Finite Group Databasebeta

← #36716094

Subgroups — #36716094

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×#19152259C211yesmaximal, characteristic
2C2×#19152080C211yesmaximal, characteristic, complemented
3C22×Q8×Dic89C211yesmaximal, characteristic, complemented
4#26023328C213yesmaximal, complemented
5#26023266C213yesmaximal, complemented
6#26023262C213yesmaximal, complemented
7Q8×#13083553C211yesmaximal, characteristic, complemented
8#26023148C213yesmaximal, complemented
9#26023090C211yesmaximal, characteristic, complemented
10#26022966C213yesmaximal, complemented
11#26022937C213yesmaximal, complemented
12#26022758C213yesmaximal, complemented
13#26022691C213yesmaximal, complemented
14#26022617C211yesmaximal, characteristic, complemented
15Q8×C22⋊C4×C89C211yesmaximal, characteristic, complemented
16C22×#13083575C2211yescharacteristic
17#19152259C4116yescomplemented
18C2×#15668001C2213yes—
19C2×#15667988C2213yes—
20C2×#15667986C2213yes—
21#19152080—28no—
22C2×Q8×Dic89—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×#15667972C2216yes—
32C2×#15667969C2213yes—
33C22×C4⋊Dic89C2213yes—
34C2×C4⋊C4×D89C2213yes—
35C22×C4×Dic89C2213yes—
36#19151500C2213yes—
37#19151499C2213yes—
38#19151496C2213yes—
39#19151495C2216yes—
40#19151494C2213yes—
41#19151490C2216yes—
42#19151487C2216yes—
43#19151487C2213yes—
44#19151487C2213yes—
45#19151480C2213yes—
46#19151480C2213yes—
47#19151475C2213yes—
48#19151475C2213yes—
49#19151472C2216yes—
50#19151472C2216yes—
51#19151470C2211yescharacteristic
52#19151470C2213yes—
53#19151467C2213yes—
54#19151455C2213yes—
55C4⋊C4×Dic89C2213yes—
56C4⋊C4×Dic89C2213yes—
57C4×#13083553C2213yes—
58C4×C4⋊Dic89C2213yes—
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×C418yescomplemented
72C22×D356C2313yes—
73C22×C22⋊D89C2313yes—
74C23×Dic89C2311yescharacteristic
75C23×Dic89C2313yes—
76C22×C4×D89C2313yes—
77C22×C4×D89C2316yes—
78#15668001C2×C4124yescomplemented
79#15667988C2×C4124yescomplemented
80#15667986C2×C4124yescomplemented
81Q8×Dic89—212no—
82Q8×Dic89—212no—
83Q8×Dic89—24no—
84Q8×Dic89C2×C418yescomplemented
85Q8×Dic89D418yescomplemented
86C4×D356—212no—
87#15667974—212no—
88C4×C22⋊D89C2×C4124yescomplemented
89#15667972—412no—
90#15667969—212no—
91C2×C4⋊Dic89—212no—
92C2×C4⋊Dic89C2313yes—
93C2×C4⋊Dic89C2316yes—
94C2×C4⋊Dic89C2313yes—
95C2×C4⋊Dic89C2313yes—
96C2×#13083553C2311yescharacteristic
97C2×#13083553C2311yescharacteristic
98C2×#13083553C2313yes—
99C2×#13083553C2313yes—
100C2×#13083553C2316yes—
101C2×#13083553C2313yes—
102C4⋊C4×D89—212no—
103C2×C22⋊Dic89C2311yescharacteristic
104C2×#13083551C2313yes—
105C2×#13083551C2313yes—
106C2×#13083551C2313yes—
107C2×#13083551C2313yes—
108C2×#13083551C2316yes—
109C2×#13083551C2316yes—
110C2×#13083551C2313yes—
111C2×C4×Dic89—212no—
112C2×C4×Dic89C2313yes—
113C2×C4×Dic89C2313yes—
114C2×C4×Dic89C2313yes—
115C2×C4×Dic89C2316yes—
116C2×C4×Dic89C2313yes—
117C2×C4×Dic89C2313yes—
118#15667919C2311yescharacteristic
119#15667919C2313yes—
120#15667919C2313yes—
121#15667919C2313yes—
122#15667919C2313yes—
123#15667919C2311yescharacteristic
124#15667919C2313yes—
125#15667919C2313yes—
126C22×Q8×C89—24no—
127C22×Q8×C89—24no—
128C22×Q8×C89C2×C418yescomplemented
129C22×Q8×C89C2313yes—
130C22×Q8×C89C2311yescharacteristic
131C23×C4×C89C2313yes—
132C4×Q8×C89—24no—
133C4×Q8×C89—24no—
134C2×C4⋊C4×C89C2313yes—
135C2×C4⋊C4×C89C2313yes—
136C2×C22⋊C4×C89C2311yescharacteristic
137C2×C22⋊C4×C89C2313yes—
138C2×C42×C89C2313yes—
139C2×C42×C89C2313yes—
140#95×C89C2313yes—
141#95×C89C2313yes—
142#95×C89C2316yes—
143C23×D89C2413yes—
144#13083575—232no—
145C2×D356—26no—
146C2×D356—26no—
147C2×D356C22×C4112yes—
148C2×C22⋊D89—26no—
149C2×C22⋊D89—26no—
150C2×C22⋊D89C22×C4112yes—
151C22×Dic89—22no—
152C22×Dic89—22no—
153C22×Dic89—22no—
154C22×Dic89—26no—
155C22×Dic89—26no—
156C22×Dic89—43no—
157C22×Dic89C2413yes—
158C22×Dic89C2411yescharacteristic
159C22×Dic89C2413yes—
160C22×Dic89C2411yescharacteristic
161C22×Dic89C2413yes—
162C22×Dic89C2411yescharacteristic
163C22×Dic89C2411yescharacteristic
164C2×C4×D89—26no—
165C2×C4×D89—26no—
166C2×C4×D89—212no—
167C2×C4×D89—46no—
168C2×C4×D89—46no—
169C2×C4×D89C22×C4112yes—
170C4⋊Dic89—26no—
171C4⋊Dic89—212no—
172C4⋊Dic89C22×C4112yes—
173C4⋊Dic89C2×D4112yes—
174#13083553—22no—
175#13083553—26no—
176#13083553—22no—
177#13083553—46no—
178#13083553—43no—
179#13083553C22×C4112yes—
180C22⋊Dic89C22×C414yes—
181#13083551—26no—
182#13083551—26no—
183#13083551—26no—
184#13083551—46no—
185#13083551—43no—
186#13083551C22×C4112yes—
187#13083551C22×C4124yes—
188C4×Dic89—26no—
189C4×Dic89—26no—
190C4×Dic89—26no—
191C4×Dic89—46no—
192C4×Dic89C22×C4112yes—
193C4×Dic89C2×D4112yes—
194C24×C89C2411yescharacteristic
195C2×Q8×C89—232no—
196C2×Q8×C89—26no—
197C2×Q8×C89—26no—
198C2×Q8×C89C22×C4112yes—
199C2×Q8×C89C22×C414yes—
200C2×Q8×C89C2×D414yes—
201C2×Q8×C89#4114yes—
202C22×C4×C89—26no—
203C22×C4×C89—26no—
204C22×C4×C89C22×C4112yes—
205C22×C4×C89C2413yes—
206C22×C4×C89C2411yescharacteristic
207C22×C4×C89C2411yescharacteristic
208C22×C4×C89C2413yes—
209C22×C4×C89C2413yes—
210C22×C4×C89C2413yes—
211C4⋊C4×C89—26no—
212C4⋊C4×C89—26no—
213C22⋊C4×C89—26no—
214C22⋊C4×C89—22no—
215C42×C89—26no—
216C42×C89—26no—
217C23×C89—21no—
218C23×C89—21no—
219C23×C89—21no—
220C23×C89—21no—
221C23×C89C23×C412yes—
222C23×C89H(5,𝔽2)12yes—
223C23×C892−1+412yes—
224C23×C89C2511yescharacteristic
225C22×D89—23no—
226C22×D89—23no—
227C22×D89—23no—
228C22×D89—23no—
229C22×D89—23no—
230C22×D89—23no—
231C22×D89C23×C416yes—
232Q8×C89—224no—
233Q8×C89C4×D4116yescomplemented
234C2×C4×C89—23no—
235C2×C4×C89—23no—
236C2×C4×C89—224no—
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—21no—
245C2×C4×C89—23no—
246C2×C4×C89—21no—
247C2×C4×C89—21no—
248C2×C4×C89—23no—
249C2×C4×C89C23×C416yes—
250C2×C4×C89C23×C416yes—
251C2×C4×C89C22×D416yes—
252C2×C4×C89C2×#4116yes—
253C22⋊D89—224no—
254C2×Dic89—28no—
255C2×Dic89—21no—
256C2×Dic89—21no—
257C2×Dic89—21no—
258C2×Dic89—23no—
259C2×Dic89—23no—
260C2×Dic89—21no—
261C2×Dic89—21no—
262C2×Dic89—21no—
263C2×Dic89—21no—
264C2×Dic89—23no—
265C2×Dic89—21no—
266C2×Dic89—23no—
267C2×Dic89—23no—
268C2×Dic89—23no—
269C2×Dic89—23no—
270C2×Dic89—412no—
271C2×Dic89C23×C412yes—
272C2×Dic89C23×C416yes—
273C2×Dic89C23×C412yes—
274C2×Dic89C22×D412yes—
275C2×Dic89C22×D416yes—
276C2×Dic892−1+412yes—
277D356—224no—
278C4×D89—224no—
279C4×D89—812no—
280C22×C89—22no—
281C22×C89—22no—
282C22×C89—22no—
283C22×C89—22no—
284C22×C89—22no—
285C22×C89—22no—
286C22×C89#52014yescomplemented
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—26no—
295C2×D89—26no—
296C2×D89—26no—
297C2×D89—43no—
298C2×D89—43no—
299C2×D89—43no—
300C2×D89—43no—
301C4×C89—26no—
302C4×C89—41no—
303C4×C89—43no—
304C4×C89—41no—
305C4×C89—43no—
306C4×C89C2×C4×D4112yes—
307Dic89—22no—
308Dic89—26no—
309Dic89—22no—
310Dic89—22no—
311Dic89—22no—
312Dic89—26no—
313Dic89—43no—
314#34294—891nomaximal, Sylow
315C2×C89—24no—
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—46no—
324C2×#548—891no—
325C2×#520—891no—
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⋊C4D8911yescharacteristic, complemented
334#2326—893no—
335#2323—893no—
336C4×#121—893no—
337#2290—893no—
338#2276—891no—
339C89#3429411yescharacteristic, complemented, Sylow
340C22×#41—891no—
341C23×Q8C2×D8911yescharacteristic
342#548—8916no—
343C2×#124—893no—
344C2×C22⋊Q8—893no—
345C2×#121—893no—
346#520—1788no—
347C2×C4×Q8—891no—
348C2×C4×Q8—891no—
349C2×C4×Q8—893no—
350C2×C4×Q8—891no—
351C2×C4×Q8—893no—
352C2×C4×Q8—1788no—
353C2×C4×Q8C2×D8911yescharacteristic
354C2×C4×Q8C2×D8911yescharacteristic
355C2×C4×D4—893no—
356C2×C4×D4—893no—
357C2×C4×D4—896no—
358C2×C4×D4—893no—
359C2×#117—893no—
360C22×C4⋊C4—893no—
361C22×C4⋊C4—893no—
362C22×C42—893no—
363#391C2×D8913yes—
364#391C2×D8913yes—
365#390—893no—
366#390—893no—
367#389—893no—
368#389—893no—
369#389—896no—
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—893no—
379#385—891no—
380#385—896no—
381#385—896no—
382#384—893no—
383#380—893no—
384#380C2×D8913yes—
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⋊C4C2×D8913yes—
392#376—893no—
393C2×#41—898no—
394C2×#41—1784no—
395C2×#41—1784no—
396C22×Q8—24no—
397C22×Q8—24no—
398C22×Q8C4×D8918yescomplemented
399C22×Q8C22×D8913yes—
400C22×Q8C22×D8911yescharacteristic
401C22×D4—893no—
402C22×D4—893no—
403C23×C4—893no—
404C23×C4—893no—
405C23×C4—891no—
406C23×C4—896no—
407C23×C4C22×D8913yes—
408#124—8924no—
409C22⋊Q8—8924no—
410#121—8924no—
411C4×Q8—24no—
412C4×Q8—24no—
413C4×Q8—898no—
414C4×Q8—898no—
415C4×Q8—17812no—
416C4×Q8—17812no—
417C4×Q8—1784no—
418C4×D4—8924no—
419C4×D4—35612no—
420C4×D4—17812no—
421C4×D4—17812no—
422#117—17812no—
423C2×C4⋊C4—893no—
424C2×C4⋊C4—893no—
425C2×C4⋊C4—893no—
426C2×C4⋊C4—896no—
427C2×C4⋊C4—893no—
428C2×C4⋊C4—893no—
429C2×C4⋊C4—893no—
430C2×C4⋊C4—896no—
431C2×C4⋊C4—891no—
432C2×C4⋊C4—891no—
433C2×C4⋊C4—17812no—
434C2×C4⋊C4—17812no—
435C2×C4⋊C4C22×D8913yes—
436C2×C4⋊C4C22×D8913yes—
437C2×C22⋊C4—891no—
438C2×C22⋊C4—893no—
439C2×C22⋊C4—896no—
440C2×C22⋊C4—896no—
441C2×C22⋊C4—893no—
442C2×C22⋊C4—893no—
443C2×C22⋊C4—893no—
444C2×C22⋊C4—893no—
445C2×C22⋊C4C22×D8913yes—
446C2×C22⋊C4C22×D8911yescharacteristic
447C2×C42—893no—
448C2×C42—893no—
449C2×C42—896no—
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—893no—
464#95C22×D8916yes—
465#95C22×D8913yes—
466#95C22×D8913yes—
467C24—893no—
468C24C23×D8911yescharacteristic
469#41—17832no—
470C2×Q8—232no—
471C2×Q8—26no—
472C2×Q8—26no—
473C2×Q8C2×C4×D89112yes—
474C2×Q8C2×C4×D8914yes—
475C2×Q8D4×D8914yes—
476C2×Q8#1308357314yes—
477C2×D4—8912no—
478C2×D4—8912no—
479C2×D4—1786no—
480C2×D4—1786no—
481C2×D4—1786no—
482C2×D4—1786no—
483C22×C4—26no—
484C22×C4—26no—
485C22×C4—891no—
486C22×C4—891no—
487C22×C4—891no—
488C22×C4—893no—
489C22×C4—891no—
490C22×C4—893no—
491C22×C4—8912no—
492C22×C4—893no—
493C22×C4—3566no—
494C22×C4—3566no—
495C22×C4—3563no—
496C22×C4—17812no—
497C22×C4—1786no—
498C22×C4—1782no—
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×D8913yes—
509C22×C4C23×D8913yes—
510C22×C4C23×D8913yes—
511C4⋊C4—26no—
512C4⋊C4—26no—
513C4⋊C4—8912no—
514C4⋊C4—8912no—
515C4⋊C4—8912no—
516C4⋊C4—3566no—
517C4⋊C4—3563no—
518C4⋊C4—1786no—
519C4⋊C4—17812no—
520C4⋊C4—1782no—
521C4⋊C4—1782no—
522C4⋊C4—1786no—
523C22⋊C4—26no—
524C22⋊C4—22no—
525C22⋊C4—894no—
526C22⋊C4—8912no—
527C22⋊C4—8924no—
528C22⋊C4—3566no—
529C22⋊C4—3563no—
530C22⋊C4—1786no—
531C22⋊C4—1786no—
532C22⋊C4—1786no—
533C42—26no—
534C42—26no—
535C42—8912no—
536C42—8912no—
537C42—3566no—
538C42—1786no—
539C42—1786no—
540C42—1786no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—896no—
546C23—1783no—
547C23—1783no—
548C23—1783no—
549C23—1783no—
550C23—1783no—
551C23—1783no—
552C23C22×C4×D8912yes—
553C23#1566805812yes—
554C23#1566806012yes—
555C23C24×D8911yescharacteristic
556Q8—224no—
557Q8#15667972116yescomplemented
558D4—17824no—
559D4—17824no—
560C2×C4—23no—
561C2×C4—23no—
562C2×C4—224no—
563C2×C4—21no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—23no—
575C2×C4—892no—
576C2×C4—896no—
577C2×C4—892no—
578C2×C4—892no—
579C2×C4—896no—
580C2×C4—892no—
581C2×C4—35612no—
582C2×C4—1783no—
583C2×C4—1783no—
584C2×C4—1783no—
585C2×C4—1788no—
586C2×C4—1781no—
587C2×C4—1781no—
588C2×C4—1781no—
589C2×C4—1783no—
590C2×C4—1783no—
591C2×C4—1781no—
592C2×C4—1781no—
593C2×C4—1781no—
594C2×C4—1781no—
595C2×C4—1783no—
596C2×C4—1781no—
597C2×C4—1783no—
598C2×C4—17824no—
599C2×C4—71212no—
600C2×C4C22×C4×D8916yes—
601C2×C4C22×C4×D8916yes—
602C2×C4C2×D4×D8916yes—
603C2×C4C2×#1308357316yes—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—3563no—
611C22—3563no—
612C22—3563no—
613C22—3563no—
614C22—1786no—
615C22—1786no—
616C22—1786no—
617C22#1915208014yescomplemented
618C22C23×C4×D8911yescharacteristic
619C22C22×D4×D8911yescharacteristic
620C22C22×#1308357311yescharacteristic
621C22C2×#1566805811yescharacteristic
622C222−1+4×D8911yescharacteristic
623C22C2×#1566806011yescharacteristic
624C22#1915243511yescharacteristic
625C4—26no—
626C4—41no—
627C4—43no—
628C4—41no—
629C4—43no—
630C4—3563no—
631C4—1782no—
632C4—1786no—
633C4—1782no—
634C4—1782no—
635C4—1782no—
636C4—1786no—
637C4C2×#15667972112yes—
638C2—24no—
639C2—3566no—
640C2C22×#1566797211yescharacteristic
641C2C2×#1915208011yescharacteristic
642C2#2603346111yescharacteristic
643C2#2603390211yescharacteristic
644C2#2603391811yescharacteristic
645C2#2603424611yescharacteristic
646C2#2603432811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingQ8×C22⋊C4×C89C2
socle, cosocleC23×C89C25
prime radicalC2×#19152259C2
hypercenterQ8×C22⋊C4D89
hypocenterC89#34294
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×#19152080C2semidirect
3. C22×Q8×Dic89C2semidirect
4. #26023328C2semidirect
5. #26023266C2semidirect
6. #26023262C2semidirect
7. Q8×#13083553C2semidirect
8. #26023148C2semidirect
9. #26023090C2semidirect
10. #26022966C2semidirect
11. #26022937C2semidirect
12. #26022758C2semidirect
13. #26022691C2semidirect
14. #26022617C2semidirect
15. Q8×C22⋊C4×C89C2semidirect
17. #19152259C4semidirect
71. C2×#13083575C2×C4semidirect
78. #15668001C2×C4semidirect
79. #15667988C2×C4semidirect
80. #15667986C2×C4semidirect
84. Q8×Dic89C2×C4semidirect
85. Q8×Dic89D4semidirect
88. C4×C22⋊D89C2×C4semidirect
128. C22×Q8×C89C2×C4semidirect
233. Q8×C89C4×D4semidirect
286. C22×C89#520semidirect
333. Q8×C22⋊C4D89semidirect
339. C89#34294semidirect
398. C22×Q8C4×D89semidirect
557. Q8#15667972semidirect
617. C22#19152080semidirect

Maximal subgroups

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

subgroupindex
C2×#191522592
C2×#191520802
C22×Q8×Dic892
#260233282
#260232662
#260232622
Q8×#130835532
#260231482
#260230902
#260229662
#260229372
#260227582
#260226912
#260226172
Q8×C22⋊C4×C892
#3429489