Finite Group Databasebeta

← #36716097

Subgroups — #36716097

Order 22784.

3112 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152255C211yesmaximal, characteristic
2C2×C4×Q8×D89C211yesmaximal, characteristic, complemented
3#26023325C213yesmaximal, complemented
4#26023264C213yesmaximal, complemented
5#26023257C213yesmaximal, complemented
6Q8×#13083553C211yesmaximal, characteristic, complemented
7C4⋊C4×C22⋊D89C213yesmaximal, complemented
8#26023093C211yesmaximal, characteristic, complemented
9Q8×#13083551C211yesmaximal, characteristic, complemented
10#26022964C213yesmaximal, complemented
11#26022934C213yesmaximal, complemented
12C4×#15667983C213yesmaximal, complemented
13#26022689C213yesmaximal, complemented
14#26022617C211yesmaximal, characteristic, complemented
15C2×#520×C89C211yesmaximal, characteristic, complemented
16C22×Q8×D89C2211yescharacteristic
17#19152255C4116yescomplemented
18C2×#15667999C2213yes—
19C2×#15667991C2213yes—
20C2×#15667983C2213yes—
21C2×Q8×Dic89C2211yescharacteristic
22C2×Q8×Dic89C2211yescharacteristic
23C2×Q8×Dic89C2211yescharacteristic
24C2×C4×Dic178C2213yes—
25C2×#15667977C2213yes—
26C4×Q8×D89—28no—
27C2×C4×C22⋊D89C2216yes—
28C2×C4×C22⋊D89C2213yes—
29C2×C4⋊C4×D89C2213yes—
30C2×C42×D89C2213yes—
31#19151504C2213yes—
32#19151503C2213yes—
33#19151501C2213yes—
34#19151501C2213yes—
35#19151496C2213yes—
36#19151494C2213yes—
37#19151493C2216yes—
38#19151492C2216yes—
39#19151486C2216yes—
40#19151486C2213yes—
41#19151486C2213yes—
42#19151479C2213yes—
43#19151479C2213yes—
44#19151474C2213yes—
45#19151474C2213yes—
46#19151471C2216yes—
47#19151471C2216yes—
48#19151469C2211yescharacteristic
49#19151469C2213yes—
50#19151467C2213yes—
51#19151458C2213yes—
52#19151455C2213yes—
53C4⋊C4×Dic89C2213yes—
54C4⋊C4×Dic89C2213yes—
55C4×#13083553C2213yes—
56C4×C4⋊Dic89C2213yes—
57C4×#13083551C2213yes—
58C4×C22⋊Dic89C2213yes—
59C4×C22⋊Dic89C2216yes—
60#19151444C2213yes—
61C22×#41×C89C2211yescharacteristic
62#520×C89—28no—
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2211yescharacteristic
65C2×C4×D4×C89C2213yes—
66C2×C4×D4×C89C2213yes—
67C2×#117×C89C2213yes—
68C22×C4⋊C4×C89C2213yes—
69C22×Dic178C2313yes—
70C2×Q8×D89—24no—
71C2×Q8×D89—24no—
72C2×Q8×D89C2×C418yescomplemented
73C22×C22⋊D89C2313yes—
74C22×C4×D89C2311yescharacteristic
75C22×C4×D89C2313yes—
76C22×C4×D89C2313yes—
77#15667999C2×C4124yescomplemented
78#15667991C2×C4124yescomplemented
79#15667983C2×C4124yescomplemented
80Q8×Dic89—24no—
81Q8×Dic89—24no—
82Q8×Dic89C2×C418yescomplemented
83C4×Dic178—212no—
84#15667977—212no—
85C4×C22⋊D89—412no—
86C4×C22⋊D89C2×C4124yescomplemented
87C2×C4⋊Dic89C2313yes—
88C2×C4⋊Dic89C2313yes—
89C2×C4⋊Dic89C2313yes—
90C2×#13083553C2311yescharacteristic
91C2×#13083553C2316yes—
92C2×#13083553C2313yes—
93C2×#13083553C2316yes—
94C2×#13083553C2313yes—
95C4⋊C4×D89—212no—
96C2×C22⋊Dic89C2313yes—
97C2×C22⋊Dic89C2313yes—
98C2×C22⋊Dic89C2316yes—
99C2×C22⋊Dic89C2316yes—
100C2×C22⋊Dic89C2313yes—
101C2×#13083551C2311yescharacteristic
102C2×#13083551C2313yes—
103C2×#13083551C2311yescharacteristic
104C2×C4×Dic89C2313yes—
105C2×C4×Dic89C2313yes—
106C2×C4×Dic89C2313yes—
107C2×C4×Dic89C2313yes—
108C2×C4×Dic89C2316yes—
109C2×C4×Dic89C2313yes—
110C42×D89—212no—
111#15667919C2311yescharacteristic
112#15667919C2313yes—
113#15667919C2313yes—
114#15667919C2313yes—
115#15667919C2311yescharacteristic
116#15667919C2313yes—
117#15667919C2316yes—
118#15667919C2313yes—
119#15667919C2313yes—
120#15667919C2313yes—
121#15667919C2313yes—
122C2×#41×C89—24no—
123C2×#41×C89—24no—
124C2×#41×C89C2×C418yescomplemented
125C22×Q8×C89C2311yescharacteristic
126C22×D4×C89C2313yes—
127C23×C4×C89C2313yes—
128C23×C4×C89C2316yes—
129C4×Q8×C89—24no—
130C4×Q8×C89D418yescomplemented
131C4×D4×C89—212no—
132C4×D4×C89—212no—
133#117×C89—212no—
134C2×C4⋊C4×C89—212no—
135C2×C4⋊C4×C89C2313yes—
136C2×C4⋊C4×C89C2311yescharacteristic
137C2×C4⋊C4×C89C2313yes—
138C2×C4⋊C4×C89C2313yes—
139C2×C22⋊C4×C89C2313yes—
140C2×C22⋊C4×C89C2313yes—
141C2×C42×C89C2313yes—
142C2×C42×C89C2313yes—
143C23×D89C2411yescharacteristic
144C2×Dic178—26no—
145C2×Dic178—26no—
146C2×Dic178C22×C4112yes—
147Q8×D89—232no—
148C2×C22⋊D89—26no—
149C2×C22⋊D89—26no—
150C2×C22⋊D89C22×C4112yes—
151C22×Dic89C2413yes—
152C22×Dic89C2411yescharacteristic
153C22×Dic89C2413yes—
154C22×Dic89C2413yes—
155C22×Dic89C2413yes—
156C22×Dic89C2411yescharacteristic
157C22×Dic89C2411yescharacteristic
158C2×C4×D89—22no—
159C2×C4×D89—26no—
160C2×C4×D89—22no—
161C2×C4×D89—26no—
162C2×C4×D89—22no—
163C2×C4×D89—26no—
164C2×C4×D89—26no—
165C2×C4×D89—43no—
166C2×C4×D89C22×C4112yes—
167C4⋊Dic89—26no—
168C4⋊Dic89—26no—
169C4⋊Dic89C22×C4112yes—
170#13083553—212no—
171#13083553—22no—
172#13083553—46no—
173#13083553—43no—
174#13083553C22×C4112yes—
175C22⋊Dic89—26no—
176C22⋊Dic89—46no—
177C22⋊Dic89—43no—
178C22⋊Dic89C22×C4112yes—
179C22⋊Dic89C22×C4124yes—
180#13083551—26no—
181#13083551—22no—
182#13083551C22×C414yes—
183C4×Dic89—26no—
184C4×Dic89—26no—
185C4×Dic89—26no—
186C4×Dic89—26no—
187C4×Dic89—46no—
188C4×Dic89C22×C4112yes—
189C24×C89C2413yes—
190#41×C89—232no—
191C2×Q8×C89C22×C414yes—
192C2×Q8×C89C2×D414yes—
193C2×Q8×C89#4114yes—
194C2×D4×C89—26no—
195C2×D4×C89—26no—
196C2×D4×C89C22×C4112yes—
197C22×C4×C89—26no—
198C22×C4×C89—26no—
199C22×C4×C89—212no—
200C22×C4×C89—46no—
201C22×C4×C89—46no—
202C22×C4×C89C22×C4112yes—
203C22×C4×C89C2411yescharacteristic
204C22×C4×C89C2413yes—
205C22×C4×C89C2411yescharacteristic
206C22×C4×C89C2411yescharacteristic
207C22×C4×C89C2413yes—
208C22×C4×C89C2413yes—
209C4⋊C4×C89—26no—
210C4⋊C4×C89—26no—
211C4⋊C4×C89—22no—
212C4⋊C4×C89C2×D4112yes—
213C22⋊C4×C89—26no—
214C22⋊C4×C89—26no—
215C42×C89—26no—
216C42×C89C2×D4112yes—
217C23×C89—23no—
218C23×C89—23no—
219C23×C89—23no—
220C23×C89—23no—
221C23×C89—23no—
222C23×C89—23no—
223C23×C89C23×C416yes—
224C23×C89C2511yescharacteristic
225C22×D89—21no—
226C22×D89—21no—
227C22×D89—21no—
228C22×D89—21no—
229C22×D89C23×C412yes—
230C22×D89H(5,𝔽2)12yes—
231C22×D892−1+412yes—
232Q8×C89C4×D4116yescomplemented
233D4×C89—224no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—23no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—23no—
240C2×C4×C89—224no—
241C2×C4×C89—21no—
242C2×C4×C89—23no—
243C2×C4×C89—23no—
244C2×C4×C89—21no—
245C2×C4×C89—21no—
246C2×C4×C89—23no—
247C2×C4×C89—812no—
248C2×C4×C89C23×C412yes—
249C2×C4×C89C23×C416yes—
250C2×C4×C89C22×D412yes—
251C2×C4×C89C22×D416yes—
252C2×C4×C89C22×D416yes—
253C2×C4×C89C2×#4116yes—
254C22⋊D89—224no—
255C2×Dic89—21no—
256C2×Dic89—23no—
257C2×Dic89—23no—
258C2×Dic89—23no—
259C2×Dic89—21no—
260C2×Dic89—21no—
261C2×Dic89—23no—
262C2×Dic89—21no—
263C2×Dic89—23no—
264C2×Dic89—21no—
265C2×Dic89—23no—
266C2×Dic89—23no—
267C2×Dic89—23no—
268C2×Dic89—21no—
269C2×Dic89—23no—
270C2×Dic89—21no—
271C2×Dic89—23no—
272C2×Dic89C23×C412yes—
273C2×Dic89C23×C416yes—
274C2×Dic89C23×C416yes—
275C2×Dic892−1+412yes—
276C4×D89—28no—
277C4×D89—224no—
278C4×D89—412no—
279Dic178—224no—
280C22×C89—26no—
281C22×C89—26no—
282C22×C89—26no—
283C22×C89—43no—
284C22×C89—43no—
285C22×C89—43no—
286C22×C89—43no—
287C22×C89C24×C411yescharacteristic
288C22×C89C23×D411yescharacteristic
289C22×C89C22×#4111yescharacteristic
290C22×C89C2×H(5,𝔽2)11yescharacteristic
291C22×C89C2×2−1+411yescharacteristic
292C22×C89C2×2−1+411yescharacteristic
293C22×C89#58511yescharacteristic
294C2×D89—22no—
295C2×D89—22no—
296C2×D89—22no—
297C2×D89—22no—
298C2×D89—22no—
299C2×D89—22no—
300C2×D89#52014yescomplemented
301C4×C89—22no—
302C4×C89—22no—
303C4×C89—26no—
304C4×C89—41no—
305C4×C89—43no—
306C4×C89C2×C4×D4112yes—
307Dic89—26no—
308Dic89—26no—
309Dic89—22no—
310Dic89—22no—
311Dic89—43no—
312Dic89—41no—
313Dic89—43no—
314#34294—891nomaximal, Sylow
315C2×C89—46no—
316C2×C89C22×C4×D411yescharacteristic
317C2×C89C2×#52011yescharacteristic
318C2×C89C4×2−1+411yescharacteristic
319C2×C89#348611yescharacteristic
320C2×C89#348811yescharacteristic
321C2×C89#351611yescharacteristic
322C2×C89#352211yescharacteristic
323D89—24no—
324C2×#548—891no—
325C2×#520D8911yescharacteristic, complemented
326C22×C4×Q8—891no—
327#2367—893no—
328#2361—893no—
329#2360—893no—
330Q8×C4⋊C4—891no—
331D4×C4⋊C4—893no—
332#2341—891no—
333Q8×C22⋊C4—891no—
334#2326—893no—
335#2323—893no—
336C4×#121—893no—
337#2290—893no—
338#2276—891no—
339C89#3429411yescharacteristic, complemented, Sylow
340C22×#41C2×D8911yescharacteristic
341C23×Q8—891no—
342#548—8916no—
343C2×#124—893no—
344C2×C22⋊Q8—893no—
345C2×#121—893no—
346#520—28no—
347C2×C4×Q8—891no—
348C2×C4×Q8—891no—
349C2×C4×Q8—891no—
350C2×C4×Q8—893no—
351C2×C4×Q8—893no—
352C2×C4×Q8—1788no—
353C2×C4×Q8C2×D8911yescharacteristic
354C2×C4×Q8C2×D8911yescharacteristic
355C2×C4×D4—893no—
356C2×C4×D4—896no—
357C2×C4×D4C2×D8913yes—
358C2×C4×D4C2×D8913yes—
359C2×#117C2×D8913yes—
360C22×C4⋊C4—893no—
361C22×C4⋊C4C2×D8913yes—
362C22×C42—893no—
363#391—893no—
364#391—893no—
365#390—893no—
366#390—893no—
367#389—896no—
368#389—893no—
369#389—893no—
370#388—896no—
371#388—896no—
372#388—893no—
373#388—893no—
374#387—893no—
375#387—893no—
376#386—893no—
377#386—893no—
378#385—896no—
379#385—896no—
380#385—893no—
381#385—891no—
382#384—893no—
383#380—893no—
384#380—893no—
385C4×C4⋊C4—893no—
386C4×C4⋊C4—893no—
387C4×C4⋊C4—893no—
388C4×C4⋊C4—893no—
389C4×C22⋊C4—893no—
390C4×C22⋊C4—896no—
391C4×C22⋊C4—893no—
392#376—893no—
393C2×#41—24no—
394C2×#41—24no—
395C2×#41C4×D8918yescomplemented
396C22×Q8—898no—
397C22×Q8—893no—
398C22×Q8—1784no—
399C22×Q8—1784no—
400C22×Q8C22×D8911yescharacteristic
401C22×D4—893no—
402C22×D4C22×D8913yes—
403C23×C4—893no—
404C23×C4—893no—
405C23×C4—891no—
406C23×C4C22×D8913yes—
407C23×C4C22×D8916yes—
408#124—8924no—
409C22⋊Q8—8924no—
410#121—8924no—
411C4×Q8—24no—
412C4×Q8—898no—
413C4×Q8—1784no—
414C4×Q8—1784no—
415C4×Q8—17812no—
416C4×Q8—17812no—
417C4×Q8C22⋊D8918yescomplemented
418C4×D4—212no—
419C4×D4—212no—
420C4×D4—8924no—
421C4×D4—35612no—
422#117—212no—
423C2×C4⋊C4—212no—
424C2×C4⋊C4—893no—
425C2×C4⋊C4—893no—
426C2×C4⋊C4—896no—
427C2×C4⋊C4—893no—
428C2×C4⋊C4—893no—
429C2×C4⋊C4—896no—
430C2×C4⋊C4—893no—
431C2×C4⋊C4—891no—
432C2×C4⋊C4—17812no—
433C2×C4⋊C4C22×D8913yes—
434C2×C4⋊C4C22×D8913yes—
435C2×C4⋊C4C22×D8911yescharacteristic
436C2×C4⋊C4C22×D8913yes—
437C2×C22⋊C4—893no—
438C2×C22⋊C4—896no—
439C2×C22⋊C4—891no—
440C2×C22⋊C4—896no—
441C2×C22⋊C4—893no—
442C2×C22⋊C4—893no—
443C2×C22⋊C4—893no—
444C2×C22⋊C4—891no—
445C2×C22⋊C4C22×D8913yes—
446C2×C22⋊C4C22×D8913yes—
447C2×C42—893no—
448C2×C42—896no—
449C2×C42—893no—
450C2×C42—893no—
451C2×C42—893no—
452C2×C42—893no—
453C2×C42—17812no—
454C2×C42C22×D8913yes—
455C2×C42C22×D8913yes—
456#95—891no—
457#95—893no—
458#95—893no—
459#95—893no—
460#95—891no—
461#95—893no—
462#95—893no—
463#95—896no—
464#95—893no—
465#95—893no—
466#95—893no—
467C24—891no—
468C24C23×D8913yes—
469#41—232no—
470C2×Q8—8912no—
471C2×Q8—17832no—
472C2×Q8—1786no—
473C2×Q8—1786no—
474C2×Q8C2×C4×D8914yes—
475C2×Q8C2×C22⋊D8914yes—
476C2×Q8#1308357414yes—
477C2×D4—26no—
478C2×D4—26no—
479C2×D4—8912no—
480C2×D4—1786no—
481C2×D4—1786no—
482C2×D4C2×C4×D89112yes—
483C22×C4—26no—
484C22×C4—26no—
485C22×C4—212no—
486C22×C4—46no—
487C22×C4—46no—
488C22×C4—891no—
489C22×C4—891no—
490C22×C4—893no—
491C22×C4—893no—
492C22×C4—8912no—
493C22×C4—891no—
494C22×C4—893no—
495C22×C4—893no—
496C22×C4—3563no—
497C22×C4—1782no—
498C22×C4—1786no—
499C22×C4—1782no—
500C22×C4—1786no—
501C22×C4—1782no—
502C22×C4—1786no—
503C22×C4—1786no—
504C22×C4C2×C4×D89112yes—
505C22×C4C23×D8911yescharacteristic
506C22×C4C23×D8913yes—
507C22×C4C23×D8911yescharacteristic
508C22×C4C23×D8911yescharacteristic
509C22×C4C23×D8913yes—
510C22×C4C23×D8913yes—
511C4⋊C4—26no—
512C4⋊C4—26no—
513C4⋊C4—22no—
514C4⋊C4—8912no—
515C4⋊C4—8912no—
516C4⋊C4—3566no—
517C4⋊C4—3563no—
518C4⋊C4—1782no—
519C4⋊C4—17812no—
520C4⋊C4—1786no—
521C4⋊C4—1786no—
522C4⋊C4C2×C22⋊D89112yes—
523C22⋊C4—26no—
524C22⋊C4—26no—
525C22⋊C4—8912no—
526C22⋊C4—8924no—
527C22⋊C4—894no—
528C22⋊C4—3566no—
529C22⋊C4—3563no—
530C22⋊C4—1786no—
531C22⋊C4—1782no—
532C22⋊C4—1786no—
533C42—26no—
534C42—8912no—
535C42—3566no—
536C42—1786no—
537C42—1786no—
538C42—1786no—
539C42—1786no—
540C42C2×C22⋊D89112yes—
541C23—23no—
542C23—23no—
543C23—23no—
544C23—23no—
545C23—23no—
546C23—23no—
547C23—892no—
548C23—892no—
549C23—892no—
550C23—1781no—
551C23—1781no—
552C23—1781no—
553C23—1781no—
554C23C22×C4×D8916yes—
555C23C24×D8911yescharacteristic
556Q8—17824no—
557Q8C4×C22⋊D89116yescomplemented
558D4—224no—
559D4—17824no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—23no—
566C2×C4—224no—
567C2×C4—21no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—23no—
573C2×C4—812no—
574C2×C4—892no—
575C2×C4—896no—
576C2×C4—896no—
577C2×C4—892no—
578C2×C4—35612no—
579C2×C4—1788no—
580C2×C4—1783no—
581C2×C4—1781no—
582C2×C4—1783no—
583C2×C4—1783no—
584C2×C4—1783no—
585C2×C4—1781no—
586C2×C4—1783no—
587C2×C4—1781no—
588C2×C4—1783no—
589C2×C4—1781no—
590C2×C4—1783no—
591C2×C4—1783no—
592C2×C4—1783no—
593C2×C4—1781no—
594C2×C4—1781no—
595C2×C4—1783no—
596C2×C4—1781no—
597C2×C4—17824no—
598C2×C4C22×C4×D8912yes—
599C2×C4C22×C4×D8916yes—
600C2×C4C22×C22⋊D8912yes—
601C2×C4C22×C22⋊D8916yes—
602C2×C4C22×C22⋊D8916yes—
603C2×C4C2×#1308357416yes—
604C22—26no—
605C22—26no—
606C22—26no—
607C22—43no—
608C22—43no—
609C22—43no—
610C22—43no—
611C22—894no—
612C22—1782no—
613C22—1782no—
614C22—1782no—
615C22—1782no—
616C22—1782no—
617C22—1782no—
618C22C23×C4×D8911yescharacteristic
619C22C23×C22⋊D8911yescharacteristic
620C22C22×#1308357411yescharacteristic
621C22H(5,𝔽2)×D8911yescharacteristic
622C222−1+4×D8911yescharacteristic
623C222−1+4×D8911yescharacteristic
624C22#1915243311yescharacteristic
625C4—22no—
626C4—22no—
627C4—26no—
628C4—41no—
629C4—43no—
630C4—3561no—
631C4—3563no—
632C4—3563no—
633C4—1786no—
634C4—1786no—
635C4—1782no—
636C4—1782no—
637C4C2×C4×C22⋊D89112yes—
638C2—46no—
639C2—1784no—
640C2C22×C4×C22⋊D8911yescharacteristic
641C2#520×D8911yescharacteristic
642C2#2603346311yescharacteristic
643C2#2603390011yescharacteristic
644C2#2603391411yescharacteristic
645C2#2603423711yescharacteristic
646C2#2603433111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC2×#520×C89C2
socle, cosocleC23×C89C25
prime radicalC2×#19152255C2
hypercenterC2×#520D89
hypocenterC89#34294
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. C2×C4×Q8×D89C2semidirect
3. #26023325C2semidirect
4. #26023264C2semidirect
5. #26023257C2semidirect
6. Q8×#13083553C2semidirect
7. C4⋊C4×C22⋊D89C2semidirect
8. #26023093C2semidirect
9. Q8×#13083551C2semidirect
10. #26022964C2semidirect
11. #26022934C2semidirect
12. C4×#15667983C2semidirect
13. #26022689C2semidirect
14. #26022617C2semidirect
15. C2×#520×C89C2semidirect
17. #19152255C4semidirect
72. C2×Q8×D89C2×C4semidirect
77. #15667999C2×C4semidirect
78. #15667991C2×C4semidirect
79. #15667983C2×C4semidirect
82. Q8×Dic89C2×C4semidirect
86. C4×C22⋊D89C2×C4semidirect
124. C2×#41×C89C2×C4semidirect
130. C4×Q8×C89D4semidirect
232. Q8×C89C4×D4semidirect
300. C2×D89#520semidirect
325. C2×#520D89semidirect
339. C89#34294semidirect
395. C2×#41C4×D89semidirect
417. C4×Q8C22⋊D89semidirect
557. Q8C4×C22⋊D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191522552
C2×C4×Q8×D892
#260233252
#260232642
#260232572
Q8×#130835532
C4⋊C4×C22⋊D892
#260230932
Q8×#130835512
#260229642
#260229342
C4×#156679832
#260226892
#260226172
C2×#520×C892
#3429489