Finite Group Databasebeta

← #25129306

Subgroups — #25129306

Order 11008.

1428 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583237C212yesmaximal, complemented
2#19136978C211yesmaximal, characteristic
3#19135795C212yesmaximal, complemented
4#19135788C212yesmaximal, complemented
5#19134908C212yesmaximal, complemented
6#19134788C211yesmaximal, characteristic, complemented
7#19134783C211yesmaximal, characteristic, complemented
8#19134530C211yesmaximal, characteristic, complemented
9#19134530C211yesmaximal, characteristic, complemented
10#19134505C211yesmaximal, characteristic
11C22×C22⋊Q8×C43C211yesmaximal, characteristic
12C2×#13063284C2211yescharacteristic
13C2×#13063284C2211yescharacteristic
14#14583237—28no—
15C2×#13063258C2212yescomplemented
16C2×#13063256C2212yescomplemented
17C2×#13063249C2212yescomplemented
18C2×#13063249C2212yescomplemented
19C2×#13063249C2212yescomplemented
20C2×#13063249C2212yescomplemented
21C2×C22⋊(C43⋊C8)C2212yes—
22C2×C22⋊(C43⋊C8)C2211yescharacteristic
23C2×C22⋊(C43⋊C8)C2211yescharacteristic
24C2×C22⋊(C43⋊C8)C2212yes—
25#14582762—21no—
26#14582759C2211yescharacteristic
27#14582759C2211yescharacteristic
28#14582759C2211yescharacteristic
29#14582759C2211yescharacteristic
30#14582755—21no—
31C4×C22⋊Dic43—21no—
32C4×C22⋊Dic43—21no—
33#14582661C2212yescomplemented
34#14582661C2212yescomplemented
35C23×Q8×C43C2211yescharacteristic
36C24×C4×C43C2211yescharacteristic
37C2×C22⋊Q8×C43—21no—
38C2×C22⋊Q8×C43—21no—
39C2×C22⋊Q8×C43—28no—
40C2×C22⋊Q8×C43C2212yes—
41C2×C22⋊Q8×C43C2212yes—
42C22×C4⋊C4×C43—21no—
43C22×C4⋊C4×C43C2211yescharacteristic
44C22×C22⋊C4×C43—21no—
45C22×C43⋊C8C2312yes—
46C22×C43⋊C8C2312yes—
47#13063284—24no—
48#13063284—42no—
49C2×C4⋊Dic43C2311yescharacteristic
50C2×C4⋊Dic43C2311yescharacteristic
51C2×C4⋊Dic43C2311yescharacteristic
52C2×C4⋊Dic43C2311yescharacteristic
53C2×C22⋊Dic43—21no—
54C2×C22⋊Dic43—21no—
55C2×C22⋊Dic43—21no—
56C2×C22⋊Dic43—21no—
57C2×C22⋊Dic43—21no—
58C2×C22⋊Dic43—21no—
59C2×C4×Dic43—21no—
60C2×C4×Dic43—21no—
61#13063258—24no—
62#13063256—24no—
63#13063249—24no—
64#13063249—24no—
65#13063249—42no—
66#13063249—42no—
67C22⋊(C43⋊C8)—24no—
68C22⋊(C43⋊C8)—22no—
69C22⋊(C43⋊C8)—22no—
70C22⋊(C43⋊C8)—42no—
71C43⋊#95—21no—
72C43⋊#95—21no—
73C43⋊#95—21no—
74C43⋊#95—21no—
75C25×C43D411yescharacteristic
76C22×Q8×C43—24no—
77C22×Q8×C43—21no—
78C22×Q8×C43—22no—
79C22×Q8×C43—22no—
80C22×Q8×C43—42no—
81C22×Q8×C43C2312yes—
82C23×C4×C43—21no—
83C23×C4×C43—21no—
84C23×C4×C43—24no—
85C23×C4×C43—22no—
86C23×C4×C43—22no—
87C23×C4×C43—42no—
88C23×C4×C43D411yescharacteristic
89C23×C4×C43D411yescharacteristic
90C23×C4×C43D411yescharacteristic
91C23×C4×C43C2311yescharacteristic
92C23×C4×C43C2311yescharacteristic
93C23×C4×C43C2311yescharacteristic
94C22⋊Q8×C43—28no—
95C22⋊Q8×C43—28no—
96C22⋊Q8×C43—24no—
97C22⋊Q8×C43—24no—
98C22⋊Q8×C43—24no—
99C22⋊Q8×C43D418yes—
100C2×C4⋊C4×C43—28no—
101C2×C4⋊C4×C43—22no—
102C2×C4⋊C4×C43—22no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—24no—
105C2×C4⋊C4×C43C2312yes—
106C2×C22⋊C4×C43—28no—
107C2×C22⋊C4×C43—22no—
108C2×C22⋊C4×C43—22no—
109C24×C43—22no—
110C24×C43—22no—
111C24×C43—22no—
112C24×C43—22no—
113C24×C43—22no—
114C24×C43—41no—
115C24×C43—41no—
116C24×C43C2×D411yescharacteristic
117C24×C43C2×D411yescharacteristic
118C24×C43C2×D411yescharacteristic
119C2×Q8×C43—24no—
120C2×Q8×C43—22no—
121C2×Q8×C43—22no—
122C2×Q8×C43—22no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43—48no—
126C2×Q8×C43—48no—
127C2×Q8×C43—42no—
128C2×Q8×C43—42no—
129C2×Q8×C43C2×D414yes—
130C22×C4×C43—22no—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—22no—
134C22×C4×C43—22no—
135C22×C4×C43—24no—
136C22×C4×C43—24no—
137C22×C4×C43—22no—
138C22×C4×C43—22no—
139C22×C4×C43—22no—
140C22×C4×C43—22no—
141C22×C4×C43—22no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43—22no—
145C22×C4×C43—22no—
146C22×C4×C43—22no—
147C22×C4×C43—22no—
148C22×C4×C43—22no—
149C22×C4×C43—41no—
150C22×C4×C43—41no—
151C22×C4×C43—41no—
152C22×C4×C43—42no—
153C22×C4×C43—42no—
154C22×C4×C43—48no—
155C22×C4×C43—48no—
156C22×C4×C43—42no—
157C22×C4×C43—42no—
158C22×C4×C43C2×D414yes—
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2×D411yescharacteristic
161C22×C4×C43C2×D411yescharacteristic
162C22×C4×C43#4112yes—
163C22×C4×C43#4112yes—
164C22×C4×C43C2411yescharacteristic
165C22×Dic43—21no—
166C22×Dic43—21no—
167C22×Dic43—21no—
168C22×Dic43—21no—
169C4⋊C4×C43—24no—
170C4⋊C4×C43—28no—
171C4⋊C4×C43—28no—
172C4⋊C4×C43—22no—
173C4⋊C4×C43C2×D414yes—
174C22⋊C4×C43—28no—
175C22⋊C4×C43—28no—
176C22⋊Dic43—42no—
177C22⋊Dic43—42no—
178C22⋊Dic43—42no—
179C22⋊Dic43—42no—
180C22⋊Dic43—42no—
181C22⋊Dic43—42no—
182C4⋊Dic43—22no—
183C4⋊Dic43—22no—
184C4⋊Dic43—22no—
185C4⋊Dic43—22no—
186C4×Dic43—42no—
187C4×Dic43—42no—
188C2×C43⋊C8—22no—
189C2×C43⋊C8—22no—
190C2×C43⋊C8—22no—
191C2×C43⋊C8—22no—
192C2×C43⋊C8—22no—
193C2×C43⋊C8—22no—
194C23×C43—21no—
195C23×C43—21no—
196C23×C43—21no—
197C23×C43—21no—
198C23×C43—28no—
199C23×C43—21no—
200C23×C43—21no—
201C23×C43—21no—
202C23×C43—21no—
203C23×C43—21no—
204C23×C43—21no—
205C23×C43—21no—
206C23×C43—21no—
207C23×C43—21no—
208C23×C43—21no—
209C23×C43—21no—
210C23×C43—21no—
211C23×C43—44no—
212C23×C43—44no—
213C23×C43—44no—
214C23×C43—44no—
215C23×C43—44no—
216C23×C43—44no—
217C23×C43#12012yes—
218C23×C43#12312yes—
219C23×C43#13512yes—
220C23×C43#13712yes—
221C23×C43#13712yes—
222C23×C43C22×D411yescharacteristic
223Q8×C43—24no—
224Q8×C43—24no—
225Q8×C43—48no—
226Q8×C43—42no—
227Q8×C43—42no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—24no—
233C2×C4×C43—24no—
234C2×C4×C43—24no—
235C2×C4×C43—24no—
236C2×C4×C43—24no—
237C2×C4×C43—24no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—44no—
242C2×C4×C43—44no—
243C2×C4×C43—44no—
244C2×C4×C43—48no—
245C2×C4×C43—48no—
246C2×C4×C43—42no—
247C2×C4×C43—42no—
248C2×C4×C43—42no—
249C2×C4×C43—42no—
250C2×C4×C43—42no—
251C2×C4×C43—42no—
252C2×C4×C43#12012yes—
253C2×C4×C43#12312yes—
254C2×C4×C43C22×D412yes—
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43H(5,𝔽2)12yes—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C43⋊C8—42no—
271C43⋊C8—42no—
272#24010—431nomaximal, Sylow
273C22×C43—22no—
274C22×C43—22no—
275C22×C43—22no—
276C22×C43—22no—
277C22×C43—22no—
278C22×C43—22no—
279C22×C43—22no—
280C22×C43—22no—
281C22×C43—22no—
282C22×C43—22no—
283C22×C43—22no—
284C22×C43—22no—
285C22×C43—416no—
286C22×C43—41no—
287C22×C43—41no—
288C22×C43—41no—
289C22×C43—41no—
290C22×C43—41no—
291C22×C43—41no—
292C22×C43—41no—
293C22×C43—41no—
294C22×C43#48514yescomplemented
295C22×C43C2×#12011yescharacteristic
296C22×C43C2×#12311yescharacteristic
297C22×C43#53411yescharacteristic
298C22×C43C2×#13511yescharacteristic
299C22×C43C2×#13711yescharacteristic
300C22×C43C2×#13711yescharacteristic
301C22×C43#57511yescharacteristic
302C4×C43—22no—
303C4×C43—22no—
304C4×C43—41no—
305C4×C43—44no—
306C4×C43—44no—
307C4×C43#54612yes—
308C4×C43#54612yes—
309Dic43—81no—
310Dic43—81no—
311Dic43—81no—
312Dic43—81no—
313C22×C22⋊Q8D4311yescharacteristic
314C2×#485—432no—
315#2280—431no—
316#2036—432no—
317#2035—432no—
318#1873—432no—
319#1853—431no—
320#1852—431no—
321#1787—431no—
322#1787—431no—
323#1779—431no—
324C2×C43—24no—
325C2×C43—42no—
326C2×C43—42no—
327C2×C43#240511yescharacteristic
328C2×C43#300211yescharacteristic
329C2×C43#300411yescharacteristic
330C2×C43C2×#48511yescharacteristic
331C2×C43#309511yescharacteristic
332C2×C43#319011yescharacteristic
333C2×C43#319611yescharacteristic
334C23×Q8C2×D4311yescharacteristic
335C24×C4C2×D4311yescharacteristic
336C2×C22⋊Q8—21no—
337C2×C22⋊Q8—21no—
338C2×C22⋊Q8—28no—
339C2×C22⋊Q8C2×D4312yes—
340C2×C22⋊Q8C2×D4312yes—
341C2×#117—431no—
342C2×#117—431no—
343C22×C4⋊C4—21no—
344C22×C4⋊C4C2×D4311yescharacteristic
345C22×C22⋊C4—21no—
346#485—868no—
347C2×#107—432no—
348C2×#106—432no—
349C2×#103—432no—
350C2×#103—432no—
351C2×#103—432no—
352C2×#103—432no—
353C2×C22⋊C8—432no—
354C2×C22⋊C8—431no—
355C2×C22⋊C8—431no—
356C2×C22⋊C8—432no—
357#381—861no—
358#380—431no—
359#380—431no—
360#380—431no—
361#380—431no—
362#379—861no—
363C4×C22⋊C4—861no—
364C4×C22⋊C4—861no—
365#340—432no—
366#340—432no—
367C43#2401011yescharacteristic, complemented, Sylow
368C25C22⋊D4311yescharacteristic
369C22×Q8—24no—
370C22×Q8—21no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—42no—
374C22×Q8C22×D4312yes—
375C23×C4—21no—
376C23×C4—21no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—42no—
381C23×C4C22⋊D4311yescharacteristic
382C23×C4C22⋊D4311yescharacteristic
383C23×C4C22⋊D4311yescharacteristic
384C23×C4C22×D4311yescharacteristic
385C23×C4C22×D4311yescharacteristic
386C23×C4C22×D4311yescharacteristic
387C22×C8—432no—
388C22×C8—432no—
389C22⋊Q8—28no—
390C22⋊Q8—28no—
391C22⋊Q8—24no—
392C22⋊Q8—24no—
393C22⋊Q8—24no—
394C22⋊Q8C22⋊D4318yes—
395#117—864no—
396#117—1722no—
397C2×C4⋊C4—28no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—431no—
403C2×C4⋊C4—431no—
404C2×C4⋊C4—431no—
405C2×C4⋊C4—431no—
406C2×C4⋊C4C22×D4312yes—
407C2×C22⋊C4—28no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—861no—
411C2×C22⋊C4—861no—
412C2×C22⋊C4—861no—
413C2×C22⋊C4—861no—
414C2×C22⋊C4—861no—
415C2×C22⋊C4—861no—
416C2×C42—861no—
417C2×C42—861no—
418#107—864no—
419#106—864no—
420#103—864no—
421#103—864no—
422#103—1722no—
423#103—1722no—
424C22⋊C8—862no—
425C22⋊C8—862no—
426C22⋊C8—864no—
427C22⋊C8—1722no—
428#95—861no—
429#95—861no—
430#95—861no—
431#95—861no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—41no—
438C24—41no—
439C24C2×C22⋊D4311yescharacteristic
440C24C2×C22⋊D4311yescharacteristic
441C24C2×C22⋊D4311yescharacteristic
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—48no—
449C2×Q8—48no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8C2×C22⋊D4314yes—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—861no—
482C22×C4—861no—
483C22×C4—861no—
484C22×C4—861no—
485C22×C4#1062190012yes—
486C22×C4#1062190012yes—
487C22×C4C2×C22⋊D4311yescharacteristic
488C22×C4C2×C22⋊D4311yescharacteristic
489C22×C4C2×C22⋊D4314yes—
490C22×C4C2×C22⋊D4311yescharacteristic
491C22×C4C23×D4311yescharacteristic
492C2×C8—862no—
493C2×C8—862no—
494C2×C8—862no—
495C2×C8—862no—
496C2×C8—862no—
497C2×C8—862no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—862no—
503C4⋊C4—862no—
504C4⋊C4—862no—
505C4⋊C4—862no—
506C4⋊C4C2×C22⋊D4314yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—1722no—
510C22⋊C4—1722no—
511C22⋊C4—1722no—
512C22⋊C4—1722no—
513C22⋊C4—1722no—
514C22⋊C4—1722no—
515C42—1722no—
516C42—1722no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—28no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23#1306329412yes—
541C23#1306330612yes—
542C23#1306334512yes—
543C23#1306335612yes—
544C23#1306335612yes—
545C23C22×C22⋊D4311yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—48no—
549Q8—42no—
550Q8—42no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—1721no—
576C2×C4—1721no—
577C2×C4—1721no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4#1306329412yes—
588C2×C4#1306330612yes—
589C2×C4C22×C22⋊D4312yes—
590C2×C4C2×#1062190011yescharacteristic
591C2×C4C2×#1062190011yescharacteristic
592C2×C4#1306337012yes—
593C8—1722no—
594C8—1722no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22#1458323714yescomplemented
617C22C2×#1306329411yescharacteristic
618C22C2×#1306330611yescharacteristic
619C22#1458345811yescharacteristic
620C22C2×#1306334511yescharacteristic
621C22C2×#1306335611yescharacteristic
622C22C2×#1306335611yescharacteristic
623C22#1458369711yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—44no—
628C4—44no—
629C4—3441no—
630C4—3441no—
631C4—3441no—
632C4—3441no—
633C4#1458354312yes—
634C4#1458354312yes—
635C2—24no—
636C2—42no—
637C2—42no—
638C2#1913791811yescharacteristic
639C2#1914381311yescharacteristic
640C2#1914382911yescharacteristic
641C2C2×#1458323711yescharacteristic
642C2#1914453911yescharacteristic
643C2#1914537611yescharacteristic
644C2#1914542511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC22×C22⋊Q8×C43C2
socleC23×C43C22×D4
prime radicalC25×C43D4
hypercenterC22×C22⋊Q8D43
hypocenterC43#24010
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14583237C2semidirect
3. #19135795C2semidirect
4. #19135788C2semidirect
5. #19134908C2semidirect
6. #19134788C2semidirect
7. #19134783C2semidirect
8. #19134530C2semidirect
9. #19134530C2semidirect
15. C2×#13063258C22semidirect
16. C2×#13063256C22semidirect
17. C2×#13063249C22semidirect
18. C2×#13063249C22semidirect
19. C2×#13063249C22semidirect
20. C2×#13063249C22semidirect
33. #14582661C22semidirect
34. #14582661C22semidirect
294. C22×C43#485semidirect
367. C43#24010semidirect
616. C22#14583237semidirect

Maximal subgroups

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

subgroupindex
C2×#145832372
#191369782
#191357952
#191357882
#191349082
#191347882
#191347832
#191345302
#191345052
C22×C22⋊Q8×C432
#2401043