Finite Group Databasebeta

← #25166514

Subgroups — #25166514

Order 11008.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583003C211yesmaximal, characteristic, complemented
2#19138430C211yesmaximal, characteristic, complemented
3#19138389C211yesmaximal, characteristic
4#19135965C211yesmaximal, characteristic
5#19135925C211yesmaximal, characteristic, complemented
6#19135916C211yesmaximal, characteristic, complemented
7#19135689C211yesmaximal, characteristic
8#19135450C211yesmaximal, characteristic
9#19135289C211yesmaximal, characteristic
10#19135285C211yesmaximal, characteristic, complemented
11#19135273C211yesmaximal, characteristic, complemented
12#19134947C211yesmaximal, characteristic
13#19134861C211yesmaximal, characteristic, complemented
14#19134463C211yesmaximal, characteristic, complemented
15C2×C43×#489C211yesmaximal, characteristic
16C22×#10621885C2211yescharacteristic
17C22×#10621883C2211yescharacteristic
18C2×Q8×Dic43C2211yescharacteristic
19C2×#13063290C2211yescharacteristic
20#14583003—24no—
21C2×#13063256C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×Dic43⋊C8C2211yescharacteristic
24C2×#13063250C2211yescharacteristic
25C2×#13063250C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063246C2211yescharacteristic
28C2×#13063235C2211yescharacteristic
29C2×C8×Dic43C2211yescharacteristic
30#14582835—21no—
31#14582832—21no—
32#14582824—21no—
33#14582818—21no—
34#14582806—21no—
35#14582806C2211yescharacteristic
36#14582805C2211yescharacteristic
37#14582804C2211yescharacteristic
38#14582802C2211yescharacteristic
39#14582802C2211yescharacteristic
40#14582798C2211yescharacteristic
41#14582797C2211yescharacteristic
42#14582795C2211yescharacteristic
43#14582787—21no—
44#14582787—21no—
45#14582780—21no—
46C4×Dic43⋊C4C2211yescharacteristic
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582654C2211yescharacteristic
52#14582653C2211yescharacteristic
53C2×C4⋊Q8×C43C2211yescharacteristic
54C2×#124×C43C2211yescharacteristic
55C43×#489—24no—
56C2×#103×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C2×#102×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C2×#97×C43C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C2×#10621885—22no—
63C2×#10621885—22no—
64C2×#10621885—22no—
65C2×#10621883—22no—
66C2×#10621883—22no—
67C2×#10621883—41no—
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2311yescharacteristic
70Q8×Dic43—22no—
71Q8×Dic43—41no—
72#13063290—22no—
73#13063290—41no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4D411yescharacteristic
83C2×Dic43⋊C4D411yescharacteristic
84C2×C22⋊Dic43—21no—
85C2×C22⋊Dic43—21no—
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43—21no—
90C2×C4×Dic43C2311yescharacteristic
91C2×C4×Dic43C2311yescharacteristic
92#13063256—22no—
93#13063256—41no—
94Dic43⋊C8—22no—
95#13063250—22no—
96#13063250—22no—
97#13063249—22no—
98#13063246—22no—
99#13063235—22no—
100C8×Dic43—22no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95D411yescharacteristic
111C43⋊#95D411yescharacteristic
112C22×Q8×C43—21no—
113C22×Q8×C43C2311yescharacteristic
114C22×D4×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C22×C8×C43C2311yescharacteristic
117C4⋊Q8×C43—22no—
118C4⋊Q8×C43—41no—
119#124×C43—22no—
120#124×C43—41no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C22⋊C4×C43—21no—
125C2×C22⋊C4×C43—21no—
126C2×C42×C43C2311yescharacteristic
127#103×C43—22no—
128#103×C43D414yes—
129#102×C43—22no—
130#102×C43—22no—
131#97×C43—22no—
132C24×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—42no—
139C2×Q8×C43—42no—
140C2×Q8×C43—81no—
141C2×Q8×C43C2×D412yes—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43—21no—
146C2×D4×C43—21no—
147C2×D4×C43—21no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2×D411yescharacteristic
153C22×C4×C43C2411yescharacteristic
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—21no—
158C22×Dic43C2×D411yescharacteristic
159C22×Dic43C2×D411yescharacteristic
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
162C2×Dic86—21no—
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86#4112yes—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43C2×D412yes—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—42no—
178C4⋊C4×C43C2×D412yes—
179C22⋊C4×C43—42no—
180C22⋊C4×C43—42no—
181C42×C43—21no—
182C42×C43—21no—
183C22⋊Dic43—42no—
184C22⋊Dic43—42no—
185C22⋊Dic43—81no—
186C22⋊Dic43—81no—
187#10621885—28no—
188#10621883—44no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—42no—
196C4⋊Dic43—81no—
197Dic43⋊C4—22no—
198Dic43⋊C4—41no—
199Dic43⋊C4—42no—
200Dic43⋊C4—81no—
201Dic43⋊C4—81no—
202C4×Dic43—21no—
203C4×Dic43—21no—
204C4×Dic43—21no—
205C4×Dic43—21no—
206C4×Dic43—42no—
207C4×Dic43—42no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8#4112yes—
213C2×C43⋊C8#4112yes—
214C23×C43—41no—
215C23×C43—41no—
216C23×C43—41no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43—41no—
221C23×C43C22×D411yescharacteristic
222Q8×C43—22no—
223Q8×C43—22no—
224Q8×C43—41no—
225Q8×C43—41no—
226Q8×C43—84no—
227D4×C43—41no—
228D4×C43—41no—
229D4×C43—41no—
230D4×C43—41no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43C22×D411yescharacteristic
247C2×C4×C43C2×#4111yescharacteristic
248C2×C4×C43C2×#4111yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43H(5,𝔽2)11yescharacteristic
251C2×C4×C432−1+411yescharacteristic
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—22no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43#13712yes—
276Dic86—22no—
277Dic86—22no—
278Dic86—41no—
279Dic86—41no—
280C8×C43—22no—
281C8×C43—41no—
282C43⋊C8—22no—
283C43⋊C8—22no—
284#26846—431nomaximal, Sylow
285C22×C43—81no—
286C22×C43—81no—
287C22×C43—81no—
288C22×C43—81no—
289C22×C43—81no—
290C22×C43—81no—
291C22×C43—81no—
292C22×C43#53211yescharacteristic
293C22×C43D4211yescharacteristic
294C22×C43#54811yescharacteristic
295C22×C43C2×#13711yescharacteristic
296C22×C43#57511yescharacteristic
297C22×C43#57711yescharacteristic
298C22×C43#57711yescharacteristic
299C4×C43—42no—
300C4×C43—41no—
301C4×C43—42no—
302C4×C43—41no—
303C4×C43—81no—
304C4×C43#52511yescharacteristic
305C4×C43#54211yescharacteristic
306C4×C43#54711yescharacteristic
307C4×C43#55111yescharacteristic
308Dic43—22no—
309Dic43—41no—
310Dic43—42no—
311Dic43—41no—
312Dic43—41no—
313Dic43—81no—
314Dic43—81no—
315Dic43—81no—
316C2×#489D4311yescharacteristic
317C2×#441—431no—
318#2451—431no—
319#2448—431no—
320#2065—431no—
321#2058—431no—
322#2056—431no—
323#2015—431no—
324#1968—431no—
325#1936—431no—
326#1935—431no—
327#1933—431no—
328#1880—431no—
329#1865—431no—
330#1770—431no—
331C2×C43—161no—
332C2×C43#246111yescharacteristic
333C2×C43#317311yescharacteristic
334C2×C43#318111yescharacteristic
335C2×C43#327511yescharacteristic
336C2×C43#327811yescharacteristic
337C2×C43#336911yescharacteristic
338C2×C43#337111yescharacteristic
339C22×Q16—431no—
340C22×SD16—431no—
341C2×C4⋊Q8C2×D4311yescharacteristic
342C2×#124C2×D4311yescharacteristic
343C2×C4×Q8—431no—
344C2×C4×Q8—431no—
345#489—24no—
346#441—864no—
347C2×#106—431no—
348C2×#106—431no—
349C2×C4⋊C8—431no—
350C2×#103—431no—
351C2×#103—431no—
352C2×#103—431no—
353C2×#103C2×D4311yescharacteristic
354C2×#103C2×D4311yescharacteristic
355C2×#102—431no—
356C2×#102C2×D4311yescharacteristic
357C2×#102C2×D4311yescharacteristic
358C2×#97—431no—
359C2×#97C2×D4311yescharacteristic
360C2×C4×C8—431no—
361#399—861no—
362#398—861no—
363#397—861no—
364#395—861no—
365#391—431no—
366#391—431no—
367#391—431no—
368#391—861no—
369#390—431no—
370#390—431no—
371#389—431no—
372#389—431no—
373#389—431no—
374#388—861no—
375#388—861no—
376#387—861no—
377C4×C4⋊C4—431no—
378#340—431no—
379#340—431no—
380#340—431no—
381#340—431no—
382#336—431no—
383#336—431no—
384C43#2684611yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×Q8—431no—
387C22×Q8C22×D4311yescharacteristic
388C22×D4C22×D4311yescharacteristic
389C2×Q16—862no—
390C2×Q16—862no—
391C2×Q16—862no—
392C2×SD16—862no—
393C2×SD16—862no—
394C2×SD16—1721no—
395C22×C8—431no—
396C22×C8—431no—
397C22×C8C22×D4311yescharacteristic
398C22×C8C22×D4311yescharacteristic
399C4⋊Q8—22no—
400C4⋊Q8—41no—
401#124—22no—
402#124—41no—
403C4×Q8—862no—
404C4×Q8—862no—
405C4×Q8—1721no—
406C4×Q8—1721no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—431no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4C22×D4311yescharacteristic
419C2×C4⋊C4C22×D4311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C22⋊C4—861no—
425C2×C22⋊C4—861no—
426C2×C42—431no—
427C2×C42—431no—
428C2×C42—861no—
429C2×C42—861no—
430C2×C42C22×D4311yescharacteristic
431#106—862no—
432#106—1721no—
433C4⋊C8—862no—
434#103—22no—
435#103—862no—
436#103—862no—
437#103—862no—
438#103C22⋊D4314yes—
439#102—22no—
440#102—22no—
441#102—862no—
442#97—22no—
443#97—862no—
444C4×C8—862no—
445#95—431no—
446#95—431no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453#95—861no—
454#95—861no—
455#95—861no—
456C24—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—81no—
465C2×Q8—432no—
466C2×Q8—861no—
467C2×Q8—861no—
468C2×Q8—861no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×Q8C2×C22⋊D4312yes—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—431no—
482C22×C4—431no—
483C22×C4—431no—
484C22×C4—431no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4D4×D4311yescharacteristic
490C22×C4D4×D4311yescharacteristic
491C22×C4C23×D4311yescharacteristic
492Q16—868no—
493SD16—1724no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—432no—
500C2×C8—432no—
501C2×C8—861no—
502C2×C8—861no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8C2×C22⋊D4312yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—862no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—3441no—
518C4⋊C4—3441no—
519C4⋊C4—3441no—
520C4⋊C4—1721no—
521C4⋊C4—1722no—
522C4⋊C4—1722no—
523C4⋊C4C2×C22⋊D4312yes—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—3441no—
527C22⋊C4—3441no—
528C22⋊C4—1722no—
529C22⋊C4—1722no—
530C42—21no—
531C42—21no—
532C42—861no—
533C42—861no—
534C42—861no—
535C42—861no—
536C42—1722no—
537C42—1722no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D4×D4311yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—84no—
551Q8—862no—
552Q8—862no—
553Q8—1721no—
554Q8—1721no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—432no—
575C2×C4—861no—
576C2×C4—861no—
577C2×C4—862no—
578C2×C4—861no—
579C2×C4—861no—
580C2×C4—861no—
581C2×C4—861no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—861no—
585C2×C4—861no—
586C2×C4—861no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4C22×C22⋊D4311yescharacteristic
599C2×C4#41×D4311yescharacteristic
600C2×C4#41×D4311yescharacteristic
601C2×C4#41×D4311yescharacteristic
602C2×C4#1306337011yescharacteristic
603C2×C4#1306337211yescharacteristic
604C8—22no—
605C8—41no—
606C8—862no—
607C8—862no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1458344811yescharacteristic
616C22D4×C22⋊D4311yescharacteristic
617C22#1458355711yescharacteristic
618C22#137×D4311yescharacteristic
619C22#1458369511yescharacteristic
620C22#1458370811yescharacteristic
621C22#1458370911yescharacteristic
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—862no—
628C4—3441no—
629C4—3441no—
630C4—3441no—
631C4—1721no—
632C4—1721no—
633C4—1721no—
634C4—1722no—
635C4#1458340611yescharacteristic
636C4#1458351311yescharacteristic
637C4#1458354811yescharacteristic
638C4#1458357911yescharacteristic
639C2—161no—
640C2#1913855811yescharacteristic
641C2#1914521111yescharacteristic
642C2#1914530711yescharacteristic
643C2#1914622611yescharacteristic
644C2#1914625511yescharacteristic
645C2#1914733211yescharacteristic
646C2#1914736211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#489C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenterC2×#489D43
hypocenterC43#26846
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#14583003C2semidirect
2. #19138430C2semidirect
5. #19135925C2semidirect
6. #19135916C2semidirect
10. #19135285C2semidirect
11. #19135273C2semidirect
13. #19134861C2semidirect
14. #19134463C2semidirect
384. C43#26846semidirect

Maximal subgroups

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

subgroupindex
C2×#145830032
#191384302
#191383892
#191359652
#191359252
#191359162
#191356892
#191354502
#191352892
#191352852
#191352732
#191349472
#191348612
#191344632
C2×C43×#4892
#2684643