Finite Group Databasebeta

← #25106349

Subgroups — #25106349

Order 11008.

1366 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583446C211yesmaximal, characteristic
2C2×#14582955C211yesmaximal, characteristic, complemented
3#19137000C211yesmaximal, characteristic, complemented
4#19135566C212yesmaximal, complemented
5#19135417C212yesmaximal, complemented
6#19134916C212yesmaximal, complemented
7#19134914C212yesmaximal, complemented
8#19134596C211yesmaximal, characteristic, complemented
9#19134595C211yesmaximal, characteristic, complemented
10#19134592C211yesmaximal, characteristic, complemented
11C43×#1802C211yesmaximal, characteristic, complemented
12C22×#10621898—21no—
13#14583446—24no—
14#14583446C418yescomplemented
15C2×#13063324C2211yescharacteristic
16C2×#13063321C2211yescharacteristic
17C2×#13063311—21no—
18C2×#13063300C2211yescharacteristic
19C2×#13063298—21no—
20C2×#13063296C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×C4×C22⋊D43—21no—
23C2×#13063283C2212yes—
24C2×C42×D43C2211yescharacteristic
25#14582955—24no—
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063247C2212yes—
28C2×#13063246C2212yes—
29C2×#13063245C2212yes—
30C2×(C2×D43)⋊C8C2212yes—
31C2×#13063235C2212yes—
32#14582769C2211yescharacteristic
33#14582764C2212yes—
34#14582756—21no—
35#14582756—21no—
36C4⋊C4×Dic43—21no—
37C4⋊C4×Dic43—21no—
38#14582662C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C2×C4×D4×C43C2211yescharacteristic
42C2×C4⋊C8×C43C2211yescharacteristic
43C2×#102×C43C2212yes—
44C43×#384C2211yescharacteristic
45C43×#340C2212yes—
46C2×#10621898—28no—
47C2×#10621898—44no—
48C2×#10621898—44no—
49C22×Dic86—21no—
50C22×D172—21no—
51C22×D172C2311yescharacteristic
52C22×C22⋊D43—21no—
53C22×C22⋊D43—21no—
54C22×C22⋊D43—21no—
55C22×C4×D43—21no—
56C22×C4×D43D411yescharacteristic
57C22×C4×D43D411yescharacteristic
58C22×C4×D43C2311yescharacteristic
59C22×C43⋊C8C2312yes—
60#13063324—22no—
61#13063324C2×C414yes—
62#13063321—22no—
63#13063321C2×C414yes—
64#13063311—28no—
65#13063300—22no—
66#13063300C2×C414yes—
67#13063298—28no—
68#13063296—22no—
69#13063296C2×C414yes—
70C4×D172—22no—
71C4×D172C2×C414yes—
72C4×C22⋊D43—28no—
73#13063283—42no—
74#13063283—42no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43C2312yes—
78C2×C4⋊Dic43C2311yescharacteristic
79C2×Dic43⋊C4—21no—
80C2×C22⋊Dic43—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—22no—
84C2×C4×Dic43—22no—
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88C42×D43C2×C414yes—
89C42×D43D414yes—
90C4⋊(C43⋊C8)—22no—
91#13063247—42no—
92#13063246—42no—
93#13063245—42no—
94(C2×D43)⋊C8—24no—
95#13063235—24no—
96C43⋊#95—22no—
97C43⋊#95—22no—
98C22×D4×C43C2311yescharacteristic
99C23×C4×C43—21no—
100C22×C8×C43C2312yes—
101C4×D4×C43—22no—
102C4×D4×C43C2×C414yes—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—21no—
105C2×C4⋊C4×C43C2312yes—
106C2×C4⋊C4×C43C2311yescharacteristic
107C2×C22⋊C4×C43—21no—
108C2×C42×C43C2311yescharacteristic
109C4⋊C8×C43—22no—
110#102×C43—42no—
111C24×C43—21no—
112C23×D43—21no—
113C23×D43C2×D411yescharacteristic
114C2×D4×C43—21no—
115C2×D4×C43—21no—
116C2×D4×C43—21no—
117C2×D4×C43—21no—
118C2×D4×C43—21no—
119C2×D4×C43C22×C412yes—
120C22×C4×C43—24no—
121C22×C4×C43—21no—
122C22×C4×C43—22no—
123C22×C4×C43—42no—
124C22×C4×C43—42no—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2411yescharacteristic
128C2×C22⋊D43—24no—
129C2×C22⋊D43—24no—
130C2×C22⋊D43—24no—
131C2×C22⋊D43—42no—
132C2×C22⋊D43—42no—
133C2×C22⋊D43—42no—
134C2×C22⋊D43—42no—
135C2×C22⋊D43—42no—
136C2×C22⋊D43—42no—
137C22×Dic43—21no—
138C22×Dic43—22no—
139C22×Dic43C2×D411yescharacteristic
140C22×Dic43C2×D411yescharacteristic
141C22×Dic43C2×D411yescharacteristic
142#10621898—432no—
143C2×D172—21no—
144C2×D172—24no—
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—42no—
150C2×D172—42no—
151C2×D172C22×C412yes—
152C2×C4×D43—22no—
153C2×C4×D43—22no—
154C2×C4×D43—21no—
155C2×C4×D43—21no—
156C2×C4×D43—21no—
157C2×C4×D43—21no—
158C2×C4×D43—24no—
159C2×C4×D43—41no—
160C2×C4×D43—41no—
161C2×C4×D43—42no—
162C2×C4×D43—42no—
163C2×C4×D43C22⋊C414yes—
164C2×C4×D43C22⋊C414yes—
165C2×C4×D43C22×C412yes—
166C2×C4×D43C2×D412yes—
167C2×Dic86—24no—
168C2×Dic86—42no—
169C2×Dic86—42no—
170C2×C8×C43—22no—
171C2×C8×C43—22no—
172C2×C8×C43—22no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—22no—
175C4⋊C4×C43—22no—
176C4⋊C4×C43—42no—
177C4⋊C4×C43—42no—
178C4⋊C4×C43C22×C412yes—
179C22⋊C4×C43—24no—
180C42×C43C22×C412yes—
181C42×C43C2×D412yes—
182C22⋊Dic43—24no—
183(C2×D43)⋊C4—24no—
184(C2×D43)⋊C4—24no—
185(C2×D43)⋊C4—44no—
186C4⋊Dic43—21no—
187C4⋊Dic43—22no—
188C4⋊Dic43—22no—
189C4⋊Dic43—42no—
190C4⋊Dic43—42no—
191C4⋊Dic43C22×C412yes—
192Dic43⋊C4—24no—
193C4×Dic43—44no—
194C4×Dic43C22⋊C414yes—
195C4×Dic43C22⋊C414yes—
196C4×Dic43C22×C412yes—
197C4×Dic43C2×D412yes—
198C2×C43⋊C8—22no—
199C2×C43⋊C8—22no—
200C2×C43⋊C8—22no—
201C23×C43—22no—
202C23×C43—41no—
203C23×C43—41no—
204C23×C43—41no—
205C23×C43—41no—
206C23×C43—41no—
207C23×C43—41no—
208C23×C43C22×D411yescharacteristic
209C22×D43—21no—
210C22×D43—21no—
211C22×D43—21no—
212C22×D43—21no—
213C22×D43—21no—
214C22×D43—21no—
215C22×D43—22no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—41no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43C2×C22⋊C412yes—
223D4×C43—22no—
224D4×C43—22no—
225D4×C43—41no—
226D4×C43—41no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—22no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—48no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43C2×C22⋊C412yes—
237C2×C4×C43C2×C22⋊C412yes—
238C2×C4×C43AGL(1,ℤ/8)12yes—
239C2×C4×C43AGL(1,ℤ/8)12yes—
240C2×C4×C43C23×C411yescharacteristic
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43H(5,𝔽2)11yescharacteristic
243C2×C4×C43H(5,𝔽2)11yescharacteristic
244C2×C4×C432−1+411yescharacteristic
245C2×C4×C432−1+411yescharacteristic
246C22⋊D43—48no—
247C22⋊D43—48no—
248C22⋊D43—48no—
249C2×Dic43—21no—
250C2×Dic43—21no—
251C2×Dic43—21no—
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—22no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—42no—
259C2×Dic43—42no—
260C2×Dic43—42no—
261C2×Dic43C2×C22⋊C412yes—
262C2×Dic43C2×C22⋊C412yes—
263C2×Dic43C2×C22⋊C412yes—
264D172—22no—
265D172—22no—
266D172—48no—
267D172—41no—
268D172—41no—
269C4×D43—22no—
270C4×D43—22no—
271C4×D43—22no—
272C4×D43—22no—
273C4×D43—28no—
274C4×D43—44no—
275C4×D43—48no—
276Dic86—48no—
277C8×C43—42no—
278C43⋊C8—42no—
279#21869—431nomaximal, Sylow
280C22×C43—42no—
281C22×C43—42no—
282C22×C43—42no—
283C22×C43—81no—
284C22×C43—81no—
285C22×C43—81no—
286C22×C43—81no—
287C22×C43C22×C22⋊C411yescharacteristic
288C22×C43#53511yescharacteristic
289C22×C43#53611yescharacteristic
290C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C43#57511yescharacteristic
293C22×C43#57511yescharacteristic
294C2×D43—22no—
295C2×D43—22no—
296C2×D43—22no—
297C2×D43—41no—
298C2×D43—41no—
299C2×D43—41no—
300C2×D43—41no—
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—81no—
305C2×D43—81no—
306C2×D43—81no—
307C2×D43—81no—
308C4×C43—22no—
309C4×C43—42no—
310C4×C43—82no—
311C4×C43#41814yes—
312C4×C43#52011yescharacteristic
313C4×C43#52011yescharacteristic
314C4×C43#53711yescharacteristic
315C4×C43#53711yescharacteristic
316Dic43—22no—
317Dic43—22no—
318Dic43—22no—
319Dic43—42no—
320Dic43—82no—
321C2×#532—431no—
322C2×#432—431no—
323#2285—431no—
324#1993—432no—
325#1961—432no—
326#1875—432no—
327#1875—432no—
328#1802—431no—
329#1802—431no—
330#1802—431no—
331#1802D4311yescharacteristic, complemented
332C2×C43—82no—
333C2×C43#231111yescharacteristic
334C2×C43C2×#41811yescharacteristic
335C2×C43#289611yescharacteristic
336C2×C43#321411yescharacteristic
337C2×C43#321411yescharacteristic
338C2×C43#323511yescharacteristic
339C2×C43#323511yescharacteristic
340D43—42no—
341D43—82no—
342C22×#41—861no—
343#532—438no—
344#532—864no—
345C2×C4⋊Q8—431no—
346C2×#127—431no—
347C2×#124—861no—
348C2×#121—431no—
349C2×#121—431no—
350C2×#121—861no—
351C2×C4×D4—431no—
352C2×C4×D4—861no—
353C2×C4×D4C2×D4311yescharacteristic
354C2×#117—432no—
355C22×C42—431no—
356#432—864no—
357C2×C4⋊C8—431no—
358C2×C4⋊C8C2×D4311yescharacteristic
359C2×#102—432no—
360C2×#102—432no—
361C2×#102—432no—
362C2×#102C2×D4312yes—
363C2×C22⋊C8—432no—
364C2×#97—432no—
365#384—431no—
366#384C2×D4311yescharacteristic
367#382—432no—
368#380—861no—
369#380—861no—
370C4×C4⋊C4—861no—
371C4×C4⋊C4—861no—
372#340—432no—
373#340—432no—
374#340—432no—
375#340C2×D4312yes—
376C43#2186911yescharacteristic, complemented, Sylow
377C2×#41—868no—
378C2×#41—1724no—
379C2×#41—1724no—
380C22×Q8—861no—
381C22×D4—431no—
382C22×D4—861no—
383C22×D4—861no—
384C22×D4—861no—
385C22×D4—861no—
386C22×D4C22×D4311yescharacteristic
387C23×C4—21no—
388C23×C4—431no—
389C23×C4—431no—
390C23×C4—431no—
391C23×C4—861no—
392C22×C8—432no—
393C22×C8C22×D4312yes—
394C4⋊Q8—434no—
395C4⋊Q8—862no—
396#127—434no—
397#127—862no—
398#124—868no—
399#121—434no—
400#121—434no—
401#121—868no—
402#121—862no—
403#121—862no—
404C4×D4—22no—
405C4×D4—434no—
406C4×D4—868no—
407C4×D4—862no—
408C4×D4C4×D4314yes—
409#117—1722no—
410#117—1722no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—432no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4C22×D4312yes—
419C2×C4⋊C4C22×D4311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—861no—
422C2×C22⋊C4—861no—
423C2×C22⋊C4—861no—
424C2×C22⋊C4—862no—
425C2×C42—431no—
426C2×C42—431no—
427C2×C42—431no—
428C2×C42—434no—
429C2×C42—434no—
430C2×C42—862no—
431C2×C42C22×D4311yescharacteristic
432C4⋊C8—22no—
433C4⋊C8—862no—
434#102—42no—
435#102—1722no—
436#102—1722no—
437#102—1722no—
438C22⋊C8—864no—
439#97—864no—
440#95—862no—
441#95—862no—
442C24—21no—
443C24—431no—
444C24—861no—
445#41—17232no—
446C2×Q8—864no—
447C2×Q8—1722no—
448C2×Q8—1722no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—432no—
455C2×D4—864no—
456C2×D4—864no—
457C2×D4—864no—
458C2×D4—864no—
459C2×D4—861no—
460C2×D4—861no—
461C2×D4—861no—
462C2×D4—861no—
463C2×D4—861no—
464C2×D4—1722no—
465C2×D4—1722no—
466C2×D4—1722no—
467C2×D4—1722no—
468C2×D4—1722no—
469C2×D4—1722no—
470C2×D4—1722no—
471C2×D4—1722no—
472C2×D4C2×C4×D4312yes—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—431no—
479C22×C4—431no—
480C22×C4—431no—
481C22×C4—434no—
482C22×C4—432no—
483C22×C4—432no—
484C22×C4—434no—
485C22×C4—862no—
486C22×C4—862no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—861no—
491C22×C4—861no—
492C22×C4—864no—
493C22×C4—862no—
494C22×C4—1721no—
495C22×C4—1721no—
496C22×C4—1722no—
497C22×C4—1722no—
498C22×C4D4×D4311yescharacteristic
499C22×C4D4×D4311yescharacteristic
500C22×C4C23×D4311yescharacteristic
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—862no—
505C2×C8—862no—
506C2×C8—862no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—432no—
513C4⋊C4—864no—
514C4⋊C4—861no—
515C4⋊C4—862no—
516C4⋊C4—862no—
517C4⋊C4—1722no—
518C4⋊C4—1722no—
519C4⋊C4C2×C4×D4312yes—
520C22⋊C4—24no—
521C22⋊C4—864no—
522C22⋊C4—864no—
523C22⋊C4—864no—
524C22⋊C4—1724no—
525C42—434no—
526C42—434no—
527C42—432no—
528C42—432no—
529C42—1724no—
530C42C2×C4×D4312yes—
531C42D4×D4312yes—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—432no—
540C23—861no—
541C23—861no—
542C23—861no—
543C23—861no—
544C23—861no—
545C23—861no—
546C23—862no—
547C23—1721no—
548C23—1721no—
549C23—1721no—
550C23—1721no—
551C23—1721no—
552C23—1721no—
553C23C2×D4×D4311yescharacteristic
554Q8—1728no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—862no—
560D4—862no—
561D4—1728no—
562D4—1728no—
563D4—1728no—
564D4—1728no—
565D4—1721no—
566D4—1721no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—432no—
577C2×C4—432no—
578C2×C4—432no—
579C2×C4—861no—
580C2×C4—861no—
581C2×C4—861no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—861no—
585C2×C4—862no—
586C2×C4—862no—
587C2×C4—868no—
588C2×C4—862no—
589C2×C4—862no—
590C2×C4—862no—
591C2×C4—1724no—
592C2×C4—1722no—
593C2×C4—1722no—
594C2×C4—1722no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1728no—
598C2×C4C22⋊C4×D4312yes—
599C2×C4C22⋊C4×D4312yes—
600C2×C4#1306334912yes—
601C2×C4#1306335212yes—
602C2×C4C22×C4×D4311yescharacteristic
603C2×C4C2×D4×D4311yescharacteristic
604C2×C4#1306337011yescharacteristic
605C2×C4#1306337111yescharacteristic
606C2×C4#1306337211yescharacteristic
607C2×C4#1306337311yescharacteristic
608C8—42no—
609C8—1722no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—862no—
618C22—862no—
619C22—862no—
620C22—3441no—
621C22—3441no—
622C22—3441no—
623C22—3441no—
624C22—1722no—
625C22—1722no—
626C22—1722no—
627C22—1721no—
628C22—1721no—
629C22—1721no—
630C22—1721no—
631C22C2×C22⋊C4×D4311yescharacteristic
632C22#1458346511yescharacteristic
633C22#1458347211yescharacteristic
634C22C2×#1306334911yescharacteristic
635C22C2×#1306335211yescharacteristic
636C22#1458369511yescharacteristic
637C22#1458369811yescharacteristic
638C4—22no—
639C4—42no—
640C4—82no—
641C4—862no—
642C4—862no—
643C4—862no—
644C4—3442no—
645C4—1722no—
646C4#1458290214yes—
647C4#1458337911yescharacteristic
648C4#1458338011yescharacteristic
649C4#1458347811yescharacteristic
650C4#1458347911yescharacteristic
651C2—82no—
652C2—3442no—
653C2—1722no—
654C2#1913718411yescharacteristic
655C2C2×#1458290211yescharacteristic
656C2#1914311111yescharacteristic
657C2#1914558811yescharacteristic
658C2#1914558911yescharacteristic
659C2#1914579111yescharacteristic
660C2#1914579211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1802C2
socleC23×C43C22×D4
prime radicalC2×#14583446C2
cosocleC22×C4×C43C24
hypercenter#1802D43
hypocenterC43#21869
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#14582955C2semidirect
3. #19137000C2semidirect
4. #19135566C2semidirect
5. #19135417C2semidirect
6. #19134916C2semidirect
7. #19134914C2semidirect
8. #19134596C2semidirect
9. #19134595C2semidirect
10. #19134592C2semidirect
11. C43×#1802C2semidirect
14. #14583446C4semidirect
331. #1802D43semidirect
376. C43#21869semidirect

Maximal subgroups

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

subgroupindex
C2×#145834462
C2×#145829552
#191370002
#191355662
#191354172
#191349162
#191349142
#191345962
#191345952
#191345922
C43×#18022
#2186943