Finite Group Databasebeta

← #36716092

Subgroups — #36716092

Order 22784.

2984 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152260C211yesmaximal, characteristic
2C2×#19152074C211yesmaximal, characteristic, complemented
3C2×C4×#13083575C211yesmaximal, characteristic, complemented
4#26023323C213yesmaximal, complemented
5#26023292C213yesmaximal, complemented
6#26023281C213yesmaximal, complemented
7#26023199C213yesmaximal, complemented
8Q8×C4⋊Dic89C211yesmaximal, characteristic, complemented
9#26023090C211yesmaximal, characteristic, complemented
10Q8×#13083551C211yesmaximal, characteristic, complemented
11#26022963C213yesmaximal, complemented
12#26022946C213yesmaximal, complemented
13C4×#15667992C213yesmaximal, complemented
14#26022696C213yesmaximal, complemented
15C89×#2276C211yesmaximal, characteristic, complemented
16C22×#13083575C2211yescharacteristic
17C22×Q8×D89C2211yescharacteristic
18#19152260C4116yescomplemented
19C2×#15668000C2213yes—
20C2×#15667992C2213yes—
21C2×#15667987C2213yes—
22#19152074—28no—
23C4×#13083575—28no—
24C2×Q8×Dic89C2211yescharacteristic
25C2×Q8×Dic89C2211yescharacteristic
26C2×C4×Dic178C2213yes—
27C2×#15667977C2213yes—
28C2×C4×D356C2213yes—
29C2×C4×D356C2213yes—
30C2×#15667974C2213yes—
31C2×#15667970C2213yes—
32C2×#15667970C2213yes—
33C2×#15667969C2213yes—
34C2×C42×D89C2213yes—
35#19151504C2213yes—
36#19151503C2213yes—
37#19151500C2216yes—
38#19151500C2213yes—
39#19151499C2213yes—
40#19151497C2213yes—
41#19151497C2213yes—
42#19151490C2213yes—
43#19151490C2216yes—
44#19151490C2211yescharacteristic
45#19151490C2216yes—
46#19151489C2216yes—
47#19151483C2213yes—
48#19151483C2213yes—
49#19151476C2213yes—
50#19151476C2213yes—
51#19151472C2216yes—
52#19151472C2213yes—
53#19151472C2213yes—
54#19151468C2213yes—
55#19151455C2213yes—
56#19151455C2213yes—
57C4×C4⋊Dic89C2213yes—
58C4×C4⋊Dic89C2216yes—
59C4×#13083551C2213yes—
60C4×#13083551C2213yes—
61C4×#13083551C2216yes—
62C2×C4×Q8×C89C2211yescharacteristic
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2211yescharacteristic
65C4×C4⋊C4×C89C2213yes—
66C4×C4⋊C4×C89C2213yes—
67C4×C4⋊C4×C89C2213yes—
68#376×C89C2213yes—
69C2×#13083575—24no—
70C2×#13083575—24no—
71C2×#13083575C2×C418yes—
72C22×Dic178C2313yes—
73C2×Q8×D89—24no—
74C2×Q8×D89—24no—
75C2×Q8×D89C2×C418yescomplemented
76C22×D356C2313yes—
77C22×D356C2313yes—
78C22×C4×D89C2313yes—
79C22×C4×D89C2313yes—
80C22×C4×D89C2313yes—
81C22×C4×D89C2313yes—
82#15668000C2×C4124yescomplemented
83#15667992C2×C4124yescomplemented
84#15667987C2×C4124yescomplemented
85Q8×Dic89—24no—
86Q8×Dic89—24no—
87C4×Dic178—212no—
88#15667977—212no—
89C4×D356—212no—
90C4×D356C2×C4124yescomplemented
91#15667974—212no—
92#15667970—212no—
93#15667970—212no—
94#15667969—212no—
95C2×C4⋊Dic89C2311yescharacteristic
96C2×C4⋊Dic89C2313yes—
97C2×C4⋊Dic89C2313yes—
98C2×C4⋊Dic89C2313yes—
99C2×C4⋊Dic89C2316yes—
100C2×C4⋊Dic89C2313yes—
101C2×#13083553C2313yes—
102C2×#13083553C2313yes—
103C2×#13083551C2311yescharacteristic
104C2×#13083551C2311yescharacteristic
105C2×#13083551C2313yes—
106C2×#13083551C2316yes—
107C2×#13083551C2313yes—
108C2×#13083551C2316yes—
109C2×#13083551C2313yes—
110C2×#13083551C2313yes—
111C2×#13083551C2316yes—
112C2×#13083551C2313yes—
113C2×#13083551C2311yescharacteristic
114C2×C4×Dic89C2316yes—
115C2×C4×Dic89C2313yes—
116C2×C4×Dic89C2313yes—
117C2×C4×Dic89C2313yes—
118C2×C4×Dic89C2311yescharacteristic
119C42×D89—212no—
120#15667919C2313yes—
121#15667919C2311yescharacteristic
122#15667919C2313yes—
123#15667919C2313yes—
124#15667919C2313yes—
125#15667919C2316yes—
126#15667919C2313yes—
127C22×Q8×C89C2311yescharacteristic
128C4×Q8×C89—24no—
129C4×Q8×C89C2×C418yescomplemented
130C4×Q8×C89D418yescomplemented
131C2×C4⋊C4×C89C2313yes—
132C2×C4⋊C4×C89C2313yes—
133C2×C4⋊C4×C89C2313yes—
134C2×C42×C89C2313yes—
135C2×C42×C89C2316yes—
136C2×C42×C89C2313yes—
137C2×C42×C89C2313yes—
138#95×C89C2311yescharacteristic
139#95×C89C2313yes—
140#95×C89C2313yes—
141#95×C89C2313yes—
142C23×D89C2411yescharacteristic
143C23×D89C2413yes—
144#13083575—232no—
145C2×Dic178—26no—
146C2×Dic178—26no—
147C2×Dic178C22×C4112yes—
148Q8×D89—232no—
149C2×D356—26no—
150C2×D356—26no—
151C2×D356—43no—
152C2×D356—43no—
153C2×D356C22×C4112yes—
154C2×D356C22×C4112yes—
155C22×Dic89C2413yes—
156C22×Dic89C2411yescharacteristic
157C22×Dic89C2413yes—
158C22×Dic89C2411yescharacteristic
159C22×Dic89C2413yes—
160C22×Dic89C2411yescharacteristic
161C2×C4×D89—26no—
162C2×C4×D89—26no—
163C2×C4×D89—26no—
164C2×C4×D89—26no—
165C2×C4×D89—26no—
166C2×C4×D89—26no—
167C2×C4×D89—26no—
168C2×C4×D89—43no—
169C2×C4×D89—43no—
170C2×C4×D89—43no—
171C2×C4×D89C22×C4112yes—
172C2×C4×D89C22×C4112yes—
173C4⋊Dic89—26no—
174C4⋊Dic89—26no—
175C4⋊Dic89—22no—
176C4⋊Dic89—43no—
177C4⋊Dic89—46no—
178C4⋊Dic89C22×C4112yes—
179#13083553—26no—
180#13083553—26no—
181#13083551—22no—
182#13083551—26no—
183#13083551—212no—
184#13083551—22no—
185#13083551—43no—
186#13083551—46no—
187#13083551—43no—
188#13083551—43no—
189#13083551C22×C414yes—
190#13083551C22×C4112yes—
191#13083551C22×C4124yes—
192C4×Dic89—26no—
193C4×Dic89—26no—
194C4×Dic89—212no—
195C4×Dic89—26no—
196C4×Dic89—22no—
197C2×Q8×C89C22×C414yes—
198C2×Q8×C89C2×D414yes—
199C2×Q8×C89#4114yes—
200C22×C4×C89C2411yescharacteristic
201C22×C4×C89C2413yes—
202C22×C4×C89C2413yes—
203C22×C4×C89C2411yescharacteristic
204C22×C4×C89C2411yescharacteristic
205C22×C4×C89C2413yes—
206C22×C4×C89C2413yes—
207C4⋊C4×C89—26no—
208C4⋊C4×C89C22×C4112yes—
209C4⋊C4×C89C2×D4112yes—
210C42×C89—26no—
211C42×C89—46no—
212C42×C89C22×C4112yes—
213C42×C89C2×D4112yes—
214C23×C89C2511yescharacteristic
215C22×D89—21no—
216C22×D89—23no—
217C22×D89—21no—
218C22×D89—23no—
219C22×D89—21no—
220C22×D89—23no—
221C22×D89—21no—
222C22×D89—23no—
223C22×D89—23no—
224C22×D89—23no—
225C22×D89—21no—
226C22×D89—21no—
227C22×D89C23×C416yes—
228C22×D89C23×C412yes—
229Q8×C89C4×D4116yes—
230C2×C4×C89—21no—
231C2×C4×C89—23no—
232C2×C4×C89—23no—
233C2×C4×C89—23no—
234C2×C4×C89—21no—
235C2×C4×C89—23no—
236C2×C4×C89—21no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—23no—
240C2×C4×C89—21no—
241C2×C4×C89—23no—
242C2×C4×C89—21no—
243C2×C4×C89—23no—
244C2×C4×C89C23×C412yes—
245C2×C4×C89C23×C416yes—
246C2×C4×C89C23×C416yes—
247C2×C4×C89C22×D412yes—
248C2×C4×C89C22×D416yes—
249C2×C4×C89C22×D416yes—
250C2×C4×C89C2×#4116yes—
251C2×Dic89—21no—
252C2×Dic89—21no—
253C2×Dic89—23no—
254C2×Dic89—21no—
255C2×Dic89—23no—
256C2×Dic89—21no—
257C2×Dic89—21no—
258C2×Dic89—23no—
259C2×Dic89—23no—
260C2×Dic89—21no—
261C2×Dic89—21no—
262C2×Dic89—23no—
263C2×Dic89—23no—
264C2×Dic89—23no—
265C2×Dic89—23no—
266C2×Dic89C23×C416yes—
267C2×Dic89C23×C412yes—
268C2×Dic892−1+412yes—
269D356—224no—
270D356—412no—
271C4×D89—224no—
272C4×D89—224no—
273C4×D89—412no—
274C4×D89—412no—
275Dic178—224no—
276C22×C89C24×C411yescharacteristic
277C22×C89C23×D411yescharacteristic
278C22×C89C22×#4111yescharacteristic
279C22×C89C2×2−1+411yescharacteristic
280C22×C89#58511yescharacteristic
281C22×C89#58511yescharacteristic
282C22×C89#58511yescharacteristic
283C2×D89—22no—
284C2×D89—26no—
285C2×D89—22no—
286C2×D89—26no—
287C2×D89—22no—
288C2×D89—26no—
289C2×D89—43no—
290C2×D89—41no—
291C2×D89—43no—
292C2×D89—41no—
293C2×D89—41no—
294C2×D89—43no—
295C2×D89—41no—
296C2×D89—43no—
297C4×C89—26no—
298C4×C89—22no—
299C4×C89—22no—
300C4×C89—26no—
301C4×C89—41no—
302C4×C89—43no—
303C4×C89C2×C4×D4112yes—
304Dic89—26no—
305Dic89—22no—
306Dic89—22no—
307Dic89—43no—
308Dic89—43no—
309Dic89—41no—
310#34293—891nomaximal, Sylow
311C2×C89C22×C4×D411yescharacteristic
312C2×C89#342811yescharacteristic
313C2×C89C4×2−1+411yescharacteristic
314C2×C89#348711yescharacteristic
315C2×C89#348911yescharacteristic
316C2×C89#351811yescharacteristic
317C2×C89#352411yescharacteristic
318D89—42no—
319D89—46no—
320C2×#548—891no—
321C2×#519—891no—
322C2×C4×#41—891no—
323#2367—893no—
324#2363—893no—
325#2362—893no—
326#2354—893no—
327Q8×C4⋊C4—891no—
328#2341—891no—
329Q8×C22⋊C4—891no—
330#2326—893no—
331#2324—893no—
332C4×C22⋊Q8—893no—
333#2290—893no—
334#2276D8911yescharacteristic, complemented
335C89#3429311yescharacteristic, complemented, Sylow
336C22×#41—891no—
337C23×Q8—891no—
338#548—8916no—
339C2×#124—893no—
340C2×C22⋊Q8—893no—
341C2×#121—893no—
342#519—1788no—
343C4×#41—1788no—
344C2×C4×Q8—891no—
345C2×C4×Q8—891no—
346C2×C4×Q8—893no—
347C2×C4×Q8—893no—
348C2×C4×Q8C2×D8911yescharacteristic
349C2×C4×Q8C2×D8911yescharacteristic
350C2×C4×Q8C2×D8911yescharacteristic
351C2×C4×D4—893no—
352C2×C4×D4—893no—
353C2×C4×D4—893no—
354C2×#117—893no—
355C2×#117—893no—
356C2×#117—893no—
357C22×C42—893no—
358#391—893no—
359#391—893no—
360#390—893no—
361#390—896no—
362#390—893no—
363#389—893no—
364#389—893no—
365#388—896no—
366#388—896no—
367#388—896no—
368#388—893no—
369#388—891no—
370#387—893no—
371#387—893no—
372#386—893no—
373#386—893no—
374#385—896no—
375#385—893no—
376#385—893no—
377#384—893no—
378#380—893no—
379#380—893no—
380C4×C4⋊C4—893no—
381C4×C4⋊C4—896no—
382C4×C4⋊C4C2×D8913yes—
383C4×C4⋊C4C2×D8913yes—
384C4×C4⋊C4C2×D8913yes—
385C4×C22⋊C4—893no—
386C4×C22⋊C4—893no—
387C4×C22⋊C4—896no—
388#376C2×D8913yes—
389C2×#41—898no—
390C2×#41—1784no—
391C2×#41—1784no—
392C22×Q8—898no—
393C22×Q8—893no—
394C22×Q8—1784no—
395C22×Q8—1784no—
396C22×Q8C22×D8911yescharacteristic
397C22×D4—893no—
398C22×D4—893no—
399C23×C4—893no—
400C23×C4—893no—
401C23×C4—893no—
402C23×C4—893no—
403#124—8924no—
404C22⋊Q8—8924no—
405#121—8924no—
406C4×Q8—24no—
407C4×Q8—1784no—
408C4×Q8—1784no—
409C4×Q8—17812no—
410C4×Q8—17812no—
411C4×Q8C4×D8918yescomplemented
412C4×Q8D35618yescomplemented
413C4×D4—8924no—
414C4×D4—17812no—
415C4×D4—17812no—
416#117—17812no—
417#117—17812no—
418#117—17812no—
419C2×C4⋊C4—893no—
420C2×C4⋊C4—893no—
421C2×C4⋊C4—896no—
422C2×C4⋊C4—893no—
423C2×C4⋊C4—893no—
424C2×C4⋊C4—893no—
425C2×C4⋊C4—893no—
426C2×C4⋊C4—891no—
427C2×C4⋊C4C22×D8913yes—
428C2×C4⋊C4C22×D8913yes—
429C2×C4⋊C4C22×D8913yes—
430C2×C22⋊C4—891no—
431C2×C22⋊C4—893no—
432C2×C22⋊C4—896no—
433C2×C22⋊C4—893no—
434C2×C22⋊C4—896no—
435C2×C22⋊C4—893no—
436C2×C22⋊C4—893no—
437C2×C22⋊C4—891no—
438C2×C22⋊C4—893no—
439C2×C22⋊C4—896no—
440C2×C22⋊C4—891no—
441C2×C42—893no—
442C2×C42—896no—
443C2×C42—893no—
444C2×C42—893no—
445C2×C42—891no—
446C2×C42—17812no—
447C2×C42C22×D8913yes—
448C2×C42C22×D8916yes—
449C2×C42C22×D8913yes—
450C2×C42C22×D8913yes—
451#95—893no—
452#95—891no—
453#95—893no—
454#95—893no—
455#95—893no—
456#95—896no—
457#95—893no—
458#95C22×D8911yescharacteristic
459#95C22×D8913yes—
460#95C22×D8913yes—
461#95C22×D8913yes—
462C24—893no—
463C24—891no—
464#41—17832no—
465C2×Q8—8912no—
466C2×Q8—17832no—
467C2×Q8—1786no—
468C2×Q8—1786no—
469C2×Q8C2×C4×D8914yes—
470C2×Q8C2×D35614yes—
471C2×Q8#1308357414yes—
472C2×D4—8912no—
473C2×D4—8912no—
474C2×D4—3563no—
475C2×D4—3563no—
476C2×D4—1786no—
477C2×D4—1786no—
478C22×C4—891no—
479C22×C4—891no—
480C22×C4—893no—
481C22×C4—8912no—
482C22×C4—8912no—
483C22×C4—891no—
484C22×C4—893no—
485C22×C4—893no—
486C22×C4—3563no—
487C22×C4—3563no—
488C22×C4—3563no—
489C22×C4—1786no—
490C22×C4—1786no—
491C22×C4—1786no—
492C22×C4—1786no—
493C22×C4—1786no—
494C22×C4—1786no—
495C22×C4—1786no—
496C22×C4C23×D8911yescharacteristic
497C22×C4C23×D8913yes—
498C22×C4C23×D8913yes—
499C22×C4C23×D8911yescharacteristic
500C22×C4C23×D8911yescharacteristic
501C22×C4C23×D8913yes—
502C22×C4C23×D8913yes—
503C4⋊C4—26no—
504C4⋊C4—8912no—
505C4⋊C4—3563no—
506C4⋊C4—3566no—
507C4⋊C4—1782no—
508C4⋊C4—1786no—
509C4⋊C4—1786no—
510C4⋊C4—1786no—
511C4⋊C4—1786no—
512C4⋊C4C2×C4×D89112yes—
513C4⋊C4C2×D356112yes—
514C22⋊C4—894no—
515C22⋊C4—8912no—
516C22⋊C4—8924no—
517C22⋊C4—3563no—
518C22⋊C4—3566no—
519C22⋊C4—3563no—
520C22⋊C4—3563no—
521C22⋊C4—1782no—
522C22⋊C4—1786no—
523C22⋊C4—1782no—
524C22⋊C4—17812no—
525C42—26no—
526C42—46no—
527C42—1782no—
528C42—1786no—
529C42—1786no—
530C42—1786no—
531C42—17812no—
532C42C2×C4×D89112yes—
533C42C2×D356112yes—
534C23—896no—
535C23—892no—
536C23—1781no—
537C23—1783no—
538C23—1781no—
539C23—1783no—
540C23—1781no—
541C23—1783no—
542C23—1783no—
543C23—1781no—
544C23—1781no—
545C23—1783no—
546C23—1783no—
547C23—1781no—
548C23C24×D8911yescharacteristic
549Q8—17824no—
550Q8C4×D356116yes—
551D4—35612no—
552D4—17824no—
553C2×C4—21no—
554C2×C4—23no—
555C2×C4—23no—
556C2×C4—23no—
557C2×C4—21no—
558C2×C4—23no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—21no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—23no—
567C2×C4—896no—
568C2×C4—892no—
569C2×C4—892no—
570C2×C4—35612no—
571C2×C4—35612no—
572C2×C4—1781no—
573C2×C4—1783no—
574C2×C4—1783no—
575C2×C4—1781no—
576C2×C4—1781no—
577C2×C4—1783no—
578C2×C4—1783no—
579C2×C4—1783no—
580C2×C4—1783no—
581C2×C4—1781no—
582C2×C4—1781no—
583C2×C4—1783no—
584C2×C4—1781no—
585C2×C4—1783no—
586C2×C4—1781no—
587C2×C4—17824no—
588C2×C4—17824no—
589C2×C4C22×C4×D8912yes—
590C2×C4C22×C4×D8916yes—
591C2×C4C22×C4×D8916yes—
592C2×C4C22×D35616yes—
593C2×C4C22×D35612yes—
594C2×C4C22×D35616yes—
595C2×C4C2×#1308357416yes—
596C22—3563no—
597C22—3561no—
598C22—3561no—
599C22—3563no—
600C22—3561no—
601C22—3563no—
602C22—3563no—
603C22—3561no—
604C22—1782no—
605C22—1786no—
606C22—1782no—
607C22—1786no—
608C22—1786no—
609C22—1782no—
610C22C23×C4×D8911yescharacteristic
611C22C23×D35611yescharacteristic
612C22C22×#1308357411yescharacteristic
613C222−1+4×D8911yescharacteristic
614C22#1915243311yescharacteristic
615C22#1915243511yescharacteristic
616C22#1915243511yescharacteristic
617C4—26no—
618C4—26no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—43no—
623C4—3561no—
624C4—3563no—
625C4—3563no—
626C4—1782no—
627C4—1782no—
628C4—1786no—
629C4C2×C4×D356112yes—
630C2—3566no—
631C2—3562no—
632C2C22×C4×D35611yescharacteristic
633C2#2603345211yescharacteristic
634C2#2603346111yescharacteristic
635C2#2603391111yescharacteristic
636C2#2603393311yescharacteristic
637C2#2603426311yescharacteristic
638C2#2603435311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2276C2
socle, cosocleC23×C89C25
prime radicalC2×#19152260C2
hypercenter#2276D89
hypocenterC89#34293
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. C2×#19152074C2semidirect
3. C2×C4×#13083575C2semidirect
4. #26023323C2semidirect
5. #26023292C2semidirect
6. #26023281C2semidirect
7. #26023199C2semidirect
8. Q8×C4⋊Dic89C2semidirect
9. #26023090C2semidirect
10. Q8×#13083551C2semidirect
11. #26022963C2semidirect
12. #26022946C2semidirect
13. C4×#15667992C2semidirect
14. #26022696C2semidirect
15. C89×#2276C2semidirect
18. #19152260C4semidirect
75. C2×Q8×D89C2×C4semidirect
82. #15668000C2×C4semidirect
83. #15667992C2×C4semidirect
84. #15667987C2×C4semidirect
90. C4×D356C2×C4semidirect
129. C4×Q8×C89C2×C4semidirect
130. C4×Q8×C89D4semidirect
334. #2276D89semidirect
335. C89#34293semidirect
411. C4×Q8C4×D89semidirect
412. C4×Q8D356semidirect

Maximal subgroups

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

subgroupindex
C2×#191522602
C2×#191520742
C2×C4×#130835752
#260233232
#260232922
#260232812
#260231992
Q8×C4⋊Dic892
#260230902
Q8×#130835512
#260229632
#260229462
C4×#156679922
#260226962
C89×#22762
#3429389