Finite Group Databasebeta

← #25103457

Subgroups — #25103457

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×#14583163C211yesmaximal, characteristic
2#19143413C211yesmaximal, characteristic, complemented
3#19143402C211yesmaximal, characteristic
4C2×#14582986C211yesmaximal, characteristic, complemented
5#19143017C211yesmaximal, characteristic, complemented
6#19135498C211yesmaximal, characteristic, complemented
7#19135256C211yesmaximal, characteristic, complemented
8#19135249C211yesmaximal, characteristic, complemented
9#19135111C211yesmaximal, characteristic, complemented
10#19135104C211yesmaximal, characteristic
11#19134655C211yesmaximal, characteristic, complemented
12#108×Dic43C211yesmaximal, characteristic, complemented
13#19134390C211yesmaximal, characteristic, complemented
14#19134386C211yesmaximal, characteristic, complemented
15C43×#1951C211yesmaximal, characteristic, complemented
16C2×#13063358C2211yescharacteristic
17C2×#13063353C2211yescharacteristic
18C22×#10621873C2211yescharacteristic
19C2×#13063333C2211yescharacteristic
20C2×#13063331C2211yescharacteristic
21#14583367C2211yescharacteristic
22C2×#13063290C2211yescharacteristic
23C2×#13063287C2211yescharacteristic
24#14583163C418yescomplemented
25#14583010—21no—
26#14583005—21no—
27#14582998—21no—
28#14582998—21no—
29#14582993—21no—
30#14582986—21no—
31#14582986—21no—
32#14582986—21no—
33#14582986—21no—
34#14582986—21no—
35#14582986—21no—
36#14582981—21no—
37C2×#13063261C2211yescharacteristic
38#14582907C2211yescharacteristic
39#14582903C2211yescharacteristic
40#14582898C2211yescharacteristic
41#14582898C2211yescharacteristic
42C2×#13063248C2211yescharacteristic
43C2×#13063245C2211yescharacteristic
44C2×#13063243C2211yescharacteristic
45C2×#13063243C2211yescharacteristic
46C2×#13063241C2211yescharacteristic
47C2×#13063241C2211yescharacteristic
48#14582856—21no—
49#14582855C2211yescharacteristic
50#14582853—21no—
51#14582851—21no—
52M16×Dic43C2211yescharacteristic
53M16×Dic43C2211yescharacteristic
54M16×Dic43C2211yescharacteristic
55C2×C8×Dic43C2211yescharacteristic
56C4×C8×D43—21no—
57#14582668C2211yescharacteristic
58#14582668C2211yescharacteristic
59#14582663C2211yescharacteristic
60#14582663C2211yescharacteristic
61C2×#108×C43C2211yescharacteristic
62C43×#428C2211yescharacteristic
63C43×#428C2211yescharacteristic
64C2×#106×C43C2211yescharacteristic
65C43×#405C2211yescharacteristic
66C43×#343C2211yescharacteristic
67C43×#343C2211yescharacteristic
68C2×#10621898C2311yescharacteristic
69C22×Dic86C2311yescharacteristic
70C22×D172C2311yescharacteristic
71C22×C4×D43—21no—
72#13063358—22no—
73#13063358C2×C414yescomplemented
74#13063353—22no—
75#13063353C2×C414yes—
76C2×Dic172—21no—
77C2×#10621873—21no—
78C2×#10621873—21no—
79C2×#10621873—21no—
80C2×#10621873—21no—
81C2×#10621873—22no—
82C2×#10621873C2×C414yescomplemented
83C2×D344—21no—
84#13063333—22no—
85#13063333—22no—
86#13063331—22no—
87#13063331C2×C414yescomplemented
88C2×#10621877C2311yescharacteristic
89C2×#10621877C2311yescharacteristic
90C2×#10621877C2311yescharacteristic
91C2×#10621872—21no—
92C2×#10621872—21no—
93C22×C43⋊C8C2311yescharacteristic
94C2×C8×D43—21no—
95C2×C8×D43—21no—
96#13063290—21no—
97#13063290—21no—
98#13063290—21no—
99#13063290—21no—
100#13063290D411yescharacteristic
101#13063290D411yescharacteristic
102#13063287—21no—
103#13063287—21no—
104#13063287—21no—
105#13063287—21no—
106#13063287D411yescharacteristic
107#13063287D411yescharacteristic
108(C22⋊D43)⋊C4—21no—
109(C22⋊D43)⋊C4—21no—
110#13063282—21no—
111C2×Dic43⋊C4—21no—
112C2×(C2×D43)⋊C4—21no—
113C2×C4×Dic43—21no—
114C2×C4×Dic43C2311yescharacteristic
115C42×D43—21no—
116#13063261—22no—
117#13063248—21no—
118#13063248—21no—
119#13063248—21no—
120#13063248—21no—
121#13063248—21no—
122#13063248—21no—
123#13063245—21no—
124#13063245—21no—
125#13063245—21no—
126#13063245—21no—
127#13063245—21no—
128#13063245—21no—
129#13063243—22no—
130#13063243C2×C414yescomplemented
131#13063241—22no—
132#13063241C2×C414yescomplemented
133#13063236—21no—
134#13063235—21no—
135#13063235—21no—
136#13063235—21no—
137C4×C43⋊C8—21no—
138C8×Dic43—21no—
139C8×Dic43—21no—
140C8×Dic43—21no—
141C8×Dic43D411yescharacteristic
142C8×Dic43D411yescharacteristic
143C8×Dic43D411yescharacteristic
144C8×Dic43D411yescharacteristic
145C2×M16×C43C2311yescharacteristic
146C2×M16×C43C2311yescharacteristic
147C2×M16×C43C2311yescharacteristic
148C22×C8×C43C2311yescharacteristic
149#117×C43C2311yescharacteristic
150C2×C4⋊C4×C43C2311yescharacteristic
151C2×C4⋊C4×C43C2311yescharacteristic
152#108×C43C2×C414yescomplemented
153#107×C43—21no—
154#107×C43—21no—
155#106×C43—21no—
156#106×C43—21no—
157#106×C43—21no—
158#106×C43—21no—
159#97×C43—21no—
160C4×C8×C43—21no—
161C23×D43—21no—
162C22×C4×C43—21no—
163C22×C4×C43—21no—
164C22×C4×C43C2411yescharacteristic
165C2×C22⋊D43—21no—
166C22×Dic43—21no—
167C22×Dic43#4111yescharacteristic
168C22×Dic43#4111yescharacteristic
169#10621898—21no—
170#10621898—21no—
171#10621898C22×C412yes—
172#10621898#4112yes—
173C2×D172—21no—
174C2×D172—21no—
175C2×D172—21no—
176C2×D172—21no—
177C2×D172—21no—
178C2×D172C22×C412yes—
179C2×D172C2×D411yescharacteristic
180C2×C4×D43—21no—
181C2×C4×D43—21no—
182C2×C4×D43—21no—
183C2×C4×D43—41no—
184C2×C4×D43—41no—
185C2×C4×D43—82no—
186C2×Dic86—21no—
187C2×Dic86—21no—
188C2×Dic86—21no—
189C2×Dic86—21no—
190C2×Dic86—21no—
191C2×Dic86C22×C412yes—
192C2×Dic86C2×D411yescharacteristic
193M16×C43—21no—
194M16×C43—21no—
195M16×C43—21no—
196M16×C43C22×C412yes—
197M16×C43C22×C412yes—
198M16×C43#4112yes—
199C2×C8×C43—21no—
200C2×C8×C43—21no—
201C2×C8×C43—21no—
202C2×C8×C43C22×C412yes—
203C2×C8×C43C2×D411yescharacteristic
204C2×C8×C43C2×D411yescharacteristic
205C2×C8×C43#4112yes—
206C4⋊C4×C43—21no—
207C4⋊C4×C43—21no—
208C4⋊C4×C43—21no—
209C4⋊C4×C43—21no—
210C4⋊C4×C43C2×D411yescharacteristic
211C4⋊C4×C43C2×D411yescharacteristic
212C22⋊C4×C43—21no—
213C42×C43—21no—
214(C2×D43)⋊C4—21no—
215(C2×D43)⋊C4—21no—
216(C2×D43)⋊C4—41no—
217(C2×D43)⋊C4—41no—
218Dic43⋊C4—21no—
219Dic43⋊C4—21no—
220Dic43⋊C4—41no—
221Dic43⋊C4—41no—
222C4×Dic43—21no—
223C4×Dic43—21no—
224C4×Dic43—41no—
225C4×Dic43—41no—
226C4×Dic43C2×D411yescharacteristic
227C4×Dic43C2×D411yescharacteristic
228C4×Dic43C2×D411yescharacteristic
229C4×Dic43C2×D411yescharacteristic
230#10621877—21no—
231#10621877—21no—
232#10621877—21no—
233#10621877—21no—
234#10621877—21no—
235#10621877C22×C412yes—
236C2×C43⋊C8—21no—
237C2×C43⋊C8—21no—
238C2×C43⋊C8—21no—
239C2×C43⋊C8—21no—
240C2×C43⋊C8—21no—
241C2×C43⋊C8C2×D411yescharacteristic
242C2×C43⋊C8C2×D411yescharacteristic
243Dic172—24no—
244D344—24no—
245#10621873—24no—
246#10621873—24no—
247#10621873—24no—
248#10621873—24no—
249#10621872—24no—
250#10621872—24no—
251C8×D43—24no—
252C8×D43—24no—
253C23×C43C2×#4111yescharacteristic
254C22×D43—21no—
255C22×D43—22no—
256C22×D43—41no—
257C22×D43—41no—
258C22×D43—81no—
259C22×D43—81no—
260C2×C4×C43—21no—
261C2×C4×C43—21no—
262C2×C4×C43—41no—
263C2×C4×C43—41no—
264C2×C4×C43—81no—
265C2×C4×C43—81no—
266C2×C4×C43C23×C411yescharacteristic
267C2×C4×C43C22×D411yescharacteristic
268C2×C4×C43C22×D411yescharacteristic
269C2×C4×C43C2×#4111yescharacteristic
270C2×C4×C43C2×#4111yescharacteristic
271C2×C4×C43C2×#4111yescharacteristic
272C22⋊D43—22no—
273C22⋊D43—22no—
274C2×Dic43—21no—
275C2×Dic43—22no—
276C2×Dic43—22no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43#12111yescharacteristic
281C2×Dic43#12111yescharacteristic
282C2×Dic43#12111yescharacteristic
283C2×Dic43#12111yescharacteristic
284D172—22no—
285D172—22no—
286D172—21no—
287D172—41no—
288D172—41no—
289D172C4×D412yes—
290C4×D43—22no—
291C4×D43—22no—
292C4×D43—42no—
293C4×D43—42no—
294C4×D43—82no—
295C4×D43—82no—
296Dic86—21no—
297Dic86—22no—
298Dic86—22no—
299Dic86—41no—
300Dic86—41no—
301Dic86C4×D412yes—
302C8×C43—22no—
303C8×C43—22no—
304C8×C43—22no—
305C8×C43C4×D412yes—
306C8×C43C4×D412yes—
307C43⋊C8—22no—
308C43⋊C8—21no—
309C43⋊C8—21no—
310C43⋊C8—41no—
311C43⋊C8—41no—
312C4×#471—431nomaximal, Sylow
313C22×C43—21no—
314C22×C43—21no—
315C22×C43C4×#4111yescharacteristic
316C22×C43C2×#12111yescharacteristic
317C22×C43#52911yescharacteristic
318C2×D43—22no—
319C2×D43—41no—
320C2×D43—41no—
321C2×D43—42no—
322C2×D43—82no—
323C2×D43—81no—
324C2×D43—81no—
325C2×D43—82no—
326C4×C43—41no—
327C4×C43—41no—
328C4×C43—81no—
329C4×C43—81no—
330C4×C43C2×C4×D411yescharacteristic
331C4×C43C2×C4×D411yescharacteristic
332C4×C43#52511yescharacteristic
333C4×C43#53211yescharacteristic
334Dic43—21no—
335Dic43—21no—
336Dic43—21no—
337Dic43—22no—
338Dic43—42no—
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#1951D4311yescharacteristic, complemented
347#1931—431no—
348#1930—431no—
349#1908—431no—
350#1907—431no—
351#1815—431no—
352C4×#108—431no—
353C4×#101—431no—
354C4×#100—431no—
355C2×C43—21no—
356C2×C43—21no—
357C2×C43C4×#12111yescharacteristic
358C2×C43C2×#47111yescharacteristic
359C2×C43#306711yescharacteristic
360D43—42no—
361D43—82no—
362C2×#137—431no—
363C2×AGL(1,ℤ/8)—431no—
364C22×SD16—431no—
365C2×#131—431no—
366C2×#131—431no—
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×#108C2×D4311yescharacteristic
385#428C2×D4311yescharacteristic
386#428C2×D4311yescharacteristic
387C2×#106C2×D4311yescharacteristic
388#419—431no—
389#418—431no—
390#417—431no—
391#417—431no—
392C2×#103—431no—
393C2×#102—431no—
394C2×#101—431no—
395C2×#101—431no—
396C2×#100—431no—
397C2×#100—431no—
398#405—431no—
399#405—861no—
400#405—861no—
401#405C2×D4311yescharacteristic
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#343C2×D4311yescharacteristic
411#343C2×D4311yescharacteristic
412#341—431no—
413#341—431no—
414C43C4×#47111yescharacteristic, complemented, Sylow
415C2×#41—431no—
416C22×Q8—431no—
417C22×D4—431no—
418C23×C4—861no—
419#137—434no—
420#137—862no—
421AGL(1,ℤ/8)—434no—
422AGL(1,ℤ/8)—862no—
423C2×Q16—861no—
424C2×SD16—434no—
425C2×SD16—861no—
426C2×SD16—861no—
427C2×SD16—861no—
428C2×SD16—861no—
429C2×SD16—862no—
430C2×D8—861no—
431#131—434no—
432#131—862no—
433#131—862no—
434#131—862no—
435C2×M16—431no—
436C2×M16—431no—
437C2×M16—431no—
438C2×M16—861no—
439C2×M16—861no—
440C2×M16C22×D4311yescharacteristic
441C2×M16C22×D4311yescharacteristic
442C2×M16C22×D4311yescharacteristic
443C22×C8—431no—
444C22×C8—861no—
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—861no—
462#117C22×D4311yescharacteristic
463C2×C4⋊C4—861no—
464C2×C4⋊C4C22×D4311yescharacteristic
465C2×C4⋊C4C22×D4311yescharacteristic
466C2×C22⋊C4—861no—
467C2×C42—431no—
468C2×C42—861no—
469C2×C42—861no—
470#108—862no—
471#108C4×D4314yescomplemented
472#107—21no—
473#107—21no—
474#106—21no—
475#106—21no—
476#106—21no—
477#106—21no—
478#103—861no—
479#103—861no—
480#103—861no—
481#103—861no—
482#103—861no—
483#103—861no—
484#102—861no—
485#102—861no—
486#102—861no—
487#102—861no—
488#102—861no—
489#102—861no—
490#101—434no—
491#101—862no—
492#100—434no—
493#100—862no—
494#97—21no—
495#97—861no—
496#97—861no—
497#97—861no—
498#97—861no—
499C4×C8—21no—
500C4×C8—431no—
501C4×C8—431no—
502C4×C8—431no—
503C4×C8—431no—
504C4×C8—861no—
505C4×C8—861no—
506C4×C8—861no—
507C4×C8—861no—
508C24—861no—
509#41—432no—
510#41—432no—
511#41—861no—
512#41—861no—
513C2×Q8—432no—
514C2×Q8—431no—
515C2×Q8—861no—
516C2×Q8—861no—
517C2×Q8—861no—
518C2×Q8—861no—
519C2×Q8—861no—
520C2×D4—432no—
521C2×D4—431no—
522C2×D4—861no—
523C2×D4—861no—
524C2×D4—861no—
525C2×D4—861no—
526C2×D4—861no—
527C2×D4—861no—
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—864no—
542SD16—864no—
543SD16—864no—
544SD16—864no—
545D8—864no—
546M16—21no—
547M16—21no—
548M16—21no—
549M16—432no—
550M16—864no—
551M16—861no—
552M16—864no—
553M16—861no—
554M16—861no—
555M16—861no—
556M16—861no—
557M16C2×C4×D4312yes—
558M16C2×C4×D4312yes—
559M16#1062190212yes—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—431no—
564C2×C8—431no—
565C2×C8—864no—
566C2×C8—861no—
567C2×C8—861no—
568C2×C8—861no—
569C2×C8—864no—
570C2×C8—861no—
571C2×C8—861no—
572C2×C8C2×C4×D4312yes—
573C2×C8D4×D4311yescharacteristic
574C2×C8D4×D4311yescharacteristic
575C2×C8#1062190212yes—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—861no—
581C4⋊C4—861no—
582C4⋊C4—1721no—
583C4⋊C4—1721no—
584C4⋊C4D4×D4311yescharacteristic
585C4⋊C4D4×D4311yescharacteristic
586C22⋊C4—21no—
587C22⋊C4—861no—
588C22⋊C4—861no—
589C22⋊C4—1721no—
590C22⋊C4—1721no—
591C42—21no—
592C42—431no—
593C42—431no—
594C42—431no—
595C42—431no—
596C42—861no—
597C42—861no—
598C42—1721no—
599C42—1721no—
600C23—861no—
601C23—862no—
602C23—3441no—
603C23—3441no—
604C23—1721no—
605C23—1721no—
606C23#41×D4311yescharacteristic
607Q8—432no—
608Q8—861no—
609Q8—862no—
610Q8—862no—
611Q8—1721no—
612Q8—1721no—
613D4—432no—
614D4—862no—
615D4—862no—
616D4—862no—
617D4—862no—
618D4—861no—
619D4—1721no—
620D4—1721no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—431no—
628C2×C4—431no—
629C2×C4—431no—
630C2×C4—431no—
631C2×C4—862no—
632C2×C4—861no—
633C2×C4—862no—
634C2×C4—862no—
635C2×C4—862no—
636C2×C4—3442no—
637C2×C4—3442no—
638C2×C4—1722no—
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×#1062190211yescharacteristic
648C2×C4C2×#1062190211yescharacteristic
649C8—22no—
650C8—22no—
651C8—22no—
652C8—862no—
653C8—861no—
654C8—861no—
655C8—1721no—
656C8—1721no—
657C8#1306328712yes—
658C8#1306328712yes—
659C22—21no—
660C22—21no—
661C22—862no—
662C22—3442no—
663C22—3442no—
664C22—3441no—
665C22—3441no—
666C22—1722no—
667C22—1721no—
668C22—1721no—
669C22C4×#1062190211yescharacteristic
670C22#121×D4311yescharacteristic
671C22#1458343411yescharacteristic
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—432no—
677C4—862no—
678C4—861no—
679C4—861no—
680C4—861no—
681C4—1722no—
682C4C4×D4×D4311yescharacteristic
683C4C2×#1306328711yescharacteristic
684C4#1458341211yescharacteristic
685C4#1458345011yescharacteristic
686C2—21no—
687C2—21no—
688C2—3442no—
689C2—1722no—
690C2#1913710311yescharacteristic
691C2#471×D4311yescharacteristic
692C2#1914431011yescharacteristic
subgroupG/H
centerC22#121×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1951C2
socleC22×C43C2×#121
prime radicalC2×#13063353C22
cosocleC22×C4×C43C24
hypercenter#1951D43
hypocenterC43C4×#471
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
2. #19143413C2semidirect
4. C2×#14582986C2semidirect
5. #19143017C2semidirect
6. #19135498C2semidirect
7. #19135256C2semidirect
8. #19135249C2semidirect
9. #19135111C2semidirect
11. #19134655C2semidirect
12. #108×Dic43C2semidirect
13. #19134390C2semidirect
14. #19134386C2semidirect
15. C43×#1951C2semidirect
24. #14583163C4semidirect
73. #13063358C2×C4semidirect
82. C2×#10621873C2×C4semidirect
87. #13063331C2×C4semidirect
130. #13063243C2×C4semidirect
132. #13063241C2×C4semidirect
152. #108×C43C2×C4semidirect
339. Dic43#471semidirect
346. #1951D43semidirect
414. C43C4×#471semidirect
471. #108C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145831632
#191434132
#191434022
C2×#145829862
#191430172
#191354982
#191352562
#191352492
#191351112
#191351042
#191346552
#108×Dic432
#191343902
#191343862
C43×#19512
C4×#47143