Finite Group Databasebeta

← #36714248

Subgroups — #36714248

Order 22784.

3232 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152260C211yesmaximal, characteristic
2C2×C4×#13083575C211yesmaximal, characteristic, complemented
3C2×C4×Q8×D89C211yesmaximal, characteristic, complemented
4#26023321C213yesmaximal, complemented
5#26023278C213yesmaximal, complemented
6#26023267C213yesmaximal, complemented
7#26023178C213yesmaximal, complemented
8#26023090C211yesmaximal, characteristic, complemented
9Q8×#13083551C211yesmaximal, characteristic, complemented
10#26022791C213yesmaximal, complemented
11#26022782C213yesmaximal, complemented
12#26022760C213yesmaximal, complemented
13C4×Q8×Dic89C211yesmaximal, characteristic, complemented
14C4×#15667974C213yesmaximal, complemented
15C89×#2351C211yesmaximal, characteristic, complemented
16C22×#13083575C2211yescharacteristic
17C22×Q8×D89C2211yescharacteristic
18#19152260C4116yescomplemented
19C2×#15668000C2213yes—
20C2×#15667992C2213yes—
21C2×#15667987C2213yes—
22C4×#13083575—28no—
23C2×Q8×Dic89C2211yescharacteristic
24C2×Q8×Dic89C2211yescharacteristic
25C2×C4×Dic178C2213yes—
26C2×#15667977C2213yes—
27C4×Q8×D89—28no—
28C2×C4×D356C2213yes—
29C2×C4×D356C2213yes—
30C2×#15667974C2213yes—
31C2×#15667974C2213yes—
32C2×#15667974C2213yes—
33C2×#15667970C2213yes—
34C2×C4⋊C4×D89C2213yes—
35C2×C42×D89C2213yes—
36C2×C42×D89C2213yes—
37#19151504C2213yes—
38#19151503C2213yes—
39#19151500C2213yes—
40#19151499C2213yes—
41#19151488C2216yes—
42#19151487C2216yes—
43#19151487C2216yes—
44#19151481C2216yes—
45#19151476C2216yes—
46#19151469C2216yes—
47#19151455C2213yes—
48#19151455C2213yes—
49C4⋊C4×Dic89C2213yes—
50C4⋊C4×Dic89C2213yes—
51C4⋊C4×Dic89C2213yes—
52C4⋊C4×Dic89C2213yes—
53C4×C4⋊Dic89C2213yes—
54C4×#13083551C2213yes—
55C4×#13083551C2213yes—
56#19151445C2213yes—
57#19151445C2216yes—
58#19151445C2213yes—
59#19151445C2213yes—
60#19151445C2211yescharacteristic
61C42×Dic89C2213yes—
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—
69C2×#13083575—24no—
70C2×#13083575—24no—
71C2×#13083575C2×C418yescomplemented
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—
82C22×C4×D89C2313yes—
83C22×C4×D89C2311yescharacteristic
84#15668000C2×C4124yescomplemented
85#15667992C2×C4124yescomplemented
86#15667987C2×C4124yescomplemented
87Q8×Dic89—24no—
88Q8×Dic89D418yescomplemented
89C4×Dic178—212no—
90#15667977—212no—
91C4×D356—212no—
92C4×D356C2×C4124yescomplemented
93#15667974—212no—
94#15667974—212no—
95#15667974—212no—
96#15667970—212no—
97C2×C4⋊Dic89C2313yes—
98C2×C4⋊Dic89C2313yes—
99C2×C4⋊Dic89C2313yes—
100C2×#13083553C2316yes—
101C4⋊C4×D89—212no—
102C2×#13083551C2313yes—
103C2×#13083551C2313yes—
104C2×#13083551C2313yes—
105C2×#13083551C2316yes—
106C2×#13083551C2316yes—
107C2×#13083551C2311yescharacteristic
108C2×#13083551C2313yes—
109C2×#13083551C2316yes—
110C2×#13083551C2311yescharacteristic
111C2×C4×Dic89C2311yescharacteristic
112C2×C4×Dic89C2313yes—
113C2×C4×Dic89C2313yes—
114C2×C4×Dic89C2313yes—
115C2×C4×Dic89C2313yes—
116C2×C4×Dic89C2311yescharacteristic
117C2×C4×Dic89C2313yes—
118C2×C4×Dic89C2313yes—
119C2×C4×Dic89C2313yes—
120C2×C4×Dic89C2313yes—
121C42×D89—212no—
122C42×D89—212no—
123#15667919C2311yescharacteristic
124#15667919C2313yes—
125#15667919C2313yes—
126#15667919C2313yes—
127#15667919C2316yes—
128#15667919C2316yes—
129C22×Q8×C89C2311yescharacteristic
130C4×Q8×C89—24no—
131C4×Q8×C89—24no—
132C4×Q8×C89C2×C418yescomplemented
133C2×C4⋊C4×C89C2313yes—
134C2×C4⋊C4×C89C2313yes—
135C2×C4⋊C4×C89C2313yes—
136C2×C4⋊C4×C89C2313yes—
137C2×C4⋊C4×C89C2313yes—
138C2×C42×C89C2313yes—
139C2×C42×C89C2313yes—
140C2×C42×C89C2313yes—
141#95×C89C2311yescharacteristic
142#95×C89C2313yes—
143#95×C89C2316yes—
144C23×D89C2413yes—
145C23×D89C2411yescharacteristic
146#13083575—232no—
147C2×Dic178—26no—
148C2×Dic178—26no—
149C2×Dic178C22×C4112yes—
150Q8×D89—232no—
151C2×D356—26no—
152C2×D356—26no—
153C2×D356—26no—
154C2×D356—26no—
155C2×D356C22×C4112yes—
156C2×D356C22×C4112yes—
157C22×Dic89C2413yes—
158C22×Dic89C2411yescharacteristic
159C22×Dic89C2413yes—
160C22×Dic89C2413yes—
161C22×Dic89C2411yescharacteristic
162C22×Dic89C2411yescharacteristic
163C2×C4×D89—26no—
164C2×C4×D89—26no—
165C2×C4×D89—26no—
166C2×C4×D89—26no—
167C2×C4×D89—22no—
168C2×C4×D89—22no—
169C2×C4×D89—26no—
170C2×C4×D89—26no—
171C2×C4×D89—26no—
172C2×C4×D89—26no—
173C2×C4×D89—26no—
174C2×C4×D89—26no—
175C2×C4×D89—26no—
176C2×C4×D89—43no—
177C2×C4×D89—43no—
178C2×C4×D89C22×C4112yes—
179C2×C4×D89C22×C4112yes—
180C2×C4×D89#4114yescomplemented
181C4⋊Dic89—26no—
182C4⋊Dic89C22×C4112yes—
183C4⋊Dic89C2×D4112yes—
184#13083553—212no—
185#13083551—26no—
186#13083551—26no—
187#13083551—212no—
188#13083551—26no—
189#13083551—46no—
190#13083551C22×C414yes—
191#13083551C22×C4112yes—
192#13083551C22×C4124yes—
193#13083551#4114yescomplemented
194C4×Dic89—26no—
195C4×Dic89—26no—
196C4×Dic89—26no—
197C4×Dic89—26no—
198C4×Dic89—22no—
199C4×Dic89—26no—
200C4×Dic89—26no—
201C4×Dic89—22no—
202C4×Dic89—43no—
203C4×Dic89C2×D4112yes—
204C2×Q8×C89C22×C414yes—
205C2×Q8×C89C2×D414yes—
206C2×Q8×C89#4114yes—
207C22×C4×C89C2413yes—
208C22×C4×C89C2413yes—
209C22×C4×C89C2413yes—
210C22×C4×C89C2411yescharacteristic
211C22×C4×C89C2411yescharacteristic
212C22×C4×C89C2411yescharacteristic
213C22×C4×C89C2413yes—
214C4⋊C4×C89—26no—
215C4⋊C4×C89—26no—
216C4⋊C4×C89—26no—
217C4⋊C4×C89—26no—
218C4⋊C4×C89C22×C4112yes—
219C42×C89—26no—
220C42×C89—26no—
221C42×C89C22×C4112yes—
222C23×C89C2511yescharacteristic
223C22×D89—21no—
224C22×D89—21no—
225C22×D89—21no—
226C22×D89—21no—
227C22×D89—23no—
228C22×D89—23no—
229C22×D89—23no—
230C22×D89—23no—
231C22×D89—23no—
232C22×D89—23no—
233C22×D89C23×C416yes—
234C22×D89C23×C412yes—
235C22×D89C2×#4112yes—
236C22×D89C2×#4112yes—
237Q8×C89C4×D4116yescomplemented
238C2×C4×C89—21no—
239C2×C4×C89—21no—
240C2×C4×C89—23no—
241C2×C4×C89—23no—
242C2×C4×C89—21no—
243C2×C4×C89—21no—
244C2×C4×C89—21no—
245C2×C4×C89—21no—
246C2×C4×C89—23no—
247C2×C4×C89—23no—
248C2×C4×C89—23no—
249C2×C4×C89—23no—
250C2×C4×C89—23no—
251C2×C4×C89—23no—
252C2×C4×C89C23×C412yes—
253C2×C4×C89C23×C416yes—
254C2×C4×C89C23×C416yes—
255C2×C4×C89C22×D416yes—
256C2×C4×C89C2×#4112yes—
257C2×C4×C89C2×#4112yes—
258C2×C4×C89C2×#4116yes—
259C2×Dic89—21no—
260C2×Dic89—23no—
261C2×Dic89—21no—
262C2×Dic89—23no—
263C2×Dic89—21no—
264C2×Dic89—23no—
265C2×Dic89—21no—
266C2×Dic89—23no—
267C2×Dic89—23no—
268C2×Dic89—23no—
269C2×Dic89—23no—
270C2×Dic89C23×C416yes—
271C2×Dic89C23×C412yes—
272C2×Dic89C22×D416yes—
273C2×Dic89C22×D412yes—
274C2×Dic89C2×#4112yes—
275C2×Dic89C2×#4112yes—
276C2×Dic892−1+412yes—
277D356—224no—
278D356—224no—
279C4×D89—224no—
280C4×D89—224no—
281C4×D89—28no—
282C4×D89—412no—
283C4×D89—412no—
284C4×D89—412no—
285Dic178—224no—
286C22×C89C24×C411yescharacteristic
287C22×C89C23×D411yescharacteristic
288C22×C89C22×#4111yescharacteristic
289C22×C89C22×#4111yescharacteristic
290C22×C89C22×#4111yescharacteristic
291C22×C89C2×2−1+411yescharacteristic
292C22×C89#58511yescharacteristic
293C2×D89—22no—
294C2×D89—22no—
295C2×D89—22no—
296C2×D89—22no—
297C2×D89—22no—
298C2×D89—22no—
299C2×D89—26no—
300C2×D89—26no—
301C2×D89—26no—
302C2×D89—43no—
303C2×D89—43no—
304C2×D89—43no—
305C2×D89—43no—
306C2×D89C4×#4114yescomplemented
307C4×C89—26no—
308C4×C89—22no—
309C4×C89—22no—
310C4×C89—22no—
311C4×C89—43no—
312C4×C89—43no—
313C4×C89C2×C4×D4112yes—
314Dic89—26no—
315Dic89—22no—
316Dic89—26no—
317Dic89—22no—
318Dic89—43no—
319Dic89#54814yescomplemented
320C4×#548—891nomaximal, Sylow
321C2×C89C22×C4×D411yescharacteristic
322C2×C89C2×C4×#4111yescharacteristic
323C2×C89C4×2−1+411yescharacteristic
324C2×C89C2×#54811yescharacteristic
325C2×C89D4×#4111yescharacteristic
326C2×C89#348011yescharacteristic
327C2×C89#348211yescharacteristic
328D89—24no—
329D89—46no—
330C2×#548—891no—
331C2×C4×#41—891no—
332C22×C4×Q8—891no—
333#2367—893no—
334#2362—893no—
335#2361—893no—
336#2352—893no—
337#2351D8911yescharacteristic, complemented
338#2341—891no—
339Q8×C22⋊C4—891no—
340C4×#124—893no—
341C4×C22⋊Q8—893no—
342C4×#121—893no—
343C42×Q8—891no—
344C42×D4—893no—
345C89C4×#54811yescharacteristic, complemented, Sylow
346C22×#41—891no—
347C23×Q8—891no—
348#548—8916no—
349C2×#124—893no—
350C2×C22⋊Q8—893no—
351C2×#121—893no—
352C4×#41—1788no—
353C2×C4×Q8—891no—
354C2×C4×Q8—891no—
355C2×C4×Q8—893no—
356C2×C4×Q8—893no—
357C2×C4×Q8—1788no—
358C2×C4×Q8C2×D8911yescharacteristic
359C2×C4×Q8C2×D8911yescharacteristic
360C2×C4×Q8C2×D8911yescharacteristic
361C2×C4×D4—893no—
362C2×C4×D4—893no—
363C2×C4×D4—893no—
364C2×C4×D4—893no—
365C2×C4×D4—893no—
366C2×#117—893no—
367C22×C4⋊C4—893no—
368C22×C42—893no—
369C22×C42—893no—
370#391—893no—
371#391—893no—
372#390—893no—
373#390—893no—
374#389C2×D8913yes—
375#389C2×D8913yes—
376#388—896no—
377#388—896no—
378#388—896no—
379#387—896no—
380#386—896no—
381#385—896no—
382#382C2×D8913yes—
383#380—893no—
384#380—893no—
385C4×C4⋊C4—893no—
386C4×C4⋊C4—893no—
387C4×C4⋊C4—893no—
388C4×C4⋊C4—893no—
389C4×C4⋊C4—893no—
390C4×C4⋊C4C2×D8913yes—
391C4×C22⋊C4—893no—
392C4×C22⋊C4—893no—
393C4×C22⋊C4—893no—
394C4×C22⋊C4—893no—
395C4×C22⋊C4—891no—
396C4×C22⋊C4—893no—
397C4×C22⋊C4—896no—
398C43—893no—
399C2×#41—898no—
400C2×#41—1784no—
401C2×#41—1784no—
402C22×Q8—898no—
403C22×Q8—893no—
404C22×Q8—1784no—
405C22×Q8—1784no—
406C22×Q8C22×D8911yescharacteristic
407C22×D4—893no—
408C22×D4—893no—
409C23×C4—893no—
410C23×C4—893no—
411C23×C4—891no—
412C23×C4—893no—
413C23×C4—893no—
414C23×C4—893no—
415#124—8924no—
416C22⋊Q8—8924no—
417#121—8924no—
418C4×Q8—24no—
419C4×Q8—24no—
420C4×Q8—898no—
421C4×Q8—17812no—
422C4×Q8—17812no—
423C4×Q8—1784no—
424C4×Q8C4×D8918yescomplemented
425C4×D4—8924no—
426C4×D4—17812no—
427C4×D4—17812no—
428C4×D4—17812no—
429C4×D4—17812no—
430#117—17812no—
431C2×C4⋊C4—893no—
432C2×C4⋊C4—893no—
433C2×C4⋊C4—896no—
434C2×C4⋊C4—893no—
435C2×C4⋊C4—17812no—
436C2×C4⋊C4C22×D8913yes—
437C2×C4⋊C4C22×D8913yes—
438C2×C4⋊C4C22×D8913yes—
439C2×C4⋊C4C22×D8913yes—
440C2×C4⋊C4C22×D8913yes—
441C2×C22⋊C4—891no—
442C2×C22⋊C4—891no—
443C2×C22⋊C4—893no—
444C2×C22⋊C4—896no—
445C2×C22⋊C4—896no—
446C2×C22⋊C4—896no—
447C2×C22⋊C4—893no—
448C2×C22⋊C4—893no—
449C2×C22⋊C4—893no—
450C2×C42—893no—
451C2×C42—893no—
452C2×C42—893no—
453C2×C42—893no—
454C2×C42—893no—
455C2×C42—893no—
456C2×C42—893no—
457C2×C42—893no—
458C2×C42—891no—
459C2×C42—891no—
460C2×C42—17812no—
461C2×C42—17812no—
462C2×C42C22×D8913yes—
463C2×C42C22×D8913yes—
464C2×C42C22×D8913yes—
465#95—891no—
466#95—893no—
467#95—893no—
468#95—893no—
469#95—896no—
470#95—896no—
471#95C22×D8911yescharacteristic
472#95C22×D8913yes—
473#95C22×D8916yes—
474C24—891no—
475C24—893no—
476#41—17832no—
477C2×Q8—8912no—
478C2×Q8—17832no—
479C2×Q8—1786no—
480C2×Q8—1786no—
481C2×Q8C2×C4×D8914yes—
482C2×Q8D4×D8914yes—
483C2×Q8#1308357514yes—
484C2×D4—8912no—
485C2×D4—8912no—
486C2×D4—1786no—
487C2×D4—1786no—
488C2×D4—1786no—
489C2×D4—1786no—
490C22×C4—891no—
491C22×C4—891no—
492C22×C4—891no—
493C22×C4—893no—
494C22×C4—893no—
495C22×C4—8912no—
496C22×C4—8912no—
497C22×C4—894no—
498C22×C4—893no—
499C22×C4—3563no—
500C22×C4—3563no—
501C22×C4—1786no—
502C22×C4—1786no—
503C22×C4—1786no—
504C22×C4—1786no—
505C22×C4—1786no—
506C22×C4—1782no—
507C22×C4—1782no—
508C22×C4—1786no—
509C22×C4—1786no—
510C22×C4—1786no—
511C22×C4—1786no—
512C22×C4—1786no—
513C22×C4—1786no—
514C22×C4C23×D8913yes—
515C22×C4C23×D8913yes—
516C22×C4C23×D8913yes—
517C22×C4C23×D8911yescharacteristic
518C22×C4C23×D8911yescharacteristic
519C22×C4C23×D8911yescharacteristic
520C22×C4C23×D8913yes—
521C4⋊C4—26no—
522C4⋊C4—26no—
523C4⋊C4—26no—
524C4⋊C4—26no—
525C4⋊C4—8912no—
526C4⋊C4—8912no—
527C4⋊C4—17812no—
528C4⋊C4—1786no—
529C4⋊C4C2×C4×D89112yes—
530C22⋊C4—894no—
531C22⋊C4—8912no—
532C22⋊C4—8924no—
533C22⋊C4—894no—
534C22⋊C4—3566no—
535C22⋊C4—17812no—
536C22⋊C4—1786no—
537C22⋊C4—1786no—
538C22⋊C4—1786no—
539C42—26no—
540C42—26no—
541C42—8912no—
542C42—3563no—
543C42—1782no—
544C42—1786no—
545C42—1786no—
546C42—1786no—
547C42—1786no—
548C42—1786no—
549C42—1782no—
550C42—1786no—
551C42C2×C4×D89112yes—
552C23—892no—
553C23—892no—
554C23—892no—
555C23—896no—
556C23—1783no—
557C23—1783no—
558C23—1783no—
559C23—1783no—
560C23—1783no—
561C23—1783no—
562C23—1781no—
563C23—1781no—
564C23—1781no—
565C23—1781no—
566C23C24×D8911yescharacteristic
567Q8—17824no—
568Q8#15667974116yescomplemented
569D4—17824no—
570D4—17824no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—23no—
580C2×C4—23no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—23no—
584C2×C4—23no—
585C2×C4—892no—
586C2×C4—892no—
587C2×C4—892no—
588C2×C4—896no—
589C2×C4—892no—
590C2×C4—896no—
591C2×C4—892no—
592C2×C4—35612no—
593C2×C4—35612no—
594C2×C4—35612no—
595C2×C4—1788no—
596C2×C4—1783no—
597C2×C4—1783no—
598C2×C4—1783no—
599C2×C4—1781no—
600C2×C4—1783no—
601C2×C4—1783no—
602C2×C4—1781no—
603C2×C4—1781no—
604C2×C4—1783no—
605C2×C4—1781no—
606C2×C4—1783no—
607C2×C4—17824no—
608C2×C4—17824no—
609C2×C4C22×C4×D8912yes—
610C2×C4C22×C4×D8916yes—
611C2×C4C22×C4×D8916yes—
612C2×C4C2×D4×D8916yes—
613C2×C4#41×D8912yes—
614C2×C4#41×D8912yes—
615C2×C4C2×#1308357516yes—
616C22—894no—
617C22—3563no—
618C22—3563no—
619C22—3563no—
620C22—3563no—
621C22—1786no—
622C22—1786no—
623C22—1786no—
624C22—1782no—
625C22—1782no—
626C22—1782no—
627C22—1782no—
628C22—1782no—
629C22—1782no—
630C22C23×C4×D8911yescharacteristic
631C22C22×D4×D8911yescharacteristic
632C22C2×#41×D8911yescharacteristic
633C22C2×#41×D8911yescharacteristic
634C22C22×#1308357511yescharacteristic
635C222−1+4×D8911yescharacteristic
636C22#1915243511yescharacteristic
637C4—26no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—43no—
642C4—43no—
643C4—894no—
644C4—3563no—
645C4—1782no—
646C4—1786no—
647C4—1786no—
648C4—1782no—
649C4C2×#15667974112yes—
650C2—3566no—
651C2—1784no—
652C2C22×#1566797411yescharacteristic
653C2C4×#41×D8911yescharacteristic
654C2#2603346111yescharacteristic
655C2#548×D8911yescharacteristic
656C2#2603374111yescharacteristic
657C2#2603385211yescharacteristic
658C2#2603387211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2351C2
socle, cosocleC23×C89C25
prime radicalC2×#19152260C2
hypercenter#2351D89
hypocenterC89C4×#548
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13083575C2semidirect
3. C2×C4×Q8×D89C2semidirect
4. #26023321C2semidirect
5. #26023278C2semidirect
6. #26023267C2semidirect
7. #26023178C2semidirect
8. #26023090C2semidirect
9. Q8×#13083551C2semidirect
10. #26022791C2semidirect
11. #26022782C2semidirect
12. #26022760C2semidirect
13. C4×Q8×Dic89C2semidirect
14. C4×#15667974C2semidirect
15. C89×#2351C2semidirect
18. #19152260C4semidirect
71. C2×#13083575C2×C4semidirect
75. C2×Q8×D89C2×C4semidirect
84. #15668000C2×C4semidirect
85. #15667992C2×C4semidirect
86. #15667987C2×C4semidirect
88. Q8×Dic89D4semidirect
92. C4×D356C2×C4semidirect
132. C4×Q8×C89C2×C4semidirect
180. C2×C4×D89#41semidirect
193. #13083551#41semidirect
237. Q8×C89C4×D4semidirect
306. C2×D89C4×#41semidirect
319. Dic89#548semidirect
337. #2351D89semidirect
345. C89C4×#548semidirect
424. C4×Q8C4×D89semidirect
568. Q8#15667974semidirect

Maximal subgroups

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

subgroupindex
C2×#191522602
C2×C4×#130835752
C2×C4×Q8×D892
#260233212
#260232782
#260232672
#260231782
#260230902
Q8×#130835512
#260227912
#260227822
#260227602
C4×Q8×Dic892
C4×#156679742
C89×#23512
C4×#54889