Finite Group Databasebeta

← #25103467

Subgroups — #25103467

Order 11008.

914 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583160C211yesmaximal, characteristic
2#19143419C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×Dic43C211yesmaximal, characteristic
4C2×#14582990C211yesmaximal, characteristic, complemented
5#131×Dic43C211yesmaximal, characteristic, complemented
6#19135495C211yesmaximal, characteristic, complemented
7#19135374C211yesmaximal, characteristic, complemented
8#19135254C211yesmaximal, characteristic, complemented
9#19135251C211yesmaximal, characteristic, complemented
10#19135110C211yesmaximal, characteristic, complemented
11#19134652C211yesmaximal, characteristic, complemented
12#19134496C211yesmaximal, characteristic, complemented
13#19134390C211yesmaximal, characteristic, complemented
14#100×Dic43C211yesmaximal, characteristic, complemented
15C43×#1907C211yesmaximal, characteristic
16C2×#13063355C2211yescharacteristic
17C22×#10621883C2211yescharacteristic
18C2×#13063332C2211yescharacteristic
19#41×Dic43C2211yescharacteristic
20C2×#13063290C2211yescharacteristic
21C2×D4×Dic43C2211yescharacteristic
22#14583160C418yescomplemented
23#14583011—21no—
24#14583003—21no—
25#14583000—21no—
26#14582996—21no—
27#14582991—21no—
28#14582990—21no—
29#14582990—21no—
30#14582990—21no—
31#14582990—21no—
32#14582990—21no—
33SD16×Dic43—21no—
34D8×Dic43—21no—
35C2×#13063260C2211yescharacteristic
36C2×#13063260C2211yescharacteristic
37#14582941C2211yescharacteristic
38#14582939C2211yescharacteristic
39C2×#13063256C2211yescharacteristic
40#14582909C2211yescharacteristic
41#14582904C2211yescharacteristic
42#14582900C2211yescharacteristic
43C2×#13063250C2211yescharacteristic
44C2×#13063243C2211yescharacteristic
45C2×#13063243C2211yescharacteristic
46C2×#13063242C2211yescharacteristic
47#14582857—21no—
48#14582857C2211yescharacteristic
49#14582854—21no—
50#14582854C2211yescharacteristic
51M16×Dic43—21no—
52M16×Dic43C2211yescharacteristic
53M16×Dic43C2211yescharacteristic
54M16×Dic43C2211yescharacteristic
55C2×C8×Dic43—21no—
56C2×C8×Dic43C2211yescharacteristic
57#14582668C2211yescharacteristic
58#14582667C2211yescharacteristic
59#14582667C2211yescharacteristic
60#14582664C2211yescharacteristic
61C2×AGL(1,ℤ/8)×C43C2211yescharacteristic
62C2×#131×C43C2211yescharacteristic
63C43×#417C2211yescharacteristic
64C2×#102×C43C2211yescharacteristic
65C2×#100×C43C2211yescharacteristic
66C43×#343C2211yescharacteristic
67C43×#341C2211yescharacteristic
68C22×Dic86C2311yescharacteristic
69C23×Dic43—21no—
70#13063355—22no—
71#13063355C2×C414yescomplemented
72C2×#10621885—21no—
73C2×#10621883—21no—
74C2×#10621883—21no—
75C2×#10621883—21no—
76C2×#10621883—22no—
77C2×#10621883C2×C414yescomplemented
78#13063332—22no—
79#13063332C2×C414yescomplemented
80C2×#10621877—21no—
81C2×#10621877C2311yescharacteristic
82C2×#10621877C2311yescharacteristic
83C2×#10621877C2311yescharacteristic
84C22×C43⋊C8—21no—
85C22×C43⋊C8C2311yescharacteristic
86Q8×Dic43D411yescharacteristic
87Q8×Dic43D411yescharacteristic
88#13063290—21no—
89#13063290—21no—
90#13063290—21no—
91#13063290—21no—
92D4×Dic43—21no—
93D4×Dic43—21no—
94D4×Dic43—21no—
95D4×Dic43—21no—
96D4×Dic43—21no—
97D4×Dic43—21no—
98D4×Dic43D411yescharacteristic
99D4×Dic43D411yescharacteristic
100#13063284—21no—
101#13063284C2311yescharacteristic
102C2×C4⋊Dic43C2311yescharacteristic
103C2×Dic43⋊C4—21no—
104C2×C22⋊Dic43—21no—
105C2×C4×Dic43—21no—
106C2×C4×Dic43—21no—
107C2×C4×Dic43C2311yescharacteristic
108#13063260—22no—
109#13063260C2×C414yescomplemented
110#13063259—21no—
111#13063258—21no—
112#13063257—21no—
113#13063256—21no—
114#13063256—21no—
115#13063256—21no—
116#13063250—21no—
117#13063250—21no—
118#13063250—21no—
119#13063250—21no—
120#13063249—21no—
121#13063246—21no—
122#13063246—21no—
123#13063246—21no—
124#13063243—22no—
125#13063243C2×C414yescomplemented
126#13063242C2×C414yescomplemented
127#13063235—21no—
128#13063235—21no—
129#13063235—21no—
130#13063235—21no—
131#13063235—21no—
132C8×Dic43—21no—
133C8×Dic43—21no—
134C8×Dic43—21no—
135C8×Dic43—21no—
136C8×Dic43—21no—
137C8×Dic43D411yescharacteristic
138C8×Dic43D411yescharacteristic
139C8×Dic43D411yescharacteristic
140C8×Dic43D411yescharacteristic
141C2×#41×C43C2311yescharacteristic
142C22×D4×C43C2311yescharacteristic
143AGL(1,ℤ/8)×C43—22no—
144AGL(1,ℤ/8)×C43C2×C414yes—
145C2×SD16×C43—21no—
146C2×D8×C43—21no—
147#131×C43—22no—
148#131×C43—22no—
149C2×M16×C43—21no—
150C2×M16×C43C2311yescharacteristic
151C2×M16×C43C2311yescharacteristic
152C2×M16×C43C2311yescharacteristic
153C22×C8×C43—21no—
154C22×C8×C43C2311yescharacteristic
155C2×C4⋊C4×C43C2311yescharacteristic
156#103×C43—21no—
157#102×C43—21no—
158#102×C43—21no—
159#102×C43—21no—
160#100×C43—22no—
161C24×C43—21no—
162#41×C43—21no—
163#41×C43—21no—
164#41×C43C22×C412yes—
165#41×C43#4112yes—
166C2×Q8×C43C2×D411yescharacteristic
167C2×D4×C43—21no—
168C2×D4×C43—21no—
169C2×D4×C43—21no—
170C2×D4×C43—21no—
171C2×D4×C43—21no—
172C2×D4×C43—21no—
173C2×D4×C43C22×C412yes—
174C2×D4×C43C2×D411yescharacteristic
175C22×C4×C43—21no—
176C22×C4×C43—21no—
177C22×C4×C43C2411yescharacteristic
178C22×Dic43—21no—
179C22×Dic43—21no—
180C22×Dic43—21no—
181C22×Dic43—21no—
182C22×Dic43—41no—
183C22×Dic43—41no—
184C22×Dic43—82no—
185C22×Dic43#4111yescharacteristic
186C22×Dic43#4111yescharacteristic
187C2×Dic86—21no—
188C2×Dic86—21no—
189C2×Dic86—21no—
190C2×Dic86—21no—
191C2×Dic86—21no—
192C2×Dic86C22×C412yes—
193SD16×C43—24no—
194D8×C43—24no—
195M16×C43—24no—
196M16×C43—21no—
197M16×C43—21no—
198M16×C43—21no—
199M16×C43—21no—
200M16×C43C22×C412yes—
201M16×C43#4112yes—
202C2×C8×C43—24no—
203C2×C8×C43—21no—
204C2×C8×C43—21no—
205C2×C8×C43—21no—
206C2×C8×C43—21no—
207C2×C8×C43—21no—
208C2×C8×C43C2×D411yescharacteristic
209C2×C8×C43C2×D411yescharacteristic
210C4⋊C4×C43—21no—
211C4⋊C4×C43—21no—
212C22⋊Dic43—21no—
213C22⋊Dic43—21no—
214C22⋊Dic43—21no—
215C22⋊Dic43—41no—
216C22⋊Dic43—41no—
217#10621885—24no—
218#10621883—24no—
219#10621883—24no—
220#10621883—24no—
221C4⋊Dic43—21no—
222C4⋊Dic43—21no—
223C4⋊Dic43—21no—
224C4⋊Dic43—21no—
225C4⋊Dic43C2×D411yescharacteristic
226C4⋊Dic43C2×D411yescharacteristic
227Dic43⋊C4—41no—
228Dic43⋊C4—41no—
229C4×Dic43—21no—
230C4×Dic43—21no—
231C4×Dic43—21no—
232C4×Dic43—41no—
233C4×Dic43—41no—
234C4×Dic43C2×D411yescharacteristic
235C4×Dic43C2×D411yescharacteristic
236C4×Dic43C2×D411yescharacteristic
237C4×Dic43C2×D411yescharacteristic
238#10621877—21no—
239#10621877—21no—
240#10621877—24no—
241#10621877—21no—
242#10621877—21no—
243#10621877C22×C412yes—
244#10621877C22×C412yes—
245C2×C43⋊C8—21no—
246C2×C43⋊C8—21no—
247C2×C43⋊C8—24no—
248C2×C43⋊C8—21no—
249C2×C43⋊C8C22×C412yes—
250C2×C43⋊C8C2×D411yescharacteristic
251C2×C43⋊C8C2×D411yescharacteristic
252C2×C43⋊C8#4112yes—
253C23×C43—21no—
254C23×C43—22no—
255C23×C43—41no—
256C23×C43—41no—
257C23×C43—81no—
258C23×C43—81no—
259C23×C43C2×#4111yescharacteristic
260Q8×C43—21no—
261Q8×C43C4×D412yes—
262D4×C43—22no—
263D4×C43—22no—
264D4×C43—22no—
265D4×C43—22no—
266D4×C43—21no—
267D4×C43—41no—
268D4×C43—41no—
269D4×C43C4×D412yes—
270C2×C4×C43—21no—
271C2×C4×C43—22no—
272C2×C4×C43—22no—
273C2×C4×C43—41no—
274C2×C4×C43—81no—
275C2×C4×C43C23×C411yescharacteristic
276C2×C4×C43C22×D411yescharacteristic
277C2×C4×C43C22×D411yescharacteristic
278C2×C4×C43C2×#4111yescharacteristic
279C2×C4×C43C2×#4111yescharacteristic
280C2×C4×C43C2×#4111yescharacteristic
281C2×Dic43—22no—
282C2×Dic43—22no—
283C2×Dic43—21no—
284C2×Dic43—21no—
285C2×Dic43—41no—
286C2×Dic43—42no—
287C2×Dic43—41no—
288C2×Dic43—41no—
289C2×Dic43—42no—
290C2×Dic43—41no—
291C2×Dic43—82no—
292C2×Dic43—82no—
293C2×Dic43—81no—
294C2×Dic43#12111yescharacteristic
295C2×Dic43#12111yescharacteristic
296C2×Dic43#12111yescharacteristic
297C2×Dic43#12111yescharacteristic
298Dic86—22no—
299Dic86—22no—
300Dic86—41no—
301Dic86—41no—
302C8×C43—22no—
303C8×C43—21no—
304C8×C43—22no—
305C8×C43—21no—
306C8×C43—41no—
307C43⋊C8—22no—
308C43⋊C8—22no—
309C43⋊C8—41no—
310C43⋊C8C4×D412yes—
311C43⋊C8C4×D412yes—
312C4×#471—431nomaximal, Sylow
313C22×C43—21no—
314C22×C43—21no—
315C22×C43—22no—
316C22×C43—41no—
317C22×C43—41no—
318C22×C43—42no—
319C22×C43—82no—
320C22×C43—82no—
321C22×C43—81no—
322C22×C43—81no—
323C22×C43C4×#4111yescharacteristic
324C22×C43C2×#12111yescharacteristic
325C22×C43#52911yescharacteristic
326C4×C43—22no—
327C4×C43—81no—
328C4×C43C2×C4×D411yescharacteristic
329C4×C43C2×C4×D411yescharacteristic
330C4×C43#52511yescharacteristic
331C4×C43#53211yescharacteristic
332Dic43—21no—
333Dic43—21no—
334Dic43—21no—
335Dic43—41no—
336Dic43—41no—
337Dic43—42no—
338Dic43—81no—
339Dic43#47112yescomplemented
340C2×#471—431no—
341C4×#137—431no—
342C4×AGL(1,ℤ/8)—431no—
343C2×C4×SD16—431no—
344C4×#131—431no—
345#1977—431no—
346#1951—431no—
347#1931—431no—
348#1930—431no—
349#1908—431no—
350#1907D4311yescharacteristic
351#1815—431no—
352C4×#108—431no—
353C4×#101—431no—
354C4×#100—431no—
355C2×C43—21no—
356C2×C43—21no—
357C2×C43—42no—
358C2×C43—82no—
359C2×C43C4×#12111yescharacteristic
360C2×C43C2×#47111yescharacteristic
361C2×C43#306711yescharacteristic
362C2×#137—431no—
363C2×AGL(1,ℤ/8)C2×D4311yescharacteristic
364C22×SD16—431no—
365C2×#131—431no—
366C2×#131C2×D4311yescharacteristic
367C4×#41—431no—
368C2×C4×Q8—431no—
369C2×C4×D4—431no—
370#471—438no—
371#442—861no—
372#441—861no—
373#440—861no—
374#440—861no—
375C4×Q16—861no—
376C4×SD16—861no—
377C4×SD16—861no—
378C4×SD16—861no—
379C4×SD16—861no—
380C4×SD16—861no—
381C4×SD16—861no—
382C4×D8—861no—
383C2×#108—431no—
384C2×#108—431no—
385#428—431no—
386#428—431no—
387C2×#106—431no—
388#419—431no—
389#418—431no—
390#417—431no—
391#417C2×D4311yescharacteristic
392C2×#103—431no—
393C2×#102C2×D4311yescharacteristic
394C2×#101—431no—
395C2×#101—431no—
396C2×#100—431no—
397C2×#100C2×D4311yescharacteristic
398#405—431no—
399#405—431no—
400#405—861no—
401#405—861no—
402C4×M16—431no—
403C4×M16—431no—
404C4×M16—431no—
405C4×M16—861no—
406C2×C4×C8—431no—
407C2×C4×C8—861no—
408#343—431no—
409#343—431no—
410#343—431no—
411#343C2×D4311yescharacteristic
412#341—431no—
413#341C2×D4311yescharacteristic
414C43C4×#47111yescharacteristic, complemented, Sylow
415C2×#41C22×D4311yescharacteristic
416C22×Q8—431no—
417C22×D4C22×D4311yescharacteristic
418C23×C4—861no—
419#137—434no—
420#137—862no—
421AGL(1,ℤ/8)—22no—
422AGL(1,ℤ/8)C4×D4314yes—
423C2×Q16—861no—
424C2×SD16—21no—
425C2×SD16—434no—
426C2×SD16—862no—
427C2×SD16—861no—
428C2×SD16—861no—
429C2×SD16—861no—
430C2×D8—21no—
431#131—22no—
432#131—22no—
433#131—434no—
434#131—862no—
435C2×M16—21no—
436C2×M16—431no—
437C2×M16—431no—
438C2×M16—431no—
439C2×M16—861no—
440C2×M16C22×D4311yescharacteristic
441C2×M16C22×D4311yescharacteristic
442C2×M16C22×D4311yescharacteristic
443C22×C8—21no—
444C22×C8—431no—
445C22×C8—861no—
446C22×C8C22×D4311yescharacteristic
447C4×Q8—431no—
448C4×Q8—431no—
449C4×Q8—861no—
450C4×Q8—861no—
451C4×Q8—861no—
452C4×Q8—861no—
453C4×D4—431no—
454C4×D4—431no—
455C4×D4—861no—
456C4×D4—861no—
457C4×D4—861no—
458C4×D4—861no—
459C4×D4—861no—
460C4×D4—861no—
461#117—431no—
462#117—861no—
463C2×C4⋊C4—431no—
464C2×C4⋊C4—861no—
465C2×C4⋊C4C22×D4311yescharacteristic
466C2×C22⋊C4—861no—
467C2×C42—431no—
468C2×C42—861no—
469C2×C42—861no—
470#108—434no—
471#108—862no—
472#107—861no—
473#107—861no—
474#106—861no—
475#106—861no—
476#106—861no—
477#106—861no—
478#103—21no—
479#103—861no—
480#103—861no—
481#103—861no—
482#103—861no—
483#103—861no—
484#102—21no—
485#102—21no—
486#102—21no—
487#102—861no—
488#102—861no—
489#102—861no—
490#101—434no—
491#101—862no—
492#100—22no—
493#100—434no—
494#97—861no—
495#97—861no—
496#97—861no—
497#97—861no—
498#97—861no—
499C4×C8—431no—
500C4×C8—431no—
501C4×C8—431no—
502C4×C8—431no—
503C4×C8—861no—
504C4×C8—861no—
505C4×C8—861no—
506C4×C8—861no—
507C4×C8—861no—
508C24—21no—
509#41—21no—
510#41—21no—
511#41C2×C4×D4312yes—
512#41#1062190012yes—
513C2×Q8—432no—
514C2×Q8—861no—
515C2×Q8—861no—
516C2×Q8—861no—
517C2×Q8—861no—
518C2×Q8—861no—
519C2×Q8D4×D4311yescharacteristic
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—21no—
526C2×D4C2×C4×D4312yes—
527C2×D4D4×D4311yescharacteristic
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—431no—
531C22×C4—431no—
532C22×C4—861no—
533C22×C4—861no—
534C22×C4—861no—
535C22×C4—861no—
536C22×C4—3442no—
537C22×C4—1721no—
538C22×C4—1721no—
539C22×C4C23×D4311yescharacteristic
540Q16—864no—
541SD16—24no—
542SD16—864no—
543SD16—864no—
544SD16—864no—
545D8—24no—
546M16—24no—
547M16—21no—
548M16—21no—
549M16—21no—
550M16—21no—
551M16—432no—
552M16—432no—
553M16—864no—
554M16—861no—
555M16—861no—
556M16—861no—
557M16—861no—
558M16C2×C4×D4312yes—
559M16#1062190012yes—
560C2×C8—24no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—21no—
565C2×C8—21no—
566C2×C8—432no—
567C2×C8—431no—
568C2×C8—431no—
569C2×C8—432no—
570C2×C8—864no—
571C2×C8—861no—
572C2×C8—861no—
573C2×C8—861no—
574C2×C8D4×D4311yescharacteristic
575C2×C8D4×D4311yescharacteristic
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—431no—
579C4⋊C4—431no—
580C4⋊C4—861no—
581C4⋊C4—861no—
582C4⋊C4—861no—
583C4⋊C4—861no—
584C4⋊C4—1721no—
585C4⋊C4—1721no—
586C22⋊C4—861no—
587C22⋊C4—861no—
588C22⋊C4—861no—
589C22⋊C4—1721no—
590C22⋊C4—1721no—
591C42—431no—
592C42—431no—
593C42—431no—
594C42—431no—
595C42—861no—
596C42—861no—
597C42—861no—
598C42—1721no—
599C42—1721no—
600C23—21no—
601C23—22no—
602C23—41no—
603C23—41no—
604C23—81no—
605C23—81no—
606C23#41×D4311yescharacteristic
607Q8—21no—
608Q8—862no—
609Q8—862no—
610Q8—1721no—
611Q8—1721no—
612Q8(C22⋊D43)⋊C412yes—
613D4—22no—
614D4—22no—
615D4—22no—
616D4—22no—
617D4—21no—
618D4—41no—
619D4—41no—
620D4(C22⋊D43)⋊C412yes—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—41no—
625C2×C4—81no—
626C2×C4—431no—
627C2×C4—431no—
628C2×C4—431no—
629C2×C4—431no—
630C2×C4—862no—
631C2×C4—862no—
632C2×C4—861no—
633C2×C4—861no—
634C2×C4—3442no—
635C2×C4—3442no—
636C2×C4—3441no—
637C2×C4—1722no—
638C2×C4—1721no—
639C2×C4—1722no—
640C2×C4—1721no—
641C2×C4—1721no—
642C2×C4—1721no—
643C2×C4C22×C4×D4311yescharacteristic
644C2×C4C2×D4×D4311yescharacteristic
645C2×C4C2×D4×D4311yescharacteristic
646C2×C4#41×D4311yescharacteristic
647C2×C4C2×#1062190011yescharacteristic
648C2×C4C2×#1062190011yescharacteristic
649C8—22no—
650C8—21no—
651C8—22no—
652C8—21no—
653C8—41no—
654C8—432no—
655C8—432no—
656C8—862no—
657C8—862no—
658C8—1721no—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22C4×#1062190011yescharacteristic
670C22#121×D4311yescharacteristic
671C22#1458342611yescharacteristic
672C4—22no—
673C4—81no—
674C4—432no—
675C4—861no—
676C4—861no—
677C4—861no—
678C4—3441no—
679C4—1722no—
680C4—1721no—
681C4—1721no—
682C4C4×D4×D4311yescharacteristic
683C4C2×(C22⋊D43)⋊C411yescharacteristic
684C4#1458340511yescharacteristic
685C4#1458344411yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—82no—
690C2#1913710511yescharacteristic
691C2#471×D4311yescharacteristic
692C2#1914431311yescharacteristic
subgroupG/H
centerC22#121×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1907C2
socleC22×C43C2×#121
prime radicalC2×AGL(1,ℤ/8)×C43C22
cosocleC22×C4×C43C24
hypercenter#1907D43
hypocenterC43C4×#471
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1907

Complements

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

NH ≅ G/Ntype
2. #19143419C2semidirect
4. C2×#14582990C2semidirect
5. #131×Dic43C2semidirect
6. #19135495C2semidirect
7. #19135374C2semidirect
8. #19135254C2semidirect
9. #19135251C2semidirect
10. #19135110C2semidirect
11. #19134652C2semidirect
12. #19134496C2semidirect
13. #19134390C2semidirect
14. #100×Dic43C2semidirect
22. #14583160C4semidirect
71. #13063355C2×C4semidirect
77. C2×#10621883C2×C4semidirect
79. #13063332C2×C4semidirect
109. #13063260C2×C4semidirect
125. #13063243C2×C4semidirect
126. #13063242C2×C4semidirect
339. Dic43#471semidirect
414. C43C4×#471semidirect

Maximal subgroups

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

subgroupindex
C2×#145831602
#191434192
AGL(1,ℤ/8)×Dic432
C2×#145829902
#131×Dic432
#191354952
#191353742
#191352542
#191352512
#191351102
#191346522
#191344962
#191343902
#100×Dic432
C43×#19072
C4×#47143