Finite Group Databasebeta

← #25097367

Subgroups — #25097367

Order 11008.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583379C211yesmaximal, characteristic
2C2×#14582942C211yesmaximal, characteristic, complemented
3C2×#14582865C211yesmaximal, characteristic
4#19137183C211yesmaximal, characteristic
5#19134793C212yesmaximal, complemented
6#19134791C212yesmaximal, complemented
7#19134786C212yesmaximal, complemented
8#19134781C212yesmaximal, complemented
9#19134316C211yesmaximal, characteristic, complemented
10C2×#14582662C211yesmaximal, characteristic, complemented
11C2×C43×#427C211yesmaximal, characteristic, complemented
12C22×#10621898C2211yescharacteristic
13C22×#10621877C2211yescharacteristic
14#14583379—24no—
15#14583379—24no—
16C2×C4×Dic86—21no—
17C2×C4×D172—21no—
18C2×(C22⋊D43)⋊C4—22no—
19C2×#13063283C2212yes—
20C22×C4⋊Dic43C2211yescharacteristic
21C22×C4⋊Dic43C2211yescharacteristic
22C2×C4⋊C4×D43C2212yes—
23#14582942—24no—
24C2×#13063258C2212yes—
25C2×#13063256C2212yes—
26#14582865C418yescomplemented
27C2×(C2×D43)⋊C8C2212yes—
28C2×(C2×D43)⋊C8C2212yes—
29#14582805—21no—
30#14582801—21no—
31#14582776—22no—
32#14582756C2212yes—
33#14582756C2212yes—
34#14582662C2212yes—
35#14582662C2212yes—
36#14582661C2212yes—
37#14582661C2212yes—
38C23×C8×C43C2211yescharacteristic
39C2×#117×C43C2211yescharacteristic
40C2×#117×C43C2211yescharacteristic
41C43×#427—24no—
42C2×#107×C43C2211yescharacteristic
43C2×#107×C43C2211yescharacteristic
44C2×#106×C43C2211yescharacteristic
45C2×#106×C43C2211yescharacteristic
46C2×#10621898—22no—
47C2×#10621898—22no—
48C2×#10621898—22no—
49C2×#10621898—22no—
50C2×#10621898C2×C414yes—
51C2×#10621898Q814yes—
52C22×Dic86D411yescharacteristic
53C22×D172D411yescharacteristic
54C22×C22⋊D43D412yes—
55C23×Dic43—21no—
56C23×Dic43—21no—
57C22×C4×D43—22no—
58C22×C4×D43C2312yes—
59C2×#10621877—22no—
60C2×#10621877—22no—
61C2×#10621877C2×C414yescomplemented
62C22×C43⋊C8C2312yes—
63C4×Dic86—28no—
64C4×D172—28no—
65(C22⋊D43)⋊C4—216no—
66#13063283—24no—
67#13063283—24no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—22no—
70C2×C4⋊Dic43—22no—
71C2×C4⋊Dic43—22no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43D414yes—
74C2×C4⋊Dic43C2312yes—
75C2×C4⋊Dic43C2312yes—
76C2×Dic43⋊C4—22no—
77C4⋊C4×D43—24no—
78C4⋊C4×D43—24no—
79C2×(C2×D43)⋊C4—22no—
80C2×(C2×D43)⋊C4—22no—
81C2×(C2×D43)⋊C4—22no—
82C2×C4×Dic43—22no—
83#13063258—24no—
84#13063256—24no—
85(C2×D43)⋊C8C2×C418yescomplemented
86(C2×D43)⋊C8C2×C418yescomplemented
87C43⋊#95—22no—
88C43⋊#95—22no—
89C23×C4×C43C2311yescharacteristic
90C22×C8×C43—22no—
91C22×C8×C43—22no—
92C22×C8×C43C2×C414yescomplemented
93C22×C8×C43C2311yescharacteristic
94C22×C8×C43C2311yescharacteristic
95#117×C43—22no—
96#117×C43—22no—
97#117×C43—22no—
98#117×C43D414yes—
99C2×C4⋊C4×C43C2311yescharacteristic
100C2×C4⋊C4×C43C2311yescharacteristic
101C2×C4⋊C4×C43C2311yescharacteristic
102C2×C4⋊C4×C43C2311yescharacteristic
103C2×C22⋊C4×C43—21no—
104C2×C22⋊C4×C43—21no—
105C2×C42×C43—21no—
106C2×C42×C43—21no—
107#107×C43—22no—
108#107×C43—22no—
109#106×C43—22no—
110#106×C43—22no—
111C24×C43C2×D411yescharacteristic
112C23×D43C2×D412yes—
113C22×C4×C43—21no—
114C22×C4×C43—21no—
115C22×C4×C43—21no—
116C22×C4×C43—21no—
117C22×C4×C43C22×C412yes—
118C22×C4×C43C2×D412yes—
119C22×C4×C43C2×D412yes—
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43C2×Q812yes—
122C22×C4×C43#4112yes—
123C22×C4×C43#4112yes—
124C22×C4×C43C2411yescharacteristic
125C2×C22⋊D43—24no—
126C2×C22⋊D43—24no—
127C2×C22⋊D43C4⋊C418yes—
128C22×Dic43—24no—
129C22×Dic43—22no—
130C22×Dic43—22no—
131C22×Dic43—42no—
132C22×Dic43—81no—
133C22×Dic43—81no—
134C22×Dic43—81no—
135C22×Dic43—81no—
136C22×Dic43C2×D412yes—
137#10621898—28no—
138#10621898—28no—
139#10621898—28no—
140#10621898—28no—
141C2×D172—22no—
142C2×D172—22no—
143C2×D172C4⋊C414yes—
144C2×C4×D43—22no—
145C2×C4×D43—22no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—28no—
149C2×C4×D43—82no—
150C2×C4×D43—82no—
151C2×C4×D43C22×C414yes—
152C2×C4×D43C2×Q814yes—
153C2×Dic86—22no—
154C2×Dic86—22no—
155C2×Dic86C4⋊C414yes—
156C2×C8×C43—28no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43C22×C412yes—
162C2×C8×C43C22×C412yes—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43C2×D412yes—
170C4⋊C4×C43C2×D412yes—
171C22⋊C4×C43—24no—
172C22⋊C4×C43—42no—
173C42×C43—24no—
174C42×C43—42no—
175(C2×D43)⋊C4—28no—
176(C2×D43)⋊C4—82no—
177(C2×D43)⋊C4—82no—
178C4⋊Dic43—24no—
179C4⋊Dic43—22no—
180C4⋊Dic43—22no—
181C4⋊Dic43—22no—
182C4⋊Dic43—42no—
183C4⋊Dic43C2×D414yes—
184Dic43⋊C4—28no—
185C4×Dic43—28no—
186#10621877—28no—
187C2×C43⋊C8—22no—
188C2×C43⋊C8—22no—
189C2×C43⋊C8C22×C414yes—
190C23×C43—21no—
191C23×C43—21no—
192C23×C43—21no—
193C23×C43—21no—
194C23×C43C2×C4⋊C412yes—
195C23×C43#12012yes—
196C23×C43#12312yes—
197C23×C43C22×D411yescharacteristic
198C22×D43—22no—
199C22×D43—22no—
200C22×D43—22no—
201C22×D43—22no—
202C22×D43—22no—
203C22×D43—22no—
204C22×D43C2×C4⋊C414yes—
205C2×C4×C43—22no—
206C2×C4×C43—22no—
207C2×C4×C43—41no—
208C2×C4×C43—41no—
209C2×C4×C43—41no—
210C2×C4×C43—41no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43C2×C4⋊C412yes—
218C2×C4×C43C4×D414yes—
219C2×C4×C43C4×D414yes—
220C2×C4×C43#12012yes—
221C2×C4×C43C22⋊Q814yes—
222C2×C4×C43C22⋊Q814yes—
223C2×C4×C43#12312yes—
224C2×C4×C43C23×C411yescharacteristic
225C2×C4×C43C22×D411yescharacteristic
226C2×C4×C43C22×D411yescharacteristic
227C2×C4×C43C22×Q811yescharacteristic
228C2×C4×C43C2×#4111yescharacteristic
229C2×C4×C43C2×#4111yescharacteristic
230C22⋊D43—216no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—24no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43—42no—
237C2×Dic43—42no—
238C2×Dic43—42no—
239C2×Dic43—84no—
240C2×Dic43—84no—
241C2×Dic43C2×C4⋊C414yes—
242D172—28no—
243C4×D43—24no—
244C4×D43—24no—
245C4×D43—24no—
246C4×D43—24no—
247C4×D43—88no—
248Dic86—28no—
249C8×C43—22no—
250C8×C43—22no—
251C43⋊C8—24no—
252#20849—431nomaximal, Sylow
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—22no—
256C22×C43—22no—
257C22×C43—22no—
258C22×C43—22no—
259C22×C43#38514yes—
260C22×C43C22×C4⋊C411yescharacteristic
261C22×C43C2×#12011yescharacteristic
262C22×C43C2×#12311yescharacteristic
263C22×C43#57611yescharacteristic
264C22×C43#57711yescharacteristic
265C22×C43#57711yescharacteristic
266C22×C43#57811yescharacteristic
267C2×D43—24no—
268C2×D43—24no—
269C2×D43—24no—
270C2×D43—42no—
271C2×D43—42no—
272C2×D43—42no—
273C2×D43—42no—
274C4×C43—42no—
275C4×C43—42no—
276C4×C43—81no—
277C4×C43—81no—
278C4×C43#38514yes—
279C4×C43C2×C4×D411yescharacteristic
280C4×C43C2×C4×D411yescharacteristic
281C4×C43C2×C22⋊Q811yescharacteristic
282C4×C43C2×C22⋊Q811yescharacteristic
283Dic43—24no—
284Dic43—44no—
285Dic43—82no—
286C2×#520—431no—
287C2×#428—431no—
288C2×#427D4311yescharacteristic, complemented
289C2×#408—431no—
290#2311—431no—
291#1854—432no—
292#1854—432no—
293#1853—432no—
294#1852—432no—
295#1736—431no—
296C2×#340—431no—
297C2×C43—24no—
298C2×C43C2×#38511yescharacteristic
299C2×C43#291211yescharacteristic
300C2×C43#291311yescharacteristic
301C2×C43#301211yescharacteristic
302C2×C43#301311yescharacteristic
303C2×C43#309611yescharacteristic
304C2×C43#309711yescharacteristic
305D43—44no—
306C22×#41—431no—
307C22×M16—431no—
308C23×C8C2×D4311yescharacteristic
309#520—864no—
310#520—864no—
311C2×C4×Q8—861no—
312C2×C4×D4—861no—
313C2×C4×D4—862no—
314C2×#117—432no—
315C2×#117C2×D4311yescharacteristic
316C2×#117C2×D4311yescharacteristic
317C22×C4⋊C4—431no—
318C22×C4⋊C4—431no—
319C22×C4⋊C4—432no—
320#428—864no—
321#427—24no—
322C2×#107—432no—
323C2×#107C2×D4311yescharacteristic
324C2×#107C2×D4311yescharacteristic
325C2×#106—432no—
326C2×#106C2×D4311yescharacteristic
327C2×#106C2×D4311yescharacteristic
328#408—438no—
329C2×C22⋊C8—432no—
330C2×C22⋊C8—432no—
331#391—861no—
332#390—861no—
333#386—862no—
334#380—432no—
335#380—432no—
336#340—432no—
337#340—432no—
338#340—432no—
339#340—432no—
340C43#2084911yescharacteristic, complemented, Sylow
341C2×#41—434no—
342C2×#41—434no—
343C2×#41—862no—
344C2×#41—862no—
345C2×#41—862no—
346C2×#41—862no—
347C22×Q8—431no—
348C22×D4—432no—
349C22×D4—431no—
350C23×C4—432no—
351C23×C4—862no—
352C23×C4—861no—
353C23×C4—861no—
354C23×C4C22×D4311yescharacteristic
355C2×M16—434no—
356C2×M16—862no—
357C2×M16—862no—
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—432no—
361C22×C8C4×D4314yescomplemented
362C22×C8C22×D4311yescharacteristic
363C22×C8C22×D4311yescharacteristic
364C4×Q8—868no—
365C4×D4—868no—
366C4×D4—8616no—
367#117—22no—
368#117—22no—
369#117—22no—
370#117—864no—
371#117—864no—
372#117C22⋊D4314yes—
373C2×C4⋊C4—434no—
374C2×C4⋊C4—432no—
375C2×C4⋊C4—432no—
376C2×C4⋊C4—864no—
377C2×C4⋊C4—862no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—862no—
380C2×C4⋊C4—864no—
381C2×C4⋊C4—862no—
382C2×C4⋊C4—862no—
383C2×C4⋊C4—861no—
384C2×C4⋊C4C22×D4311yescharacteristic
385C2×C4⋊C4C22×D4311yescharacteristic
386C2×C4⋊C4C22×D4311yescharacteristic
387C2×C4⋊C4C22×D4311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—862no—
391C2×C22⋊C4—862no—
392C2×C22⋊C4—862no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—862no—
396#107—22no—
397#107—22no—
398#107—864no—
399#106—22no—
400#106—22no—
401#106—864no—
402C22⋊C8—438no—
403C22⋊C8—438no—
404#95—862no—
405#95—862no—
406C24—432no—
407C24D4×D4311yescharacteristic
408#41—868no—
409#41—868no—
410#41—868no—
411#41—868no—
412C2×Q8—434no—
413C2×Q8—862no—
414C2×Q8—862no—
415C2×D4—438no—
416C2×D4—434no—
417C2×D4—864no—
418C2×D4—864no—
419C2×D4—862no—
420C2×D4—862no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—432no—
426C22×C4—434no—
427C22×C4—434no—
428C22×C4—868no—
429C22×C4—862no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—862no—
433C22×C4—864no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—3442no—
437C22×C4—3442no—
438C22×C4—3441no—
439C22×C4—3441no—
440C22×C4—3441no—
441C22×C4—3441no—
442C22×C4—1722no—
443C22×C4C2×C4×D4312yes—
444C22×C4#1062189812yes—
445C22×C4D4×D4311yescharacteristic
446C22×C4D4×D4312yes—
447C22×C4Q8×D4312yes—
448C22×C4#1062190212yes—
449C22×C4C2×C22⋊D4312yes—
450C22×C4C23×D4311yescharacteristic
451M16—868no—
452C2×C8—28no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—434no—
458C2×C8—862no—
459C2×C8—862no—
460C2×C8C2×C4×D4312yes—
461C2×C8C2×C4×D4312yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—434no—
469C4⋊C4—868no—
470C4⋊C4—864no—
471C4⋊C4—862no—
472C4⋊C4—862no—
473C4⋊C4—862no—
474C4⋊C4—1722no—
475C4⋊C4C2×C22⋊D4312yes—
476C4⋊C4C2×C22⋊D4312yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—868no—
480C22⋊C4—3442no—
481C22⋊C4—3442no—
482C42—24no—
483C42—42no—
484C42—868no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—434no—
490C23—862no—
491C23—862no—
492C23—862no—
493C23—862no—
494C23—862no—
495C23—862no—
496C23C4⋊C4×D4312yes—
497C23#1306329312yes—
498C23#1306331012yes—
499C23C2×D4×D4311yescharacteristic
500Q8—868no—
501D4—8616no—
502D4—868no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—434no—
516C2×C4—862no—
517C2×C4—862no—
518C2×C4—864no—
519C2×C4—864no—
520C2×C4—864no—
521C2×C4—864no—
522C2×C4—864no—
523C2×C4—3448no—
524C2×C4—3444no—
525C2×C4—3444no—
526C2×C4—1722no—
527C2×C4—1722no—
528C2×C4—1722no—
529C2×C4—1722no—
530C2×C4—1722no—
531C2×C4C4⋊C4×D4312yes—
532C2×C4C4×C22⋊D4314yes—
533C2×C4#1306328714yes—
534C2×C4#1306329312yes—
535C2×C4#1306330114yes—
536C2×C4#1306330314yes—
537C2×C4#1306331012yes—
538C2×C4C22×C4×D4311yescharacteristic
539C2×C4C2×D4×D4311yescharacteristic
540C2×C4C22×C22⋊D4311yescharacteristic
541C2×C4C2×Q8×D4311yescharacteristic
542C2×C4C2×#1062189811yescharacteristic
543C2×C4C2×#1062190211yescharacteristic
544C8—22no—
545C8—22no—
546C8—864no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—864no—
554C22—864no—
555C22—864no—
556C22—1722no—
557C22—1722no—
558C22—1722no—
559C22—1722no—
560C22#1458277014yes—
561C22C2×C4⋊C4×D4311yescharacteristic
562C22C2×#1306329311yescharacteristic
563C22C2×#1306331011yescharacteristic
564C22#1458370111yescharacteristic
565C22#1458370611yescharacteristic
566C22#1458370611yescharacteristic
567C22#1458371311yescharacteristic
568C4—42no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—864no—
573C4—3442no—
574C4—1724no—
575C4#1458277014yes—
576C4C2×C4×C22⋊D4311yescharacteristic
577C4C2×#1306328711yescharacteristic
578C4C2×#1306330111yescharacteristic
579C4C2×#1306330311yescharacteristic
580C2—24no—
581C2—1724no—
582C2C2×#1458277011yescharacteristic
583C2#1914318711yescharacteristic
584C2#1914319211yescharacteristic
585C2#1914389511yescharacteristic
586C2#1914390611yescharacteristic
587C2#1914455611yescharacteristic
588C2#1914456711yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#427C2
socleC23×C43C22×D4
prime radicalC22×#10621898C22
cosocleC22×C4×C43C24
hypercenterC2×#427D43
hypocenterC43#20849
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#14582942C2semidirect
5. #19134793C2semidirect
6. #19134791C2semidirect
7. #19134786C2semidirect
8. #19134781C2semidirect
9. #19134316C2semidirect
10. C2×#14582662C2semidirect
11. C2×C43×#427C2semidirect
26. #14582865C4semidirect
61. C2×#10621877C2×C4semidirect
85. (C2×D43)⋊C8C2×C4semidirect
86. (C2×D43)⋊C8C2×C4semidirect
92. C22×C8×C43C2×C4semidirect
288. C2×#427D43semidirect
340. C43#20849semidirect
361. C22×C8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145833792
C2×#145829422
C2×#145828652
#191371832
#191347932
#191347912
#191347862
#191347812
#191343162
C2×#145826622
C2×C43×#4272
#2084943