Finite Group Databasebeta

← #36716045

Subgroups — #36716045

Order 22784.

2912 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
1C2×Q8×D356C211yesmaximal, characteristic
2C2×#19152074C211yesmaximal, characteristic, complemented
3C2×C4×Q8×D89C211yesmaximal, characteristic, complemented
4#26023308C213yesmaximal, complemented
5#26023278C213yesmaximal, complemented
6#26023256C213yesmaximal, complemented
7#26023178C213yesmaximal
8Q8×#13083551C211yesmaximal, characteristic, complemented
9Q8×#13083551C211yesmaximal, characteristic, complemented
10#26022969C213yesmaximal, complemented
11#26022932C213yesmaximal, complemented
12#26022782C213yesmaximal, complemented
13#26022693C213yesmaximal
14#26022617C211yesmaximal, characteristic, complemented
15C89×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D89C2211yescharacteristic
17C22×Q8×D89C2211yescharacteristic
18Q8×D356C4116yescomplemented
19C2×#15668012C2213yes—
20C2×#15667992C2213yes—
21C2×#15667992C2213yes—
22#19152074—28no—
23C2×Q8×Dic89C2211yescharacteristic
24C2×Q8×Dic89C2211yescharacteristic
25C2×C4×Dic178C2213yes—
26C2×C4×Dic178C2213yes—
27C2×#15667977C2213yes—
28C2×#15667977C2213yes—
29C4×Q8×D89—28no—
30C2×C4×D356C2213yes—
31C2×#15667970C2213yes—
32C2×#15667969C2213yes—
33C2×C4⋊C4×D89C2213yes—
34C2×C42×D89C2213yes—
35#19151504C2213yes—
36#19151504C2213yes—
37#19151503C2213yes—
38#19151503C2213yes—
39#19151494C2216yes—
40#19151488C2213yes—
41#19151488C2213yes—
42#19151487C2216yes—
43#19151481C2216yes—
44#19151481C2216yes—
45#19151481C2213yes—
46#19151481C2213yes—
47#19151481C2216yes—
48#19151476C2213yes—
49#19151476C2211yescharacteristic
50#19151469C2213yes—
51#19151469C2213yes—
52#19151455C2213yes—
53#19151455C2213yes—
54C4⋊C4×Dic89C2213yes—
55C4⋊C4×Dic89C2213yes—
56C4⋊C4×Dic89C2216yes—
57C4×C4⋊Dic89C2213yes—
58C4×#13083551C2213yes—
59C4×#13083551C2213yes—
60#19151445C2216yes—
61#19151444C2213yes—
62C2×C4×Q8×C89C2211yescharacteristic
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2211yescharacteristic
65#389×C89C2213yes—
66#389×C89C2213yes—
67#382×C89C2213yes—
68C4×C4⋊C4×C89C2213yes—
69C22×Dic178C2313yes—
70C22×Dic178C2313yes—
71C2×Q8×D89—24no—
72C2×Q8×D89—24no—
73C2×Q8×D89—24no—
74C2×Q8×D89—24no—
75C2×Q8×D89C2×C418yes—
76C2×Q8×D89C2×C418yescomplemented
77C22×D356C2311yescharacteristic
78C22×C4×D89C2311yescharacteristic
79C22×C4×D89C2313yes—
80C22×C4×D89C2313yes—
81C22×C4×D89C2313yes—
82#15668012C2×C4124yescomplemented
83#15667992C2×C4124yes—
84#15667992C2×C4124yescomplemented
85Q8×Dic89—24no—
86Q8×Dic89D418yescomplemented
87C4×Dic178—212no—
88C4×Dic178—212no—
89#15667977—212no—
90#15667977—212no—
91C4×D356C2×C4124yes—
92#15667970—212no—
93#15667969—212no—
94C2×C4⋊Dic89C2313yes—
95C2×C4⋊Dic89C2313yes—
96C2×C4⋊Dic89C2316yes—
97C2×C4⋊Dic89C2313yes—
98C2×#13083553C2313yes—
99C2×#13083553C2313yes—
100C2×#13083553C2316yes—
101C4⋊C4×D89—212no—
102C2×#13083551C2313yes—
103C2×#13083551C2311yescharacteristic
104C2×#13083551C2311yescharacteristic
105C2×#13083551C2313yes—
106C2×#13083551C2311yescharacteristic
107C2×#13083551C2313yes—
108C2×#13083551C2313yes—
109C2×#13083551C2313yes—
110C2×C4×Dic89C2313yes—
111C2×C4×Dic89C2316yes—
112C2×C4×Dic89C2313yes—
113C2×C4×Dic89C2313yes—
114C2×C4×Dic89C2316yes—
115C2×C4×Dic89C2313yes—
116C42×D89—212no—
117#15667919C2311yescharacteristic
118#15667919C2313yes—
119#15667919C2313yes—
120#15667919C2313yes—
121#15667919C2316yes—
122#15667919C2313yes—
123#15667919C2313yes—
124#15667919C2313yes—
125#15667919C2313yes—
126#15667919C2316yes—
127C22×Q8×C89C2311yescharacteristic
128C4×Q8×C89—24no—
129C4×Q8×C89—24no—
130C4×Q8×C89C2×C418yescomplemented
131C2×C4⋊C4×C89C2313yes—
132C2×C4⋊C4×C89C2313yes—
133C2×C4⋊C4×C89C2313yes—
134C2×C4⋊C4×C89C2313yes—
135C2×C4⋊C4×C89C2313yes—
136C2×C42×C89C2313yes—
137C2×C42×C89C2313yes—
138C2×C42×C89C2313yes—
139#95×C89C2311yescharacteristic
140#95×C89C2313yes—
141#95×C89C2316yes—
142C23×D89C2411yescharacteristic
143C23×D89C2411yescharacteristic
144C2×Dic178—26no—
145C2×Dic178—26no—
146C2×Dic178—26no—
147C2×Dic178—26no—
148C2×Dic178C22×C4112yes—
149C2×Dic178C22×C4112yes—
150Q8×D89—232no—
151Q8×D89—232no—
152C2×D356—22no—
153C2×D356—22no—
154C2×D356C22×C414yes—
155C22×Dic89C2411yescharacteristic
156C22×Dic89C2413yes—
157C22×Dic89C2413yes—
158C22×Dic89C2413yes—
159C22×Dic89C2411yescharacteristic
160C22×Dic89C2413yes—
161C2×C4×D89—22no—
162C2×C4×D89—26no—
163C2×C4×D89—26no—
164C2×C4×D89—26no—
165C2×C4×D89—26no—
166C2×C4×D89—22no—
167C2×C4×D89—22no—
168C2×C4×D89—26no—
169C2×C4×D89—26no—
170C2×C4×D89—43no—
171C2×C4×D89C22×C4112yes—
172C2×C4×D89C22×C4112yes—
173C4⋊Dic89—26no—
174C4⋊Dic89C22×C4112yes—
175C4⋊Dic89C22×C4124yes—
176C4⋊Dic89C2×D4112yes—
177#13083553—26no—
178#13083553—26no—
179#13083553—212no—
180#13083551—26no—
181#13083551—22no—
182#13083551—22no—
183#13083551—22no—
184#13083551—26no—
185#13083551—43no—
186#13083551C22×C4112yes—
187#13083551C22×C4112yes—
188C4×Dic89—26no—
189C4×Dic89—212no—
190C4×Dic89—26no—
191C4×Dic89—26no—
192C4×Dic89—46no—
193C4×Dic89C2×D4112yes—
194C2×Q8×C89C22×C414yes—
195C2×Q8×C89C2×D414yes—
196C2×Q8×C89#4114yes—
197C22×C4×C89C2413yes—
198C22×C4×C89C2413yes—
199C22×C4×C89C2411yescharacteristic
200C22×C4×C89C2411yescharacteristic
201C22×C4×C89C2413yes—
202C22×C4×C89C2411yescharacteristic
203C22×C4×C89C2413yes—
204C4⋊C4×C89—26no—
205C4⋊C4×C89—26no—
206C4⋊C4×C89—26no—
207C4⋊C4×C89—43no—
208C4⋊C4×C89C22×C4112yes—
209C42×C89—26no—
210C42×C89—26no—
211C42×C89C22×C4112yes—
212C23×C89C2511yescharacteristic
213C22×D89—21no—
214C22×D89—21no—
215C22×D89—21no—
216C22×D89—21no—
217C22×D89—21no—
218C22×D89—21no—
219C22×D89—21no—
220C22×D89—21no—
221C22×D89—21no—
222C22×D89—21no—
223C22×D89C23×C412yes—
224C22×D89C23×C412yes—
225C22×D892−1+412yes—
226C22×D892−1+412yes—
227Q8×C89C4×D4116yes—
228C2×C4×C89—21no—
229C2×C4×C89—23no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—23no—
234C2×C4×C89—21no—
235C2×C4×C89—23no—
236C2×C4×C89—23no—
237C2×C4×C89—23no—
238C2×C4×C89—23no—
239C2×C4×C89—21no—
240C2×C4×C89—23no—
241C2×C4×C89—21no—
242C2×C4×C89—23no—
243C2×C4×C89C23×C412yes—
244C2×C4×C89C23×C416yes—
245C2×C4×C89C23×C416yes—
246C2×C4×C89C22×D416yes—
247C2×C4×C89C2×#4116yes—
248C2×C4×C892−1+412yes—
249C2×Dic89—23no—
250C2×Dic89—23no—
251C2×Dic89—23no—
252C2×Dic89—23no—
253C2×Dic89—21no—
254C2×Dic89—23no—
255C2×Dic89—21no—
256C2×Dic89—23no—
257C2×Dic89—21no—
258C2×Dic89—23no—
259C2×Dic89—23no—
260C2×Dic89—21no—
261C2×Dic89—21no—
262C2×Dic89—23no—
263C2×Dic89C23×C416yes—
264C2×Dic89C23×C416yes—
265C2×Dic89C22×D412yes—
266C2×Dic89C22×D416yes—
267D356—28no—
268C4×D89—28no—
269C4×D89—224no—
270C4×D89—224no—
271C4×D89—412no—
272Dic178—224no—
273Dic178—224no—
274C22×C89C24×C411yescharacteristic
275C22×C89C23×D411yescharacteristic
276C22×C89C22×#4111yescharacteristic
277C22×C89C2×2−1+411yescharacteristic
278C22×C89C2×2−1+411yescharacteristic
279C22×C89C2×2−1+411yescharacteristic
280C22×C89#58511yescharacteristic
281C2×D89—22no—
282C2×D89—22no—
283C2×D89—22no—
284C2×D89—22no—
285C2×D89—22no—
286C2×D89—22no—
287C2×D89—22no—
288C2×D89—22no—
289C2×D89—22no—
290C2×D89—41no—
291C2×D89—41no—
292C2×D89—41no—
293C2×D89—41no—
294C2×D89#51914yescomplemented
295C4×C89—22no—
296C4×C89—26no—
297C4×C89—22no—
298C4×C89—43no—
299C4×C89—43no—
300C4×C89—41no—
301C4×C89C2×C4×D4112yes—
302Dic89—26no—
303Dic89—26no—
304Dic89—22no—
305Dic89—26no—
306Dic89—43no—
307Dic89—41no—
308#34291—891nomaximal, Sylow
309C2×C89C22×C4×D411yescharacteristic
310C2×C89C2×#51911yescharacteristic
311C2×C89C4×2−1+411yescharacteristic
312C2×C89#348611yescharacteristic
313C2×C89#348911yescharacteristic
314C2×C89#351611yescharacteristic
315C2×C89#352311yescharacteristic
316D89—24no—
317D89—42no—
318C2×D4×Q8—891no—
319C2×#519—891no—
320C22×C4×Q8—891no—
321#2365—893no—
322#2362—893no—
323#2360—893no—
324#2352—893no—
325#2351D8911yescharacteristic, complemented
326Q8×C22⋊C4—891no—
327Q8×C22⋊C4—891no—
328#2327—893no—
329#2323—893no—
330C4×C22⋊Q8—893no—
331#2290—893no—
332#2276—891no—
333C89#3429111yescharacteristic, complemented, Sylow
334C23×Q8—891no—
335C23×Q8—891no—
336D4×Q8—8916no—
337C2×C4⋊Q8—893no—
338C2×C22⋊Q8—893no—
339C2×C22⋊Q8—893no—
340#519—1788no—
341C2×C4×Q8—891no—
342C2×C4×Q8—891no—
343C2×C4×Q8—893no—
344C2×C4×Q8—893no—
345C2×C4×Q8—893no—
346C2×C4×Q8—893no—
347C2×C4×Q8—1788no—
348C2×C4×Q8C2×D8911yescharacteristic
349C2×C4×Q8C2×D8911yescharacteristic
350C2×C4×Q8C2×D8911yescharacteristic
351C2×C4×D4—893no—
352C2×#117—893no—
353C2×#117—893no—
354C22×C4⋊C4—893no—
355C22×C42—893no—
356#391—893no—
357#391—893no—
358#391—893no—
359#391—893no—
360#389—896no—
361#389C2×D8913yes—
362#389C2×D8913yes—
363#388—893no—
364#388—893no—
365#388—896no—
366#387—896no—
367#387—893no—
368#387—896no—
369#387—896no—
370#387—893no—
371#386—893no—
372#386—891no—
373#385—893no—
374#385—893no—
375#382C2×D8913yes—
376#380—893no—
377#380—893no—
378C4×C4⋊C4—893no—
379C4×C4⋊C4—893no—
380C4×C4⋊C4—896no—
381C4×C4⋊C4—893no—
382C4×C4⋊C4C2×D8913yes—
383C4×C22⋊C4—893no—
384C4×C22⋊C4—893no—
385C4×C22⋊C4—896no—
386#376—893no—
387C22×Q8—898no—
388C22×Q8—898no—
389C22×Q8—893no—
390C22×Q8—893no—
391C22×Q8—1784no—
392C22×Q8—1784no—
393C22×Q8—1784no—
394C22×Q8—1784no—
395C22×Q8C22×D8911yescharacteristic
396C22×D4—891no—
397C23×C4—893no—
398C23×C4—893no—
399C23×C4—893no—
400C23×C4—891no—
401C4⋊Q8—8924no—
402C22⋊Q8—8924no—
403C22⋊Q8—8924no—
404C4×Q8—24no—
405C4×Q8—24no—
406C4×Q8—898no—
407C4×Q8—17812no—
408C4×Q8—17812no—
409C4×Q8—17812no—
410C4×Q8—17812no—
411C4×Q8—1784no—
412C4×Q8C4×D8918yescomplemented
413C4×D4—8924no—
414#117—17812no—
415#117—17812no—
416C2×C4⋊C4—893no—
417C2×C4⋊C4—893no—
418C2×C4⋊C4—896no—
419C2×C4⋊C4—896no—
420C2×C4⋊C4—893no—
421C2×C4⋊C4—893no—
422C2×C4⋊C4—893no—
423C2×C4⋊C4—17812no—
424C2×C4⋊C4C22×D8913yes—
425C2×C4⋊C4C22×D8913yes—
426C2×C4⋊C4C22×D8913yes—
427C2×C4⋊C4C22×D8913yes—
428C2×C4⋊C4C22×D8913yes—
429C2×C22⋊C4—893no—
430C2×C22⋊C4—893no—
431C2×C22⋊C4—893no—
432C2×C22⋊C4—891no—
433C2×C22⋊C4—893no—
434C2×C22⋊C4—893no—
435C2×C22⋊C4—891no—
436C2×C22⋊C4—891no—
437C2×C42—893no—
438C2×C42—896no—
439C2×C42—896no—
440C2×C42—893no—
441C2×C42—893no—
442C2×C42—893no—
443C2×C42—17812no—
444C2×C42C22×D8913yes—
445C2×C42C22×D8913yes—
446C2×C42C22×D8913yes—
447#95—891no—
448#95—893no—
449#95—893no—
450#95—893no—
451#95—893no—
452#95—896no—
453#95—893no—
454#95—893no—
455#95—893no—
456#95—896no—
457#95C22×D8916yes—
458#95C22×D8911yescharacteristic
459#95C22×D8913yes—
460C24—891no—
461C24—891no—
462C2×Q8—8912no—
463C2×Q8—8912no—
464C2×Q8—17832no—
465C2×Q8—17832no—
466C2×Q8—1786no—
467C2×Q8—1786no—
468C2×Q8—1786no—
469C2×Q8—1786no—
470C2×Q8C2×C4×D8914yes—
471C2×Q8D4×D8914yes—
472C2×Q8#1308357514yes—
473C2×D4—894no—
474C2×D4—1782no—
475C2×D4—1782no—
476C22×C4—891no—
477C22×C4—893no—
478C22×C4—893no—
479C22×C4—893no—
480C22×C4—8912no—
481C22×C4—8912no—
482C22×C4—891no—
483C22×C4—893no—
484C22×C4—3563no—
485C22×C4—1786no—
486C22×C4—1782no—
487C22×C4—1786no—
488C22×C4—1782no—
489C22×C4—1786no—
490C22×C4—1786no—
491C22×C4—1786no—
492C22×C4—1786no—
493C22×C4—1782no—
494C22×C4C23×D8913yes—
495C22×C4C23×D8913yes—
496C22×C4C23×D8911yescharacteristic
497C22×C4C23×D8911yescharacteristic
498C22×C4C23×D8913yes—
499C22×C4C23×D8911yescharacteristic
500C22×C4C23×D8913yes—
501C4⋊C4—26no—
502C4⋊C4—26no—
503C4⋊C4—26no—
504C4⋊C4—43no—
505C4⋊C4—8912no—
506C4⋊C4—8924no—
507C4⋊C4—8912no—
508C4⋊C4—17812no—
509C4⋊C4—1786no—
510C4⋊C4—1786no—
511C4⋊C4—1786no—
512C4⋊C4C2×C4×D89112yes—
513C22⋊C4—8912no—
514C22⋊C4—8912no—
515C22⋊C4—3563no—
516C22⋊C4—1782no—
517C22⋊C4—1786no—
518C22⋊C4—1786no—
519C22⋊C4—1782no—
520C22⋊C4—1782no—
521C42—26no—
522C42—26no—
523C42—8912no—
524C42—3566no—
525C42—1786no—
526C42—1786no—
527C42—17812no—
528C42—1786no—
529C42C2×C4×D89112yes—
530C23—892no—
531C23—892no—
532C23—892no—
533C23—892no—
534C23—1781no—
535C23—1781no—
536C23—1781no—
537C23—1781no—
538C23—1781no—
539C23—1781no—
540C23—1781no—
541C23—1781no—
542C23—1781no—
543C23—1781no—
544C23C24×D8911yescharacteristic
545Q8—17824no—
546Q8—17824no—
547Q8#15667974116yes—
548D4—1788no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—23no—
554C2×C4—23no—
555C2×C4—21no—
556C2×C4—23no—
557C2×C4—23no—
558C2×C4—21no—
559C2×C4—23no—
560C2×C4—23no—
561C2×C4—23no—
562C2×C4—23no—
563C2×C4—21no—
564C2×C4—896no—
565C2×C4—896no—
566C2×C4—892no—
567C2×C4—896no—
568C2×C4—35612no—
569C2×C4—1788no—
570C2×C4—1781no—
571C2×C4—1783no—
572C2×C4—1783no—
573C2×C4—1781no—
574C2×C4—1781no—
575C2×C4—1783no—
576C2×C4—1783no—
577C2×C4—1783no—
578C2×C4—1783no—
579C2×C4—1783no—
580C2×C4—1783no—
581C2×C4—1781no—
582C2×C4—1781no—
583C2×C4—1783no—
584C2×C4—17824no—
585C2×C4—17824no—
586C2×C4C22×C4×D8912yes—
587C2×C4C22×C4×D8916yes—
588C2×C4C22×C4×D8916yes—
589C2×C4C2×D4×D8916yes—
590C2×C4C2×#1308357516yes—
591C2×C4#1566805912yes—
592C22—894no—
593C22—3561no—
594C22—3561no—
595C22—3561no—
596C22—3561no—
597C22—1782no—
598C22—1782no—
599C22—1782no—
600C22—1782no—
601C22—1782no—
602C22—1782no—
603C22—1782no—
604C22—1782no—
605C22—1782no—
606C22C23×C4×D8911yescharacteristic
607C22C22×D4×D8911yescharacteristic
608C22C22×#1308357511yescharacteristic
609C222−1+4×D8911yescharacteristic
610C222−1+4×D8911yescharacteristic
611C22C2×#1566805911yescharacteristic
612C22#1915243411yescharacteristic
613C4—22no—
614C4—26no—
615C4—22no—
616C4—43no—
617C4—41no—
618C4—43no—
619C4—3563no—
620C4—3561no—
621C4—1786no—
622C4—1786no—
623C4—1786no—
624C4—1782no—
625C4C2×#15667974112yes—
626C2—3562no—
627C2—1784no—
628C2C22×#1566797411yescharacteristic
629C2#519×D8911yescharacteristic
630C2C4×#1566805911yescharacteristic
631C2#2603389811yescharacteristic
632C2#2603393111yescharacteristic
633C2#2603424811yescharacteristic
634C2#2603434911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2351C2
socle, cosocleC23×C89C25
prime radicalC22×D356C23
hypercenter#2351D89
hypocenterC89#34291
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×#19152074C2semidirect
3. C2×C4×Q8×D89C2semidirect
4. #26023308C2semidirect
5. #26023278C2semidirect
6. #26023256C2semidirect
8. Q8×#13083551C2semidirect
9. Q8×#13083551C2semidirect
10. #26022969C2semidirect
11. #26022932C2semidirect
12. #26022782C2semidirect
14. #26022617C2semidirect
15. C89×#2351C2semidirect
18. Q8×D356C4semidirect
76. C2×Q8×D89C2×C4semidirect
82. #15668012C2×C4semidirect
84. #15667992C2×C4semidirect
86. Q8×Dic89D4semidirect
130. C4×Q8×C89C2×C4semidirect
294. C2×D89#519semidirect
325. #2351D89semidirect
333. C89#34291semidirect
412. C4×Q8C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D3562
C2×#191520742
C2×C4×Q8×D892
#260233082
#260232782
#260232562
#260231782
Q8×#130835512
#260229692
#260229322
#260227822
#260226932
#260226172
C89×#23512
#3429189