Finite Group Databasebeta

← #25124251

Subgroups — #25124251

Order 11008.

1442 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583477C211yesmaximal, characteristic
2C2×#14582917C211yesmaximal, characteristic, complemented
3#19143115C211yesmaximal, characteristic, complemented
4#19143114C211yesmaximal, characteristic, complemented
5#19137218C211yesmaximal, characteristic, complemented
6#19135823C214yesmaximal, complemented
7#19135818C214yesmaximal, complemented
8C2×#14582667C211yesmaximal, characteristic, complemented
9C2×C43×#421C211yesmaximal, characteristic, complemented
10C22×#10621898C2211yescharacteristic
11C22×#10621877C2211yescharacteristic
12#14583477—22no—
13#14583477—22no—
14#14583477—22no—
15#14583477C2214yes—
16C2×#13063304—21no—
17C2×#13063301—21no—
18C2×#13063299—21no—
19C2×#13063297—22no—
20C2×#13063296—21no—
21C2×#13063284C2211yescharacteristic
22C22×C4⋊Dic43C2211yescharacteristic
23#14582917—24no—
24C2×#13063253C2214yescomplemented
25#14582909—21no—
26#14582908—21no—
27#14582905—21no—
28#14582904—21no—
29#14582901C2212yescomplemented
30#14582901C2212yescomplemented
31#14582900C2212yescomplemented
32#14582900C2212yescomplemented
33#14582896—22no—
34#14582895—22no—
35#14582798—22no—
36#14582774—22no—
37#14582769C2212yescomplemented
38#14582765C2212yescomplemented
39#14582667C2214yescomplemented
40C22×#41×C43C2211yescharacteristic
41C22×M16×C43C2211yescharacteristic
42C2×#117×C43C2211yescharacteristic
43C43×#421—24no—
44C2×#104×C43C2214yescomplemented
45C2×#10621898—21no—
46C2×#10621898—21no—
47C2×#10621898—22no—
48C2×#10621898—22no—
49C2×#10621898—22no—
50C2×#10621898—22no—
51C2×#10621898D412yes—
52C2×#10621898C2312yes—
53C22×Dic86D411yescharacteristic
54C22×D172D411yescharacteristic
55C22×C22⋊D43—21no—
56C22×C22⋊D43—22no—
57C23×Dic43D412yes—
58C22×C4×D43—21no—
59C2×#10621877—21no—
60C2×#10621877—22no—
61C2×#10621877—22no—
62C2×#10621877C2312yes—
63C22×C43⋊C8—21no—
64#13063304—24no—
65#13063304—24no—
66#13063301—24no—
67#13063301—24no—
68#13063299—24no—
69#13063299—24no—
70#13063297—28no—
71#13063297—28no—
72#13063296—24no—
73#13063296—24no—
74#13063284—21no—
75#13063284—21no—
76#13063284—21no—
77#13063284—21no—
78C2×C4⋊Dic43—21no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43—21no—
82C2×C4⋊Dic43—24no—
83C2×C4⋊Dic43D412yes—
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×C4⋊Dic43C2312yes—
87C2×Dic43⋊C4—22no—
88C2×C22⋊Dic43—22no—
89C2×C22⋊Dic43—21no—
90C2×(C2×D43)⋊C4—22no—
91C2×C4×Dic43—21no—
92#13063253—28no—
93#13063250—22no—
94#13063250—22no—
95#13063249—22no—
96#13063249—22no—
97#13063247—22no—
98#13063247—22no—
99#13063246—22no—
100#13063246—22no—
101C43⋊#95—24no—
102C2×#41×C43—21no—
103C2×#41×C43—21no—
104C2×#41×C43—22no—
105C2×#41×C43—22no—
106C2×#41×C43—22no—
107C2×#41×C43—22no—
108C2×#41×C43D412yes—
109C2×#41×C43C2312yes—
110C22×Q8×C43D411yescharacteristic
111C22×D4×C43—21no—
112C22×D4×C43D411yescharacteristic
113C23×C4×C43—21no—
114C23×C4×C43C2311yescharacteristic
115C2×M16×C43—21no—
116C2×M16×C43—22no—
117C2×M16×C43—22no—
118C2×M16×C43C2312yes—
119C22×C8×C43—21no—
120#117×C43—22no—
121#117×C43D414yescomplemented
122C2×C4⋊C4×C43—21no—
123C2×C22⋊C4×C43—21no—
124C2×C42×C43C2312yes—
125#104×C43—28no—
126C24×C43—21no—
127C24×C43C2×D411yescharacteristic
128C23×D43—21no—
129#41×C43—24no—
130#41×C43—24no—
131#41×C43—22no—
132#41×C43—22no—
133#41×C43—22no—
134#41×C43—22no—
135#41×C43—42no—
136#41×C43—42no—
137#41×C43—42no—
138#41×C43—42no—
139C2×Q8×C43—21no—
140C2×Q8×C43—22no—
141C2×Q8×C43—22no—
142C2×Q8×C43C2×D412yes—
143C2×D4×C43—21no—
144C2×D4×C43—22no—
145C2×D4×C43—22no—
146C2×D4×C43—22no—
147C2×D4×C43—22no—
148C2×D4×C43—42no—
149C2×D4×C43—42no—
150C2×D4×C43C2×D412yes—
151C22×C4×C43—22no—
152C22×C4×C43—22no—
153C22×C4×C43—21no—
154C22×C4×C43—22no—
155C22×C4×C43—42no—
156C22×C4×C43—42no—
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2×D411yescharacteristic
161C22×C4×C43C2×D412yes—
162C22×C4×C43C2×D411yescharacteristic
163C22×C4×C43#4112yes—
164C22×C4×C43#4112yes—
165C22×C4×C43#4112yes—
166C22×C4×C43C2411yescharacteristic
167C2×C22⋊D43—28no—
168C2×C22⋊D43—22no—
169C2×C22⋊D43—22no—
170C2×C22⋊D43—42no—
171C2×C22⋊D43—42no—
172C2×C22⋊D43—84no—
173C22×Dic43—22no—
174C22×Dic43—21no—
175C22×Dic43—22no—
176C22×Dic43—21no—
177C22×Dic43—21no—
178C22×Dic43—82no—
179C22×Dic43C2×D414yes—
180#10621898—24no—
181#10621898—24no—
182#10621898—22no—
183#10621898—22no—
184#10621898—22no—
185#10621898—22no—
186#10621898—42no—
187#10621898—42no—
188#10621898—42no—
189#10621898—42no—
190C2×D172—21no—
191C2×D172—22no—
192C2×D172—22no—
193C2×D172C2×D412yes—
194C2×C4×D43—22no—
195C2×C4×D43—22no—
196C2×C4×D43—42no—
197C2×C4×D43—42no—
198C2×Dic86—22no—
199C2×Dic86—22no—
200C2×Dic86—21no—
201C2×Dic86C2×D412yes—
202M16×C43—22no—
203M16×C43—22no—
204M16×C43—42no—
205C2×C8×C43—22no—
206C2×C8×C43—42no—
207C4⋊C4×C43—24no—
208C22⋊C4×C43—24no—
209C42×C43—22no—
210C42×C43C2×D414yescomplemented
211C22⋊Dic43—28no—
212C22⋊Dic43—41no—
213C22⋊Dic43—41no—
214(C2×D43)⋊C4—28no—
215C4⋊Dic43—22no—
216C4⋊Dic43—22no—
217C4⋊Dic43—22no—
218C4⋊Dic43—22no—
219C4⋊Dic43—21no—
220C4⋊Dic43—21no—
221C4⋊Dic43—21no—
222C4⋊Dic43—21no—
223C4⋊Dic43—84no—
224Dic43⋊C4—28no—
225C4×Dic43—41no—
226C4×Dic43—41no—
227#10621877—22no—
228#10621877—22no—
229#10621877—42no—
230C2×C43⋊C8—22no—
231C2×C43⋊C8—42no—
232C23×C43—22no—
233C23×C43—41no—
234C23×C43—41no—
235C23×C43—42no—
236C23×C43—42no—
237C23×C43—42no—
238C23×C43#12011yescharacteristic
239C23×C43#12011yescharacteristic
240C23×C43#12112yes—
241C23×C43#12412yes—
242C23×C43C22×D411yescharacteristic
243C22×D43—22no—
244C22×D43—42no—
245C22×D43—42no—
246C22×D43—42no—
247Q8×C43—22no—
248Q8×C43—22no—
249Q8×C43—42no—
250D4×C43—22no—
251D4×C43—22no—
252D4×C43—42no—
253D4×C43—42no—
254D4×C43—44no—
255D4×C43—42no—
256C2×C4×C43—22no—
257C2×C4×C43—24no—
258C2×C4×C43—42no—
259C2×C4×C43—42no—
260C2×C4×C43—42no—
261C2×C4×C43—42no—
262C2×C4×C43—42no—
263C2×C4×C43—44no—
264C2×C4×C43#12011yescharacteristic
265C2×C4×C43#12011yescharacteristic
266C2×C4×C43#12112yes—
267C2×C4×C43#12112yes—
268C2×C4×C43#12112yes—
269C2×C4×C43#12112yes—
270C2×C4×C43#12112yes—
271C2×C4×C43#12312yes—
272C2×C4×C43#12312yes—
273C2×C4×C43#12312yes—
274C2×C4×C43#12312yes—
275C2×C4×C43#12412yes—
276C2×C4×C43C22×D411yescharacteristic
277C2×C4×C43C22×D411yescharacteristic
278C2×C4×C43C22×D411yescharacteristic
279C2×C4×C43C2×#4111yescharacteristic
280C2×C4×C43C2×#4111yescharacteristic
281C2×C4×C43C2×#4111yescharacteristic
282C22⋊D43—42no—
283C22⋊D43—42no—
284C22⋊D43—44no—
285C22⋊D43—88no—
286C2×Dic43—22no—
287C2×Dic43—24no—
288C2×Dic43—44no—
289C2×Dic43—42no—
290C2×Dic43—44no—
291C2×Dic43—42no—
292C2×Dic43—42no—
293C2×Dic43—41no—
294C2×Dic43—41no—
295C2×Dic43—82no—
296D172—22no—
297D172—22no—
298D172—42no—
299C4×D43—42no—
300C4×D43—42no—
301C4×D43—44no—
302Dic86—22no—
303Dic86—22no—
304Dic86—42no—
305C8×C43—42no—
306C43⋊C8—42no—
307#23499—431nomaximal, Sylow
308C22×C43—21no—
309C22×C43—21no—
310C22×C43—21no—
311C22×C43—21no—
312C22×C43—21no—
313C22×C43—21no—
314C22×C43—42no—
315C22×C43—42no—
316C22×C43—42no—
317C22×C43—42no—
318C22×C43—42no—
319C22×C43—84no—
320C22×C43#39412yes—
321C22×C43#39412yes—
322C22×C43C2×#12011yescharacteristic
323C22×C43C2×#12111yescharacteristic
324C22×C43C2×#12411yescharacteristic
325C2×D43—42no—
326C2×D43—42no—
327C2×D43—42no—
328C2×D43—84no—
329C4×C43—44no—
330C4×C43—42no—
331C4×C43#39412yes—
332C4×C43#39412yes—
333C4×C43C2×#12111yescharacteristic
334C4×C43C2×#12111yescharacteristic
335C4×C43C2×#12311yescharacteristic
336C4×C43C2×#12311yescharacteristic
337Dic43—42no—
338Dic43—44no—
339Dic43—81no—
340Dic43—81no—
341C2×#537—431no—
342C2×#421—431no—
343C2×#421D4311yescharacteristic, complemented
344#2896—431no—
345#2896—431no—
346#2317—431no—
347#2040—434no—
348#2039—434no—
349C2×#343—431no—
350C2×C43—21no—
351C2×C43—22no—
352C2×C43—21no—
353C2×C43—22no—
354C2×C43—82no—
355C2×C43C2×#39411yescharacteristic
356C2×C43#301911yescharacteristic
357C2×C43#302011yescharacteristic
358D43—82no—
359C22×#41—431no—
360C22×#41C2×D4311yescharacteristic
361C22×M16—431no—
362C22×M16C2×D4311yescharacteristic
363#537—434no—
364#537—862no—
365#537—862no—
366#537—862no—
367C2×C22⋊Q8—861no—
368C2×C22⋊Q8—861no—
369C2×#121—861no—
370C2×#121—862no—
371C2×#121—861no—
372C2×#117—431no—
373C2×#117C2×D4311yescharacteristic
374C22×C4⋊C4—431no—
375#421—24no—
376#421—864no—
377C2×#104—434no—
378C2×#104C2×D4314yescomplemented
379#419—861no—
380#419—861no—
381#418—861no—
382#418—861no—
383#417—432no—
384#417—432no—
385#417—432no—
386#417—432no—
387#416—862no—
388#416—862no—
389#389—862no—
390#385—862no—
391#384—432no—
392#382—432no—
393#343—434no—
394C43#2349911yescharacteristic, complemented, Sylow
395C2×#41—21no—
396C2×#41—21no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—432no—
402C2×#41—432no—
403C2×#41—861no—
404C2×#41—861no—
405C2×#41—862no—
406C2×#41—862no—
407C2×#41—862no—
408C2×#41—862no—
409C2×#41D17212yes—
410C2×#41C22×D4312yes—
411C22×Q8—431no—
412C22×Q8D17211yescharacteristic
413C22×D4—21no—
414C22×D4—431no—
415C22×D4—862no—
416C22×D4—861no—
417C22×D4D17211yescharacteristic
418C23×C4—21no—
419C23×C4—432no—
420C23×C4—861no—
421C23×C4C22×D4311yescharacteristic
422C2×M16—21no—
423C2×M16—22no—
424C2×M16—22no—
425C2×M16—432no—
426C2×M16—861no—
427C2×M16—862no—
428C2×M16—862no—
429C2×M16C22×D4312yes—
430C22×C8—21no—
431C22×C8—861no—
432C22⋊Q8—864no—
433C22⋊Q8—864no—
434C22⋊Q8—864no—
435C22⋊Q8—864no—
436#121—864no—
437#121—864no—
438#121—868no—
439#121—864no—
440#121—868no—
441#121—864no—
442#117—22no—
443#117—861no—
444#117—861no—
445#117—861no—
446#117—861no—
447#117C22⋊D4314yescomplemented
448C2×C4⋊C4—21no—
449C2×C4⋊C4—432no—
450C2×C4⋊C4—432no—
451C2×C4⋊C4—431no—
452C2×C4⋊C4—431no—
453C2×C4⋊C4—862no—
454C2×C4⋊C4—861no—
455C2×C4⋊C4—861no—
456C2×C4⋊C4—861no—
457C2×C4⋊C4—861no—
458C2×C4⋊C4—864no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—862no—
461C2×C22⋊C4—862no—
462C2×C22⋊C4—861no—
463C2×C42—861no—
464C2×C42C22×D4312yes—
465#104—28no—
466#104—868no—
467#103—862no—
468#103—862no—
469#103—862no—
470#103—862no—
471#102—862no—
472#102—862no—
473#102—862no—
474#102—862no—
475#95—864no—
476C24—21no—
477C24—861no—
478C24D4×D4311yescharacteristic
479#41—24no—
480#41—24no—
481#41—22no—
482#41—22no—
483#41—22no—
484#41—22no—
485#41—42no—
486#41—42no—
487#41—42no—
488#41—42no—
489#41—864no—
490#41—864no—
491#41—862no—
492#41—862no—
493#41—862no—
494#41—862no—
495#41—1722no—
496#41—1722no—
497#41—1722no—
498#41—1722no—
499C2×Q8—21no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—432no—
503C2×Q8—861no—
504C2×Q8—862no—
505C2×Q8—862no—
506C2×Q8C2×D17212yes—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4—21no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—42no—
513C2×D4—42no—
514C2×D4—432no—
515C2×D4—868no—
516C2×D4—862no—
517C2×D4—862no—
518C2×D4—861no—
519C2×D4—862no—
520C2×D4—862no—
521C2×D4—3444no—
522C2×D4—1722no—
523C2×D4—1722no—
524C2×D4C2×D17212yes—
525C22×C4—21no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—42no—
530C22×C4—42no—
531C22×C4—434no—
532C22×C4—862no—
533C22×C4—862no—
534C22×C4—861no—
535C22×C4—862no—
536C22×C4—862no—
537C22×C4—861no—
538C22×C4—861no—
539C22×C4—3442no—
540C22×C4—1722no—
541C22×C4—1722no—
542C22×C4C2×D17211yescharacteristic
543C22×C4C2×D17211yescharacteristic
544C22×C4#1062189812yes—
545C22×C4D4×D4311yescharacteristic
546C22×C4D4×D4311yescharacteristic
547C22×C4D4×D4311yescharacteristic
548C22×C4#1062190012yes—
549C22×C4#1062190012yes—
550C22×C4C2×C22⋊D4312yes—
551C22×C4C23×D4311yescharacteristic
552M16—22no—
553M16—22no—
554M16—42no—
555M16—862no—
556M16—862no—
557M16—1722no—
558C2×C8—22no—
559C2×C8—42no—
560C2×C8—862no—
561C2×C8—1722no—
562C4⋊C4—24no—
563C4⋊C4—868no—
564C4⋊C4—862no—
565C4⋊C4—862no—
566C4⋊C4—862no—
567C4⋊C4—862no—
568C4⋊C4—861no—
569C4⋊C4—861no—
570C4⋊C4—861no—
571C4⋊C4—861no—
572C4⋊C4—3444no—
573C22⋊C4—24no—
574C22⋊C4—868no—
575C22⋊C4—868no—
576C22⋊C4—1721no—
577C22⋊C4—1721no—
578C42—22no—
579C42—1721no—
580C42—1721no—
581C42C2×C22⋊D4314yescomplemented
582C23—22no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—862no—
589C23—1722no—
590C23—1722no—
591C23—1722no—
592C23#1306329511yescharacteristic
593C23#1306329511yescharacteristic
594C23#1306329712yes—
595C23#1306331112yes—
596C23C2×D4×D4311yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—42no—
600Q8—862no—
601Q8—862no—
602Q8—1722no—
603D4—22no—
604D4—22no—
605D4—44no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—862no—
610D4—862no—
611D4—3448no—
612D4—1724no—
613D4—1722no—
614D4—1722no—
615D4—1722no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—864no—
625C2×C4—862no—
626C2×C4—3442no—
627C2×C4—1722no—
628C2×C4—1722no—
629C2×C4—1721no—
630C2×C4—1721no—
631C2×C4—1724no—
632C2×C4—1722no—
633C2×C4—1724no—
634C2×C4—1722no—
635C2×C4—1722no—
636C2×C4—1724no—
637C2×C4#1306329511yescharacteristic
638C2×C4#1306329511yescharacteristic
639C2×C4#1306329612yes—
640C2×C4#1306329612yes—
641C2×C4#1306329712yes—
642C2×C4#1306329912yes—
643C2×C4#1306329912yes—
644C2×C4#1306330712yes—
645C2×C4#1306330712yes—
646C2×C4#1306330912yes—
647C2×C4#1306330912yes—
648C2×C4#1306331112yes—
649C2×C4C2×D4×D4311yescharacteristic
650C2×C4C22×C22⋊D4311yescharacteristic
651C2×C4C22×D17211yescharacteristic
652C2×C4C2×#1062190011yescharacteristic
653C2×C4C2×#1062190011yescharacteristic
654C2×C4C2×#1062189811yescharacteristic
655C8—42no—
656C8—1722no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—84no—
669C22—3444no—
670C22—1722no—
671C22—1722no—
672C22—1722no—
673C22#1458281512yes—
674C22#1458281512yes—
675C22C2×#1306329511yescharacteristic
676C22C2×#1306329711yescharacteristic
677C22C2×#1306331111yescharacteristic
678C4—44no—
679C4—42no—
680C4—3441no—
681C4—3441no—
682C4—1722no—
683C4—1724no—
684C4#1458281512yes—
685C4#1458281512yes—
686C4C2×#1306329611yescharacteristic
687C4C2×#1306329911yescharacteristic
688C4C2×#1306330711yescharacteristic
689C4C2×#1306330911yescharacteristic
690C2—21no—
691C2—22no—
692C2—22no—
693C2—21no—
694C2—82no—
695C2—3442no—
696C2C2×#1458281511yescharacteristic
697C2#1914395411yescharacteristic
698C2#1914396111yescharacteristic
subgroupG/H
centerC22C2×#13063295
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#421C2
socleC22×C43C2×#120
prime radicalC2×#14583477C2
hypercenterC2×#421D43
hypocenterC43#23499
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#14582917C2semidirect
3. #19143115C2semidirect
4. #19143114C2semidirect
5. #19137218C2semidirect
6. #19135823C2semidirect
7. #19135818C2semidirect
8. C2×#14582667C2semidirect
9. C2×C43×#421C2semidirect
24. C2×#13063253C22semidirect
29. #14582901C22semidirect
30. #14582901C22semidirect
31. #14582900C22semidirect
32. #14582900C22semidirect
37. #14582769C22semidirect
38. #14582765C22semidirect
39. #14582667C22semidirect
44. C2×#104×C43C22semidirect
121. #117×C43D4semidirect
210. C42×C43C2×D4semidirect
343. C2×#421D43semidirect
378. C2×#104C2×D43semidirect
394. C43#23499semidirect
447. #117C22⋊D43semidirect
581. C42C2×C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834772
C2×#145829172
#191431152
#191431142
#191372182
#191358232
#191358182
C2×#145826672
C2×C43×#4212
#2349943