Finite Group Databasebeta

← #36719620

Subgroups — #36719620

Order 22784.

2132 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152305C211yesmaximal, characteristic
2#26023412C214yesmaximal, complemented
3#26023344C212yesmaximal, complemented
4#26023341C211yesmaximal, characteristic, complemented
5#26023279C212yesmaximal, complemented
6#26023200C212yesmaximal
7#26023164C211yesmaximal, characteristic, complemented
8#26023123C212yesmaximal, complemented
9#26022989C212yesmaximal, complemented
10#26022930C212yesmaximal, complemented
11#26022927C214yesmaximal, complemented
12#26022889C212yesmaximal
13#26022868C212yesmaximal, complemented
14#26022819C211yesmaximal, characteristic, complemented
15#26022726C211yesmaximal, characteristic, complemented
16#26022717C211yesmaximal, characteristic
17C89×#2354C211yesmaximal, characteristic
18#19152305C4116yescomplemented
19C2×#15668012C2211yescharacteristic
20C2×#15668003C2212yes—
21C2×#15667991C2212yes—
22C2×#15667989C2214yes—
23C2×C4×Dic178C2211yescharacteristic
24C2×#15667977C2212yes—
25C2×#15667977C2212yes—
26C2×D4×Dic89C2212yes—
27C2×#15667971C2212yes—
28C2×#15667968C2214yes—
29C22×C4⋊Dic89C2212yes—
30#19151503C2211yescharacteristic
31#19151503C2211yescharacteristic
32#19151503C2212yes—
33#19151502C2214yes—
34#19151501C2211yescharacteristic
35#19151501C2211yescharacteristic
36#19151497C2211yescharacteristic
37#19151497C2211yescharacteristic
38#19151496C2214yes—
39#19151496C2212yes—
40#19151496C2214yes—
41#19151495C2214yes—
42#19151494C2212yes—
43#19151494C2211yescharacteristic
44#19151494C2211yescharacteristic
45#19151494C2212yes—
46#19151494C2212yes—
47#19151494C2212yes—
48#19151493C2214yes—
49#19151492C2214yes—
50#19151492C2214yes—
51#19151491C2212yes—
52#19151491C2212yes—
53#19151486C2214yes—
54#19151486C2214yes—
55#19151485C2214yes—
56#19151482C2214yes—
57#19151481C2214yes—
58#19151481C2214yes—
59#19151480C2212yes—
60#19151480C2212yes—
61#19151467C2212yes—
62#19151459C2214yes—
63#19151456C2214yes—
64#19151456C2214yes—
65C4⋊C4×Dic89C2212yes—
66C4⋊C4×Dic89C2212yes—
67C4⋊C4×Dic89C2212yes—
68C4⋊C4×Dic89C2212yes—
69C4×#13083553C2214yes—
70C4×#13083553C2212yes—
71C4×C4⋊Dic89C2212yes—
72C4×C4⋊Dic89C2211yescharacteristic
73C4×C22⋊Dic89C2212yes—
74#19151444C2212yes—
75C42×Dic89C2211yescharacteristic
76C2×C4×D4×C89C2211yescharacteristic
77C2×#117×C89C2212yes—
78#390×C89C2212yes—
79#388×C89C2214yes—
80#384×C89C2211yescharacteristic
81#380×C89C2212yes—
82C4×C4⋊C4×C89C2211yescharacteristic
83C4×C4⋊C4×C89C2212yes—
84C22×Dic178C2311yescharacteristic
85C22×Dic178C2312yes—
86C23×Dic89C2314yes—
87#15668012C2×C418yes—
88#15668003C2×C4116yes—
89#15667991C2×C4116yes—
90#15667989C2×C4132yes—
91C4×Dic178C2×C418yes—
92#15667977—44no—
93#15667977—44no—
94D4×Dic89C2×C4116yes—
95#15667971—44no—
96#15667968—48no—
97C2×C4⋊Dic89C2×C4116yes—
98C2×C4⋊Dic89C2311yescharacteristic
99C2×C4⋊Dic89C2311yescharacteristic
100C2×C4⋊Dic89C2312yes—
101C2×C4⋊Dic89C2312yes—
102C2×C4⋊Dic89C2312yes—
103C2×C4⋊Dic89C2312yes—
104C2×C4⋊Dic89C2314yes—
105C2×C4⋊Dic89C2311yescharacteristic
106C2×#13083553C2312yes—
107C2×#13083553C2312yes—
108C2×#13083553C2314yes—
109C2×#13083553C2314yes—
110C2×#13083553C2314yes—
111C2×#13083553C2314yes—
112C2×#13083553C2314yes—
113C2×#13083553C2312yes—
114C2×C22⋊Dic89C2312yes—
115C2×C22⋊Dic89C2312yes—
116C2×C22⋊Dic89C2312yes—
117C2×C22⋊Dic89C2312yes—
118C2×C22⋊Dic89C2314yes—
119C2×C4×Dic89C2311yescharacteristic
120C2×C4×Dic89C2311yescharacteristic
121C2×C4×Dic89C2311yescharacteristic
122C2×C4×Dic89C2311yescharacteristic
123C2×C4×Dic89C2311yescharacteristic
124C2×C4×Dic89C2312yes—
125C2×C4×Dic89C2314yes—
126C2×C4×Dic89C2311yescharacteristic
127C2×C4×Dic89C2314yes—
128C2×C4×Dic89C2314yes—
129C2×C4×Dic89C2312yes—
130#15667919C2312yes—
131#15667919C2312yes—
132#15667919C2312yes—
133#15667919C2312yes—
134#15667919C2312yes—
135#15667919C2312yes—
136#15667919C2314yes—
137#15667919C2312yes—
138#15667919C2312yes—
139#15667919C2312yes—
140#15667919C2314yes—
141#15667919C2314yes—
142#15667919C2314yes—
143#15667919C2314yes—
144C22×D4×C89C2311yescharacteristic
145C23×C4×C89C2312yes—
146C4×D4×C89C2×C418yes—
147#117×C89—44no—
148C2×C4⋊C4×C89C2311yescharacteristic
149C2×C4⋊C4×C89C2312yes—
150C2×C4⋊C4×C89C2311yescharacteristic
151C2×C4⋊C4×C89C2312yes—
152C2×C4⋊C4×C89C2312yes—
153C2×C4⋊C4×C89C2311yescharacteristic
154C2×C22⋊C4×C89C2314yes—
155C2×C22⋊C4×C89C2314yes—
156C2×C22⋊C4×C89C2312yes—
157C2×C42×C89C2312yes—
158C2×C42×C89C2314yes—
159C2×C42×C89C2311yescharacteristic
160#95×C89C2314yes—
161#95×C89C2312yes—
162C2×Dic178—22no—
163C2×Dic178—22no—
164C2×Dic178—42no—
165C2×Dic178—42no—
166C2×Dic178C22×C414yes—
167C2×Dic178C22×C418yes—
168C22×Dic89—44no—
169C22×Dic89—44no—
170C22×Dic89C22×C4116yes—
171C22×Dic89C2411yescharacteristic
172C22×Dic89C2412yes—
173C22×Dic89C2411yescharacteristic
174C22×Dic89C2411yescharacteristic
175C22×Dic89C2412yes—
176C22×Dic89C2412yes—
177C22×Dic89C2414yes—
178C22×Dic89C2412yes—
179C22×Dic89C2411yescharacteristic
180C4⋊Dic89—22no—
181C4⋊Dic89—22no—
182C4⋊Dic89—42no—
183C4⋊Dic89—42no—
184C4⋊Dic89C22×C418yes—
185C4⋊Dic89C22×C414yes—
186C4⋊Dic89C22×C4116yes—
187C4⋊Dic89C22×C418yes—
188#13083553—24no—
189#13083553—44no—
190#13083553—44no—
191#13083553—44no—
192#13083553—42no—
193#13083553—44no—
194#13083553C22×C418yes—
195#13083553C22×C4116yes—
196C22⋊Dic89—24no—
197C22⋊Dic89—42no—
198C22⋊Dic89—42no—
199C22⋊Dic89—42no—
200C22⋊Dic89C22×C4116yes—
201C4×Dic89—22no—
202C4×Dic89—24no—
203C4×Dic89—22no—
204C4×Dic89—22no—
205C4×Dic89—22no—
206C4×Dic89—28no—
207C4×Dic89—22no—
208C4×Dic89—44no—
209C4×Dic89—44no—
210C4×Dic89—41no—
211C4×Dic89C22×C418yes—
212C24×C89C2412yes—
213C2×D4×C89—22no—
214C2×D4×C89—22no—
215C2×D4×C89C22×C414yes—
216C22×C4×C89—24no—
217C22×C4×C89—24no—
218C22×C4×C89C22×C418yes—
219C22×C4×C89C2412yes—
220C22×C4×C89C2412yes—
221C22×C4×C89C2414yes—
222C22×C4×C89C2411yescharacteristic
223C22×C4×C89C2412yes—
224C22×C4×C89C2411yescharacteristic
225C22×C4×C89C2411yescharacteristic
226C4⋊C4×C89—24no—
227C4⋊C4×C89—22no—
228C4⋊C4×C89—24no—
229C4⋊C4×C89—22no—
230C4⋊C4×C89—42no—
231C4⋊C4×C89C22×C414yes—
232C22⋊C4×C89—44no—
233C22⋊C4×C89—44no—
234C22⋊C4×C89C22×C418yes—
235C42×C89—24no—
236C42×C89—44no—
237C42×C89C22×C414yes—
238C23×C89—22no—
239C23×C89—22no—
240C23×C89—22no—
241C23×C89—22no—
242C23×C89—22no—
243C23×C89—22no—
244C23×C89C23×C414yes—
245C23×C89C2511yescharacteristic
246D4×C89—28no—
247C2×C4×C89—22no—
248C2×C4×C89—21no—
249C2×C4×C89—22no—
250C2×C4×C89—21no—
251C2×C4×C89—216no—
252C2×C4×C89—22no—
253C2×C4×C89—24no—
254C2×C4×C89—22no—
255C2×C4×C89—24no—
256C2×C4×C89—22no—
257C2×C4×C89—22no—
258C2×C4×C89—22no—
259C2×C4×C89—24no—
260C2×C4×C89—22no—
261C2×C4×C89C23×C412yes—
262C2×C4×C89C23×C414yes—
263C2×C4×C89C23×C412yes—
264C2×C4×C89C23×C412yes—
265C2×C4×C89H(5,𝔽2)12yes—
266C2×C4×C892−1+412yes—
267C2×C4×C892−1+412yes—
268C2×C4×C892−1+412yes—
269C2×Dic89—24no—
270C2×Dic89—22no—
271C2×Dic89—22no—
272C2×Dic89—21no—
273C2×Dic89—21no—
274C2×Dic89—22no—
275C2×Dic89—22no—
276C2×Dic89—24no—
277C2×Dic89—22no—
278C2×Dic89—21no—
279C2×Dic89—21no—
280C2×Dic89—22no—
281C2×Dic89—21no—
282C2×Dic89—22no—
283C2×Dic89—21no—
284C2×Dic89—21no—
285C2×Dic89—21no—
286C2×Dic89—21no—
287C2×Dic89—22no—
288C2×Dic89—22no—
289C2×Dic89—21no—
290C2×Dic89—21no—
291C2×Dic89—22no—
292C2×Dic89—416no—
293C2×Dic89C23×C414yes—
294C2×Dic89C23×C414yes—
295C2×Dic89C23×C418yes—
296C2×Dic89H(5,𝔽2)12yes—
297Dic178—28no—
298Dic178—48no—
299C22×C89—24no—
300C22×C89—24no—
301C22×C89—24no—
302C22×C89—42no—
303C22×C89—42no—
304C22×C89—42no—
305C22×C89—42no—
306C22×C89C24×C411yescharacteristic
307C22×C89C2×H(5,𝔽2)11yescharacteristic
308C22×C89C2×H(5,𝔽2)11yescharacteristic
309C22×C89C2×2−1+411yescharacteristic
310C22×C89C2×2−1+411yescharacteristic
311C22×C89C2×2−1+411yescharacteristic
312C22×C89#58511yescharacteristic
313C4×C89—24no—
314C4×C89—22no—
315C4×C89—42no—
316C4×C89—42no—
317C4×C89—44no—
318C4×C89#51914yes—
319C4×C89#52014yes—
320Dic89—24no—
321Dic89—28no—
322Dic89—24no—
323Dic89—22no—
324Dic89—42no—
325Dic89—41no—
326Dic89—41no—
327Dic89—42no—
328Dic89—41no—
329#34484—891nomaximal, Sylow
330C2×C89—44no—
331C2×C89C2×#51911yescharacteristic
332C2×C89C2×#52011yescharacteristic
333C2×C89C4×H(5,𝔽2)11yescharacteristic
334C2×C89#354411yescharacteristic
335C2×C89#354711yescharacteristic
336C2×C89#354811yescharacteristic
337C2×C89#355511yescharacteristic
338C2×#554—891no—
339#2378—894no—
340#2370—892no—
341#2369—891no—
342#2362—892no—
343#2354—892no—
344#2354D8911yescharacteristic
345Q8×C4⋊C4—891no—
346#2344—892no—
347#2330—892no—
348#2323—894no—
349#2323—892no—
350#2319—892no—
351#2316—892no—
352C4×C4⋊Q8—891no—
353#2295—891no—
354#2293—891no—
355C89#3448411yescharacteristic, complemented, Sylow
356#554—8916no—
357C2×C4⋊Q8—891no—
358C2×#125—892no—
359C2×C22⋊Q8—892no—
360C2×C22⋊Q8—894no—
361C2×C4×Q8—892no—
362C2×C4×Q8—892no—
363C2×C4×Q8—891no—
364C2×C4×D4—892no—
365C2×C4×D4C2×D8911yescharacteristic
366C2×#117—892no—
367C2×#117—894no—
368C2×#117C2×D8912yes—
369C22×C4⋊C4—892no—
370#391—892no—
371#391—891no—
372#391—891no—
373#391—894no—
374#390—891no—
375#390—891no—
376#390C2×D8912yes—
377#389—892no—
378#389—892no—
379#389—891no—
380#389—892no—
381#389—892no—
382#389—894no—
383#389—891no—
384#389—891no—
385#389—891no—
386#389—894no—
387#389—894no—
388#389—892no—
389#389—894no—
390#388—894no—
391#388—894no—
392#388—892no—
393#388—894no—
394#388—894no—
395#388—892no—
396#388C2×D8914yes—
397#387—894no—
398#387—892no—
399#387—894no—
400#387—894no—
401#387—894no—
402#387—892no—
403#384—892no—
404#384C2×D8911yescharacteristic
405#381—894no—
406#380—894no—
407#380—894no—
408#380C2×D8912yes—
409C4×C4⋊C4—892no—
410C4×C4⋊C4—892no—
411C4×C4⋊C4—892no—
412C4×C4⋊C4—892no—
413C4×C4⋊C4—894no—
414C4×C4⋊C4—892no—
415C4×C4⋊C4—891no—
416C4×C4⋊C4—892no—
417C4×C4⋊C4C2×D8912yes—
418C4×C4⋊C4C2×D8911yescharacteristic
419C4×C22⋊C4—892no—
420#376—892no—
421C43—891no—
422C22×Q8—892no—
423C22×Q8—891no—
424C22×D4C22×D8911yescharacteristic
425C23×C4—894no—
426C23×C4C22×D8912yes—
427C4⋊Q8—898no—
428#125—8916no—
429C22⋊Q8—8916no—
430C22⋊Q8—8932no—
431C4×Q8—898no—
432C4×Q8—3564no—
433C4×Q8—3564no—
434C4×D4—8916no—
435C4×D4C4×D8918yes—
436#117—44no—
437#117—3568no—
438#117—3564no—
439C2×C4⋊C4—8916no—
440C2×C4⋊C4—891no—
441C2×C4⋊C4—894no—
442C2×C4⋊C4—892no—
443C2×C4⋊C4—892no—
444C2×C4⋊C4—892no—
445C2×C4⋊C4—894no—
446C2×C4⋊C4—894no—
447C2×C4⋊C4—892no—
448C2×C4⋊C4—892no—
449C2×C4⋊C4—894no—
450C2×C4⋊C4—894no—
451C2×C4⋊C4—892no—
452C2×C4⋊C4—894no—
453C2×C4⋊C4—892no—
454C2×C4⋊C4—891no—
455C2×C4⋊C4—891no—
456C2×C4⋊C4C22×D8911yescharacteristic
457C2×C4⋊C4C22×D8912yes—
458C2×C4⋊C4C22×D8911yescharacteristic
459C2×C4⋊C4C22×D8912yes—
460C2×C4⋊C4C22×D8912yes—
461C2×C4⋊C4C22×D8911yescharacteristic
462C2×C22⋊C4—894no—
463C2×C22⋊C4—892no—
464C2×C22⋊C4—892no—
465C2×C22⋊C4—892no—
466C2×C22⋊C4—892no—
467C2×C22⋊C4C22×D8914yes—
468C2×C22⋊C4C22×D8914yes—
469C2×C22⋊C4C22×D8912yes—
470C2×C42—892no—
471C2×C42—894no—
472C2×C42—894no—
473C2×C42—891no—
474C2×C42—891no—
475C2×C42—892no—
476C2×C42—894no—
477C2×C42—891no—
478C2×C42—891no—
479C2×C42—891no—
480C2×C42—891no—
481C2×C42C22×D8912yes—
482C2×C42C22×D8914yes—
483C2×C42C22×D8911yescharacteristic
484#95—892no—
485#95—892no—
486#95—892no—
487#95—892no—
488#95—892no—
489#95—892no—
490#95—892no—
491#95—894no—
492#95—894no—
493#95—894no—
494#95—894no—
495#95—894no—
496#95—892no—
497#95—892no—
498#95C22×D8914yes—
499#95C22×D8912yes—
500C24C23×D8912yes—
501C2×Q8—898no—
502C2×Q8—894no—
503C2×Q8—3562no—
504C2×Q8—3562no—
505C2×Q8—1782no—
506C2×Q8—1782no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4C2×C4×D8914yes—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—8916no—
513C22×C4—891no—
514C22×C4—892no—
515C22×C4—894no—
516C22×C4—892no—
517C22×C4—891no—
518C22×C4—892no—
519C22×C4—891no—
520C22×C4—892no—
521C22×C4—891no—
522C22×C4—3564no—
523C22×C4—3564no—
524C22×C4C2×C4×D8918yes—
525C22×C4C23×D8912yes—
526C22×C4C23×D8912yes—
527C22×C4C23×D8914yes—
528C22×C4C23×D8911yescharacteristic
529C22×C4C23×D8912yes—
530C22×C4C23×D8911yescharacteristic
531C22×C4C23×D8911yescharacteristic
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—894no—
538C4⋊C4—8916no—
539C4⋊C4—898no—
540C4⋊C4—898no—
541C4⋊C4—898no—
542C4⋊C4—8916no—
543C4⋊C4—3564no—
544C4⋊C4—3562no—
545C4⋊C4—3562no—
546C4⋊C4—3564no—
547C4⋊C4—3564no—
548C4⋊C4—3562no—
549C4⋊C4—3564no—
550C4⋊C4—1784no—
551C4⋊C4—1782no—
552C4⋊C4—1782no—
553C4⋊C4C2×C4×D8914yes—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—8916no—
557C22⋊C4—3562no—
558C22⋊C4—3562no—
559C22⋊C4—3562no—
560C22⋊C4—1784no—
561C22⋊C4C2×C4×D8918yes—
562C42—24no—
563C42—44no—
564C42—898no—
565C42—3564no—
566C42—3564no—
567C42—3561no—
568C42—1788no—
569C42—1782no—
570C42—1782no—
571C42—1784no—
572C42—1782no—
573C42—1782no—
574C42—1782no—
575C42C2×C4×D8914yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C22×C4×D8914yes—
583C23C24×D8911yescharacteristic
584Q8—3568no—
585Q8—1788no—
586D4—28no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—216no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—894no—
602C2×C4—894no—
603C2×C4—898no—
604C2×C4—892no—
605C2×C4—35616no—
606C2×C4—1782no—
607C2×C4—1781no—
608C2×C4—1782no—
609C2×C4—1781no—
610C2×C4—1781no—
611C2×C4—1782no—
612C2×C4—1781no—
613C2×C4—1781no—
614C2×C4—1781no—
615C2×C4—1781no—
616C2×C4—1782no—
617C2×C4—1782no—
618C2×C4—1781no—
619C2×C4—1781no—
620C2×C4—1782no—
621C2×C4—1784no—
622C2×C4—1782no—
623C2×C4—1782no—
624C2×C4—1781no—
625C2×C4—1781no—
626C2×C4—1782no—
627C2×C4—1782no—
628C2×C4—1784no—
629C2×C4C22×C4×D8912yes—
630C2×C4C22×C4×D8914yes—
631C2×C4C22×C4×D8912yes—
632C2×C4C22×C4×D8912yes—
633C2×C4#1566805712yes—
634C2×C4#1566805912yes—
635C2×C4#1566805912yes—
636C2×C4#1566805912yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C23×C4×D8911yescharacteristic
645C22H(5,𝔽2)×D8911yescharacteristic
646C22C2×#1566805711yescharacteristic
647C22C2×#1566805911yescharacteristic
648C22C2×#1566805911yescharacteristic
649C22C2×#1566805911yescharacteristic
650C22#1915243511yescharacteristic
651C4—24no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—3561no—
657C4—3562no—
658C4—3561no—
659C4—3562no—
660C4—3561no—
661C4—1784no—
662C4—1784no—
663C4—1788no—
664C4—1782no—
665C4#1915207514yes—
666C4#1915207714yes—
667C2—44no—
668C2C2×#1915207511yescharacteristic
669C2C2×#1915207711yescharacteristic
670C2#2603345811yescharacteristic
671C2#2603458711yescharacteristic
672C2#2603462911yescharacteristic
673C2#2603464511yescharacteristic
674C2#2603472811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2354C2
socle, cosocleC23×C89C25
prime radicalC22×D4×C89C23
hypercenter#2354D89
hypocenterC89#34484
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #26023412C2semidirect
3. #26023344C2semidirect
4. #26023341C2semidirect
5. #26023279C2semidirect
7. #26023164C2semidirect
8. #26023123C2semidirect
9. #26022989C2semidirect
10. #26022930C2semidirect
11. #26022927C2semidirect
13. #26022868C2semidirect
14. #26022819C2semidirect
15. #26022726C2semidirect
18. #19152305C4semidirect
355. C89#34484semidirect

Maximal subgroups

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

subgroupindex
C2×#191523052
#260234122
#260233442
#260233412
#260232792
#260232002
#260231642
#260231232
#260229892
#260229302
#260229272
#260228892
#260228682
#260228192
#260227262
#260227172
C89×#23542
#3448489