Finite Group Databasebeta

← #25097830

Subgroups — #25097830

Order 11008.

1954 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583409C211yesmaximal, characteristic
2C2×#117×D43C211yesmaximal, characteristic, complemented
3C2×#14582897C212yesmaximal, complemented
4C2×#14582861C211yesmaximal, characteristic, complemented
5#19134918C211yesmaximal, characteristic, complemented
6#19134914C212yesmaximal, complemented
7#19134907C211yesmaximal, characteristic, complemented
8#19134906C212yesmaximal, complemented
9#19134905C211yesmaximal, characteristic, complemented
10#19134313C212yesmaximal, complemented
11C43×#1875C211yesmaximal, characteristic, complemented
12C22×#10621900C2211yescharacteristic
13C23×C4×D43C2211yescharacteristic
14C22×#10621872C2212yes—
15#14583409—24no—
16#14583409C418yescomplemented
17C2×#13063307—21no—
18C2×#13063305C2211yescharacteristic
19C2×#13063304C2211yescharacteristic
20C2×#13063296C2211yescharacteristic
21C2×#13063293—21no—
22C2×D4×Dic43—21no—
23C2×(C22⋊D43)⋊C4—21no—
24C2×#13063284C2211yescharacteristic
25C2×#13063283C2212yes—
26C2×#13063283C2211yescharacteristic
27C2×#13063282—21no—
28C2×#13063281—21no—
29#117×D43—28no—
30C2×C4⋊C4×D43C2212yes—
31C2×C22⋊C4×D43—21no—
32C2×C42×D43—21no—
33#14582897—28no—
34C2×#13063250C2212yescomplemented
35C2×#13063248C2212yescomplemented
36C2×#13063246C2212yescomplemented
37#14582861—24no—
38C2×C22⋊(C43⋊C8)C2211yescharacteristic
39C2×(C2×D43)⋊C8C2212yes—
40#14582662C2212yescomplemented
41#14582661C2212yescomplemented
42#14582661C2212yescomplemented
43C2×#121×C43C2211yescharacteristic
44C2×#117×C43C2211yescharacteristic
45C2×#102×C43C2212yescomplemented
46C2×C22⋊C8×C43C2211yescharacteristic
47C43×#340C2212yescomplemented
48C24×D43D411yescharacteristic
49C2×#10621900—24no—
50C2×#10621900—22no—
51C2×#10621900—24no—
52C2×#10621900C2×C414yes—
53C22×Dic86C2311yescharacteristic
54C22×C22⋊D43—21no—
55C22×C22⋊D43—21no—
56C23×Dic43—21no—
57C23×Dic43D411yescharacteristic
58C22×C4×D43—24no—
59C22×C4×D43—24no—
60C22×C4×D43—22no—
61C22×C4×D43C2×C414yes—
62C22×C4×D43D411yescharacteristic
63C22×C4×D43D411yescharacteristic
64C22×C4×D43D414yes—
65C22×C4×D43C2311yescharacteristic
66C22×C4×D43C2311yescharacteristic
67C2×#10621872—24no—
68C2×#10621872—24no—
69C2×#10621872—24no—
70C22×C43⋊C8C2312yes—
71#13063307—28no—
72#13063305—22no—
73#13063305C2×C414yescomplemented
74#13063304—22no—
75#13063304C2×C414yescomplemented
76#13063296—22no—
77#13063296C2×C414yescomplemented
78#13063293—28no—
79D4×Dic43—28no—
80(C22⋊D43)⋊C4—28no—
81#13063284—24no—
82#13063283—22no—
83#13063283—24no—
84#13063283—24no—
85#13063283C2×C414yes—
86#13063282—28no—
87#13063281—28no—
88C2×C4⋊Dic43—21no—
89C2×C4⋊Dic43C2311yescharacteristic
90C2×C4⋊Dic43C2312yes—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—22no—
93C4⋊C4×D43—24no—
94C4⋊C4×D43D418yescomplemented
95C2×C22⋊Dic43—21no—
96C2×C22⋊Dic43—21no—
97C2×C22⋊Dic43—21no—
98C2×(C2×D43)⋊C4—22no—
99C2×(C2×D43)⋊C4—21no—
100C2×(C2×D43)⋊C4—21no—
101C22⋊C4×D43—28no—
102C2×C4×Dic43—21no—
103C2×C4×Dic43—22no—
104C2×C4×Dic43—21no—
105C42×D43—28no—
106#13063250—24no—
107#13063248—24no—
108#13063246—24no—
109C22⋊(C43⋊C8)—22no—
110(C2×D43)⋊C8—24no—
111C22×D4×C43—21no—
112C22×D4×C43C2311yescharacteristic
113C23×C4×C43C2311yescharacteristic
114C22×C8×C43C2312yes—
115#121×C43—22no—
116#121×C43C2×C414yescomplemented
117#117×C43—24no—
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2312yes—
120C2×C22⋊C4×C43—21no—
121C2×C22⋊C4×C43—21no—
122C2×C42×C43—21no—
123#102×C43—24no—
124C22⋊C8×C43—22no—
125C24×C43—21no—
126C24×C43C2×D411yescharacteristic
127C23×D43—22no—
128C23×D43—22no—
129C23×D43—42no—
130C23×D43—42no—
131C23×D43C22⋊C414yes—
132C23×D43C2×D411yescharacteristic
133C23×D43C2×D411yescharacteristic
134C2×D4×C43—21no—
135C2×D4×C43—24no—
136C2×D4×C43—22no—
137C2×D4×C43—22no—
138C2×D4×C43—44no—
139C2×D4×C43C22×C412yes—
140C22×C4×C43—22no—
141C22×C4×C43—22no—
142C22×C4×C43—22no—
143C22×C4×C43—21no—
144C22×C4×C43C22×C412yes—
145C22×C4×C43C2×D412yes—
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2411yescharacteristic
148C2×C22⋊D43—24no—
149C2×C22⋊D43—24no—
150C2×C22⋊D43—44no—
151C2×C22⋊D43—42no—
152C2×C22⋊D43—42no—
153C22×Dic43—22no—
154C22×Dic43—22no—
155C22×Dic43—24no—
156C22×Dic43—22no—
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—42no—
160C22×Dic43—42no—
161C22×Dic43C22⋊C414yes—
162C22×Dic43C2×D411yescharacteristic
163C22×Dic43C2×D411yescharacteristic
164#10621900—216no—
165#10621900—48no—
166C2×C4×D43—22no—
167C2×C4×D43—22no—
168C2×C4×D43—22no—
169C2×C4×D43—22no—
170C2×C4×D43—216no—
171C2×C4×D43—216no—
172C2×C4×D43—22no—
173C2×C4×D43—28no—
174C2×C4×D43—22no—
175C2×C4×D43—44no—
176C2×C4×D43—44no—
177C2×C4×D43C22⋊C414yes—
178C2×C4×D43C22⋊C414yes—
179C2×C4×D43C22×C412yes—
180C2×C4×D43C22×C412yes—
181C2×C4×D43C2×D412yes—
182C2×C4×D43C2×D412yes—
183C2×C4×D43C2×D414yes—
184C2×C4×D43#4114yes—
185C2×Dic86—22no—
186C2×Dic86—22no—
187C2×Dic86—21no—
188C2×Dic86C22×C412yes—
189C2×C8×C43—22no—
190C2×C8×C43—22no—
191C2×C8×C43—22no—
192C4⋊C4×C43—22no—
193C4⋊C4×C43—21no—
194C4⋊C4×C43C22×C412yes—
195C4⋊C4×C43C2×D414yes—
196C22⋊C4×C43—24no—
197C22⋊C4×C43—24no—
198C42×C43—24no—
199C22⋊Dic43—24no—
200C22⋊Dic43—24no—
201C22⋊Dic43—24no—
202(C2×D43)⋊C4—28no—
203(C2×D43)⋊C4—24no—
204(C2×D43)⋊C4—24no—
205C4⋊Dic43—22no—
206C4⋊Dic43—21no—
207C4⋊Dic43—24no—
208C4⋊Dic43C22×C412yes—
209C4⋊Dic43C2×D414yes—
210Dic43⋊C4—24no—
211Dic43⋊C4—28no—
212C4×Dic43—28no—
213C4×Dic43—24no—
214C4×Dic43—24no—
215C2×C43⋊C8—22no—
216C2×C43⋊C8—22no—
217C2×C43⋊C8—22no—
218#10621872—48no—
219C23×C43—21no—
220C23×C43—21no—
221C23×C43—22no—
222C23×C43—22no—
223C23×C43—22no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43—42no—
227C23×C43—42no—
228C23×C43C2×C22⋊C412yes—
229C23×C43C22×D411yescharacteristic
230C22×D43—22no—
231C22×D43—22no—
232C22×D43—21no—
233C22×D43—21no—
234C22×D43—22no—
235C22×D43—22no—
236C22×D43—28no—
237C22×D43—44no—
238C22×D43—44no—
239C22×D43—48no—
240C22×D43—48no—
241C22×D43C2×C22⋊C412yes—
242C22×D43C2×C22⋊C412yes—
243C22×D43#12012yes—
244C22×D43#12312yes—
245D4×C43—24no—
246D4×C43—48no—
247D4×C43—42no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—24no—
251C2×C4×C43—24no—
252C2×C4×C43—22no—
253C2×C4×C43—24no—
254C2×C4×C43—42no—
255C2×C4×C43—42no—
256C2×C4×C43—42no—
257C2×C4×C43C2×C22⋊C412yes—
258C2×C4×C43C23×C411yescharacteristic
259C2×C4×C43C22×D412yes—
260C2×C4×C43C22×D411yescharacteristic
261C2×C4×C43C2×#4112yes—
262C22⋊D43—48no—
263C22⋊D43—48no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—28no—
267C2×Dic43—22no—
268C2×Dic43—22no—
269C2×Dic43—24no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—42no—
273C2×Dic43—42no—
274C2×Dic43—42no—
275C2×Dic43—48no—
276C2×Dic43C2×C22⋊C412yes—
277C2×Dic43C2×C22⋊C412yes—
278C2×Dic43#12012yes—
279C2×Dic43#12312yes—
280C4×D43—28no—
281C4×D43—24no—
282C4×D43—24no—
283C4×D43—28no—
284C4×D43—416no—
285C4×D43C4×D418yes—
286C4×D43#12118yes—
287Dic86—24no—
288Dic86—42no—
289C8×C43—42no—
290C43⋊C8—42no—
291#20910—431nomaximal, Sylow
292C22×C43—22no—
293C22×C43—22no—
294C22×C43—22no—
295C22×C43—42no—
296C22×C43—42no—
297C22×C43—42no—
298C22×C43—42no—
299C22×C43—42no—
300C22×C43—82no—
301C22×C43—82no—
302C22×C43C22×C22⋊C411yescharacteristic
303C22×C43C2×#12011yescharacteristic
304C22×C43C2×#12311yescharacteristic
305C22×C43#57512yes—
306C22×C43#57512yes—
307C2×D43—24no—
308C2×D43—22no—
309C2×D43—22no—
310C2×D43—24no—
311C2×D43—22no—
312C2×D43—22no—
313C2×D43—22no—
314C2×D43—416no—
315C2×D43—42no—
316C2×D43—42no—
317C2×D43#38614yes—
318C4×C43—22no—
319C4×C43—42no—
320C4×C43—44no—
321C4×C43C2×C4×D412yes—
322C4×C43C2×#12112yes—
323Dic43—22no—
324Dic43—42no—
325Dic43—44no—
326Dic43—42no—
327Dic43#38614yes—
328C2×#525—431no—
329C22×#117—431no—
330C2×#417—432no—
331C2×#407—431no—
332#1875—432no—
333#1875—431no—
334#1875D4311yescharacteristic, complemented
335#1873—431no—
336#1873—432no—
337#1873—431no—
338#1736—432no—
339C2×C43—42no—
340C2×C43—82no—
341C2×C43C2×#38611yescharacteristic
342C2×C43#289611yescharacteristic
343C2×C43#289611yescharacteristic
344C2×C43#300512yes—
345C2×C43#309312yes—
346D43—24no—
347D43—42no—
348C22×#41—431no—
349C24×C4—431no—
350C22×M16—432no—
351#525—438no—
352#525—864no—
353C2×#123—861no—
354C2×C22⋊Q8—431no—
355C2×C22⋊Q8—431no—
356C2×#121—431no—
357C2×#121C2×D4311yescharacteristic
358C2×#120—861no—
359C2×C4×D4—861no—
360C2×C4×D4—861no—
361C2×#117—431no—
362C2×#117—431no—
363C2×#117—432no—
364C2×#117—861no—
365C2×#117—861no—
366C2×#117—868no—
367C2×#117C2×D4311yescharacteristic
368C22×C4⋊C4—432no—
369C22×C22⋊C4—861no—
370C22×C42—861no—
371#417—868no—
372C2×#103—432no—
373C2×#103—432no—
374C2×#102—432no—
375C2×#102C2×D4312yescomplemented
376#407—864no—
377C2×C22⋊C8—431no—
378C2×C22⋊C8—432no—
379C2×C22⋊C8C2×D4311yescharacteristic
380#340—432no—
381#340—432no—
382#340—432no—
383#340C2×D4312yescomplemented
384C43#2091011yescharacteristic, complemented, Sylow
385C25—431no—
386C2×#41—434no—
387C2×#41—862no—
388C2×#41—864no—
389C2×#41—864no—
390C22×Q8—431no—
391C22×D4—21no—
392C22×D4—861no—
393C22×D4—861no—
394C22×D4C22×D4311yescharacteristic
395C23×C4—431no—
396C23×C4—431no—
397C23×C4—434no—
398C23×C4—431no—
399C23×C4—431no—
400C23×C4—434no—
401C23×C4—431no—
402C23×C4—864no—
403C23×C4—864no—
404C23×C4—862no—
405C23×C4—861no—
406C23×C4C22×D4311yescharacteristic
407C2×M16—864no—
408C2×M16—864no—
409C2×M16—864no—
410C22×C8—432no—
411C22×C8C22×D4312yes—
412#123—868no—
413C22⋊Q8—434no—
414C22⋊Q8—434no—
415C22⋊Q8—862no—
416C22⋊Q8—862no—
417#121—22no—
418#121—434no—
419#121—862no—
420#121C4×D4314yescomplemented
421#120—868no—
422C4×D4—868no—
423C4×D4—868no—
424#117—24no—
425#117—434no—
426#117—868no—
427#117—868no—
428#117—864no—
429#117—862no—
430#117—864no—
431#117—864no—
432C2×C4⋊C4—432no—
433C2×C4⋊C4—431no—
434C2×C4⋊C4—438no—
435C2×C4⋊C4—864no—
436C2×C4⋊C4—862no—
437C2×C4⋊C4—861no—
438C2×C4⋊C4—861no—
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C4⋊C4C22×D4312yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—868no—
444C2×C22⋊C4—862no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C22⋊C4—861no—
448C2×C22⋊C4—861no—
449C2×C22⋊C4—861no—
450C2×C42—21no—
451C2×C42—868no—
452C2×C42—862no—
453C2×C42—861no—
454C2×C42—861no—
455#103—864no—
456#103—864no—
457#102—24no—
458#102—864no—
459C22⋊C8—22no—
460C22⋊C8—864no—
461C22⋊C8—862no—
462C24—21no—
463C24—431no—
464C24—431no—
465C24—434no—
466C24—862no—
467C24—862no—
468C24—1722no—
469C24—1722no—
470C24D4×D4311yescharacteristic
471#41—8616no—
472#41—1728no—
473C2×Q8—432no—
474C2×Q8—861no—
475C2×Q8—862no—
476C2×Q8—862no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—44no—
482C2×D4—864no—
483C2×D4—864no—
484C2×D4—1722no—
485C2×D4—1722no—
486C2×D4—1724no—
487C2×D4C2×C4×D4312yes—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—431no—
493C22×C4—432no—
494C22×C4—434no—
495C22×C4—432no—
496C22×C4—434no—
497C22×C4—434no—
498C22×C4—431no—
499C22×C4—434no—
500C22×C4—434no—
501C22×C4—432no—
502C22×C4—432no—
503C22×C4—862no—
504C22×C4—862no—
505C22×C4—8616no—
506C22×C4—8616no—
507C22×C4—862no—
508C22×C4—862no—
509C22×C4—868no—
510C22×C4—862no—
511C22×C4—862no—
512C22×C4—864no—
513C22×C4—862no—
514C22×C4—862no—
515C22×C4—862no—
516C22×C4—861no—
517C22×C4—861no—
518C22×C4—1724no—
519C22×C4—1724no—
520C22×C4—1722no—
521C22×C4—1722no—
522C22×C4C2×C4×D4312yes—
523C22×C4D4×D4311yescharacteristic
524C22×C4D4×D4312yes—
525C22×C4C23×D4311yescharacteristic
526M16—1728no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—862no—
531C2×C8—862no—
532C2×C8—862no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—432no—
536C4⋊C4—434no—
537C4⋊C4—868no—
538C4⋊C4—864no—
539C4⋊C4—864no—
540C4⋊C4—862no—
541C4⋊C4—861no—
542C4⋊C4C2×C4×D4312yes—
543C4⋊C4D4×D4314yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—864no—
547C22⋊C4—864no—
548C22⋊C4—868no—
549C22⋊C4—864no—
550C22⋊C4—864no—
551C22⋊C4—864no—
552C42—24no—
553C42—864no—
554C42—864no—
555C42—868no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—432no—
566C23—432no—
567C23—432no—
568C23—432no—
569C23—868no—
570C23—862no—
571C23—862no—
572C23—861no—
573C23—861no—
574C23—862no—
575C23—862no—
576C23—1724no—
577C23—1724no—
578C23—1728no—
579C23—1728no—
580C23C22⋊C4×D4312yes—
581C23C2×D4×D4311yescharacteristic
582Q8—864no—
583Q8—1722no—
584D4—24no—
585D4—48no—
586D4—42no—
587D4—1728no—
588D4—1728no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—432no—
599C2×C4—432no—
600C2×C4—432no—
601C2×C4—438no—
602C2×C4—438no—
603C2×C4—432no—
604C2×C4—868no—
605C2×C4—868no—
606C2×C4—868no—
607C2×C4—861no—
608C2×C4—861no—
609C2×C4—864no—
610C2×C4—864no—
611C2×C4—862no—
612C2×C4—862no—
613C2×C4—864no—
614C2×C4—17216no—
615C2×C4—1728no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4—1722no—
619C2×C4—1722no—
620C2×C4—1722no—
621C2×C4C22⋊C4×D4312yes—
622C2×C4C22×C4×D4311yescharacteristic
623C2×C4C2×D4×D4312yes—
624C2×C4C2×D4×D4311yescharacteristic
625C2×C4#41×D4312yes—
626C8—42no—
627C8—1722no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—82no—
637C22—82no—
638C22—434no—
639C22—864no—
640C22—864no—
641C22—862no—
642C22—862no—
643C22—862no—
644C22—862no—
645C22—862no—
646C22—17216no—
647C22—1722no—
648C22—1722no—
649C22C2×C22⋊C4×D4311yescharacteristic
650C22#120×D4311yescharacteristic
651C22#123×D4311yescharacteristic
652C22#1458369512yes—
653C22#1458369612yes—
654C4—22no—
655C4—44no—
656C4—42no—
657C4—434no—
658C4—862no—
659C4—1724no—
660C4—1722no—
661C4—1722no—
662C4C4×D4×D4312yes—
663C4#121×D4312yes—
664C2—42no—
665C2—82no—
666C2—864no—
667C2—1722no—
668C2#386×D4311yescharacteristic
669C2#1914310911yescharacteristic
670C2#1914311011yescharacteristic
671C2#1914383312yes—
672C2#1914452312yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1875C2
socleC23×C43C22×D4
prime radicalC2×#14583409C2
cosocleC22×C4×C43C24
hypercenter#1875D43
hypocenterC43#20910
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. C2×#117×D43C2semidirect
3. C2×#14582897C2semidirect
4. C2×#14582861C2semidirect
5. #19134918C2semidirect
6. #19134914C2semidirect
7. #19134907C2semidirect
8. #19134906C2semidirect
9. #19134905C2semidirect
10. #19134313C2semidirect
11. C43×#1875C2semidirect
16. #14583409C4semidirect
34. C2×#13063250C22semidirect
35. C2×#13063248C22semidirect
36. C2×#13063246C22semidirect
40. #14582662C22semidirect
41. #14582661C22semidirect
42. #14582661C22semidirect
45. C2×#102×C43C22semidirect
47. C43×#340C22semidirect
73. #13063305C2×C4semidirect
75. #13063304C2×C4semidirect
77. #13063296C2×C4semidirect
94. C4⋊C4×D43D4semidirect
116. #121×C43C2×C4semidirect
334. #1875D43semidirect
375. C2×#102C2×D43semidirect
383. #340C2×D43semidirect
384. C43#20910semidirect
420. #121C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834092
C2×#117×D432
C2×#145828972
C2×#145828612
#191349182
#191349142
#191349072
#191349062
#191349052
#191343132
C43×#18752
#2091043