
400#95—893no—
401#95—891no—
402#95—896no—
403#95—893no—
404#95—893no—
405#95C22×D8916yes—
406C24—891no—
407C2×Q8—8912no—
408C2×Q8—17832no—
409C2×Q8—1786no—
410C2×Q8—1786no—
411C2×Q8C2×C4×D8914yes—
412C2×Q8#1308357414yes—
413C2×Q8#1308357414yes—
414C22×C4—893no—
415C22×C4—891no—
416C22×C4—893no—
417C22×C4—8912no—
418C22×C4—891no—
419C22×C4—891no—
420C22×C4—893no—
421C22×C4—893no—
422C22×C4—1786no—
423C22×C4—1786no—
424C22×C4C23×D8913yes—
425C22×C4C23×D8913yes—
426C22×C4C23×D8911yescharacteristic
427C22×C4C23×D8913yes—
428C22×C4C23×D8911yescharacteristic
429C22×C4C23×D8911yescharacteristic
430C22×C4C23×D8913yes—
431C4⋊C4—26no—
432C4⋊C4—26no—
433C4⋊C4—8912no—
434C4⋊C4—8912no—
435C4⋊C4—8912no—
436C4⋊C4—3563no—
437C4⋊C4—3563no—
438C4⋊C4—3566no—
439C4⋊C4—1786no—
440C4⋊C4—1786no—
441C4⋊C4—1782no—
442C4⋊C4—1786no—
443C4⋊C4C2×C4×D89112yes—
444C22⋊C4—8912no—
445C22⋊C4—894no—
446C22⋊C4—3563no—
447C22⋊C4—3563no—
448C22⋊C4—1782no—
449C22⋊C4—1782no—
450C42—26no—
451C42—26no—
452C42—26no—
453C42—22no—
454C42—43no—
455C42—43no—
456C42—8912no—
457C42—8924no—
458C42—3566no—
459C42—1786no—
460C42—1786no—
461C42C2×C4×D89112yes—
462C23—892no—
463C23—1781no—
464C23—1781no—
465C23—1781no—
466C23—1781no—
467C23—1781no—
468C23—1781no—
469C23C24×D8911yescharacteristic
470Q8—17824no—
471Q8#15667969116yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—23no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—23no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—892no—
487C2×C4—896no—
488C2×C4—896no—
489C2×C4—1781no—
490C2×C4—1781no—
491C2×C4—1783no—
492C2×C4—1783no—
493C2×C4—1781no—
494C2×C4—1781no—
495C2×C4—1781no—
496C2×C4—1783no—
497C2×C4—1783no—
498C2×C4—1783no—
499C2×C4—1783no—
500C2×C4—1781no—
501C2×C4—1783no—
502C2×C4—1783no—
503C2×C4—1781no—
504C2×C4—1783no—
505C2×C4—1781no—
506C2×C4—1783no—
507C2×C4—17824no—
508C2×C4C22×C4×D8916yes—
509C2×C4C22×C4×D8912yes—
510C2×C4C22×C4×D8916yes—
511C2×C4C2×#1308357416yes—
512C2×C4C2×#1308357416yes—
513C2×C4#1566805912yes—
514C2×C4#1566806012yes—
515C22—3561no—
516C22—3561no—
517C22—3561no—
518C22—3561no—
519C22—1782no—
520C22—1782no—
521C22—1782no—
522C22C23×C4×D8911yescharacteristic
523C22C22×#1308357411yescharacteristic
524C22C22×#1308357411yescharacteristic
525C22C2×#1566805911yescharacteristic
526C22C2×#1566806011yescharacteristic
527C22#1915243411yescharacteristic
528C22#1915243411yescharacteristic
529C4—26no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—43no—
534C4—43no—
535C4—3563no—
536C4—3561no—
537C4—3561no—
538C4—3563no—
539C4—1786no—
540C4—1782no—
541C4—1786no—
542C4C2×#15667969112yes—
543C2—3562no—
544C2C22×#1566796911yescharacteristic
545C2C4×#1566805911yescharacteristic
546C2C4×#1566806011yescharacteristic
547C2#2603424511yescharacteristic
548C2#2603426911yescharacteristic
549C2#2603434811yescharacteristic
550C2#2603436111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC42×Q8×C89C2
socle, cosocleC23×C89C25
prime radicalC23×D89C24
hypercenterC42×Q8D89
hypocenterC89#34262
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
2. #26023213C2semidirect
3. #26023205C2semidirect
4. #26023170C2semidirect
5. Q8×#13083553C2semidirect
10. #26022898C2semidirect
13. #26022729C2semidirect
14. #26022617C2semidirect
15. C42×Q8×C89C2semidirect
17. #19152074C4semidirect
290. C42×Q8D89semidirect
291. C89#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#191520742
#260232132
#260232052
#260231702
Q8×#130835532
#260231412
#260231332
Q8×#130835512
#260228982
#260228572
#260227492
#260227292
#260226172
C42×Q8×C892
#3426289