Finite Group Databasebeta

← C4×#126×D89 #36712905

Subgroups — C4×#126×D89 #36712905

Order 22784.

2838 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#126×D89C211yesmaximal, characteristic
2#26022806C213yesmaximal, complemented
3C4×#15668007C213yesmaximal, complemented
4#126×Dic89C211yesmaximal, characteristic, complemented
5C4×#15668006C211yesmaximal, characteristic, complemented
6#26022802C213yesmaximal, complemented
7C4×#15668005C213yesmaximal, complemented
8#26022800C211yesmaximal, characteristic, complemented
9#388×D89C213yesmaximal
10#387×D89C213yesmaximal, complemented
11#383×D89C211yesmaximal, characteristic, complemented
12C4×C4⋊C4×D89C213yesmaximal, complemented
13C4×C22⋊C4×D89C213yesmaximal
14C43×D89C211yesmaximal, characteristic, complemented
15C4×#126×C89C211yesmaximal, characteristic, complemented
16C23×C4×D89C2211yescharacteristic
17C2×#15668007C2213yes—
18C2×#15668006C2211yescharacteristic
19C2×#15668005C2213yes—
20#126×D89C4116yesdirect product factor
21C2×C4⋊C4×D89C2213yes—
22C2×C4⋊C4×D89C2213yes—
23C2×C22⋊C4×D89C2213yes—
24C2×C22⋊C4×D89C2213yes—
25C2×C42×D89C2213yes—
26C2×C42×D89C2213yes—
27C2×C42×D89C2213yes—
28C2×C42×D89C2211yescharacteristic
29#19151492C2213yes—
30#19151491C2213yes—
31#19151490C2213yes—
32#19151489C2213yes—
33#19151488C2213yes—
34#19151487C2213yes—
35#19151486C2213yes—
36#19151485C2213yescomplemented
37#19151484C2213yescomplemented
38#19151483C2213yescomplemented
39#19151482C2213yescomplemented
40#19151481C2213yescomplemented
41#19151480C2213yescomplemented
42#19151479C2213yescomplemented
43#19151466C2211yescharacteristic, complemented
44#19151465C2213yescomplemented
45#19151465C2213yescomplemented
46C4⋊C4×Dic89C2213yescomplemented
47C4⋊C4×Dic89C2213yescomplemented
48C4×#13083553C2213yescomplemented
49C4×#13083553C2213yescomplemented
50C4×C4⋊Dic89C2213yescomplemented
51#19151449C2213yescomplemented
52#19151449C2213yescomplemented
53C22⋊C4×Dic89C2213yes—
54C22⋊C4×Dic89C2213yes—
55C4×#13083551C2213yes—
56C4×#13083551C2213yes—
57C4×C22⋊Dic89C2213yes—
58#19151445C2213yes—
59#19151445C2213yes—
60#95×D89C2213yes—
61#95×D89C2213yes—
62#95×D89C2213yes—
63#95×D89C2213yes—
64C42×Dic89C2211yescharacteristic, complemented
65C42×Dic89C2213yescomplemented
66C42×Dic89C2213yescomplemented
67C2×#126×C89C2211yescharacteristic
68#388×C89C2213yes—
69#387×C89C2213yescomplemented
70#383×C89C2211yescharacteristic, complemented
71C4×C4⋊C4×C89C2213yescomplemented
72C4×C22⋊C4×C89C2213yes—
73C43×C89C2211yescharacteristic, complemented
74C24×D89C2311yescharacteristic
75C23×Dic89C2311yescharacteristic
76C22×C4×D89—212no—
77C22×C4×D89C2311yescharacteristic
78C22×C4×D89C2311yescharacteristic
79C22×C4×D89C2311yescharacteristic
80C22×C4×D89C2313yes—
81C22×C4×D89C2313yes—
82C22×C4×D89C2313yes—
83C22×C4×D89C2313yes—
84C22×C4×D89C2311yescharacteristic
85#15668007C2×C4124yescomplemented
86#15668006C2×C418yescomplemented
87#15668005C2×C4124yescomplemented
88C2×C4⋊Dic89C2313yes—
89C2×C4⋊Dic89C2313yes—
90C2×#13083553C2313yes—
91C2×#13083553C2313yes—
92C2×#13083553C2313yes—
93C2×#13083553C2313yes—
94C4⋊C4×D89—212no—
95C4⋊C4×D89C2×C4124yescomplemented
96C2×C22⋊Dic89C2313yes—
97C2×C22⋊Dic89C2313yes—
98C2×#13083551C2313yes—
99C2×#13083551C2313yes—
100C2×#13083551C2313yes—
101C2×#13083551C2313yes—
102C22⋊C4×D89—212no—
103C22⋊C4×D89C2×C4124yes—
104C2×C4×Dic89C2313yes—
105C2×C4×Dic89C2313yes—
106C2×C4×Dic89C2313yes—
107C2×C4×Dic89C2313yes—
108C2×C4×Dic89C2313yes—
109C2×C4×Dic89C2313yes—
110C2×C4×Dic89C2313yes—
111C2×C4×Dic89C2313yes—
112C2×C4×Dic89C2313yes—
113C2×C4×Dic89C2313yes—
114C42×D89—212no—
115C42×D89—212no—
116C42×D89—212no—
117C42×D89C2×C418yescomplemented
118#15667919C2313yes—
119#15667919C2313yes—
120#15667919C2313yes—
121#15667919C2313yes—
122#15667919C2313yes—
123#15667919C2313yes—
124#15667919C2313yes—
125#15667919C2313yes—
126#15667919C2313yes—
127#15667919C2313yes—
128#15667919C2313yes—
129#15667919C2313yes—
130C23×C4×C89C2311yescharacteristic
131#126×C89C2×C418yescomplemented
132C2×C4⋊C4×C89C2313yes—
133C2×C4⋊C4×C89C2313yes—
134C2×C22⋊C4×C89C2313yes—
135C2×C22⋊C4×C89C2313yes—
136C2×C42×C89C2313yes—
137C2×C42×C89C2313yes—
138C2×C42×C89C2313yes—
139C2×C42×C89C2311yescharacteristic
140#95×C89C2313yes—
141#95×C89C2313yes—
142#95×C89C2313yes—
143#95×C89C2313yes—
144C23×D89—26no—
145C23×D89—26no—
146C23×D89C22×C414yes—
147C23×D89C2411yescharacteristic
148C23×D89C2411yescharacteristic
149C22×Dic89—26no—
150C22×Dic89C2411yescharacteristic
151C22×Dic89C2413yes—
152C22×Dic89C2413yes—
153C22×Dic89C2413yes—
154C22×Dic89C2413yes—
155C22×Dic89C2411yescharacteristic
156C2×C4×D89—26no—
157C2×C4×D89—26no—
158C2×C4×D89—26no—
159C2×C4×D89—26no—
160C2×C4×D89—26no—
161C2×C4×D89—26no—
162C2×C4×D89—26no—
163C2×C4×D89—26no—
164C2×C4×D89—26no—
165C2×C4×D89—26no—
166C2×C4×D89—26no—
167C2×C4×D89—26no—
168C2×C4×D89—416no—
169C2×C4×D89C22×C4112yes—
170C2×C4×D89C22×C4112yes—
171C2×C4×D89#41112yes—
172C2×C4×D89#41112yes—
173C4⋊Dic89—26no—
174C4⋊Dic89C22×C4112yescomplemented
175#13083553—26no—
176#13083553—26no—
177#13083553C22×C4112yescomplemented
178#13083553C22×C4112yescomplemented
179C22⋊Dic89—26no—
180C22⋊Dic89C22×C4112yes—
181#13083551—26no—
182#13083551—26no—
183#13083551C22×C4112yes—
184#13083551C22×C4112yes—
185C4×Dic89—26no—
186C4×Dic89—26no—
187C4×Dic89—26no—
188C4×Dic89—26no—
189C4×Dic89—26no—
190C4×Dic89—26no—
191C4×Dic89—26no—
192C4×Dic89—26no—
193C4×Dic89—26no—
194C4×Dic89C22×C4112yescomplemented
195C24×C89C2411yescharacteristic
196C22×C4×C89—26no—
197C22×C4×C89C2411yescharacteristic
198C22×C4×C89C2413yes—
199C22×C4×C89C2413yes—
200C22×C4×C89C2413yes—
201C22×C4×C89C2413yes—
202C22×C4×C89C2411yescharacteristic
203C4⋊C4×C89—26no—
204C4⋊C4×C89C22×C4112yescomplemented
205C22⋊C4×C89—26no—
206C22⋊C4×C89C22×C4112yes—
207C42×C89—26no—
208C42×C89—26no—
209C42×C89—26no—
210C42×C89C22×C414yescomplemented
211C23×C89—23no—
212C23×C89—23no—
213C23×C89C23×C412yes—
214C23×C89C2511yescharacteristic
215C22×D89—23no—
216C22×D89—23no—
217C22×D89—224no—
218C22×D89—44no—
219C22×D89—412no—
220C22×D89C23×C412yes—
221C22×D89C23×C412yes—
222C22×D89C2×#4116yes—
223C22×D89C2×#4116yes—
224C2×C4×C89—23no—
225C2×C4×C89—23no—
226C2×C4×C89—23no—
227C2×C4×C89—23no—
228C2×C4×C89—23no—
229C2×C4×C89—23no—
230C2×C4×C89—23no—
231C2×C4×C89—23no—
232C2×C4×C89—23no—
233C2×C4×C89—23no—
234C2×C4×C89—44no—
235C2×C4×C89C23×C416yes—
236C2×C4×C89C23×C416yes—
237C2×C4×C89C2×#4116yes—
238C2×C4×C89C2×#4116yes—
239C2×Dic89—23no—
240C2×Dic89—23no—
241C2×Dic89—23no—
242C2×Dic89—23no—
243C2×Dic89—23no—
244C2×Dic89—23no—
245C2×Dic89—23no—
246C2×Dic89—23no—
247C2×Dic89—23no—
248C2×Dic89—23no—
249C2×Dic89—44no—
250C2×Dic89C23×C416yes—
251C2×Dic89C23×C416yes—
252C2×Dic89C2×#4116yes—
253C2×Dic89C2×#4116yes—
254C4×D89—224no—
255C4×D89—224no—
256C4×D89—224no—
257C4×D89—44no—
258C4×D89—412no—
259C4×D89—44no—
260C4×D89—44no—
261C4×D89#126116yesdirect product factor
262C22×C89—26no—
263C22×C89—41no—
264C22×C89—43no—
265C22×C89C24×C411yescharacteristic
266C22×C89C22×#4113yes—
267C22×C89C22×#4113yes—
268C2×D89—26no—
269C2×D89—41no—
270C2×D89—43no—
271C2×D89—416no—
272C2×D89C4×#41112yes—
273C2×D89C2×#12614yes—
274C2×D89#529112yes—
275C4×C89—26no—
276C4×C89—26no—
277C4×C89—26no—
278C4×C89—41no—
279C4×C89—43no—
280C4×C89C2×#12614yescomplemented
281Dic89—26no—
282Dic89—26no—
283Dic89—26no—
284Dic89—41no—
285Dic89—43no—
286Dic89C2×#12614yescomplemented
287C2×C4×#126—891nomaximal, Sylow
288C2×C89—42no—
289C2×C89C2×C4×#4113yes—
290C2×C89C22×#12611yescharacteristic
291C2×C89C2×#52913yes—
292D89—42no—
293D89C4×#12618yesdirect product factor
294C22×#126—891no—
295C4×#126—893no—
296C4×#126—893no—
297C4×#126—891no—
298C4×#126—893no—
299C4×#126—893no—
300C4×#126—891no—
301C4×#126—891no—
302C4×#126D8911yescharacteristic, direct product factor
303C2×#388—893no—
304C2×#387—893no—
305C2×#383—891no—
306C2×C4×C4⋊C4—893no—
307C2×C4×C22⋊C4—893no—
308C2×C43—891no—
309C89C2×C4×#12611yescharacteristic, complemented, Sylow
310C24×C4—891no—
311C2×#126—8916no—
312C2×#126—893no—
313C2×#126—893no—
314C2×#126—891no—
315C2×#126C2×D8911yescharacteristic
316C22×C4⋊C4—893no—
317C22×C4⋊C4—893no—
318C22×C22⋊C4—893no—
319C22×C22⋊C4—893no—
320C22×C42—891no—
321C22×C42—893no—
322C22×C42—893no—
323C22×C42—893no—
324#388—893no—
325#388—893no—
326#388—893no—
327#388—893no—
328#388—893no—
329#388—893no—
330#388—893no—
331#388C2×D8913yes—
332#387—893no—
333#387—893no—
334#387—893no—
335#387—893no—
336#387—893no—
337#387—893no—
338#387—893no—
339#387C2×D8913yescomplemented
340#383—891no—
341#383—893no—
342#383—893no—
343#383C2×D8911yescharacteristic, complemented
344C4×C4⋊C4—893no—
345C4×C4⋊C4—893no—
346C4×C4⋊C4—893no—
347C4×C4⋊C4—893no—
348C4×C4⋊C4—893no—
349C4×C4⋊C4—893no—
350C4×C4⋊C4—893no—
351C4×C4⋊C4C2×D8913yescomplemented
352C4×C22⋊C4—893no—
353C4×C22⋊C4—893no—
354C4×C22⋊C4—893no—
355C4×C22⋊C4—893no—
356C4×C22⋊C4—893no—
357C4×C22⋊C4—893no—
358C4×C22⋊C4—893no—
359C4×C22⋊C4C2×D8913yes—
360C2×#95—893no—
361C2×#95—893no—
362C2×#95—893no—
363C2×#95—893no—
364C43—893no—
365C43—893no—
366C43—891no—
367C43C2×D8911yescharacteristic, complemented
368C25—891no—
369C23×C4—891no—
370C23×C4—893no—
371C23×C4—893no—
372C23×C4—893no—
373C23×C4—891no—
374C23×C4—893no—
375C23×C4—891no—
376C23×C4—891no—
377C23×C4—891no—
378C23×C4—17812no—
379C23×C4C22×D8911yescharacteristic
380#126—8924no—
381#126—8924no—
382#126—898no—
383#126C4×D8918yesdirect product factor
384C2×C4⋊C4—893no—
385C2×C4⋊C4—893no—
386C2×C4⋊C4—893no—
387C2×C4⋊C4—8924no—
388C2×C4⋊C4—893no—
389C2×C4⋊C4—893no—
390C2×C4⋊C4—893no—
391C2×C4⋊C4—17812no—
392C2×C4⋊C4C22×D8913yes—
393C2×C4⋊C4C22×D8913yes—
394C2×C22⋊C4—893no—
395C2×C22⋊C4—893no—
396C2×C22⋊C4—893no—
397C2×C22⋊C4—8924no—
398C2×C22⋊C4—893no—
399C2×C22⋊C4—893no—
400C2×C22⋊C4—893no—
401C2×C22⋊C4—17812no—
402C2×C22⋊C4C22×D8913yes—
403C2×C22⋊C4C22×D8913yes—
404C2×C42—893no—
405C2×C42—898no—
406C2×C42—893no—
407C2×C42—893no—
408C2×C42—893no—
409C2×C42—893no—
410C2×C42—893no—
411C2×C42—893no—
412C2×C42—893no—
413C2×C42—893no—
414C2×C42—893no—
415C2×C42—17812no—
416C2×C42—17812no—
417C2×C42—17812no—
418C2×C42C22×D8913yes—
419C2×C42C22×D8913yes—
420C2×C42C22×D8913yes—
421C2×C42C22×D8911yescharacteristic
422#95—893no—
423#95—893no—
424#95—893no—
425#95—893no—
426#95—893no—
427#95—893no—
428#95—893no—
429#95—893no—
430#95—893no—
431#95—893no—
432#95—893no—
433#95—893no—
434#95C22×D8913yes—
435#95C22×D8913yes—
436#95C22×D8913yes—
437#95C22×D8913yes—
438C24—891no—
439C24—894no—
440C24—891no—
441C24—1786no—
442C24—1786no—
443C24C23×D8911yescharacteristic
444C22×C4—26no—
445C22×C4—891no—
446C22×C4—8912no—
447C22×C4—893no—
448C22×C4—893no—
449C22×C4—893no—
450C22×C4—8912no—
451C22×C4—8912no—
452C22×C4—8912no—
453C22×C4—891no—
454C22×C4—893no—
455C22×C4—35616no—
456C22×C4—1786no—
457C22×C4—1786no—
458C22×C4—1786no—
459C22×C4—1786no—
460C22×C4—1786no—
461C22×C4—1786no—
462C22×C4—1786no—
463C22×C4—1786no—
464C22×C4—1786no—
465C22×C4—1786no—
466C22×C4—1786no—
467C22×C4—1786no—
468C22×C4—1786no—
469C22×C4C23×D8911yescharacteristic
470C22×C4C23×D8913yes—
471C22×C4C23×D8913yes—
472C22×C4C23×D8913yes—
473C22×C4C23×D8913yes—
474C22×C4C23×D8911yescharacteristic
475C4⋊C4—26no—
476C4⋊C4—8912no—
477C4⋊C4—8912no—
478C4⋊C4—8912no—
479C4⋊C4—1786no—
480C4⋊C4—1786no—
481C4⋊C4—1786no—
482C4⋊C4C2×C4×D89112yescomplemented
483C22⋊C4—26no—
484C22⋊C4—8912no—
485C22⋊C4—8912no—
486C22⋊C4—8912no—
487C22⋊C4—1786no—
488C22⋊C4—1786no—
489C22⋊C4—1786no—
490C22⋊C4C2×C4×D89112yes—
491C42—26no—
492C42—26no—
493C42—26no—
494C42—8912no—
495C42—1786no—
496C42—1786no—
497C42—1786no—
498C42—1786no—
499C42—1786no—
500C42—1786no—
501C42—1786no—
502C42—1786no—
503C42—1786no—
504C42C2×C4×D8914yescomplemented
505C23—23no—
506C23—23no—
507C23—892no—
508C23—896no—
509C23—896no—
510C23—892no—
511C23—3564no—
512C23—35612no—
513C23—17824no—
514C23—1783no—
515C23—1783no—
516C23C22×C4×D8912yes—
517C23C24×D8911yescharacteristic
518C2×C4—23no—
519C2×C4—23no—
520C2×C4—23no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—23no—
528C2×C4—44no—
529C2×C4—896no—
530C2×C4—8916no—
531C2×C4—896no—
532C2×C4—896no—
533C2×C4—896no—
534C2×C4—3564no—
535C2×C4—3564no—
536C2×C4—35612no—
537C2×C4—3564no—
538C2×C4—3564no—
539C2×C4—17824no—
540C2×C4—17824no—
541C2×C4—17824no—
542C2×C4—1783no—
543C2×C4—1783no—
544C2×C4—1783no—
545C2×C4—1783no—
546C2×C4—1783no—
547C2×C4—1783no—
548C2×C4—1783no—
549C2×C4—1783no—
550C2×C4—1783no—
551C2×C4—1783no—
552C2×C4C22×C4×D8916yes—
553C2×C4C22×C4×D8916yes—
554C2×C4#41×D8916yes—
555C2×C4#41×D8916yes—
556C22—26no—
557C22—41no—
558C22—43no—
559C22—8912no—
560C22—894no—
561C22—8912no—
562C22—35616no—
563C22—3561no—
564C22—3563no—
565C22—1786no—
566C22C23×C4×D8911yescharacteristic
567C22C2×#41×D8913yes—
568C22C2×#41×D8913yes—
569C4—26no—
570C4—26no—
571C4—26no—
572C4—41no—
573C4—43no—
574C4—894no—
575C4—3561no—
576C4—3563no—
577C4—1786no—
578C4—1786no—
579C4—1786no—
580C4#126×D8914yesdirect product factor
581C2—42no—
582C2—898no—
583C2—3562no—
584C2C4×#41×D8913yes—
585C2C2×#126×D8911yescharacteristic
586C2#529×D8913yes—
subgroupG/H
centerC22×C4C23×D89
commutatorC22×C89C24×C4
FrattiniC23C24×D89
FittingC4×#126×C89C2
socle, cosocleC23×C89C25
prime radicalC24×D89C23
hypercenterC4×#126D89
hypocenterC89C2×C4×#126
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C4×#126

Complements

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

NH ≅ G/Ntype
2. #26022806C2semidirect
3. C4×#15668007C2semidirect
4. #126×Dic89C2semidirect
5. C4×#15668006C2semidirect
6. #26022802C2semidirect
7. C4×#15668005C2semidirect
8. #26022800C2semidirect
10. #387×D89C2semidirect
11. #383×D89C2semidirect
12. C4×C4⋊C4×D89C2semidirect
14. C43×D89C2semidirect
15. C4×#126×C89C2semidirect
20. #126×D89C4direct (N × H)
36. #19151485C22semidirect
37. #19151484C22semidirect
38. #19151483C22semidirect
39. #19151482C22semidirect
40. #19151481C22semidirect
41. #19151480C22semidirect
42. #19151479C22semidirect
43. #19151466C22semidirect
44. #19151465C22semidirect
45. #19151465C22semidirect
46. C4⋊C4×Dic89C22semidirect
47. C4⋊C4×Dic89C22semidirect
48. C4×#13083553C22semidirect
49. C4×#13083553C22semidirect
50. C4×C4⋊Dic89C22semidirect
51. #19151449C22semidirect
52. #19151449C22semidirect
64. C42×Dic89C22semidirect
65. C42×Dic89C22semidirect
66. C42×Dic89C22semidirect
69. #387×C89C22semidirect
70. #383×C89C22semidirect
71. C4×C4⋊C4×C89C22semidirect
73. C43×C89C22semidirect
85. #15668007C2×C4semidirect
86. #15668006C2×C4semidirect
87. #15668005C2×C4semidirect
95. C4⋊C4×D89C2×C4semidirect
117. C42×D89C2×C4semidirect
131. #126×C89C2×C4semidirect
174. C4⋊Dic89C22×C4semidirect
177. #13083553C22×C4semidirect
178. #13083553C22×C4semidirect
194. C4×Dic89C22×C4semidirect
204. C4⋊C4×C89C22×C4semidirect
210. C42×C89C22×C4semidirect
261. C4×D89#126direct (N × H)
280. C4×C89C2×#126semidirect
286. Dic89C2×#126semidirect
293. D89C4×#126direct (N × H)
302. C4×#126D89direct (N × H)
309. C89C2×C4×#126semidirect
339. #387C2×D89semidirect
343. #383C2×D89semidirect
351. C4×C4⋊C4C2×D89semidirect
367. C43C2×D89semidirect
383. #126C4×D89direct (N × H)
482. C4⋊C4C2×C4×D89semidirect
504. C42C2×C4×D89semidirect
580. C4#126×D89direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#126×D892
#260228062
C4×#156680072
#126×Dic892
C4×#156680062
#260228022
C4×#156680052
#260228002
#388×D892
#387×D892
#383×D892
C4×C4⋊C4×D892
C4×C22⋊C4×D892
C43×D892
C4×#126×C892
C2×C4×#12689