Finite Group Databasebeta

← #25098314

Subgroups — #25098314

Order 11008.

2238 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D43C211yesmaximal, characteristic, complemented
2C22×#13063300C211yesmaximal, characteristic
3C2×#14582905C211yesmaximal, characteristic, complemented
4C2×#14582903C211yesmaximal, characteristic, complemented
5#19136974C211yesmaximal, characteristic, complemented
6#19135032C214yesmaximal, complemented
7#19135022C214yesmaximal, complemented
8#19134314C211yesmaximal, characteristic, complemented
9C43×#1736C211yesmaximal, characteristic, complemented
10C23×D172—21no—
11C23×D172C2211yescharacteristic
12C23×C4×D43C2211yescharacteristic
13C22×#10621877C2211yescharacteristic
14C22×#10621872C2212yescomplemented
15C2×M16×D43—24no—
16C2×M16×D43—24no—
17C2×M16×D43—24no—
18C22×C8×D43C2212yescomplemented
19C2×#13063300—24no—
20C2×#13063300—21no—
21C2×#13063300—21no—
22C2×#13063300C418yescomplemented
23C2×#13063300C2212yes—
24C2×#13063300C2212yes—
25C2×#13063284C2211yescharacteristic
26C22×(C2×D43)⋊C4—21no—
27#14582905—24no—
28#14582903—24no—
29C2×#13063247C2214yescomplemented
30C2×#13063245C2214yescomplemented
31#14582760—21no—
32#14582756C2212yescomplemented
33#14582756C2212yescomplemented
34#14582754—21no—
35C4×(C2×D43)⋊C4—21no—
36C4×(C2×D43)⋊C4—21no—
37#14582661C2214yescomplemented
38C22×M16×C43C2211yescharacteristic
39C2×#117×C43C2211yescharacteristic
40C22×C4⋊C4×C43C2211yescharacteristic
41C43×#340C2214yescomplemented
42C24×D43—21no—
43C24×D43D411yescharacteristic
44C22×D172—22no—
45C22×D172—22no—
46C22×D172—22no—
47C22×D172—22no—
48C22×D172—28no—
49C22×D172—22no—
50C22×D172—22no—
51C22×D172—21no—
52C22×D172—22no—
53C22×D172—48no—
54C22×D172C2×C414yes—
55C22×D172C2312yes—
56C23×Dic43D411yescharacteristic
57C22×C4×D43—22no—
58C22×C4×D43—22no—
59C22×C4×D43—22no—
60C22×C4×D43—22no—
61C22×C4×D43C2×C414yes—
62C22×C4×D43D411yescharacteristic
63C22×C4×D43D411yescharacteristic
64C22×C4×D43D414yes—
65C22×C4×D43C2311yescharacteristic
66C22×C4×D43C2311yescharacteristic
67C2×#10621877—22no—
68C2×#10621877—22no—
69C2×#10621877D414yescomplemented
70C2×#10621872—24no—
71C2×#10621872—24no—
72C2×#10621872—24no—
73M16×D43—232no—
74C22×C43⋊C8C2312yes—
75C2×C8×D43—24no—
76C2×C8×D43—24no—
77C2×C8×D43—24no—
78#13063300—28no—
79#13063300—28no—
80#13063300—24no—
81#13063300—24no—
82#13063300C2×C418yes—
83#13063300C2×C418yes—
84#13063284—22no—
85#13063284—22no—
86C2×C4⋊Dic43C2312yes—
87C2×C22⋊Dic43—21no—
88C2×(C2×D43)⋊C4—28no—
89C2×(C2×D43)⋊C4—22no—
90C2×(C2×D43)⋊C4—22no—
91C2×(C2×D43)⋊C4—22no—
92C2×(C2×D43)⋊C4—22no—
93C2×C4×Dic43—21no—
94#13063247—28no—
95#13063245—28no—
96C43⋊#95—22no—
97C43⋊#95—22no—
98C23×C4×C43—21no—
99C23×C4×C43C2311yescharacteristic
100C2×M16×C43—22no—
101C2×M16×C43—22no—
102C2×M16×C43D414yescomplemented
103C22×C8×C43C2312yes—
104#117×C43—22no—
105#117×C43—22no—
106C2×C4⋊C4×C43—21no—
107C2×C4⋊C4×C43—22no—
108C2×C4⋊C4×C43C2×C414yes—
109C2×C4⋊C4×C43C2312yes—
110C2×C4⋊C4×C43C2312yes—
111C2×C22⋊C4×C43—21no—
112C2×C42×C43—21no—
113C24×C43C2×D411yescharacteristic
114C23×D43—22no—
115C23×D43—22no—
116C23×D43—24no—
117C23×D43—24no—
118C23×D43—22no—
119C23×D43—42no—
120C23×D43—44no—
121C23×D43—44no—
122C23×D43—42no—
123C23×D43—42no—
124C23×D43C22⋊C414yes—
125C23×D43C2×D411yescharacteristic
126C23×D43C2×D411yescharacteristic
127C22×C4×C43—24no—
128C22×C4×C43—22no—
129C22×C4×C43—22no—
130C22×C4×C43—44no—
131C22×C4×C43C22×C412yes—
132C22×C4×C43C2×D412yes—
133C22×C4×C43C2×D412yes—
134C22×C4×C43C2×D412yes—
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43#4112yes—
137C22×C4×C43#4112yes—
138C22×C4×C43C2411yescharacteristic
139C22×Dic43—22no—
140C22×Dic43—22no—
141C22×Dic43—22no—
142C22×Dic43C22⋊C414yes—
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C2×D172—24no—
146C2×D172—22no—
147C2×D172—28no—
148C2×D172—28no—
149C2×D172—28no—
150C2×D172—28no—
151C2×D172—22no—
152C2×D172—22no—
153C2×D172—22no—
154C2×D172—22no—
155C2×D172—48no—
156C2×D172—432no—
157C2×D172—44no—
158C2×D172—42no—
159C2×D172—42no—
160C2×D172—44no—
161C2×D172—48no—
162C2×D172C22×C414yes—
163C2×C4×D43—28no—
164C2×C4×D43—28no—
165C2×C4×D43—28no—
166C2×C4×D43—22no—
167C2×C4×D43—22no—
168C2×C4×D43—22no—
169C2×C4×D43—22no—
170C2×C4×D43—28no—
171C2×C4×D43—21no—
172C2×C4×D43—21no—
173C2×C4×D43—21no—
174C2×C4×D43—21no—
175C2×C4×D43—21no—
176C2×C4×D43—21no—
177C2×C4×D43—21no—
178C2×C4×D43—21no—
179C2×C4×D43C22⋊C414yes—
180C2×C4×D43C22⋊C414yes—
181C2×C4×D43C22×C412yes—
182C2×C4×D43C22×C412yes—
183C2×C4×D43C2×D412yes—
184C2×C4×D43C2×D412yes—
185M16×C43—28no—
186C2×C8×C43—22no—
187C2×C8×C43—22no—
188C2×C8×C43C2×D414yescomplemented
189C4⋊C4×C43—24no—
190C4⋊C4×C43—22no—
191C4⋊C4×C43—22no—
192C4⋊C4×C43—22no—
193C4⋊C4×C43C22×C414yes—
194C22⋊C4×C43—42no—
195C42×C43—42no—
196C22⋊Dic43—42no—
197(C2×D43)⋊C4—28no—
198(C2×D43)⋊C4—28no—
199(C2×D43)⋊C4—44no—
200(C2×D43)⋊C4—44no—
201C4⋊Dic43—22no—
202C4⋊Dic43—22no—
203C4×Dic43—42no—
204#10621877—28no—
205C2×C43⋊C8—22no—
206C2×C43⋊C8—22no—
207C2×C43⋊C8C2×D414yescomplemented
208#10621872—216no—
209C8×D43—216no—
210C23×C43—22no—
211C23×C43—22no—
212C23×C43C2×C22⋊C412yes—
213C23×C43#12112yes—
214C23×C43#12112yes—
215C23×C43C22×D411yescharacteristic
216C22×D43—21no—
217C22×D43—21no—
218C22×D43—22no—
219C22×D43—21no—
220C22×D43—21no—
221C22×D43—22no—
222C22×D43—22no—
223C22×D43—28no—
224C22×D43—28no—
225C22×D43—24no—
226C22×D43—44no—
227C22×D43—48no—
228C22×D43—48no—
229C22×D43—48no—
230C22×D43—48no—
231C22×D43—42no—
232C22×D43—42no—
233C22×D43—44no—
234C22×D43—44no—
235C22×D43—48no—
236C22×D43—88no—
237C22×D43—88no—
238C22×D43C2×C22⋊C412yes—
239C22×D43C2×C22⋊C412yes—
240C22×D43AGL(1,ℤ/8)14yes—
241C2×C4×C43—24no—
242C2×C4×C43—48no—
243C2×C4×C43—44no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43—42no—
247C2×C4×C43C2×C22⋊C412yes—
248C2×C4×C43C4×D414yes—
249C2×C4×C43C4×D414yes—
250C2×C4×C43#12014yes—
251C2×C4×C43#12112yes—
252C2×C4×C43#12112yes—
253C2×C4×C43#12314yes—
254C2×C4×C43C23×C411yescharacteristic
255C2×C4×C43C22×D411yescharacteristic
256C2×C4×C43C22×D411yescharacteristic
257C2×C4×C43C22×D411yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×C4×C43C2×#4111yescharacteristic
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—28no—
265C2×Dic43—42no—
266C2×Dic43—42no—
267C2×Dic43—42no—
268C2×Dic43C2×C22⋊C412yes—
269C2×Dic43C2×C22⋊C412yes—
270D172—24no—
271D172—24no—
272D172—416no—
273D172—416no—
274D172—42no—
275D172—42no—
276D172—48no—
277C4×D43—22no—
278C4×D43—22no—
279C4×D43—22no—
280C4×D43—22no—
281C4×D43—22no—
282C4×D43—22no—
283C4×D43—22no—
284C4×D43—22no—
285C4×D43—28no—
286C4×D43—28no—
287C8×C43—24no—
288C43⋊C8—24no—
289#20973—431nomaximal, Sylow
290C22×C43—22no—
291C22×C43—22no—
292C22×C43—24no—
293C22×C43—24no—
294C22×C43#38614yes—
295C22×C43C22×C22⋊C411yescharacteristic
296C22×C43C2×#12111yescharacteristic
297C22×C43C2×#12111yescharacteristic
298C22×C43C2×AGL(1,ℤ/8)12yes—
299C22×C43#57512yes—
300C2×D43—22no—
301C2×D43—22no—
302C2×D43—24no—
303C2×D43—24no—
304C2×D43—22no—
305C2×D43—22no—
306C2×D43—22no—
307C2×D43—416no—
308C2×D43—42no—
309C2×D43—42no—
310C2×D43—44no—
311C2×D43—44no—
312C2×D43—44no—
313C2×D43—816no—
314C2×D43—84no—
315C2×D43—84no—
316C2×D43#41814yescomplemented
317C4×C43—44no—
318C4×C43—82no—
319C4×C43#38614yes—
320C4×C43C2×C4×D411yescharacteristic
321C4×C43C2×C4×D411yescharacteristic
322C4×C43C2×#12011yescharacteristic
323C4×C43C2×#12311yescharacteristic
324Dic43—22no—
325Dic43—22no—
326Dic43—82no—
327C23×M16—431no—
328C22×#121—431no—
329C2×#418—431no—
330C2×#418—431no—
331#2280—431no—
332#1895—434no—
333#1893—434no—
334#1736—431no—
335#1736D4311yescharacteristic, complemented
336C2×C43—24no—
337C2×C43C2×#38611yescharacteristic
338C2×C43C2×#41811yescharacteristic
339C2×C43#289611yescharacteristic
340C2×C43#305512yes—
341C2×C43#305512yes—
342D43—24no—
343D43—42no—
344D43—84no—
345C23×D4—431no—
346C23×D4—861no—
347C24×C4—431no—
348C22×M16—431no—
349C22×M16—432no—
350C22×M16—864no—
351C22×M16—864no—
352C22×M16—864no—
353C22×M16C2×D4311yescharacteristic
354C23×C8—432no—
355C2×#121—432no—
356C2×#121—432no—
357C2×#121—438no—
358C2×#121—864no—
359C2×#121—861no—
360C2×#121—861no—
361C2×#117—431no—
362C2×#117C2×D4311yescharacteristic
363C22×C4⋊C4C2×D4311yescharacteristic
364C22×C22⋊C4—861no—
365#418—864no—
366#418—864no—
367C2×#102—434no—
368C2×#102—434no—
369#381—861no—
370#380—432no—
371#380—432no—
372#379—861no—
373C4×C22⋊C4—861no—
374C4×C22⋊C4—861no—
375#340—434no—
376#340C2×D4314yescomplemented
377C43#2097311yescharacteristic, complemented, Sylow
378C25—431no—
379C25—861no—
380C22×D4—434no—
381C22×D4—432no—
382C22×D4—868no—
383C22×D4—862no—
384C22×D4—862no—
385C22×D4—862no—
386C22×D4—862no—
387C22×D4—862no—
388C22×D4—861no—
389C22×D4—862no—
390C22×D4—862no—
391C22×D4—1728no—
392C23×C4—21no—
393C23×C4—431no—
394C23×C4—434no—
395C23×C4—434no—
396C23×C4—431no—
397C23×C4—431no—
398C23×C4—431no—
399C23×C4—431no—
400C23×C4—862no—
401C23×C4—862no—
402C23×C4—862no—
403C23×C4—862no—
404C23×C4C22×D4311yescharacteristic
405C2×M16—22no—
406C2×M16—22no—
407C2×M16—434no—
408C2×M16—864no—
409C2×M16—864no—
410C2×M16—8632no—
411C2×M16—862no—
412C2×M16—862no—
413C2×M16—864no—
414C2×M16D17214yescomplemented
415C22×C8—432no—
416C22×C8—864no—
417C22×C8—864no—
418C22×C8—864no—
419C22×C8C22×D4312yes—
420#121—438no—
421#121—438no—
422#121—868no—
423#121—868no—
424#121—864no—
425#121—864no—
426#117—22no—
427#117—22no—
428#117—862no—
429#117—862no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—432no—
433C2×C4⋊C4C4×D4314yes—
434C2×C4⋊C4C22×D4312yes—
435C2×C4⋊C4C22×D4312yes—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—868no—
438C2×C22⋊C4—862no—
439C2×C22⋊C4—862no—
440C2×C22⋊C4—862no—
441C2×C22⋊C4—862no—
442C2×C22⋊C4—861no—
443C2×C42—21no—
444C2×C42—861no—
445#102—868no—
446#102—868no—
447#95—862no—
448#95—862no—
449C24—431no—
450C24—431no—
451C24—434no—
452C24—864no—
453C24—862no—
454C24—862no—
455C24—864no—
456C24—862no—
457C24—1722no—
458C24—1724no—
459C24—1724no—
460C24—1722no—
461C24—1722no—
462C24D4×D4311yescharacteristic
463C2×D4—434no—
464C2×D4—868no—
465C2×D4—868no—
466C2×D4—864no—
467C2×D4—868no—
468C2×D4—868no—
469C2×D4—862no—
470C2×D4—862no—
471C2×D4—862no—
472C2×D4—862no—
473C2×D4—862no—
474C2×D4—1728no—
475C2×D4—17232no—
476C2×D4—1728no—
477C2×D4—1724no—
478C2×D4—1722no—
479C2×D4—1722no—
480C2×D4—1724no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—434no—
486C22×C4—431no—
487C22×C4—431no—
488C22×C4—434no—
489C22×C4—434no—
490C22×C4—432no—
491C22×C4—432no—
492C22×C4—432no—
493C22×C4—432no—
494C22×C4—862no—
495C22×C4—862no—
496C22×C4—868no—
497C22×C4—868no—
498C22×C4—868no—
499C22×C4—862no—
500C22×C4—862no—
501C22×C4—862no—
502C22×C4—862no—
503C22×C4—868no—
504C22×C4—861no—
505C22×C4—861no—
506C22×C4—861no—
507C22×C4—861no—
508C22×C4—861no—
509C22×C4—861no—
510C22×C4—861no—
511C22×C4—861no—
512C22×C4—862no—
513C22×C4C2×C4×D4312yes—
514C22×C4C2×D17212yes—
515C22×C4#1062189812yes—
516C22×C4D4×D4311yescharacteristic
517C22×C4D4×D4312yes—
518C22×C4D4×D4312yes—
519C22×C4#1062190212yes—
520C22×C4C23×D4311yescharacteristic
521M16—28no—
522M16—8616no—
523M16—868no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—434no—
527C2×C8—8616no—
528C2×C8—862no—
529C2×C8—862no—
530C2×C8C2×D17214yescomplemented
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—862no—
536C4⋊C4—862no—
537C4⋊C4C2×C4×D4314yes—
538C22⋊C4—42no—
539C22⋊C4—868no—
540C22⋊C4—868no—
541C22⋊C4—1724no—
542C22⋊C4—1724no—
543C22⋊C4—1722no—
544C42—42no—
545C42—1722no—
546C23—22no—
547C23—22no—
548C23—432no—
549C23—434no—
550C23—432no—
551C23—868no—
552C23—861no—
553C23—861no—
554C23—862no—
555C23—861no—
556C23—861no—
557C23—862no—
558C23—862no—
559C23—868no—
560C23—864no—
561C23—3448no—
562C23—3448no—
563C23—1724no—
564C23—1728no—
565C23—1728no—
566C23—1728no—
567C23—1728no—
568C23—1722no—
569C23—1722no—
570C23—1724no—
571C23—1724no—
572C23—1728no—
573C23C22⋊C4×D4312yes—
574C23#1306329812yes—
575C23#1306330012yes—
576C23C2×D4×D4311yescharacteristic
577D4—864no—
578D4—864no—
579D4—17216no—
580D4—17216no—
581D4—1722no—
582D4—1722no—
583D4—1728no—
584C2×C4—24no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—432no—
591C2×C4—432no—
592C2×C4—861no—
593C2×C4—861no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—868no—
597C2×C4—862no—
598C2×C4—862no—
599C2×C4—862no—
600C2×C4—862no—
601C2×C4—862no—
602C2×C4—862no—
603C2×C4—862no—
604C2×C4—862no—
605C2×C4—868no—
606C2×C4—868no—
607C2×C4—1722no—
608C2×C4—1722no—
609C2×C4—1722no—
610C2×C4C22⋊C4×D4312yes—
611C2×C4#1306328714yes—
612C2×C4C4×D17214yes—
613C2×C4#1306329514yes—
614C2×C4#1306329812yes—
615C2×C4#1306330012yes—
616C2×C4#1306331014yes—
617C2×C4C22×C4×D4311yescharacteristic
618C2×C4C2×D4×D4311yescharacteristic
619C2×C4C2×D4×D4311yescharacteristic
620C2×C4C22×D17211yescharacteristic
621C2×C4C2×#1062189811yescharacteristic
622C2×C4C2×#1062190211yescharacteristic
623C8—24no—
624C8—864no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—434no—
630C22—864no—
631C22—864no—
632C22—862no—
633C22—862no—
634C22—862no—
635C22—862no—
636C22—862no—
637C22—34416no—
638C22—3444no—
639C22—3444no—
640C22—17216no—
641C22—1724no—
642C22—1724no—
643C22—1724no—
644C22—1722no—
645C22—1722no—
646C22#1458277714yes—
647C22C2×C22⋊C4×D4311yescharacteristic
648C22C2×#1306329811yescharacteristic
649C22C2×#1306330011yescharacteristic
650C22AGL(1,ℤ/8)×D4312yes—
651C22#1458369812yes—
652C4—44no—
653C4—82no—
654C4—862no—
655C4—862no—
656C4—3442no—
657C4#1458277714yes—
658C4C2×#1306328711yescharacteristic
659C4C2×C4×D17211yescharacteristic
660C4C2×#1306329511yescharacteristic
661C4C2×#1306331011yescharacteristic
662C2—24no—
663C2—864no—
664C2—3444no—
665C2—1722no—
666C2C2×#1458277711yescharacteristic
667C2#418×D4311yescharacteristic
668C2#1914311311yescharacteristic
669C2#1914421512yes—
670C2#1914421712yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1736C2
socleC23×C43C22×D4
prime radicalC22×#13063300C2
cosocleC22×C4×C43C24
hypercenter#1736D43
hypocenterC43#20973
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C22×M16×D43C2semidirect
3. C2×#14582905C2semidirect
4. C2×#14582903C2semidirect
5. #19136974C2semidirect
6. #19135032C2semidirect
7. #19135022C2semidirect
8. #19134314C2semidirect
9. C43×#1736C2semidirect
14. C22×#10621872C22semidirect
18. C22×C8×D43C22semidirect
22. C2×#13063300C4semidirect
29. C2×#13063247C22semidirect
30. C2×#13063245C22semidirect
32. #14582756C22semidirect
33. #14582756C22semidirect
37. #14582661C22semidirect
41. C43×#340C22semidirect
69. C2×#10621877D4semidirect
102. C2×M16×C43D4semidirect
188. C2×C8×C43C2×D4semidirect
207. C2×C43⋊C8C2×D4semidirect
316. C2×D43#418semidirect
335. #1736D43semidirect
376. #340C2×D43semidirect
377. C43#20973semidirect
414. C2×M16D172semidirect
530. C2×C8C2×D172semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D432
C22×#130633002
C2×#145829052
C2×#145829032
#191369742
#191350322
#191350222
#191343142
C43×#17362
#2097343