Finite Group Databasebeta

← #25129606

Subgroups — #25129606

Order 11008.

1504 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583227C211yesmaximal, characteristic, complemented
2C22×#13063246C211yesmaximal, characteristic, complemented
3#19138903C211yesmaximal, characteristic, complemented
4#19135831C212yesmaximal, complemented
5#19135781C212yesmaximal, complemented
6#19135019C211yesmaximal, characteristic, complemented
7#19134888C211yesmaximal, characteristic, complemented
8#19134856C212yesmaximal, complemented
9#19134535C212yesmaximal, complemented
10#19134526C211yesmaximal, characteristic, complemented
11C2×C43×#546C211yesmaximal, characteristic
12C23×C43⋊C8C2211yescharacteristic
13C2×#13063306—21no—
14C22×C4⋊Dic43C2211yescharacteristic
15#14583227—24no—
16C2×#13063256C2212yescomplemented
17C2×#13063249C2212yescomplemented
18C2×#13063249C2212yescomplemented
19C2×#13063246—24no—
20C2×#13063246C2212yes—
21C2×#13063246C2212yes—
22C2×#13063246C2212yes—
23C2×#13063246C2212yes—
24C2×C22⋊(C43⋊C8)C2211yescharacteristic
25C2×C22⋊(C43⋊C8)C2212yescomplemented
26#14582829—21no—
27#14582824—21no—
28#14582797C2212yescomplemented
29#14582760—21no—
30#14582759C2211yescharacteristic
31#14582759C2212yescomplemented
32#14582661C2212yescomplemented
33#14582654C2212yescomplemented
34C22×#41×C43C2211yescharacteristic
35C23×D4×C43C2211yescharacteristic
36C43×#546—24no—
37C43×#546—24no—
38C2×#124×C43C2211yescharacteristic
39C2×#123×C43—21no—
40C2×C22⋊Q8×C43C2211yescharacteristic
41C2×#121×C43—21no—
42C2×#121×C43C2211yescharacteristic
43C2×#120×C43—21no—
44C2×C4×D4×C43C2211yescharacteristic
45C2×C4×D4×C43C2211yescharacteristic
46C22×C22⋊C4×C43—21no—
47C23×Dic43—21no—
48C22×C43⋊C8—22no—
49C22×C43⋊C8—42no—
50C22×C43⋊C8C2312yes—
51C22×C43⋊C8C2312yes—
52#13063306—44no—
53C2×C4⋊Dic43—22no—
54C2×C4⋊Dic43—22no—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43C2312yes—
57C2×C4⋊Dic43C2312yes—
58C2×Dic43⋊C4—22no—
59C2×C22⋊Dic43—22no—
60C2×C22⋊Dic43—21no—
61#13063256—24no—
62#13063249—24no—
63#13063249—42no—
64#13063246—24no—
65#13063246—24no—
66#13063246—24no—
67#13063246—42no—
68C22⋊(C43⋊C8)—22no—
69C22⋊(C43⋊C8)—42no—
70C43⋊#95—22no—
71C43⋊#95—21no—
72C43⋊#95—22no—
73C25×C43—21no—
74C2×#41×C43—22no—
75C2×#41×C43—22no—
76C2×#41×C43—24no—
77C2×#41×C43—42no—
78C22×Q8×C43C2311yescharacteristic
79C22×D4×C43—24no—
80C22×D4×C43—21no—
81C22×D4×C43—21no—
82C22×D4×C43—21no—
83C22×D4×C43—22no—
84C22×D4×C43—42no—
85C22×D4×C43D411yescharacteristic
86C22×D4×C43D414yes—
87C22×D4×C43C2311yescharacteristic
88C22×D4×C43C2311yescharacteristic
89C23×C4×C43—21no—
90C23×C4×C43—21no—
91C23×C4×C43C2311yescharacteristic
92C23×C4×C43C2311yescharacteristic
93#124×C43—22no—
94#124×C43—22no—
95#123×C43—28no—
96C22⋊Q8×C43—22no—
97C22⋊Q8×C43D414yes—
98#121×C43—28no—
99#121×C43—22no—
100#121×C43—22no—
101#120×C43—28no—
102C4×D4×C43—22no—
103C4×D4×C43—22no—
104C4×D4×C43—22no—
105C4×D4×C43—22no—
106C2×C4⋊C4×C43—21no—
107C2×C4⋊C4×C43D411yescharacteristic
108C2×C4⋊C4×C43C2311yescharacteristic
109C2×C22⋊C4×C43—28no—
110C2×C22⋊C4×C43—21no—
111C2×C22⋊C4×C43—21no—
112C2×C22⋊C4×C43—21no—
113C2×C22⋊C4×C43—21no—
114C2×C22⋊C4×C43—21no—
115C2×C22⋊C4×C43D411yescharacteristic
116C2×C22⋊C4×C43D411yescharacteristic
117C2×C42×C43C2311yescharacteristic
118C24×C43—21no—
119C24×C43—21no—
120C24×C43—24no—
121C24×C43—44no—
122C24×C43—42no—
123C24×C43—42no—
124C24×C43—82no—
125C24×C43C2×D411yescharacteristic
126C24×C43C2×D411yescharacteristic
127#41×C43—48no—
128#41×C43—44no—
129#41×C43—44no—
130C2×Q8×C43—21no—
131C2×Q8×C43—22no—
132C2×Q8×C43—22no—
133C2×Q8×C43C2×D412yes—
134C2×D4×C43—24no—
135C2×D4×C43—24no—
136C2×D4×C43—24no—
137C2×D4×C43—21no—
138C2×D4×C43—21no—
139C2×D4×C43—22no—
140C2×D4×C43—22no—
141C2×D4×C43—22no—
142C2×D4×C43—22no—
143C2×D4×C43—28no—
144C2×D4×C43—22no—
145C2×D4×C43—22no—
146C2×D4×C43—44no—
147C2×D4×C43—44no—
148C2×D4×C43—44no—
149C2×D4×C43—48no—
150C2×D4×C43—82no—
151C2×D4×C43C2×D412yes—
152C2×D4×C43C2×D412yes—
153C2×D4×C43#4114yes—
154C22×C4×C43—22no—
155C22×C4×C43—22no—
156C22×C4×C43—24no—
157C22×C4×C43—24no—
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43—22no—
161C22×C4×C43—22no—
162C22×C4×C43—21no—
163C22×C4×C43—21no—
164C22×C4×C43—44no—
165C22×C4×C43—82no—
166C22×C4×C43C2×D412yes—
167C22×C4×C43C2×D412yes—
168C22×C4×C43C2×D411yescharacteristic
169C22×C4×C43C2×D411yescharacteristic
170C22×C4×C43C2×D411yescharacteristic
171C22×C4×C43C2×D411yescharacteristic
172C22×C4×C43C2411yescharacteristic
173C22×Dic43—22no—
174C22×Dic43—22no—
175C22×Dic43—42no—
176C22×Dic43—42no—
177C22×Dic43—42no—
178C4⋊C4×C43—24no—
179C4⋊C4×C43—22no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43C2×D412yes—
182C22⋊C4×C43—24no—
183C22⋊C4×C43—24no—
184C22⋊C4×C43—24no—
185C22⋊C4×C43—24no—
186C22⋊C4×C43—22no—
187C22⋊C4×C43—22no—
188C22⋊C4×C43—24no—
189C42×C43—21no—
190C42×C43—21no—
191C22⋊Dic43—44no—
192C22⋊Dic43—42no—
193C4⋊Dic43—22no—
194C4⋊Dic43—22no—
195C4⋊Dic43—22no—
196C4⋊Dic43—22no—
197C4⋊Dic43—42no—
198Dic43⋊C4—44no—
199C2×C43⋊C8—22no—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C2×C43⋊C8—22no—
203C2×C43⋊C8—22no—
204C2×C43⋊C8—22no—
205C2×C43⋊C8—44no—
206C23×C43—22no—
207C23×C43—22no—
208C23×C43—22no—
209C23×C43—24no—
210C23×C43—22no—
211C23×C43—21no—
212C23×C43—22no—
213C23×C43—21no—
214C23×C43—21no—
215C23×C43—22no—
216C23×C43—48no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43—42no—
220C23×C43—42no—
221C23×C43—41no—
222C23×C43—41no—
223C23×C43—42no—
224C23×C43—48no—
225C23×C43—84no—
226C23×C43—88no—
227C23×C43—88no—
228C23×C43#12012yes—
229C23×C43#13512yes—
230C23×C43AGL(1,ℤ/8)12yes—
231C23×C43C22×D411yescharacteristic
232Q8×C43—24no—
233Q8×C43—41no—
234Q8×C43—41no—
235D4×C43—22no—
236D4×C43—22no—
237D4×C43—22no—
238D4×C43—24no—
239D4×C43—48no—
240D4×C43—48no—
241D4×C43—44no—
242D4×C43—42no—
243D4×C43—84no—
244D4×C43#12314yescomplemented
245C2×C4×C43—22no—
246C2×C4×C43—22no—
247C2×C4×C43—22no—
248C2×C4×C43—24no—
249C2×C4×C43—21no—
250C2×C4×C43—21no—
251C2×C4×C43—22no—
252C2×C4×C43—21no—
253C2×C4×C43—22no—
254C2×C4×C43—22no—
255C2×C4×C43—21no—
256C2×C4×C43—21no—
257C2×C4×C43—22no—
258C2×C4×C43—48no—
259C2×C4×C43—42no—
260C2×C4×C43—42no—
261C2×C4×C43—41no—
262C2×C4×C43—41no—
263C2×C4×C43—42no—
264C2×C4×C43—84no—
265C2×C4×C43#12012yes—
266C2×C4×C43C22×D411yescharacteristic
267C2×C4×C43C22×D411yescharacteristic
268C2×C4×C43C2×#4112yes—
269C2×C4×C43H(5,𝔽2)12yes—
270C2×Dic43—42no—
271C2×Dic43—42no—
272C2×Dic43—42no—
273C2×Dic43—42no—
274C2×Dic43—42no—
275C2×Dic43—42no—
276C2×Dic43—84no—
277C43⋊C8—42no—
278C43⋊C8—42no—
279#24038—431nomaximal, Sylow
280C22×C43—22no—
281C22×C43—22no—
282C22×C43—24no—
283C22×C43—24no—
284C22×C43—24no—
285C22×C43—42no—
286C22×C43—42no—
287C22×C43—44no—
288C22×C43—44no—
289C22×C43—41no—
290C22×C43—42no—
291C22×C43—41no—
292C22×C43—42no—
293C22×C43—41no—
294C22×C43—42no—
295C22×C43—42no—
296C22×C43—42no—
297C22×C43—816no—
298C22×C43—82no—
299C22×C43—82no—
300C22×C43#44914yes—
301C22×C43C2×#12011yescharacteristic
302C22×C43#54612yes—
303C22×C43C2×#13511yescharacteristic
304C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C43#57712yes—
306C4×C43—22no—
307C4×C43—41no—
308C4×C43—42no—
309C4×C43—41no—
310C4×C43—41no—
311C4×C43—42no—
312C4×C43C2×#12311yescharacteristic
313C4×C43#53411yescharacteristic
314C4×C43#54612yes—
315Dic43—82no—
316Dic43—82no—
317C2×#546D4311yescharacteristic
318C2×#483—431no—
319C22×#102—431no—
320#2486—431no—
321#2042—432no—
322#2034—432no—
323#1892—431no—
324#1869—431no—
325#1864—432no—
326#1788—432no—
327#1786—431no—
328C2×C43—24no—
329C2×C43—44no—
330C2×C43—81no—
331C2×C43—82no—
332C2×C43#240511yescharacteristic
333C2×C43C2×#44911yescharacteristic
334C2×C43#300911yescharacteristic
335C2×C43#329612yes—
336C2×C43#330612yes—
337C22×#41C2×D4311yescharacteristic
338C23×D4C2×D4311yescharacteristic
339C23×C8—431no—
340#546—24no—
341#546—24no—
342C2×#124C2×D4311yescharacteristic
343C2×#123—21no—
344C2×#123—861no—
345C2×C22⋊Q8C2×D4311yescharacteristic
346C2×#121—21no—
347C2×#121C2×D4311yescharacteristic
348C2×#120—21no—
349C2×C4×D4C2×D4311yescharacteristic
350C2×C4×D4C2×D4311yescharacteristic
351C22×C4⋊C4—431no—
352C22×C22⋊C4—21no—
353#483—864no—
354C2×#106—432no—
355C2×#103—432no—
356C2×#103—432no—
357C2×#102—432no—
358C2×#102—432no—
359C2×#102—432no—
360C2×#102—432no—
361C2×#102—864no—
362C2×C22⋊C8—431no—
363C2×C22⋊C8—432no—
364#398—861no—
365#397—861no—
366#389—432no—
367#381—861no—
368#380—431no—
369#380—432no—
370#340—432no—
371#336—432no—
372C43#2403811yescharacteristic, complemented, Sylow
373C25—21no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—24no—
377C2×#41—42no—
378C22×Q8C22×D4311yescharacteristic
379C22×D4—24no—
380C22×D4—21no—
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—22no—
384C22×D4—42no—
385C22×D4C22⋊D4311yescharacteristic
386C22×D4C22⋊D4314yes—
387C22×D4C22×D4311yescharacteristic
388C22×D4C22×D4311yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—861no—
392C23×C4C22×D4311yescharacteristic
393C23×C4C22×D4311yescharacteristic
394C22×C8—432no—
395C22×C8—432no—
396C22×C8—862no—
397C22×C8—1722no—
398#124—22no—
399#124—22no—
400#123—28no—
401#123—1724no—
402C22⋊Q8—22no—
403C22⋊Q8C22⋊D4314yes—
404#121—28no—
405#121—22no—
406#121—22no—
407#120—28no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—432no—
414C2×C4⋊C4—432no—
415C2×C4⋊C4—862no—
416C2×C4⋊C4—862no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—862no—
419C2×C4⋊C4C22⋊D4311yescharacteristic
420C2×C4⋊C4C22×D4311yescharacteristic
421C2×C22⋊C4—28no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—861no—
428C2×C22⋊C4—862no—
429C2×C22⋊C4C22⋊D4311yescharacteristic
430C2×C22⋊C4C22⋊D4311yescharacteristic
431C2×C42C22×D4311yescharacteristic
432#106—864no—
433#103—864no—
434#103—1722no—
435#102—864no—
436#102—864no—
437#102—864no—
438#102—1722no—
439C22⋊C8—862no—
440C22⋊C8—1722no—
441#95—862no—
442#95—861no—
443#95—862no—
444C24—21no—
445C24—21no—
446C24—24no—
447C24—44no—
448C24—42no—
449C24—42no—
450C24—82no—
451C24C2×C22⋊D4311yescharacteristic
452C24C2×C22⋊D4311yescharacteristic
453#41—48no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8C2×C22⋊D4312yes—
460C2×D4—28no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—44no—
473C2×D4—48no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—82no—
477C2×D4#1062190014yes—
478C2×D4C2×C22⋊D4312yes—
479C2×D4C2×C22⋊D4312yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—862no—
493C22×C4—862no—
494C22×C4—1722no—
495C22×C4—1722no—
496C22×C4—1722no—
497C22×C4C2×C22⋊D4311yescharacteristic
498C22×C4C2×C22⋊D4311yescharacteristic
499C22×C4C2×C22⋊D4311yescharacteristic
500C22×C4C2×C22⋊D4311yescharacteristic
501C22×C4C2×C22⋊D4312yes—
502C22×C4C2×C22⋊D4312yes—
503C22×C4C23×D4311yescharacteristic
504C2×C8—862no—
505C2×C8—862no—
506C2×C8—862no—
507C2×C8—862no—
508C2×C8—862no—
509C2×C8—862no—
510C2×C8—1724no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—862no—
515C4⋊C4—862no—
516C4⋊C4—862no—
517C4⋊C4—862no—
518C4⋊C4—1724no—
519C4⋊C4—1722no—
520C4⋊C4C2×C22⋊D4312yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—1724no—
529C22⋊C4—1722no—
530C42—21no—
531C42—21no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—48no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—48no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—88no—
553C23—88no—
554C23#1306329412yes—
555C23#1306334512yes—
556C23#1306335012yes—
557C23C22×C22⋊D4311yescharacteristic
558Q8—24no—
559Q8—41no—
560Q8—41no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—24no—
565D4—48no—
566D4—48no—
567D4—44no—
568D4—42no—
569D4—84no—
570D4#1306330614yescomplemented
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—3444no—
592C2×C4—1722no—
593C2×C4—1722no—
594C2×C4—1722no—
595C2×C4—1722no—
596C2×C4—1722no—
597C2×C4—1722no—
598C2×C4#1306329412yes—
599C2×C4C22×C22⋊D4311yescharacteristic
600C2×C4C22×C22⋊D4311yescharacteristic
601C2×C4C2×#1062190012yes—
602C2×C4#1306337012yes—
603C8—1722no—
604C8—1722no—
605C22—24no—
606C22—22no—
607C22—24no—
608C22—22no—
609C22—24no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—816no—
623C22—82no—
624C22—82no—
625C22#1458304414yes—
626C22C2×#1306329411yescharacteristic
627C22#1458354312yes—
628C22C2×#1306334511yescharacteristic
629C22C2×#1306335011yescharacteristic
630C22#1458371012yes—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—3442no—
638C4—3442no—
639C4C2×#1306330611yescharacteristic
640C4#1458345811yescharacteristic
641C4#1458354312yes—
642C2—24no—
643C2—44no—
644C2—81no—
645C2—82no—
646C2#1913791811yescharacteristic
647C2C2×#1458304411yescharacteristic
648C2#1914387211yescharacteristic
649C2#1914643012yes—
650C2#1914655612yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
Fitting, prime radicalC2×C43×#546C2
socleC23×C43C22×D4
hypercenterC2×#546D43
hypocenterC43#24038
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#14583227C2semidirect
2. C22×#13063246C2semidirect
3. #19138903C2semidirect
4. #19135831C2semidirect
5. #19135781C2semidirect
6. #19135019C2semidirect
7. #19134888C2semidirect
8. #19134856C2semidirect
9. #19134535C2semidirect
10. #19134526C2semidirect
16. C2×#13063256C22semidirect
17. C2×#13063249C22semidirect
18. C2×#13063249C22semidirect
25. C2×C22⋊(C43⋊C8)C22semidirect
28. #14582797C22semidirect
31. #14582759C22semidirect
32. #14582661C22semidirect
33. #14582654C22semidirect
244. D4×C43#123semidirect
372. C43#24038semidirect
570. D4#13063306semidirect

Maximal subgroups

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

subgroupindex
C2×#145832272
C22×#130632462
#191389032
#191358312
#191357812
#191350192
#191348882
#191348562
#191345352
#191345262
C2×C43×#5462
#2403843