Finite Group Databasebeta

← #36715365

Subgroups — #36715365

Order 22784.

2858 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152074C211yesmaximal, characteristic
2C2×#19152074C211yesmaximal, characteristic
3C2×C4×Q8×D89C211yesmaximal, characteristic
4#26023173C211yesmaximal, characteristic, complemented
5#26023172C213yesmaximal, complemented
6#26023171C213yesmaximal, complemented
7#26023170C211yesmaximal, characteristic, complemented
8#26023166C213yesmaximal, complemented
9Q8×#13083553C211yesmaximal, characteristic, complemented
10#26023164C213yesmaximal, complemented
11#26022874C213yesmaximal
12#26022873C213yesmaximal
13#26022733C213yesmaximal
14#26022732C213yesmaximal
15Q8×C4⋊C4×C89C211yesmaximal, characteristic, complemented
16C22×Q8×D89C2211yescharacteristic
17#19152074—28no—
18#19152074C4116yescomplemented
19C2×Q8×Dic89C2211yescharacteristic
20C2×Q8×Dic89C2211yescharacteristic
21C2×Q8×Dic89C2211yescharacteristic
22C2×C4×Dic178C2213yes—
23C2×C4×Dic178C2213yes—
24C2×C4×Dic178C2213yes—
25C2×#15667977C2213yes—
26C2×#15667977C2213yes—
27C2×#15667977C2213yes—
28C4×Q8×D89—28no—
29C2×#15667970C2213yes—
30C2×#15667970C2213yes—
31C2×#15667969C2213yes—
32C2×#15667969C2213yes—
33C2×C4⋊C4×D89C2213yes—
34C2×C42×D89C2213yes—
35#19151497C2213yes—
36#19151497C2213yes—
37#19151496C2213yes—
38#19151496C2213yes—
39#19151495C2213yes—
40#19151495C2213yes—
41#19151494C2213yes—
42#19151494C2213yes—
43#19151493C2216yes—
44#19151493C2213yes—
45#19151493C2216yes—
46#19151493C2213yes—
47#19151472C2213yes—
48#19151472C2213yes—
49#19151469C2211yescharacteristic
50#19151469C2213yes—
51#19151469C2216yes—
52#19151467C2213yes—
53#19151467C2216yes—
54#19151464C2213yes—
55#19151463C2213yes—
56#19151462C2216yes—
57C4⋊C4×Dic89C2216yes—
58C4×#13083553C2213yes—
59C4×#13083553C2213yes—
60C4×C4⋊Dic89C2213yes—
61#19151449C2216yes—
62C2×C4×Q8×C89C2211yescharacteristic
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2211yescharacteristic
65#389×C89C2213yes—
66#389×C89C2213yes—
67#384×C89C2213yes—
68C4×C4⋊C4×C89C2213yes—
69C22×Dic178C2313yes—
70C2×Q8×D89C2×C418yes—
71C2×Q8×D89D418yes—
72C2×Q8×D89Q818yes—
73C22×C4×D89C2311yescharacteristic
74C22×C4×D89C2313yes—
75C22×C4×D89C2313yes—
76Q8×Dic89—24no—
77Q8×Dic89—24no—
78Q8×Dic89C2×C418yescomplemented
79C4×Dic178—212no—
80C4×Dic178—212no—
81C4×Dic178C2×C4124yescomplemented
82#15667977—212no—
83#15667977—212no—
84#15667977C2×C4124yescomplemented
85#15667970—212no—
86#15667970C2×C4124yes—
87#15667969—212no—
88#15667969C2×C4124yes—
89C2×C4⋊Dic89C2313yes—
90C2×C4⋊Dic89C2313yes—
91C2×C4⋊Dic89C2313yes—
92C2×C4⋊Dic89C2313yes—
93C2×C4⋊Dic89C2313yes—
94C2×#13083553C2313yes—
95C2×#13083553C2311yescharacteristic
96C2×#13083553C2313yes—
97C2×#13083553C2316yes—
98C2×#13083553C2313yes—
99C2×#13083553C2316yes—
100C2×#13083553C2313yes—
101C2×#13083553C2313yes—
102C2×#13083553C2313yes—
103C2×#13083553C2313yes—
104C4⋊C4×D89—212no—
105C2×#13083551C2313yes—
106C2×#13083551C2311yescharacteristic
107C2×#13083551C2311yescharacteristic
108C2×#13083551C2313yes—
109C2×C4×Dic89C2313yes—
110C2×C4×Dic89C2316yes—
111C2×C4×Dic89C2313yes—
112C2×C4×Dic89C2313yes—
113C2×C4×Dic89C2313yes—
114C2×C4×Dic89C2316yes—
115C2×C4×Dic89C2313yes—
116C42×D89—212no—
117#15667919C2316yes—
118#15667919C2311yescharacteristic
119#15667919C2313yes—
120#15667919C2311yescharacteristic
121#15667919C2313yes—
122#15667919C2313yes—
123#15667919C2313yes—
124#15667919C2316yes—
125C22×Q8×C89C2311yescharacteristic
126C4×Q8×C89—24no—
127C4×Q8×C89—24no—
128C4×Q8×C89C2×C418yescomplemented
129C2×C4⋊C4×C89C2311yescharacteristic
130C2×C4⋊C4×C89C2313yes—
131C2×C4⋊C4×C89C2313yes—
132C2×C4⋊C4×C89C2313yes—
133C2×C4⋊C4×C89C2316yes—
134C2×C4⋊C4×C89C2313yes—
135C2×C42×C89C2313yes—
136C2×C42×C89C2313yes—
137C2×C42×C89C2313yes—
138#95×C89C2313yes—
139#95×C89C2313yes—
140C23×D89C2411yescharacteristic
141C2×Dic178C22×C4112yes—
142C2×Dic178C2×D4112yes—
143C2×Dic178C2×Q8112yes—
144Q8×D89C4⋊C4164yescomplemented
145C22×Dic89C2413yes—
146C22×Dic89C2413yes—
147C22×Dic89C2411yescharacteristic
148C22×Dic89C2411yescharacteristic
149C22×Dic89C2413yes—
150C22×Dic89C2411yescharacteristic
151C22×Dic89C2413yes—
152C2×C4×D89—22no—
153C2×C4×D89—22no—
154C2×C4×D89—22no—
155C2×C4×D89—26no—
156C2×C4×D89—43no—
157C2×C4×D89—43no—
158C2×C4×D89C22×C4112yes—
159C2×C4×D89C2×D4112yes—
160C2×C4×D89C2×Q8112yes—
161C4⋊Dic89—26no—
162C4⋊Dic89—26no—
163C4⋊Dic89—26no—
164C4⋊Dic89—43no—
165C4⋊Dic89C22×C4112yes—
166#13083553—26no—
167#13083553—26no—
168#13083553—26no—
169#13083553—212no—
170#13083553—22no—
171#13083553—43no—
172#13083553—46no—
173#13083553—43no—
174#13083553C22×C4112yes—
175#13083553C22×C4112yes—
176#13083551—22no—
177#13083551—26no—
178#13083551C22×C414yes—
179#13083551C22×C4112yes—
180C4×Dic89—26no—
181C4×Dic89—26no—
182C4×Dic89—212no—
183C4×Dic89—26no—
184C4×Dic89—26no—
185C4×Dic89C22×C4112yes—
186C4×Dic89C22×C4124yes—
187C2×Q8×C89C22×C414yes—
188C2×Q8×C89C2×D414yes—
189C2×Q8×C89C2×Q814yes—
190C22×C4×C89C2413yes—
191C22×C4×C89C2411yescharacteristic
192C22×C4×C89C2413yes—
193C22×C4×C89C2413yes—
194C22×C4×C89C2411yescharacteristic
195C22×C4×C89C2411yescharacteristic
196C22×C4×C89C2413yes—
197C4⋊C4×C89—26no—
198C4⋊C4×C89—26no—
199C4⋊C4×C89—22no—
200C4⋊C4×C89—43no—
201C4⋊C4×C89—46no—
202C4⋊C4×C89C22×C4112yes—
203C42×C89—26no—
204C42×C89—26no—
205C42×C89C22×C4112yes—
206C23×C89C2511yescharacteristic
207C22×D89—21no—
208C22×D89—21no—
209C22×D89—21no—
210C22×D89—21no—
211C22×D89C23×C412yes—
212C22×D89C22×D412yes—
213C22×D89C22×Q812yes—
214Q8×C89C2×C4⋊C4116yes—
215C2×C4×C89—21no—
216C2×C4×C89—21no—
217C2×C4×C89—23no—
218C2×C4×C89—21no—
219C2×C4×C89—21no—
220C2×C4×C89—23no—
221C2×C4×C89—21no—
222C2×C4×C89—23no—
223C2×C4×C89—23no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—23no—
227C2×C4×C89—23no—
228C2×C4×C89—23no—
229C2×C4×C89—23no—
230C2×C4×C89C23×C412yes—
231C2×C4×C89C23×C416yes—
232C2×C4×C89C23×C416yes—
233C2×C4×C89C22×D416yes—
234C2×C4×C89C22×Q816yes—
235C2×C4×C892−1+412yes—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—23no—
239C2×Dic89—21no—
240C2×Dic89—21no—
241C2×Dic89—23no—
242C2×Dic89—21no—
243C2×Dic89—23no—
244C2×Dic89—23no—
245C2×Dic89—23no—
246C2×Dic89—23no—
247C2×Dic89—21no—
248C2×Dic89—23no—
249C2×Dic89—23no—
250C2×Dic89—21no—
251C2×Dic89C23×C412yes—
252C2×Dic89C23×C416yes—
253C2×Dic89C23×C416yes—
254C2×Dic89C22×D416yes—
255C2×Dic89C22×Q816yes—
256C2×Dic892−1+412yes—
257C4×D89—28no—
258C4×D89—412no—
259C4×D89C2×C4⋊C4148yes—
260Dic178C2×C4⋊C4148yes—
261C22×C89C24×C411yescharacteristic
262C22×C89C23×D411yescharacteristic
263C22×C89C23×Q811yescharacteristic
264C22×C89C2×2−1+411yescharacteristic
265C22×C89C2×2−1+411yescharacteristic
266C22×C89#58511yescharacteristic
267C22×C89#58511yescharacteristic
268C2×D89—22no—
269C2×D89—22no—
270C2×D89—22no—
271C2×D89—22no—
272C2×D89—22no—
273C2×D89—22no—
274C2×D89C22×C4⋊C414yes—
275C4×C89—26no—
276C4×C89—22no—
277C4×C89—22no—
278C4×C89—41no—
279C4×C89—43no—
280C4×C89—43no—
281C4×C89C22×C4⋊C4112yes—
282Dic89—22no—
283Dic89—26no—
284Dic89—22no—
285Dic89—43no—
286Dic89—43no—
287Dic89—41no—
288Dic89C22×C4⋊C4112yes—
289#34249—891nomaximal, Sylow
290C2×C89C23×C4⋊C411yescharacteristic
291C2×C89C4×2−1+411yescharacteristic
292C2×C89C4×2−1+411yescharacteristic
293C2×C89#348611yescharacteristic
294C2×C89#348711yescharacteristic
295C2×C89#350211yescharacteristic
296C2×C89#350211yescharacteristic
297D89—24no—
298C2×#519—891no—
299C2×#519—891no—
300C22×C4×Q8—891no—
301#2351—893no—
302#2351—893no—
303#2351—891no—
304#2351—891no—
305Q8×C4⋊C4—893no—
306Q8×C4⋊C4—893no—
307Q8×C4⋊C4—891no—
308Q8×C4⋊C4D8911yescharacteristic, complemented
309#2317—893no—
310#2317—893no—
311#2296—893no—
312#2296—893no—
313C89#3424911yescharacteristic, complemented, Sylow
314C23×Q8—891no—
315#519—8916no—
316#519—1788no—
317C2×C4×Q8—891no—
318C2×C4×Q8—891no—
319C2×C4×Q8—891no—
320C2×C4×Q8—893no—
321C2×C4×Q8—893no—
322C2×C4×Q8—893no—
323C2×C4×Q8—893no—
324C2×C4×Q8—893no—
325C2×C4×Q8—893no—
326C2×C4×Q8—1788no—
327C2×C4×Q8C2×D8911yescharacteristic
328C2×C4×Q8C2×D8911yescharacteristic
329C2×C4×Q8C2×D8911yescharacteristic
330C2×#117—893no—
331C2×#117—893no—
332C2×#117—893no—
333C2×#117—893no—
334C22×C4⋊C4—893no—
335C22×C42—893no—
336#389—896no—
337#389—896no—
338#389—893no—
339#389—893no—
340#389—893no—
341#389—893no—
342#389—893no—
343#389—893no—
344#389—893no—
345#389—893no—
346#389—893no—
347#389—893no—
348#389C2×D8913yes—
349#389C2×D8913yes—
350#385—896no—
351#385—893no—
352#385—891no—
353#385—893no—
354#385—893no—
355#384—893no—
356#384—896no—
357#384C2×D8913yes—
358#382—893no—
359#382—893no—
360#382—896no—
361C4×C4⋊C4—896no—
362C4×C4⋊C4—893no—
363C4×C4⋊C4—893no—
364C4×C4⋊C4—893no—
365C4×C4⋊C4—896no—
366C4×C4⋊C4C2×D8913yes—
367C22×Q8—898no—
368C22×Q8—898no—
369C22×Q8—898no—
370C22×Q8—893no—
371C22×Q8C22×D8911yescharacteristic
372C23×C4—893no—
373C23×C4—893no—
374C23×C4—891no—
375C4×Q8—24no—
376C4×Q8—24no—
377C4×Q8—898no—
378C4×Q8—8924no—
379C4×Q8—8924no—
380C4×Q8—17812no—
381C4×Q8—17812no—
382C4×Q8—17812no—
383C4×Q8—17812no—
384C4×Q8—1784no—
385C4×Q8—1784no—
386C4×Q8C4×D8918yescomplemented
387#117—8924no—
388#117—8924no—
389#117—17812no—
390#117—17812no—
391C2×C4⋊C4—893no—
392C2×C4⋊C4—893no—
393C2×C4⋊C4—893no—
394C2×C4⋊C4—893no—
395C2×C4⋊C4—896no—
396C2×C4⋊C4—893no—
397C2×C4⋊C4—893no—
398C2×C4⋊C4—896no—
399C2×C4⋊C4—893no—
400C2×C4⋊C4—893no—
401C2×C4⋊C4—893no—
402C2×C4⋊C4—893no—
403C2×C4⋊C4—891no—
404C2×C4⋊C4—893no—
405C2×C4⋊C4—893no—
406C2×C4⋊C4—17812no—
407C2×C4⋊C4C22×D8913yes—
408C2×C4⋊C4C22×D8913yes—
409C2×C4⋊C4C22×D8911yescharacteristic
410C2×C4⋊C4C22×D8916yes—
411C2×C4⋊C4C22×D8913yes—
412C2×C4⋊C4C22×D8913yes—
413C2×C22⋊C4—891no—
414C2×C22⋊C4—893no—
415C2×C22⋊C4—893no—
416C2×C22⋊C4—891no—
417C2×C42—896no—
418C2×C42—893no—
419C2×C42—896no—
420C2×C42—893no—
421C2×C42—893no—
422C2×C42—893no—
423C2×C42—893no—
424C2×C42—17812no—
425C2×C42C22×D8913yes—
426C2×C42C22×D8913yes—
427C2×C42C22×D8913yes—
428#95—893no—
429#95—891no—
430#95—891no—
431#95—896no—
432#95—893no—
433#95—893no—
434#95—893no—
435#95—896no—
436#95C22×D8913yes—
437#95C22×D8913yes—
438C24—891no—
439C2×Q8—8964no—
440C2×Q8—8912no—
441C2×Q8—8912no—
442C2×Q8—8912no—
443C2×Q8C2×C4×D8914yes—
444C2×Q8D4×D8914yes—
445C2×Q8Q8×D8914yes—
446C22×C4—893no—
447C22×C4—8912no—
448C22×C4—8912no—
449C22×C4—8912no—
450C22×C4—891no—
451C22×C4—891no—
452C22×C4—893no—
453C22×C4—893no—
454C22×C4—893no—
455C22×C4—891no—
456C22×C4—3563no—
457C22×C4—3563no—
458C22×C4—1782no—
459C22×C4—1786no—
460C22×C4—1782no—
461C22×C4—1782no—
462C22×C4C23×D8911yescharacteristic
463C22×C4C23×D8913yes—
464C22×C4C23×D8913yes—
465C22×C4C23×D8911yescharacteristic
466C22×C4C23×D8913yes—
467C22×C4C23×D8911yescharacteristic
468C22×C4C23×D8913yes—
469C4⋊C4—26no—
470C4⋊C4—26no—
471C4⋊C4—22no—
472C4⋊C4—46no—
473C4⋊C4—43no—
474C4⋊C4—8912no—
475C4⋊C4—8912no—
476C4⋊C4—8912no—
477C4⋊C4—3563no—
478C4⋊C4—3566no—
479C4⋊C4—3563no—
480C4⋊C4—3563no—
481C4⋊C4—17812no—
482C4⋊C4—1786no—
483C4⋊C4—1786no—
484C4⋊C4—1782no—
485C4⋊C4—1786no—
486C4⋊C4—1786no—
487C4⋊C4—1786no—
488C4⋊C4—1786no—
489C4⋊C4C2×C4×D89112yes—
490C22⋊C4—8912no—
491C22⋊C4—894no—
492C22⋊C4—1782no—
493C22⋊C4—1786no—
494C42—26no—
495C42—26no—
496C42—8924no—
497C42—8912no—
498C42—1786no—
499C42—1786no—
500C42—17812no—
501C42—1786no—
502C42—1786no—
503C42C2×C4×D89112yes—
504C23—892no—
505C23—892no—
506C23—892no—
507C23—1781no—
508C23—1781no—
509C23—1781no—
510C23—1781no—
511C23C24×D8911yescharacteristic
512Q8—8948no—
513Q8C4⋊C4×D89116yes—
514C2×C4—21no—
515C2×C4—23no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—23no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—23no—
528C2×C4—23no—
529C2×C4—896no—
530C2×C4—896no—
531C2×C4—896no—
532C2×C4—8948no—
533C2×C4—892no—
534C2×C4—896no—
535C2×C4—892no—
536C2×C4—35612no—
537C2×C4—1788no—
538C2×C4—1783no—
539C2×C4—1781no—
540C2×C4—1783no—
541C2×C4—1783no—
542C2×C4—1783no—
543C2×C4—1781no—
544C2×C4—1783no—
545C2×C4—1781no—
546C2×C4—1783no—
547C2×C4—1781no—
548C2×C4—1781no—
549C2×C4—1783no—
550C2×C4—1781no—
551C2×C4—1781no—
552C2×C4—1783no—
553C2×C4C22×C4×D8916yes—
554C2×C4C22×C4×D8912yes—
555C2×C4C22×C4×D8916yes—
556C2×C4C2×D4×D8916yes—
557C2×C4C2×Q8×D8916yes—
558C2×C4#1566806012yes—
559C22—894no—
560C22—1782no—
561C22—1782no—
562C22—1782no—
563C22—1782no—
564C22—1782no—
565C22—1782no—
566C22C23×C4×D8911yescharacteristic
567C22C22×D4×D8911yescharacteristic
568C22C22×Q8×D8911yescharacteristic
569C222−1+4×D8911yescharacteristic
570C22C2×#1566806011yescharacteristic
571C22#1915243311yescharacteristic
572C22#1915243411yescharacteristic
573C4—26no—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—43no—
578C4—43no—
579C4—8912no—
580C4—3563no—
581C4—3563no—
582C4—3561no—
583C4—1782no—
584C4—1786no—
585C4—1782no—
586C4C2×C4⋊C4×D89112yes—
587C2—1784no—
588C2C22×C4⋊C4×D8911yescharacteristic
589C2#2603346311yescharacteristic
590C2C4×#1566806011yescharacteristic
591C2#2603390211yescharacteristic
592C2#2603390511yescharacteristic
593C2#2603405911yescharacteristic
594C2#2603406111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingQ8×C4⋊C4×C89C2
socle, cosocleC23×C89C25
prime radicalC23×D89C24
hypercenterQ8×C4⋊C4D89
hypocenterC89#34249
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #26023173C2semidirect
5. #26023172C2semidirect
6. #26023171C2semidirect
7. #26023170C2semidirect
8. #26023166C2semidirect
9. Q8×#13083553C2semidirect
10. #26023164C2semidirect
15. Q8×C4⋊C4×C89C2semidirect
18. #19152074C4semidirect
78. Q8×Dic89C2×C4semidirect
81. C4×Dic178C2×C4semidirect
84. #15667977C2×C4semidirect
128. C4×Q8×C89C2×C4semidirect
144. Q8×D89C4⋊C4semidirect
308. Q8×C4⋊C4D89semidirect
313. C89#34249semidirect
386. C4×Q8C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191520742
C2×C4×Q8×D892
#260231732
#260231722
#260231712
#260231702
#260231662
Q8×#130835532
#260231642
#260228742
#260228732
#260227332
#260227322
Q8×C4⋊C4×C892
#3424989