Finite Group Databasebeta

← #25123446

Subgroups — #25123446

Order 11008.

2170 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063356C211yesmaximal, characteristic, complemented
2C2×#14583410C211yesmaximal, characteristic
3C2×#14582898C211yesmaximal, characteristic, complemented
4C2×#14582908C211yesmaximal, characteristic, complemented
5C2×#14582863C211yesmaximal, characteristic, complemented
6#19135603C214yesmaximal, complemented
7#19135596C214yesmaximal, complemented
8#19134314C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8×C43C211yesmaximal, characteristic, complemented
10C22×#10621898C2211yescharacteristic
11C2×#13063356—24no—
12C2×#13063356—24no—
13C2×#13063356—24no—
14C22×#10621885C2212yescomplemented
15C22×#10621884C2212yescomplemented
16C22×#10621877C2211yescharacteristic
17C22×#10621877C2211yescharacteristic
18#14583410—24no—
19#14583410—24no—
20C2×#13063308—21no—
21C2×#13063304C2212yes—
22C2×#13063299C2212yes—
23C2×#13063294—21no—
24C2×C4×C22⋊D43—22no—
25C2×#13063284C2211yescharacteristic
26#14582908—24no—
27#14582898—24no—
28C2×#13063249C2214yescomplemented
29C2×#13063248C2212yescomplemented
30C2×#13063245C2212yescomplemented
31#14582863—24no—
32C2×C22⋊(C43⋊C8)C2214yes—
33#14582661C2214yescomplemented
34C23×Q8×C43C2211yescharacteristic
35C24×C4×C43C2211yescharacteristic
36C2×C22⋊Q8×C43—24no—
37C2×C22⋊Q8×C43—22no—
38C2×C22⋊Q8×C43—24no—
39C2×C22⋊Q8×C43C2214yes—
40C22×C4⋊C4×C43—21no—
41C22×C4⋊C4×C43C2211yescharacteristic
42C22×C22⋊C4×C43—21no—
43C2×#10621898—22no—
44C2×#10621898—22no—
45C2×#10621898—22no—
46C2×#10621898—22no—
47C2×#10621898—22no—
48C2×#10621898D414yes—
49C22×Dic86C2311yescharacteristic
50C22×D172C2311yescharacteristic
51C22×C22⋊D43—22no—
52C22×C22⋊D43—21no—
53C22×C4×D43—21no—
54#13063356—232no—
55C2×#10621885—24no—
56C2×#10621885—24no—
57C2×#10621885—24no—
58C2×#10621884—24no—
59C2×#10621884—24no—
60C2×#10621884—24no—
61C2×#10621877—22no—
62C2×#10621877—22no—
63C2×#10621877—22no—
64C2×#10621877—22no—
65C2×#10621877—22no—
66C2×#10621877D414yescomplemented
67C22×C43⋊C8C2312yes—
68C22×C43⋊C8C2312yes—
69#13063308—28no—
70#13063304—24no—
71#13063304—24no—
72#13063299—24no—
73#13063299—24no—
74#13063294—28no—
75C4×C22⋊D43—216no—
76#13063284—22no—
77#13063284D414yes—
78C2×C4⋊Dic43C2312yes—
79C2×Dic43⋊C4—22no—
80C2×C22⋊Dic43—21no—
81C2×C22⋊Dic43—22no—
82C2×(C2×D43)⋊C4—22no—
83C2×C4×Dic43—21no—
84#13063249—28no—
85#13063248—24no—
86#13063245—24no—
87C22⋊(C43⋊C8)—28no—
88C25×C43D411yescharacteristic
89C22×Q8×C43—22no—
90C22×Q8×C43—21no—
91C22×Q8×C43—22no—
92C22×Q8×C43—22no—
93C22×Q8×C43—22no—
94C22×Q8×C43—22no—
95C22×Q8×C43D414yescomplemented
96C22×Q8×C43C2312yes—
97C23×C4×C43—21no—
98C23×C4×C43—21no—
99C23×C4×C43—22no—
100C23×C4×C43—22no—
101C23×C4×C43—22no—
102C23×C4×C43—22no—
103C23×C4×C43—22no—
104C23×C4×C43—22no—
105C23×C4×C43D411yescharacteristic
106C23×C4×C43D412yes—
107C23×C4×C43C2312yes—
108C23×C4×C43C2311yescharacteristic
109C22⋊Q8×C43—216no—
110C22⋊Q8×C43—28no—
111C22⋊Q8×C43—28no—
112C2×C4⋊C4×C43—28no—
113C2×C4⋊C4×C43—24no—
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43—22no—
116C2×C4⋊C4×C43D414yescomplemented
117C2×C4⋊C4×C43C2312yes—
118C2×C22⋊C4×C43—28no—
119C2×C22⋊C4×C43—24no—
120C24×C43—22no—
121C24×C43—24no—
122C24×C43—22no—
123C24×C43—22no—
124C24×C43—42no—
125C24×C43C2×D412yes—
126C24×C43C2×D411yescharacteristic
127C23×D43—21no—
128C2×Q8×C43—24no—
129C2×Q8×C43—22no—
130C2×Q8×C43—28no—
131C2×Q8×C43—28no—
132C2×Q8×C43—22no—
133C2×Q8×C43—22no—
134C2×Q8×C43—22no—
135C2×Q8×C43—22no—
136C2×Q8×C43—42no—
137C2×Q8×C43—44no—
138C2×Q8×C43—42no—
139C2×Q8×C43—44no—
140C2×Q8×C43C2×D414yes—
141C22×C4×C43—24no—
142C22×C4×C43—24no—
143C22×C4×C43—24no—
144C22×C4×C43—22no—
145C22×C4×C43—22no—
146C22×C4×C43—22no—
147C22×C4×C43—22no—
148C22×C4×C43—22no—
149C22×C4×C43—22no—
150C22×C4×C43—24no—
151C22×C4×C43—24no—
152C22×C4×C43—22no—
153C22×C4×C43—22no—
154C22×C4×C43—22no—
155C22×C4×C43—22no—
156C22×C4×C43—22no—
157C22×C4×C43—44no—
158C22×C4×C43—44no—
159C22×C4×C43—44no—
160C22×C4×C43—44no—
161C22×C4×C43—44no—
162C22×C4×C43—42no—
163C22×C4×C43—42no—
164C22×C4×C43C2×D412yes—
165C22×C4×C43C2×D412yes—
166C22×C4×C43C2×D412yes—
167C22×C4×C43C2×D412yes—
168C22×C4×C43C2×D412yes—
169C22×C4×C43C2×D412yes—
170C22×C4×C43C2×D411yescharacteristic
171C22×C4×C43#4112yes—
172C22×C4×C43C2411yescharacteristic
173C2×C22⋊D43—28no—
174C2×C22⋊D43—24no—
175C2×C22⋊D43—44no—
176C2×C22⋊D43—42no—
177C2×C22⋊D43—42no—
178C2×C22⋊D43—44no—
179C22×Dic43—22no—
180C22×Dic43—21no—
181#10621898—28no—
182#10621898—28no—
183#10621898—44no—
184#10621898—44no—
185C2×D172—21no—
186C2×D172—21no—
187C2×D172—21no—
188C2×D172—21no—
189C2×D172—21no—
190C2×D172C2×D412yes—
191C2×C4×D43—24no—
192C2×C4×D43—42no—
193C2×C4×D43—42no—
194C2×Dic86—21no—
195C2×Dic86—21no—
196C2×Dic86—21no—
197C2×Dic86—21no—
198C2×Dic86—21no—
199C2×Dic86C2×D412yes—
200C4⋊C4×C43—216no—
201C4⋊C4×C43—24no—
202C4⋊C4×C43—22no—
203C4⋊C4×C43C2×D414yes—
204C22⋊C4×C43—216no—
205C22⋊Dic43—28no—
206C22⋊Dic43—24no—
207#10621885—216no—
208#10621884—216no—
209(C2×D43)⋊C4—28no—
210C4⋊Dic43—22no—
211C4⋊Dic43C2×D414yes—
212Dic43⋊C4—28no—
213C4×Dic43—24no—
214#10621877—28no—
215#10621877—44no—
216C2×C43⋊C8—22no—
217C2×C43⋊C8—22no—
218C2×C43⋊C8—22no—
219C2×C43⋊C8—22no—
220C2×C43⋊C8—22no—
221C2×C43⋊C8C2×D414yes—
222C23×C43—22no—
223C23×C43—22no—
224C23×C43—28no—
225C23×C43—22no—
226C23×C43—22no—
227C23×C43—22no—
228C23×C43—22no—
229C23×C43—22no—
230C23×C43—22no—
231C23×C43—48no—
232C23×C43—48no—
233C23×C43—44no—
234C23×C43—44no—
235C23×C43#12012yes—
236C23×C43#12012yes—
237C23×C43#12112yes—
238C23×C43#13714yescomplemented
239C23×C43C22×D411yescharacteristic
240C22×D43—22no—
241C22×D43—41no—
242C22×D43—41no—
243C22×D43—42no—
244C22×D43—42no—
245Q8×C43—24no—
246Q8×C43—24no—
247Q8×C43—42no—
248Q8×C43—42no—
249Q8×C43—48no—
250C2×C4×C43—22no—
251C2×C4×C43—24no—
252C2×C4×C43—24no—
253C2×C4×C43—22no—
254C2×C4×C43—22no—
255C2×C4×C43—48no—
256C2×C4×C43—42no—
257C2×C4×C43—42no—
258C2×C4×C43—44no—
259C2×C4×C43—44no—
260C2×C4×C43—48no—
261C2×C4×C43—42no—
262C2×C4×C43—42no—
263C2×C4×C43—42no—
264C2×C4×C43—42no—
265C2×C4×C43—48no—
266C2×C4×C43#12014yes—
267C2×C4×C43#12012yes—
268C2×C4×C43#12012yes—
269C2×C4×C43#12114yes—
270C2×C4×C43#12114yes—
271C2×C4×C43#12112yes—
272C2×C4×C43#12714yes—
273C2×C4×C43C22×D411yescharacteristic
274C2×C4×C43C22×D411yescharacteristic
275C2×C4×C43C22×D411yescharacteristic
276C2×C4×C43C22×D411yescharacteristic
277C2×C4×C43C22×D411yescharacteristic
278C2×C4×C43C2×#4111yescharacteristic
279C22⋊D43—416no—
280C22⋊D43—48no—
281C2×Dic43—24no—
282C2×Dic43—22no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—42no—
286C2×Dic43—42no—
287D172—22no—
288D172—22no—
289D172—41no—
290D172—41no—
291C4×D43—48no—
292Dic86—22no—
293Dic86—22no—
294Dic86—41no—
295Dic86—41no—
296C43⋊C8—24no—
297C43⋊C8—42no—
298#23392—431nomaximal, Sylow
299C22×C43—22no—
300C22×C43—22no—
301C22×C43—24no—
302C22×C43—24no—
303C22×C43—24no—
304C22×C43—24no—
305C22×C43—24no—
306C22×C43—416no—
307C22×C43—42no—
308C22×C43—42no—
309C22×C43—42no—
310C22×C43—42no—
311C22×C43#39214yes—
312C22×C43C2×#12011yescharacteristic
313C22×C43C2×#12011yescharacteristic
314C22×C43C2×#12111yescharacteristic
315C22×C43C2×#13712yes—
316C22×C43#57512yes—
317C2×D43—42no—
318C2×D43—44no—
319C2×D43—81no—
320C2×D43—81no—
321C2×D43—82no—
322C4×C43—42no—
323C4×C43—44no—
324C4×C43—44no—
325C4×C43#39214yes—
326C4×C43C2×#12011yescharacteristic
327C4×C43C2×#12111yescharacteristic
328C4×C43C2×#12111yescharacteristic
329C4×C43C2×#12711yescharacteristic
330Dic43—42no—
331Dic43—44no—
332C22×#137—431no—
333C2×#525—431no—
334C22×C22⋊Q8D4311yescharacteristic, complemented
335C2×#417—431no—
336C2×#419—431no—
337C2×#407—431no—
338#2001—434no—
339#2000—434no—
340#1736—431no—
341C2×C43—24no—
342C2×C43—44no—
343C2×C43C2×#39211yescharacteristic
344C2×C43#300411yescharacteristic
345C2×C43#300511yescharacteristic
346C2×C43#300712yes—
347C2×C43#305612yes—
348D43—82no—
349C22×#41—431no—
350C23×Q8C2×D4311yescharacteristic
351C24×C4C2×D4311yescharacteristic
352C2×#137—864no—
353C2×#137—864no—
354C2×#137—864no—
355C22×Q16—432no—
356C22×SD16—432no—
357C22×M16—431no—
358C22×M16—431no—
359#525—864no—
360#525—864no—
361C2×#123—861no—
362C2×C22⋊Q8—24no—
363C2×C22⋊Q8—22no—
364C2×C22⋊Q8—24no—
365C2×C22⋊Q8—432no—
366C2×C22⋊Q8C2×D4314yes—
367C2×#121—432no—
368C2×#120—861no—
369C2×C4×D4—862no—
370C2×#117—431no—
371C22×C4⋊C4—21no—
372C22×C4⋊C4C2×D4311yescharacteristic
373C22×C22⋊C4—21no—
374#419—864no—
375#417—864no—
376C2×#103—434no—
377C2×#103—432no—
378C2×#102—432no—
379#407—864no—
380C2×C22⋊C8—434no—
381#340—434no—
382C43#2339211yescharacteristic, complemented, Sylow
383C25C22⋊D4311yescharacteristic
384C2×#41—434no—
385C2×#41—862no—
386C2×#41—862no—
387C2×#41—862no—
388C2×#41—862no—
389C2×#41—862no—
390C22×Q8—22no—
391C22×Q8—21no—
392C22×Q8—22no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8—431no—
397C22×Q8C22⋊D4314yescomplemented
398C22×Q8C22×D4312yes—
399C22×D4—431no—
400C22×D4—862no—
401C22×D4—861no—
402C23×C4—21no—
403C23×C4—21no—
404C23×C4—22no—
405C23×C4—22no—
406C23×C4—22no—
407C23×C4—22no—
408C23×C4—22no—
409C23×C4—22no—
410C23×C4—861no—
411C23×C4C22⋊D4311yescharacteristic
412C23×C4C22⋊D4312yes—
413C23×C4C22×D4312yes—
414C23×C4C22×D4311yescharacteristic
415#137—8632no—
416C2×Q16—864no—
417C2×Q16—864no—
418C2×Q16—864no—
419C2×SD16—864no—
420C2×SD16—864no—
421C2×SD16—864no—
422C2×M16—434no—
423C2×M16—862no—
424C2×M16—862no—
425C2×M16—862no—
426C2×M16—862no—
427C2×M16—862no—
428C22×C8—432no—
429C22×C8—432no—
430#123—868no—
431C22⋊Q8—216no—
432C22⋊Q8—28no—
433C22⋊Q8—28no—
434C22⋊Q8—864no—
435C22⋊Q8—864no—
436#121—864no—
437#121—864no—
438#120—868no—
439C4×D4—8616no—
440#117—434no—
441#117—862no—
442C2×C4⋊C4—28no—
443C2×C4⋊C4—24no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—22no—
446C2×C4⋊C4—432no—
447C2×C4⋊C4—862no—
448C2×C4⋊C4C22⋊D4314yescomplemented
449C2×C4⋊C4C22×D4312yes—
450C2×C22⋊C4—28no—
451C2×C22⋊C4—24no—
452C2×C22⋊C4—861no—
453C2×C22⋊C4—862no—
454C2×C22⋊C4—862no—
455C2×C42—861no—
456#103—868no—
457#103—864no—
458#102—864no—
459C22⋊C8—868no—
460C24—22no—
461C24—24no—
462C24—22no—
463C24—22no—
464C24—42no—
465C24—861no—
466C24C2×C22⋊D4312yes—
467C24C2×C22⋊D4311yescharacteristic
468#41—868no—
469#41—868no—
470#41—1724no—
471#41—1724no—
472C2×Q8—24no—
473C2×Q8—22no—
474C2×Q8—28no—
475C2×Q8—28no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—42no—
481C2×Q8—44no—
482C2×Q8—42no—
483C2×Q8—44no—
484C2×Q8—432no—
485C2×Q8—861no—
486C2×Q8—861no—
487C2×Q8—861no—
488C2×Q8—861no—
489C2×Q8—861no—
490C2×Q8C2×C22⋊D4314yes—
491C2×D4—432no—
492C2×D4—868no—
493C2×D4—864no—
494C2×D4—861no—
495C2×D4—861no—
496C2×D4—861no—
497C2×D4—861no—
498C2×D4—861no—
499C2×D4—1724no—
500C2×D4—1722no—
501C2×D4—1722no—
502C2×D4—1724no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—44no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—44no—
525C22×C4—44no—
526C22×C4—862no—
527C22×C4—861no—
528C22×C4—864no—
529C22×C4—1722no—
530C22×C4—1722no—
531C22×C4D4×D4312yes—
532C22×C4D4×D4312yes—
533C22×C4D4×D4312yes—
534C22×C4#1062190012yes—
535C22×C4C2×C22⋊D4312yes—
536C22×C4C2×C22⋊D4311yescharacteristic
537C22×C4C2×C22⋊D4312yes—
538C22×C4C2×C22⋊D4312yes—
539C22×C4C23×D4311yescharacteristic
540Q16—8616no—
541SD16—8616no—
542M16—868no—
543M16—1724no—
544C2×C8—434no—
545C2×C8—862no—
546C2×C8—862no—
547C2×C8—862no—
548C2×C8—862no—
549C2×C8—862no—
550C4⋊C4—216no—
551C4⋊C4—24no—
552C4⋊C4—22no—
553C4⋊C4—434no—
554C4⋊C4—868no—
555C4⋊C4—862no—
556C4⋊C4C2×C22⋊D4314yes—
557C22⋊C4—216no—
558C22⋊C4—868no—
559C22⋊C4—868no—
560C22⋊C4—864no—
561C42—864no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—48no—
572C23—48no—
573C23—44no—
574C23—44no—
575C23—862no—
576C23—1721no—
577C23—1721no—
578C23—1722no—
579C23—1722no—
580C23#1306329312yes—
581C23#1306329412yes—
582C23#1306329712yes—
583C23#1306335614yescomplemented
584C23C22×C22⋊D4311yescharacteristic
585Q8—24no—
586Q8—24no—
587Q8—48no—
588Q8—42no—
589Q8—42no—
590Q8—862no—
591Q8—862no—
592Q8—1721no—
593Q8—1721no—
594D4—862no—
595D4—862no—
596D4—17216no—
597D4—1728no—
598D4—1721no—
599D4—1721no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—864no—
617C2×C4—862no—
618C2×C4—1721no—
619C2×C4—1721no—
620C2×C4—1722no—
621C2×C4—1722no—
622C2×C4—1728no—
623C2×C4#1306329312yes—
624C2×C4#1306329314yes—
625C2×C4#1306329412yes—
626C2×C4#1306329614yes—
627C2×C4#1306329712yes—
628C2×C4#1306329714yes—
629C2×C4#1306332114yes—
630C2×C4C2×D4×D4311yescharacteristic
631C2×C4C2×D4×D4311yescharacteristic
632C2×C4C2×D4×D4311yescharacteristic
633C2×C4C22×C22⋊D4311yescharacteristic
634C2×C4C22×C22⋊D4311yescharacteristic
635C2×C4C2×#1062190011yescharacteristic
636C8—864no—
637C8—1722no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—3442no—
651C22—3441no—
652C22—3441no—
653C22—1724no—
654C22—1722no—
655C22#1458280714yes—
656C22C2×#1306329311yescharacteristic
657C22C2×#1306329411yescharacteristic
658C22C2×#1306329711yescharacteristic
659C22C2×#1306335612yes—
660C22#1458369712yes—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—1724no—
665C4—1722no—
666C4#1458280714yes—
667C4C2×#1306329311yescharacteristic
668C4C2×#1306329611yescharacteristic
669C4C2×#1306329711yescharacteristic
670C4C2×#1306332111yescharacteristic
671C2—24no—
672C2—44no—
673C2—3442no—
674C2C2×#1458280711yescharacteristic
675C2#1914382911yescharacteristic
676C2#1914383711yescharacteristic
677C2#1914385312yes—
678C2#1914422112yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC22×C22⋊Q8×C43C2
socleC23×C43C22×D4
prime radicalC2×#14583410C2
hypercenterC22×C22⋊Q8D43
hypocenterC43#23392
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13063356C2semidirect
3. C2×#14582898C2semidirect
4. C2×#14582908C2semidirect
5. C2×#14582863C2semidirect
6. #19135603C2semidirect
7. #19135596C2semidirect
8. #19134314C2semidirect
9. C22×C22⋊Q8×C43C2semidirect
14. C22×#10621885C22semidirect
15. C22×#10621884C22semidirect
28. C2×#13063249C22semidirect
29. C2×#13063248C22semidirect
30. C2×#13063245C22semidirect
33. #14582661C22semidirect
66. C2×#10621877D4semidirect
95. C22×Q8×C43D4semidirect
116. C2×C4⋊C4×C43D4semidirect
238. C23×C43#137semidirect
334. C22×C22⋊Q8D43semidirect
382. C43#23392semidirect
397. C22×Q8C22⋊D43semidirect
448. C2×C4⋊C4C22⋊D43semidirect
583. C23#13063356semidirect

Maximal subgroups

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

subgroupindex
C22×#130633562
C2×#145834102
C2×#145828982
C2×#145829082
C2×#145828632
#191356032
#191355962
#191343142
C22×C22⋊Q8×C432
#2339243