Finite Group Databasebeta

← #25974990

Subgroups — #25974990

Order 11008.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19148375C211yesmaximal, characteristic, complemented
2#19146760C214yesmaximal, complemented
3#19146685C214yesmaximal, complemented
4#19146640C214yesmaximal, complemented
5#19146565C214yesmaximal, complemented
6C2×#14583269C211yesmaximal, characteristic, complemented
7#19144564C212yesmaximal, complemented
8#19144413C211yesmaximal, characteristic, complemented
9#19144288C212yesmaximal, complemented
10#19144082C211yesmaximal, characteristic, complemented
11#19143907C212yesmaximal, complemented
12#19143393C211yesmaximal, characteristic, complemented
13#19143313C211yesmaximal, characteristic, complemented
14#19143125C212yesmaximal, complemented
15C43×#3512C211yesmaximal, characteristic, complemented
16C2×#13063372C2212yes—
17C2×#13063355C2212yes—
18C2×#13063345C2211yescharacteristic
19C2×#13063332C2212yes—
20D4×Dic86C2214yes—
21#14583554C2214yes—
22#14583549—24no—
23#14583470C2212yes—
24#14583469—22no—
25#14583447—21no—
26C2×#13063325C2211yescharacteristic
27C4⋊Q8×D43—21no—
28C4×#10621898C2211yescharacteristic
29#14583269C2214yes—
30#14583269C2214yes—
31#14583240C2214yes—
32#14583240C2214yes—
33#14583234C2214yes—
34#14583229C2214yes—
35#14583203C2214yes—
36#14583196C2214yes—
37#14583155C2214yes—
38#14583143C2214yes—
39#14583138C2214yes—
40#14583126C2214yes—
41#14583109C2214yes—
42#14583104C2214yes—
43#14583059C2214yes—
44#14583054C2214yes—
45#14583042C2214yes—
46#14583037C2214yes—
47#14583009C2212yes—
48#14583004C2212yes—
49#14582999C2212yes—
50#14582997C2212yes—
51#14582971C2214yes—
52#14582971C2214yes—
53#14582939C2212yes—
54#14582935C2211yescharacteristic
55#14582923C2211yescharacteristic
56#14582907C2214yes—
57#14582907C2212yes—
58#14582898C2214yes—
59#14582898C2212yes—
60#14582895C2211yescharacteristic
61#14582893C2214yes—
62#14582893C2211yescharacteristic
63C2×#13063248C2214yes—
64C2×#13063248C2212yes—
65C2×#13063248C2212yes—
66#14582866C2212yes—
67C2×#13063236C2211yescharacteristic
68C43×#553—24no—
69C43×#553C2214yes—
70C43×#552C2214yes—
71C43×#541—21no—
72C43×#540—21no—
73C43×#537C2212yes—
74C43×#532C2211yescharacteristic
75C43×#529—22no—
76C2×#125×C43C2211yescharacteristic
77C43×#520C2212yes—
78C4×#41×C43C2211yescharacteristic
79#13063372—28no—
80#13063372—44no—
81C2×#10621898C2314yes—
82C2×#10621898C2311yescharacteristic
83C2×#10621900—22no—
84C2×#10621900—24no—
85C22×Dic86C2312yes—
86C22×Dic86C2314yes—
87C2×Q8×D43—22no—
88#13063355—44no—
89#13063345—24no—
90C2×#10621885C2311yescharacteristic
91C2×#10621885C2314yes—
92C2×#10621883C2311yescharacteristic
93C2×#10621883C2314yes—
94C2×#10621884C2311yescharacteristic
95C2×#10621882C2311yescharacteristic
96#13063332—44no—
97C2×#10621877C2312yes—
98C2×#10621877C2314yes—
99C22×C43⋊C8C2311yescharacteristic
100C22×C43⋊C8C2311yescharacteristic
101C22×C43⋊C8C2314yes—
102#13063325—21no—
103#13063325—21no—
104#13063325C2314yes—
105#13063324—24no—
106#13063323—22no—
107#13063312—22no—
108#13063307—24no—
109#13063307—24no—
110#13063306—24no—
111#13063304C2314yes—
112#13063304C2314yes—
113#13063302—24no—
114#13063302—24no—
115#13063296—24no—
116C4×Dic86C2312yes—
117C4×D172C2312yes—
118D4×Dic43—24no—
119C4×C22⋊D43—22no—
120#13063284C2311yescharacteristic
121#13063282—21no—
122C2×C4⋊Dic43—22no—
123C4⋊C4×D43—24no—
124C42×D43—21no—
125#13063258C2314yes—
126#13063258C2312yes—
127#13063256C2314yes—
128#13063256C2312yes—
129C4⋊(C43⋊C8)C2314yes—
130#13063249C2312yes—
131#13063248C2314yes—
132#13063248C2314yes—
133#13063248C2314yes—
134#13063248C2314yes—
135#13063248C2314yes—
136#13063248C2314yes—
137#13063248C2314yes—
138#13063248C2312yes—
139#13063246C2312yes—
140#13063245C2314yes—
141#13063245C2312yes—
142C22⋊(C43⋊C8)C2314yes—
143C22⋊(C43⋊C8)C2314yes—
144#13063236C2312yes—
145#13063236C2312yes—
146C2×#41×C43C2311yescharacteristic
147C2×#41×C43C2312yes—
148C4⋊Q8×C43D411yescharacteristic
149#127×C43D411yescharacteristic
150#126×C43—24no—
151#126×C43—24no—
152#125×C43—24no—
153#125×C43—21no—
154#125×C43—21no—
155#125×C43C2314yes—
156#124×C43D412yes—
157#123×C43—24no—
158#123×C43—24no—
159C22⋊Q8×C43—22no—
160C22⋊Q8×C43C2314yes—
161#121×C43—22no—
162#121×C43—24no—
163#121×C43—24no—
164#121×C43D414yes—
165#121×C43C2314yes—
166C4×Q8×C43—22no—
167C4×Q8×C43C2312yes—
168C4×D4×C43—22no—
169C4×D4×C43—24no—
170C4×D4×C43—22no—
171C4×D4×C43—24no—
172C4×D4×C43C2312yes—
173C4×D4×C43C2314yes—
174#117×C43—24no—
175#117×C43—21no—
176#117×C43C2311yescharacteristic
177#117×C43C2314yes—
178C2×C4⋊C4×C43—22no—
179C2×C4⋊C4×C43C2312yes—
180C2×C4⋊C4×C43C2314yes—
181C2×C42×C43—21no—
182C2×C42×C43C2311yescharacteristic
183#41×C43—24no—
184#41×C43—24no—
185#41×C43—22no—
186#41×C43—22no—
187C2×Q8×C43C2×D412yes—
188C2×Q8×C43C2411yescharacteristic
189C2×D4×C43—21no—
190C2×D4×C43—24no—
191C2×D4×C43C2×D414yes—
192C2×D4×C43C2×D412yes—
193C2×D4×C43C2×D414yes—
194C2×D4×C43C2411yescharacteristic
195C22×C4×C43—21no—
196C22×C4×C43—22no—
197C22×C4×C43—22no—
198C22×C4×C43—24no—
199C22×C4×C43C2×D412yes—
200C22×C4×C43C2414yes—
201C22×C4×C43C2411yescharacteristic
202C2×C22⋊D43—21no—
203C2×C22⋊D43—24no—
204C22×Dic43—22no—
205C22×Dic43—24no—
206Q8×D43—48no—
207#10621900—48no—
208#10621900—416no—
209#10621898—22no—
210#10621898—28no—
211#10621898—44no—
212#10621898#4114yes—
213C2×D172C2411yescharacteristic
214C2×C4×D43—21no—
215C2×C4×D43—22no—
216C2×C4×D43—24no—
217C2×Dic86—24no—
218C2×Dic86—24no—
219C2×Dic86—28no—
220C2×Dic86—24no—
221C2×Dic86—22no—
222C2×Dic86—44no—
223C2×Dic86C2414yes—
224C2×Dic86C2411yescharacteristic
225C4⋊C4×C43—24no—
226C4⋊C4×C43—24no—
227C4⋊C4×C43—22no—
228C4⋊C4×C43—24no—
229C4⋊C4×C43—22no—
230C4⋊C4×C43—24no—
231C4⋊C4×C43C2×D414yes—
232C4⋊C4×C43C2412yes—
233C4⋊C4×C43C2414yes—
234C22⋊C4×C43—22no—
235C22⋊C4×C43—24no—
236C22⋊C4×C43—24no—
237C22⋊C4×C43—21no—
238C22⋊C4×C43—24no—
239C22⋊C4×C43C2×D414yes—
240C22⋊C4×C43C2×D414yes—
241C42×C43—22no—
242C42×C43—21no—
243C42×C43—24no—
244C42×C43C2×D411yescharacteristic
245C42×C43C2×D411yescharacteristic
246C42×C43C2412yes—
247C22⋊Dic43—21no—
248C22⋊Dic43—24no—
249C22⋊Dic43—24no—
250#10621885—22no—
251#10621885—44no—
252#10621884—22no—
253#10621883—22no—
254#10621883—44no—
255#10621882—22no—
256(C2×D43)⋊C4—22no—
257C4⋊Dic43—24no—
258C4⋊Dic43—24no—
259C4⋊Dic43C2412yes—
260Dic43⋊C4—24no—
261Dic43⋊C4—24no—
262Dic43⋊C4—22no—
263C4×Dic43—21no—
264C4×Dic43—22no—
265#10621877—24no—
266#10621877—44no—
267C2×C43⋊C8—22no—
268C2×C43⋊C8—22no—
269C2×C43⋊C8—44no—
270C2×C43⋊C8C2411yescharacteristic
271C2×C43⋊C8C2411yescharacteristic
272C2×C43⋊C8C2412yes—
273C2×C43⋊C8C2414yes—
274C23×C43—21no—
275C23×C43C22×D414yes—
276C23×C43C22×D411yescharacteristic
277C22×D43—21no—
278Q8×C43—24no—
279Q8×C43—21no—
280Q8×C43—21no—
281D4×C43—28no—
282D4×C43—24no—
283D4×C43—21no—
284D4×C43—21no—
285D4×C43—44no—
286D4×C43—42no—
287D4×C43—84no—
288C2×C4×C43—24no—
289C2×C4×C43—24no—
290C2×C4×C43—22no—
291C2×C4×C43—21no—
292C2×C4×C43—24no—
293C2×C4×C43—42no—
294C2×C4×C43—42no—
295C2×C4×C43—82no—
296C2×C4×C43—84no—
297C2×C4×C43—82no—
298C2×C4×C43C22×D412yes—
299C2×C4×C43C22×D414yes—
300C2×C4×C43C22×D412yes—
301C2×C4×C43C22×D411yescharacteristic
302C2×C4×C43C2×#4112yes—
303C2×C4×C43H(5,𝔽2)12yes—
304C2×C4×C43C2511yescharacteristic
305C22⋊D43—24no—
306C22⋊D43—48no—
307C2×Dic43—21no—
308C2×Dic43—22no—
309C2×Dic43—24no—
310C2×Dic43—48no—
311C2×Dic43—44no—
312D172—21no—
313D172C2×#4112yes—
314C4×D43—24no—
315C4×D43—48no—
316C4×D43—44no—
317Dic86—21no—
318Dic86—24no—
319Dic86—24no—
320Dic86—48no—
321Dic86—44no—
322Dic86C2×#4112yes—
323C43⋊C8—21no—
324C43⋊C8—21no—
325C43⋊C8—22no—
326C43⋊C8—24no—
327#61224—431nomaximal, Sylow
328C22×C43—22no—
329C22×C43—24no—
330C22×C43—24no—
331C22×C43—24no—
332C22×C43—42no—
333C22×C43—41no—
334C22×C43#54612yes—
335C22×C43C23×D411yescharacteristic
336C2×D43—22no—
337C2×D43—42no—
338C4×C43—22no—
339C4×C43—22no—
340C4×C43—24no—
341C4×C43—42no—
342C4×C43—44no—
343C4×C43—41no—
344C4×C43#54612yes—
345C4×C43C22×#4111yescharacteristic
346C4×C43C2×H(5,𝔽2)11yescharacteristic
347Dic43—22no—
348Dic43—42no—
349Dic43—44no—
350#3512D4311yescharacteristic, complemented
351#3499—431no—
352#3320—434no—
353#3315—434no—
354#3312—434no—
355#3307—434no—
356C2×#491—431no—
357#3097—432no—
358#3080—431no—
359#3063—432no—
360#3039—431no—
361#3013—432no—
362#2943—431no—
363#2932—431no—
364#2898—432no—
365C2×C43—22no—
366C2×C43—44no—
367C2×C43—81no—
368C2×C43C2×#54611yescharacteristic
369C2×C43#358612yes—
370D43—42no—
371C2×2−1+4—432no—
372C2×#137—432no—
373C2×#135—431no—
374C2×#131—432no—
375#553—24no—
376#553C2×D4314yes—
377#552C2×D4314yes—
378D4×Q8—434no—
379#547—434no—
380#547—864no—
381#541—21no—
382#540—21no—
383#537C2×D4312yes—
384#536—432no—
385#536—862no—
386#532—861no—
387#532C2×D4311yescharacteristic
388C2×C4⋊Q8—431no—
389C2×C4⋊Q8—861no—
390#529—22no—
391C2×#125C2×D4311yescharacteristic
392#520C2×D4312yes—
393C4×#41—431no—
394C4×#41C2×D4311yescharacteristic
395#491—434no—
396#491—434no—
397#485—434no—
398#485—434no—
399#484—434no—
400#483—434no—
401#479—434no—
402#478—434no—
403#470—434no—
404#468—434no—
405#467—434no—
406#465—434no—
407#462—434no—
408#461—434no—
409#452—434no—
410#451—434no—
411#449—434no—
412#448—434no—
413#442—432no—
414#441—432no—
415#440—432no—
416#440—432no—
417#435—434no—
418#435—434no—
419#428—432no—
420#427—431no—
421#423—431no—
422#419—432no—
423#419—434no—
424#417—432no—
425#417—434no—
426#416—431no—
427#416—431no—
428#416—434no—
429C2×#103—432no—
430C2×#103—432no—
431C2×#103—434no—
432#408—432no—
433C2×#97—431no—
434C43#6122411yescharacteristic, complemented, Sylow
4352−1+4—868no—
4362−1+4—1724no—
437C2×#41—431no—
438C2×#41—434no—
439C2×#41—862no—
440C2×#41—864no—
441C2×#41C22×D4311yescharacteristic
442C2×#41C22×D4312yes—
443C22×Q8—432no—
444C22×Q8—434no—
445C22×Q8—862no—
446#137—1724no—
447#135—864no—
448C2×Q16—431no—
449C2×Q16—434no—
450C2×SD16—431no—
451C2×SD16—431no—
452C2×SD16—434no—
453C2×D8—431no—
454#131—1724no—
455C2×M16—432no—
456C2×M16—434no—
457C22×C8—431no—
458C22×C8—431no—
459C22×C8—434no—
460C4⋊Q8—434no—
461C4⋊Q8—862no—
462C4⋊Q8—864no—
463C4⋊Q8—861no—
464C4⋊Q8—861no—
465C4⋊Q8C22⋊D4311yescharacteristic
466#127C22⋊D4311yescharacteristic
467#126—24no—
468#126—24no—
469#125—24no—
470#125—21no—
471#125—21no—
472#125C22×D4314yes—
473#124—862no—
474#124C22⋊D4312yes—
475#123—24no—
476#123—24no—
477#123—864no—
478#123—864no—
479#123—864no—
480C22⋊Q8—22no—
481C22⋊Q8—434no—
482C22⋊Q8—434no—
483C22⋊Q8—864no—
484C22⋊Q8—864no—
485C22⋊Q8C22×D4314yes—
486#121—22no—
487#121—24no—
488#121—24no—
489#121—864no—
490#121C22⋊D4314yes—
491#121C22×D4314yes—
492C4×Q8—22no—
493C4×Q8—432no—
494C4×Q8C22×D4312yes—
495C4×D4—22no—
496C4×D4—24no—
497C4×D4—22no—
498C4×D4—24no—
499C4×D4—432no—
500C4×D4—864no—
501C4×D4—862no—
502C4×D4C22×D4312yes—
503C4×D4C22×D4314yes—
504#117—24no—
505#117—21no—
506#117—431no—
507#117—861no—
508#117C22×D4311yescharacteristic
509#117C22×D4314yes—
510C2×C4⋊C4—22no—
511C2×C4⋊C4—862no—
512C2×C4⋊C4—864no—
513C2×C4⋊C4C22×D4312yes—
514C2×C4⋊C4C22×D4314yes—
515C2×C42—21no—
516C2×C42—861no—
517C2×C42C22×D4311yescharacteristic
518#107—432no—
519#107—434no—
520#106—432no—
521#106—434no—
522C4⋊C8—434no—
523#103—432no—
524#103—432no—
525#103—434no—
526#103—434no—
527#103—434no—
528#103—434no—
529#103—434no—
530#103—434no—
531#103—434no—
532#102—432no—
533#102—432no—
534#102—434no—
535C22⋊C8—434no—
536C22⋊C8—434no—
537#97—432no—
538#97—432no—
539#41—24no—
540#41—24no—
541#41—22no—
542#41—22no—
543#41—434no—
544#41—868no—
545#41—862no—
546#41—1724no—
547#41—1728no—
548#41—17216no—
549C2×Q8—431no—
550C2×Q8—434no—
551C2×Q8—868no—
552C2×Q8—864no—
553C2×Q8—862no—
554C2×Q8—864no—
555C2×Q8—864no—
556C2×Q8—1724no—
557C2×Q8—1728no—
558C2×Q8C2×C22⋊D4312yes—
559C2×Q8C23×D4311yescharacteristic
560C2×D4—21no—
561C2×D4—24no—
562C2×D4—431no—
563C2×D4—861no—
564C2×D4—864no—
565C2×D4C2×C22⋊D4314yes—
566C2×D4C2×C22⋊D4312yes—
567C2×D4C2×C22⋊D4314yes—
568C2×D4C23×D4311yescharacteristic
569C22×C4—22no—
570C22×C4—24no—
571C22×C4—21no—
572C22×C4—22no—
573C22×C4—861no—
574C22×C4—862no—
575C22×C4—864no—
576C22×C4—862no—
577C22×C4—864no—
578C22×C4C2×C22⋊D4312yes—
579C22×C4C23×D4314yes—
580C22×C4C23×D4311yescharacteristic
581Q16—862no—
582Q16—1724no—
583SD16—862no—
584SD16—862no—
585SD16—1724no—
586D8—862no—
587M16—864no—
588M16—1724no—
589C2×C8—431no—
590C2×C8—431no—
591C2×C8—432no—
592C2×C8—434no—
593C2×C8—862no—
594C2×C8—862no—
595C2×C8—1724no—
596C4⋊C4—24no—
597C4⋊C4—24no—
598C4⋊C4—22no—
599C4⋊C4—24no—
600C4⋊C4—22no—
601C4⋊C4—24no—
602C4⋊C4—432no—
603C4⋊C4—864no—
604C4⋊C4—864no—
605C4⋊C4—864no—
606C4⋊C4—864no—
607C4⋊C4—862no—
608C4⋊C4C2×C22⋊D4314yes—
609C4⋊C4C23×D4312yes—
610C4⋊C4C23×D4314yes—
611C22⋊C4—24no—
612C22⋊C4—21no—
613C22⋊C4—24no—
614C22⋊C4—22no—
615C22⋊C4—24no—
616C22⋊C4—861no—
617C22⋊C4—864no—
618C22⋊C4—864no—
619C22⋊C4—862no—
620C22⋊C4C2×C22⋊D4314yes—
621C22⋊C4C2×C22⋊D4314yes—
622C42—22no—
623C42—21no—
624C42—24no—
625C42—861no—
626C42—862no—
627C42C2×C22⋊D4311yescharacteristic
628C42C2×C22⋊D4311yescharacteristic
629C42C23×D4312yes—
630C23—21no—
631C23—861no—
632C23C22×C22⋊D4314yes—
633C23C22×C22⋊D4311yescharacteristic
634Q8—24no—
635Q8—21no—
636Q8—21no—
637Q8—432no—
638Q8—861no—
639Q8—864no—
640Q8—864no—
641Q8—1728no—
642Q8—1724no—
643D4—28no—
644D4—21no—
645D4—21no—
646D4—24no—
647D4—44no—
648D4—42no—
649D4—84no—
650D4—432no—
651D4—864no—
652D4—861no—
653D4—1728no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—22no—
657C2×C4—21no—
658C2×C4—24no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—82no—
662C2×C4—84no—
663C2×C4—82no—
664C2×C4—861no—
665C2×C4—864no—
666C2×C4—862no—
667C2×C4—864no—
668C2×C4—1728no—
669C2×C4—1724no—
670C2×C4—1728no—
671C2×C4—1724no—
672C2×C4C22×C22⋊D4312yes—
673C2×C4C22×C22⋊D4314yes—
674C2×C4C22×C22⋊D4312yes—
675C2×C4C22×C22⋊D4311yescharacteristic
676C2×C4#41×D4312yes—
677C2×C4#1306337112yes—
678C2×C4C24×D4311yescharacteristic
679C8—861no—
680C8—861no—
681C8—862no—
682C8—864no—
683C22—22no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—41no—
688C22—42no—
689C22—862no—
690C22—1722no—
691C22#1458353712yes—
692C22C23×C22⋊D4311yescharacteristic
693C4—24no—
694C4—22no—
695C4—22no—
696C4—41no—
697C4—42no—
698C4—44no—
699C4—862no—
700C4—1724no—
701C4—1722no—
702C4#1458353712yes—
703C4C2×#41×D4311yescharacteristic
704C4C2×#1306337111yescharacteristic
705C2—22no—
706C2—44no—
707C2—81no—
708C2—1722no—
709C2C2×#1458353711yescharacteristic
710C2#1914938112yes—
subgroupG/H
centerC22C23×C22⋊D43
commutator, cosocleC2×C4×C43C25
FrattiniC2×C4C24×D43
FittingC43×#3512C2
socleC22×C43C23×D4
hypercenter#3512D43
hypocenterC43#61224
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ #532 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #19148375C2semidirect
2. #19146760C2semidirect
3. #19146685C2semidirect
4. #19146640C2semidirect
5. #19146565C2semidirect
6. C2×#14583269C2semidirect
7. #19144564C2semidirect
8. #19144413C2semidirect
9. #19144288C2semidirect
10. #19144082C2semidirect
11. #19143907C2semidirect
12. #19143393C2semidirect
13. #19143313C2semidirect
14. #19143125C2semidirect
15. C43×#3512C2semidirect
350. #3512D43semidirect
434. C43#61224semidirect

Maximal subgroups

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

subgroupindex
#191483752
#191467602
#191466852
#191466402
#191465652
C2×#145832692
#191445642
#191444132
#191442882
#191440822
#191439072
#191433932
#191433132
#191431252
C43×#35122
#6122443