Finite Group Databasebeta

← #25123444

Subgroups — #25123444

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×#13063355C211yesmaximal, characteristic, complemented
2C22×#13063304C211yesmaximal, characteristic, complemented
3C2×#14582900C211yesmaximal, characteristic, complemented
4C2×#14582907C211yesmaximal, characteristic, complemented
5C2×#14582863C211yesmaximal, characteristic, complemented
6#19135603C214yesmaximal, complemented
7#19135598C214yesmaximal, complemented
8#19134316C211yesmaximal, characteristic, complemented
9C2×C43×#525C211yesmaximal, characteristic
10C23×Dic86C2211yescharacteristic
11C2×#13063355—24no—
12C2×#13063355—24no—
13C2×#13063355—24no—
14C22×#10621885C2212yescomplemented
15C22×#10621883C2212yescomplemented
16C22×#10621877C2211yescharacteristic
17C22×#10621877C2211yescharacteristic
18C2×#13063304—24no—
19C2×#13063304—24no—
20C2×#13063304—22no—
21C2×#13063304C2214yes—
22C22×C4⋊Dic43C2211yescharacteristic
23C22×Dic43⋊C4—21no—
24C22×C22⋊Dic43—21no—
25#14582907—24no—
26#14582900—24no—
27C2×#13063249C2212yescomplemented
28C2×#13063248C2214yescomplemented
29C2×#13063246C2212yescomplemented
30#14582863—24no—
31C2×C22⋊(C43⋊C8)C2214yes—
32#14582661C2214yescomplemented
33C22×#41×C43C2211yescharacteristic
34C24×C4×C43C2211yescharacteristic
35C43×#525—24no—
36C43×#525—24no—
37C2×#123×C43—21no—
38C2×C22⋊Q8×C43C2212yes—
39C2×#121×C43C2212yes—
40C2×#120×C43—21no—
41C2×C4×D4×C43—22no—
42C2×#117×C43C2211yescharacteristic
43C22×Dic86—22no—
44C22×Dic86—22no—
45C22×Dic86—22no—
46C22×Dic86—22no—
47C22×Dic86—22no—
48C22×Dic86—21no—
49C22×Dic86D414yescomplemented
50C22×Dic86C2312yes—
51C23×Dic43—21no—
52C23×Dic43—21no—
53#13063355—232no—
54C2×#10621885—24no—
55C2×#10621885—24no—
56C2×#10621885—24no—
57C2×#10621883—24no—
58C2×#10621883—24no—
59C2×#10621883—24no—
60C2×#10621877—22no—
61C2×#10621877—22no—
62C2×#10621877—22no—
63C2×#10621877—22no—
64C2×#10621877—22no—
65C2×#10621877D414yescomplemented
66C22×C43⋊C8C2312yes—
67C22×C43⋊C8C2312yes—
68#13063304—216no—
69#13063304—28no—
70#13063304—28no—
71C2×C4⋊Dic43—22no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43D414yescomplemented
74C2×C4⋊Dic43C2312yes—
75C2×Dic43⋊C4—28no—
76C2×Dic43⋊C4—24no—
77C2×C22⋊Dic43—28no—
78C2×C22⋊Dic43—24no—
79#13063249—24no—
80#13063248—28no—
81#13063246—24no—
82C22⋊(C43⋊C8)—28no—
83C25×C43D411yescharacteristic
84C2×#41×C43—22no—
85C2×#41×C43—22no—
86C2×#41×C43—22no—
87C2×#41×C43—22no—
88C2×#41×C43—22no—
89C2×#41×C43D414yes—
90C22×Q8×C43C2311yescharacteristic
91C22×D4×C43—22no—
92C22×D4×C43—21no—
93C22×D4×C43C2311yescharacteristic
94C23×C4×C43—21no—
95C23×C4×C43—22no—
96C23×C4×C43—22no—
97C23×C4×C43—22no—
98C23×C4×C43—22no—
99C23×C4×C43—22no—
100C23×C4×C43—22no—
101C23×C4×C43D412yes—
102C23×C4×C43D411yescharacteristic
103C23×C4×C43C2312yes—
104C23×C4×C43C2311yescharacteristic
105#123×C43—28no—
106C22⋊Q8×C43—24no—
107C22⋊Q8×C43—24no—
108#121×C43—24no—
109#121×C43—24no—
110#120×C43—28no—
111C4×D4×C43—216no—
112#117×C43—22no—
113#117×C43D414yes—
114C2×C4⋊C4×C43—22no—
115C2×C4⋊C4×C43C2312yes—
116C2×C22⋊C4×C43—21no—
117C2×C22⋊C4×C43—22no—
118C2×C22⋊C4×C43—22no—
119C2×C42×C43—21no—
120C24×C43—22no—
121C24×C43—24no—
122C24×C43—22no—
123C24×C43—22no—
124C24×C43—21no—
125C24×C43—42no—
126C24×C43C2×D412yes—
127C24×C43C2×D411yescharacteristic
128#41×C43—28no—
129#41×C43—28no—
130#41×C43—44no—
131#41×C43—44no—
132C2×Q8×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43C2×D412yes—
138C2×D4×C43—28no—
139C2×D4×C43—21no—
140C2×D4×C43—24no—
141C2×D4×C43—21no—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43—44no—
146C2×D4×C43—42no—
147C2×D4×C43—42no—
148C2×D4×C43—44no—
149C2×D4×C43C2×D412yes—
150C22×C4×C43—24no—
151C22×C4×C43—24no—
152C22×C4×C43—24no—
153C22×C4×C43—24no—
154C22×C4×C43—22no—
155C22×C4×C43—22no—
156C22×C4×C43—22no—
157C22×C4×C43—22no—
158C22×C4×C43—22no—
159C22×C4×C43—22no—
160C22×C4×C43—22no—
161C22×C4×C43—22no—
162C22×C4×C43—22no—
163C22×C4×C43—22no—
164C22×C4×C43—21no—
165C22×C4×C43—44no—
166C22×C4×C43—42no—
167C22×C4×C43—44no—
168C22×C4×C43—44no—
169C22×C4×C43—42no—
170C22×C4×C43—44no—
171C22×C4×C43C2×D412yes—
172C22×C4×C43C2×D412yes—
173C22×C4×C43C2×D412yes—
174C22×C4×C43C2×D412yes—
175C22×C4×C43C2×D412yes—
176C22×C4×C43C2×D412yes—
177C22×C4×C43C2×D411yescharacteristic
178C22×C4×C43#4112yes—
179C22×C4×C43C2411yescharacteristic
180C22×Dic43—24no—
181C22×Dic43—24no—
182C22×Dic43—22no—
183C22×Dic43—22no—
184C22×Dic43—44no—
185C22×Dic43—42no—
186C22×Dic43—42no—
187C2×Dic86—28no—
188C2×Dic86—28no—
189C2×Dic86—22no—
190C2×Dic86—22no—
191C2×Dic86—22no—
192C2×Dic86—22no—
193C2×Dic86—24no—
194C2×Dic86—22no—
195C2×Dic86—42no—
196C2×Dic86—44no—
197C2×Dic86—42no—
198C2×Dic86—44no—
199C2×Dic86C2×D414yes—
200C4⋊C4×C43—28no—
201C4⋊C4×C43—22no—
202C4⋊C4×C43C2×D414yes—
203C22⋊C4×C43—28no—
204C22⋊C4×C43—28no—
205C22⋊C4×C43—24no—
206C42×C43—24no—
207C22⋊Dic43—216no—
208#10621885—216no—
209#10621883—216no—
210C4⋊Dic43—24no—
211C4⋊Dic43—22no—
212C4⋊Dic43C2×D414yes—
213Dic43⋊C4—216no—
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—22no—
232C23×C43—48no—
233C23×C43—48no—
234C23×C43—44no—
235C23×C43—44no—
236C23×C43—41no—
237C23×C43—41no—
238C23×C43—42no—
239C23×C43—42no—
240C23×C43#12012yes—
241C23×C43#12012yes—
242C23×C43#12112yes—
243C23×C43#13714yescomplemented
244C23×C43C22×D411yescharacteristic
245Q8×C43—22no—
246Q8×C43—22no—
247Q8×C43—41no—
248Q8×C43—41no—
249D4×C43—22no—
250D4×C43—22no—
251D4×C43—416no—
252D4×C43—48no—
253D4×C43—41no—
254D4×C43—41no—
255C2×C4×C43—22no—
256C2×C4×C43—24no—
257C2×C4×C43—22no—
258C2×C4×C43—22no—
259C2×C4×C43—22no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—42no—
263C2×C4×C43—42no—
264C2×C4×C43—44no—
265C2×C4×C43—42no—
266C2×C4×C43—42no—
267C2×C4×C43—42no—
268C2×C4×C43—42no—
269C2×C4×C43—48no—
270C2×C4×C43—48no—
271C2×C4×C43#12014yes—
272C2×C4×C43#12012yes—
273C2×C4×C43#12012yes—
274C2×C4×C43#12114yes—
275C2×C4×C43#12114yes—
276C2×C4×C43#12112yes—
277C2×C4×C43#12714yes—
278C2×C4×C43C22×D411yescharacteristic
279C2×C4×C43C22×D411yescharacteristic
280C2×C4×C43C22×D411yescharacteristic
281C2×C4×C43C22×D411yescharacteristic
282C2×C4×C43C22×D411yescharacteristic
283C2×C4×C43C2×#4111yescharacteristic
284C2×Dic43—24no—
285C2×Dic43—24no—
286C2×Dic43—48no—
287C2×Dic43—42no—
288C2×Dic43—42no—
289C2×Dic43—44no—
290C2×Dic43—48no—
291Dic86—24no—
292Dic86—24no—
293Dic86—42no—
294Dic86—42no—
295Dic86—48no—
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—42no—
312C22×C43—44no—
313C22×C43—81no—
314C22×C43—81no—
315C22×C43—82no—
316C22×C43#39214yes—
317C22×C43C2×#12011yescharacteristic
318C22×C43C2×#12011yescharacteristic
319C22×C43C2×#12111yescharacteristic
320C22×C43C2×#13712yes—
321C22×C43#57512yes—
322C4×C43—42no—
323C4×C43—42no—
324C4×C43—44no—
325C4×C43#39214yes—
326C4×C43C2×#12011yescharacteristic
327C4×C43C2×#12111yescharacteristic
328C4×C43C2×#12111yescharacteristic
329C4×C43C2×#12711yescharacteristic
330Dic43—44no—
331Dic43—44no—
332C22×#137—431no—
333C2×#525D4311yescharacteristic
334C22×C22⋊Q8—431no—
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×C43—82no—
344C2×C43C2×#39211yescharacteristic
345C2×C43#300411yescharacteristic
346C2×C43#300511yescharacteristic
347C2×C43#300712yes—
348C2×C43#305612yes—
349C22×#41C2×D4311yescharacteristic
350C23×Q8—431no—
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—24no—
360#525—24no—
361C2×#123—21no—
362C2×C22⋊Q8—434no—
363C2×C22⋊Q8—864no—
364C2×C22⋊Q8—864no—
365C2×C22⋊Q8—862no—
366C2×C22⋊Q8C2×D4312yes—
367C2×#121C2×D4312yes—
368C2×#120—21no—
369C2×C4×D4—22no—
370C2×#117C2×D4311yescharacteristic
371C22×C4⋊C4—431no—
372C22×C4⋊C4—861no—
373C22×C22⋊C4—861no—
374#419—864no—
375#417—864no—
376C2×#103—432no—
377C2×#103—434no—
378C2×#102—432no—
379#407—864no—
380C2×C22⋊C8—434no—
381#340—434no—
382C43#2339211yescharacteristic, complemented, Sylow
383C25C22⋊D4311yescharacteristic
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41C22⋊D4314yes—
390C22×Q8—434no—
391C22×Q8—432no—
392C22×Q8—862no—
393C22×Q8—861no—
394C22×Q8—862no—
395C22×Q8—862no—
396C22×Q8—862no—
397C22×Q8—862no—
398C22×Q8C22×D4311yescharacteristic
399C22×D4—22no—
400C22×D4—21no—
401C22×D4C22×D4311yescharacteristic
402C23×C4—22no—
403C23×C4—22no—
404C23×C4—22no—
405C23×C4—22no—
406C23×C4—21no—
407C23×C4—22no—
408C23×C4—22no—
409C23×C4—861no—
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—28no—
431C22⋊Q8—24no—
432C22⋊Q8—24no—
433C22⋊Q8—868no—
434C22⋊Q8—868no—
435C22⋊Q8—8616no—
436#121—24no—
437#121—24no—
438#120—28no—
439C4×D4—216no—
440#117—22no—
441#117C22⋊D4314yes—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—434no—
444C2×C4⋊C4—432no—
445C2×C4⋊C4—868no—
446C2×C4⋊C4—862no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4—864no—
449C2×C4⋊C4C22×D4312yes—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—22no—
452C2×C22⋊C4—22no—
453C2×C22⋊C4—868no—
454C2×C22⋊C4—864no—
455C2×C42—21no—
456#103—864no—
457#103—868no—
458#102—864no—
459C22⋊C8—868no—
460C24—22no—
461C24—24no—
462C24—22no—
463C24—22no—
464C24—21no—
465C24—42no—
466C24C2×C22⋊D4312yes—
467C24C2×C22⋊D4311yescharacteristic
468#41—28no—
469#41—28no—
470#41—44no—
471#41—44no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—434no—
478C2×Q8—864no—
479C2×Q8—862no—
480C2×Q8—868no—
481C2×Q8—868no—
482C2×Q8—862no—
483C2×Q8—862no—
484C2×Q8—862no—
485C2×Q8—862no—
486C2×Q8—1722no—
487C2×Q8—1724no—
488C2×Q8—1722no—
489C2×Q8—1724no—
490C2×Q8C2×C22⋊D4312yes—
491C2×D4—28no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—44no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—44no—
502C2×D4C2×C22⋊D4312yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—44no—
519C22×C4—42no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—44no—
524C22×C4—864no—
525C22×C4—864no—
526C22×C4—862no—
527C22×C4—862no—
528C22×C4—1724no—
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—28no—
551C4⋊C4—22no—
552C4⋊C4—434no—
553C4⋊C4—8616no—
554C4⋊C4—864no—
555C4⋊C4—862no—
556C4⋊C4C2×C22⋊D4314yes—
557C22⋊C4—28no—
558C22⋊C4—28no—
559C22⋊C4—24no—
560C22⋊C4—8616no—
561C42—24no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—48no—
573C23—48no—
574C23—44no—
575C23—44no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23#1306329312yes—
581C23#1306329412yes—
582C23#1306329712yes—
583C23#1306335514yescomplemented
584C23C22×C22⋊D4311yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589Q8—864no—
590Q8—864no—
591Q8—1728no—
592Q8—1722no—
593Q8—1722no—
594D4—22no—
595D4—22no—
596D4—416no—
597D4—48no—
598D4—41no—
599D4—41no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—864no—
617C2×C4—864no—
618C2×C4—1728no—
619C2×C4—1728no—
620C2×C4—1722no—
621C2×C4—1722no—
622C2×C4—1724no—
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—44no—
651C22—42no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22#1458280714yes—
656C22C2×#1306329311yescharacteristic
657C22C2×#1306329411yescharacteristic
658C22C2×#1306329711yescharacteristic
659C22C2×#1306335512yes—
660C22#1458369712yes—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—1724no—
665C4—1724no—
666C4#1458280714yes—
667C4C2×#1306329311yescharacteristic
668C4C2×#1306329611yescharacteristic
669C4C2×#1306329711yescharacteristic
670C4C2×#1306332111yescharacteristic
671C2—24no—
672C2—44no—
673C2—82no—
674C2C2×#1458280711yescharacteristic
675C2#1914382811yescharacteristic
676C2#1914383511yescharacteristic
677C2#1914385512yes—
678C2#1914422712yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
Fitting, prime radicalC2×C43×#525C2
socleC23×C43C22×D4
hypercenterC2×#525D43
hypocenterC43#23392
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C22×#13063355C2semidirect
2. C22×#13063304C2semidirect
3. C2×#14582900C2semidirect
4. C2×#14582907C2semidirect
5. C2×#14582863C2semidirect
6. #19135603C2semidirect
7. #19135598C2semidirect
8. #19134316C2semidirect
14. C22×#10621885C22semidirect
15. C22×#10621883C22semidirect
27. C2×#13063249C22semidirect
28. C2×#13063248C22semidirect
29. C2×#13063246C22semidirect
32. #14582661C22semidirect
49. C22×Dic86D4semidirect
65. C2×#10621877D4semidirect
73. C2×C4⋊Dic43D4semidirect
243. C23×C43#137semidirect
382. C43#23392semidirect
583. C23#13063355semidirect

Maximal subgroups

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

subgroupindex
C22×#130633552
C22×#130633042
C2×#145829002
C2×#145829072
C2×#145828632
#191356032
#191355982
#191343162
C2×C43×#5252
#2339243