Finite Group Databasebeta

← #36716093

Subgroups — #36716093

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×#19152259C211yesmaximal, characteristic
2C2×#19152076C211yesmaximal, characteristic, complemented
3C2×C4×#13083575C211yesmaximal, characteristic, complemented
4#26023328C213yesmaximal, complemented
5#26023288C213yesmaximal, complemented
6#26023284C213yesmaximal, complemented
7#26023196C213yesmaximal, complemented
8Q8×#13083553C211yesmaximal, characteristic, complemented
9#26023090C211yesmaximal, characteristic, complemented
10#26022966C213yesmaximal, complemented
11#26022941C213yesmaximal, complemented
12#26022790C213yesmaximal, complemented
13#26022698C213yesmaximal, complemented
14#26022617C211yesmaximal, characteristic, complemented
15Q8×C22⋊C4×C89C211yesmaximal, characteristic, complemented
16C22×#13083575C2211yescharacteristic
17#19152259C4116yescomplemented
18C2×#15668001C2213yes—
19C2×#15667988C2213yes—
20C2×#15667986C2213yes—
21#19152076—28no—
22C4×#13083575—28no—
23C2×Q8×Dic89C2213yes—
24C2×Q8×Dic89C2213yes—
25C2×Q8×Dic89C2211yescharacteristic
26C2×Q8×Dic89C2211yescharacteristic
27C2×Q8×Dic89C2211yescharacteristic
28C2×C4×D356C2213yes—
29C2×#15667974C2213yes—
30C2×C4×C22⋊D89C2213yes—
31C2×#15667971C2213yes—
32C2×#15667971C2213yes—
33C2×#15667970C2213yes—
34C2×C42×D89C2213yes—
35#19151500C2213yes—
36#19151499C2213yes—
37#19151498C2216yes—
38#19151496C2213yes—
39#19151494C2213yes—
40#19151492C2213yes—
41#19151492C2211yescharacteristic
42#19151490C2216yes—
43#19151487C2216yes—
44#19151487C2216yes—
45#19151485C2213yes—
46#19151485C2213yes—
47#19151474C2213yes—
48#19151474C2213yes—
49#19151472C2216yes—
50#19151472C2213yes—
51#19151472C2213yes—
52#19151467C2213yes—
53#19151455C2213yes—
54C4⋊C4×Dic89C2213yes—
55C4⋊C4×Dic89C2213yes—
56C4×#13083553C2213yes—
57C4×C4⋊Dic89C2213yes—
58#19151449C2216yes—
59C4×#13083551C2213yes—
60#19151445C2216yes—
61#19151444C2213yes—
62C23×Q8×C89C2211yescharacteristic
63C2×C4×Q8×C89C2211yescharacteristic
64C2×C4×Q8×C89C2211yescharacteristic
65#391×C89C2213yes—
66#391×C89C2213yes—
67#380×C89C2213yes—
68C4×C22⋊C4×C89C2213yes—
69C2×#13083575—24no—
70C2×#13083575—24no—
71C2×#13083575C2×C418yes—
72C22×D356C2313yes—
73C22×C22⋊D89C2313yes—
74C22×C4×D89C2313yes—
75C22×C4×D89C2313yes—
76C22×C4×D89C2313yes—
77#15668001C2×C4124yescomplemented
78#15667988C2×C4124yescomplemented
79#15667986C2×C4124yescomplemented
80Q8×Dic89—212no—
81Q8×Dic89—212no—
82Q8×Dic89—24no—
83Q8×Dic89C2×C418yescomplemented
84Q8×Dic89D418yescomplemented
85C4×D356—212no—
86#15667974—212no—
87C4×C22⋊D89C2×C4124yescomplemented
88#15667971—212no—
89#15667971—212no—
90#15667970—212no—
91C2×C4⋊Dic89C2313yes—
92C2×C4⋊Dic89C2313yes—
93C2×C4⋊Dic89C2313yes—
94C2×C4⋊Dic89C2313yes—
95C2×C4⋊Dic89C2313yes—
96C2×#13083553C2311yescharacteristic
97C2×#13083553C2316yes—
98C2×#13083553C2313yes—
99C2×#13083553C2313yes—
100C2×C22⋊Dic89C2311yescharacteristic
101C2×C22⋊Dic89C2313yes—
102C2×C22⋊Dic89C2311yescharacteristic
103C2×#13083551C2316yes—
104C2×#13083551C2316yes—
105C2×#13083551C2313yes—
106C2×#13083551C2313yes—
107C2×#13083551C2316yes—
108C2×#13083551C2313yes—
109C2×C4×Dic89C2316yes—
110C2×C4×Dic89C2313yes—
111C2×C4×Dic89C2313yes—
112C2×C4×Dic89C2311yescharacteristic
113C2×C4×Dic89C2316yes—
114C2×C4×Dic89C2313yes—
115C2×C4×Dic89C2313yes—
116C42×D89—212no—
117#15667919C2311yescharacteristic
118#15667919C2313yes—
119#15667919C2313yes—
120#15667919C2313yes—
121#15667919C2313yes—
122#15667919C2311yescharacteristic
123#15667919C2313yes—
124#15667919C2313yes—
125C22×Q8×C89—24no—
126C22×Q8×C89—24no—
127C22×Q8×C89C2×C418yescomplemented
128C22×Q8×C89C2313yes—
129C22×Q8×C89C2311yescharacteristic
130C23×C4×C89C2313yes—
131C4×Q8×C89—24no—
132C4×Q8×C89—24no—
133C2×C4⋊C4×C89C2313yes—
134C2×C4⋊C4×C89C2313yes—
135C2×C22⋊C4×C89C2311yescharacteristic
136C2×C22⋊C4×C89C2313yes—
137C2×C42×C89C2313yes—
138C2×C42×C89C2313yes—
139#95×C89C2313yes—
140#95×C89C2313yes—
141#95×C89C2316yes—
142C23×D89C2413yes—
143#13083575—232no—
144C2×D356—26no—
145C2×D356—26no—
146C2×D356C22×C4112yes—
147C2×C22⋊D89—43no—
148C2×C22⋊D89—43no—
149C2×C22⋊D89C22×C4112yes—
150C22×Dic89C2411yescharacteristic
151C22×Dic89C2413yes—
152C22×Dic89C2411yescharacteristic
153C22×Dic89C2413yes—
154C22×Dic89C2411yescharacteristic
155C22×Dic89C2411yescharacteristic
156C22×Dic89C2413yes—
157C2×C4×D89—26no—
158C2×C4×D89—26no—
159C2×C4×D89—26no—
160C2×C4×D89—26no—
161C2×C4×D89—26no—
162C2×C4×D89—43no—
163C2×C4×D89—43no—
164C2×C4×D89—43no—
165C2×C4×D89C22×C4112yes—
166C4⋊Dic89—26no—
167C4⋊Dic89—26no—
168C4⋊Dic89—26no—
169C4⋊Dic89C22×C4112yes—
170C4⋊Dic89C2×D4112yes—
171#13083553—22no—
172#13083553—43no—
173#13083553—46no—
174#13083553C22×C4112yes—
175C22⋊Dic89—26no—
176C22⋊Dic89—22no—
177C22⋊Dic89C22×C414yes—
178#13083551—212no—
179#13083551—43no—
180#13083551—46no—
181#13083551—43no—
182#13083551C22×C4112yes—
183#13083551C22×C4124yes—
184C4×Dic89—26no—
185C4×Dic89—26no—
186C4×Dic89—212no—
187C4×Dic89—22no—
188C4×Dic89—46no—
189C4×Dic89C22×C4112yes—
190C4×Dic89C2×D4112yes—
191C24×C89C2411yescharacteristic
192C2×Q8×C89—232no—
193C2×Q8×C89—26no—
194C2×Q8×C89—26no—
195C2×Q8×C89C22×C4112yes—
196C2×Q8×C89C22×C414yes—
197C2×Q8×C89C2×D414yes—
198C2×Q8×C89#4114yes—
199C22×C4×C89—26no—
200C22×C4×C89—26no—
201C22×C4×C89C22×C4112yes—
202C22×C4×C89C2413yes—
203C22×C4×C89C2411yescharacteristic
204C22×C4×C89C2413yes—
205C22×C4×C89C2413yes—
206C22×C4×C89C2411yescharacteristic
207C22×C4×C89C2413yes—
208C4⋊C4×C89—26no—
209C4⋊C4×C89—26no—
210C22⋊C4×C89—22no—
211C22⋊C4×C89—43no—
212C42×C89—26no—
213C42×C89—26no—
214C23×C89—21no—
215C23×C89—21no—
216C23×C89—21no—
217C23×C89—21no—
218C23×C89—21no—
219C23×C89—21no—
220C23×C89C23×C412yes—
221C23×C89C2511yescharacteristic
222C22×D89—23no—
223C22×D89—23no—
224C22×D89—23no—
225C22×D89—23no—
226C22×D89—23no—
227C22×D89—23no—
228C22×D89C23×C416yes—
229Q8×C89—224no—
230Q8×C89C4×D4116yes—
231C2×C4×C89—21no—
232C2×C4×C89—23no—
233C2×C4×C89—23no—
234C2×C4×C89—21no—
235C2×C4×C89—224no—
236C2×C4×C89—23no—
237C2×C4×C89—21no—
238C2×C4×C89—23no—
239C2×C4×C89—21no—
240C2×C4×C89—23no—
241C2×C4×C89—23no—
242C2×C4×C89—23no—
243C2×C4×C89—21no—
244C2×C4×C89—23no—
245C2×C4×C89C23×C416yes—
246C2×C4×C89C23×C416yes—
247C2×C4×C89C22×D416yes—
248C2×C4×C89C2×#4116yes—
249C2×C4×C892−1+412yes—
250C22⋊D89—412no—
251C2×Dic89—21no—
252C2×Dic89—21no—
253C2×Dic89—21no—
254C2×Dic89—23no—
255C2×Dic89—23no—
256C2×Dic89—23no—
257C2×Dic89—21no—
258C2×Dic89—23no—
259C2×Dic89—21no—
260C2×Dic89—21no—
261C2×Dic89—23no—
262C2×Dic89—21no—
263C2×Dic89—21no—
264C2×Dic89—21no—
265C2×Dic89—23no—
266C2×Dic89—23no—
267C2×Dic89C23×C412yes—
268C2×Dic89C23×C416yes—
269C2×Dic89C23×C412yes—
270C2×Dic89C22×D416yes—
271C2×Dic89C22×D412yes—
272D356—224no—
273C4×D89—224no—
274C4×D89—412no—
275C4×D89—412no—
276C22×C89—22no—
277C22×C89—22no—
278C22×C89—22no—
279C22×C89—41no—
280C22×C89—41no—
281C22×C89—41no—
282C22×C89—41no—
283C22×C89C24×C411yescharacteristic
284C22×C89C23×D411yescharacteristic
285C22×C89C22×#4111yescharacteristic
286C22×C89C2×2−1+411yescharacteristic
287C22×C89#58511yescharacteristic
288C22×C89#58511yescharacteristic
289C22×C89#58511yescharacteristic
290C2×D89—26no—
291C2×D89—26no—
292C2×D89—26no—
293C2×D89—43no—
294C2×D89—43no—
295C2×D89—43no—
296C2×D89—43no—
297C4×C89—26no—
298C4×C89—22no—
299C4×C89—43no—
300C4×C89—41no—
301C4×C89—43no—
302C4×C89C2×C4×D4112yes—
303Dic89—22no—
304Dic89—26no—
305Dic89—22no—
306Dic89—22no—
307Dic89—26no—
308Dic89—43no—
309Dic89—41no—
310#34293—891nomaximal, Sylow
311C2×C89—42no—
312C2×C89C22×C4×D411yescharacteristic
313C2×C89#342811yescharacteristic
314C2×C89C4×2−1+411yescharacteristic
315C2×C89#348711yescharacteristic
316C2×C89#348911yescharacteristic
317C2×C89#351811yescharacteristic
318C2×C89#352411yescharacteristic
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⋊C4D8911yescharacteristic, complemented
330#2326—893no—
331#2324—893no—
332C4×C22⋊Q8—893no—
333#2290—893no—
334#2276—891no—
335C89#3429311yescharacteristic, complemented, Sylow
336C22×#41—891no—
337C23×Q8C2×D8911yescharacteristic
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—891no—
348C2×C4×Q8—893no—
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#391C2×D8913yes—
359#391C2×D8913yes—
360#390—893no—
361#390—896no—
362#390—893no—
363#389—893no—
364#389—893no—
365#388—893no—
366#388—891no—
367#388—896no—
368#388—896no—
369#388—896no—
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#380C2×D8913yes—
380C4×C4⋊C4—893no—
381C4×C4⋊C4—893no—
382C4×C4⋊C4—893no—
383C4×C4⋊C4—893no—
384C4×C4⋊C4—896no—
385C4×C22⋊C4—893no—
386C4×C22⋊C4—896no—
387C4×C22⋊C4C2×D8913yes—
388#376—893no—
389C2×#41—898no—
390C2×#41—1784no—
391C2×#41—1784no—
392C22×Q8—24no—
393C22×Q8—24no—
394C22×Q8C4×D8918yescomplemented
395C22×Q8C22×D8913yes—
396C22×Q8C22×D8911yescharacteristic
397C22×D4—893no—
398C22×D4—893no—
399C23×C4—893no—
400C23×C4—893no—
401C23×C4—893no—
402C23×C4C22×D8913yes—
403#124—8924no—
404C22⋊Q8—8924no—
405#121—8924no—
406C4×Q8—24no—
407C4×Q8—24no—
408C4×Q8—898no—
409C4×Q8—898no—
410C4×Q8—17812no—
411C4×Q8—17812no—
412C4×Q8—1784no—
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—893no—
422C2×C4⋊C4—893no—
423C2×C4⋊C4—896no—
424C2×C4⋊C4—893no—
425C2×C4⋊C4—893no—
426C2×C4⋊C4—893no—
427C2×C4⋊C4—891no—
428C2×C4⋊C4C22×D8913yes—
429C2×C4⋊C4C22×D8913yes—
430C2×C22⋊C4—891no—
431C2×C22⋊C4—893no—
432C2×C22⋊C4—896no—
433C2×C22⋊C4—896no—
434C2×C22⋊C4—893no—
435C2×C22⋊C4—893no—
436C2×C22⋊C4—891no—
437C2×C22⋊C4—893no—
438C2×C22⋊C4—896no—
439C2×C22⋊C4C22×D8911yescharacteristic
440C2×C22⋊C4C22×D8913yes—
441C2×C42—893no—
442C2×C42—893no—
443C2×C42—896no—
444C2×C42—893no—
445C2×C42—896no—
446C2×C42—893no—
447C2×C42—891no—
448C2×C42—17812no—
449C2×C42C22×D8913yes—
450C2×C42C22×D8913yes—
451#95—891no—
452#95—893no—
453#95—893no—
454#95—893no—
455#95—891no—
456#95—893no—
457#95—893no—
458#95—893no—
459#95C22×D8913yes—
460#95C22×D8913yes—
461#95C22×D8916yes—
462C24—893no—
463C24C23×D8911yescharacteristic
464#41—17832no—
465C2×Q8—232no—
466C2×Q8—26no—
467C2×Q8—26no—
468C2×Q8C2×C4×D89112yes—
469C2×Q8C2×C4×D8914yes—
470C2×Q8D4×D8914yes—
471C2×Q8#1308357314yes—
472C2×D4—8912no—
473C2×D4—8912no—
474C2×D4—3563no—
475C2×D4—3563no—
476C2×D4—1786no—
477C2×D4—1786no—
478C22×C4—26no—
479C22×C4—26no—
480C22×C4—891no—
481C22×C4—891no—
482C22×C4—891no—
483C22×C4—893no—
484C22×C4—893no—
485C22×C4—8912no—
486C22×C4—891no—
487C22×C4—893no—
488C22×C4—3563no—
489C22×C4—3563no—
490C22×C4—3563no—
491C22×C4—1786no—
492C22×C4—1786no—
493C22×C4—1786no—
494C22×C4—1786no—
495C22×C4—1786no—
496C22×C4C2×C4×D89112yes—
497C22×C4C23×D8911yescharacteristic
498C22×C4C23×D8913yes—
499C22×C4C23×D8913yes—
500C22×C4C23×D8913yes—
501C22×C4C23×D8911yescharacteristic
502C22×C4C23×D8913yes—
503C4⋊C4—26no—
504C4⋊C4—26no—
505C4⋊C4—8912no—
506C4⋊C4—8912no—
507C4⋊C4—8912no—
508C4⋊C4—3563no—
509C4⋊C4—3566no—
510C4⋊C4—1786no—
511C4⋊C4—1786no—
512C4⋊C4—1786no—
513C4⋊C4—1782no—
514C22⋊C4—22no—
515C22⋊C4—43no—
516C22⋊C4—894no—
517C22⋊C4—8912no—
518C22⋊C4—8924no—
519C22⋊C4—3566no—
520C22⋊C4—3563no—
521C22⋊C4—3563no—
522C22⋊C4—17812no—
523C22⋊C4—1782no—
524C22⋊C4—1786no—
525C42—26no—
526C42—26no—
527C42—8912no—
528C42—8912no—
529C42—3566no—
530C42—1782no—
531C42—1786no—
532C42—17812no—
533C42—1786no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—896no—
541C23—1783no—
542C23—1783no—
543C23—1783no—
544C23—1783no—
545C23—1783no—
546C23—1783no—
547C23C22×C4×D8912yes—
548C23C24×D8911yescharacteristic
549Q8—224no—
550Q8#15667972116yes—
551D4—35612no—
552D4—17824no—
553C2×C4—21no—
554C2×C4—23no—
555C2×C4—23no—
556C2×C4—21no—
557C2×C4—224no—
558C2×C4—23no—
559C2×C4—21no—
560C2×C4—23no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—23no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—23no—
567C2×C4—892no—
568C2×C4—896no—
569C2×C4—892no—
570C2×C4—892no—
571C2×C4—896no—
572C2×C4—35612no—
573C2×C4—35612no—
574C2×C4—1781no—
575C2×C4—1783no—
576C2×C4—1781no—
577C2×C4—1783no—
578C2×C4—1781no—
579C2×C4—1783no—
580C2×C4—1781no—
581C2×C4—1781no—
582C2×C4—1781no—
583C2×C4—1783no—
584C2×C4—1781no—
585C2×C4—1781no—
586C2×C4—1783no—
587C2×C4—1783no—
588C2×C4—1783no—
589C2×C4—1781no—
590C2×C4—17824no—
591C2×C4C22×C4×D8916yes—
592C2×C4C22×C4×D8916yes—
593C2×C4C2×D4×D8916yes—
594C2×C4C2×#1308357316yes—
595C2×C4#1566806012yes—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—3563no—
604C22—3563no—
605C22—3563no—
606C22—3563no—
607C22—1786no—
608C22—1786no—
609C22—1786no—
610C22C23×C4×D8911yescharacteristic
611C22C22×D4×D8911yescharacteristic
612C22C22×#1308357311yescharacteristic
613C22C2×#1566806011yescharacteristic
614C22#1915243411yescharacteristic
615C22#1915243511yescharacteristic
616C22#1915243511yescharacteristic
617C4—22no—
618C4—26no—
619C4—41no—
620C4—43no—
621C4—43no—
622C4—3561no—
623C4—3563no—
624C4—1786no—
625C4—1782no—
626C4—1786no—
627C4—1782no—
628C4—1782no—
629C4C2×#15667972112yes—
630C2—42no—
631C2—3566no—
632C2C22×#1566797211yescharacteristic
633C2#2603345411yescharacteristic
634C2C4×#1566806011yescharacteristic
635C2#2603390911yescharacteristic
636C2#2603392911yescharacteristic
637C2#2603426411yescharacteristic
638C2#2603435411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingQ8×C22⋊C4×C89C2
socle, cosocleC23×C89C25
prime radicalC2×#19152259C2
hypercenterQ8×C22⋊C4D89
hypocenterC89#34293
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#19152076C2semidirect
3. C2×C4×#13083575C2semidirect
4. #26023328C2semidirect
5. #26023288C2semidirect
6. #26023284C2semidirect
7. #26023196C2semidirect
8. Q8×#13083553C2semidirect
9. #26023090C2semidirect
10. #26022966C2semidirect
11. #26022941C2semidirect
12. #26022790C2semidirect
13. #26022698C2semidirect
14. #26022617C2semidirect
15. Q8×C22⋊C4×C89C2semidirect
17. #19152259C4semidirect
77. #15668001C2×C4semidirect
78. #15667988C2×C4semidirect
79. #15667986C2×C4semidirect
83. Q8×Dic89C2×C4semidirect
84. Q8×Dic89D4semidirect
87. C4×C22⋊D89C2×C4semidirect
127. C22×Q8×C89C2×C4semidirect
329. Q8×C22⋊C4D89semidirect
335. C89#34293semidirect
394. C22×Q8C4×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191522592
C2×#191520762
C2×C4×#130835752
#260233282
#260232882
#260232842
#260231962
Q8×#130835532
#260230902
#260229662
#260229412
#260227902
#260226982
#260226172
Q8×C22⋊C4×C892
#3429389