Finite Group Databasebeta

← #25105673

Subgroups — #25105673

Order 11008.

1466 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D43C211yesmaximal, characteristic
2C2×#14582862C211yesmaximal, characteristic
3#19136977C211yesmaximal, characteristic
4#19135029C211yesmaximal, characteristic, complemented
5#19135028C211yesmaximal, characteristic, complemented
6#19135026C211yesmaximal, characteristic, complemented
7#19134906C212yesmaximal, complemented
8#19134904C212yesmaximal, complemented
9#19134684C212yesmaximal, complemented
10#19134528C212yesmaximal, complemented
11C43×#1894C211yesmaximal, characteristic, complemented
12C22×Q8×D43C2211yescharacteristic
13C23×C4×D43C2211yescharacteristic
14C22×#10621877C2211yescharacteristic
15C2×#13063305C2211yescharacteristic
16C2×#13063304C2211yescharacteristic
17C2×#13063303—21no—
18C2×#13063302—21no—
19C2×#13063301C2211yescharacteristic
20C22⋊Q8×D43—24no—
21C22⋊Q8×D43C418yescomplemented
22C2×#13063283C2212yes—
23C2×C4⋊C4×D43—21no—
24C2×C4⋊C4×D43C2211yescharacteristic
25C2×C22⋊C4×D43—21no—
26C2×#13063250C2212yescomplemented
27C2×#13063249C2212yescomplemented
28C2×#13063248C2212yescomplemented
29#14582862—24no—
30C2×(C2×D43)⋊C8C2212yes—
31C2×(C2×D43)⋊C8C2212yes—
32#14582761—21no—
33#14582759C2211yescharacteristic
34#14582756C2212yes—
35#14582754—21no—
36C22⋊C4×Dic43—21no—
37#14582746—21no—
38#14582662C2212yescomplemented
39#14582661C2212yescomplemented
40#14582661C2212yescomplemented
41C22×M16×C43C2211yescharacteristic
42C2×C22⋊Q8×C43C2211yescharacteristic
43C2×#103×C43C2212yescomplemented
44C43×#380C2211yescharacteristic
45C43×#340C2212yescomplemented
46C24×D43D411yescharacteristic
47C22×Dic86—21no—
48C22×Dic86C2311yescharacteristic
49C2×Q8×D43—22no—
50C2×Q8×D43—42no—
51C2×Q8×D43—42no—
52C2×Q8×D43C2×C414yes—
53C23×Dic43D411yescharacteristic
54C22×C4×D43—21no—
55C22×C4×D43—21no—
56C22×C4×D43—42no—
57C22×C4×D43—42no—
58C22×C4×D43C2×C414yes—
59C22×C4×D43D411yescharacteristic
60C22×C4×D43D411yescharacteristic
61C22×C4×D43D414yes—
62C22×C4×D43C2311yescharacteristic
63C22×C4×D43C2311yescharacteristic
64C2×#10621877—22no—
65C2×#10621877—42no—
66C22×C43⋊C8C2312yes—
67#13063305—22no—
68#13063305C2×C414yes—
69#13063304—22no—
70#13063304C2×C414yes—
71#13063303—28no—
72#13063302—28no—
73#13063301—22no—
74#13063301C2×C414yes—
75#13063283—42no—
76#13063283—42no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43C2312yes—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C4⋊C4×D43—28no—
84C4⋊C4×D43—22no—
85C4⋊C4×D43C2×C414yes—
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×(C2×D43)⋊C4—22no—
91C2×(C2×D43)⋊C4—22no—
92C22⋊C4×D43—28no—
93C2×C4×Dic43—22no—
94#13063250—42no—
95#13063249—42no—
96#13063248—42no—
97(C2×D43)⋊C8—24no—
98(C2×D43)⋊C8—24no—
99C43⋊#95—21no—
100C43⋊#95—22no—
101C22×Q8×C43C2311yescharacteristic
102C23×C4×C43C2311yescharacteristic
103C2×M16×C43—22no—
104C2×M16×C43—42no—
105C22×C8×C43C2312yes—
106C22⋊Q8×C43—22no—
107C22⋊Q8×C43C2×C414yes—
108C2×C4⋊C4×C43—21no—
109C2×C4⋊C4×C43C2312yes—
110C2×C4⋊C4×C43C2311yescharacteristic
111C2×C22⋊C4×C43—21no—
112C2×C22⋊C4×C43—21no—
113#103×C43—42no—
114#95×C43—21no—
115C24×C43C2×D411yescharacteristic
116C23×D43—24no—
117C23×D43—42no—
118C23×D43—41no—
119C23×D43—41no—
120C23×D43C22⋊C414yes—
121C23×D43C2×D411yescharacteristic
122C23×D43C2×D411yescharacteristic
123C2×Q8×C43—21no—
124C2×Q8×C43—22no—
125C2×Q8×C43—22no—
126C2×Q8×C43C22×C412yes—
127C22×C4×C43—22no—
128C22×C4×C43—22no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—22no—
132C22×C4×C43C22×C412yes—
133C22×C4×C43C2×D412yes—
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2411yescharacteristic
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—22no—
139C22×Dic43—42no—
140C22×Dic43C22⋊C414yes—
141C22×Dic43C2×D411yescharacteristic
142C22×Dic43C2×D411yescharacteristic
143Q8×D43—48no—
144Q8×D43—44no—
145Q8×D43—44no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—22no—
149C2×C4×D43—22no—
150C2×C4×D43—24no—
151C2×C4×D43—24no—
152C2×C4×D43—42no—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×C4×D43—42no—
156C2×C4×D43—44no—
157C2×C4×D43—48no—
158C2×C4×D43—44no—
159C2×C4×D43—44no—
160C2×C4×D43C22⋊C414yes—
161C2×C4×D43C22⋊C414yes—
162C2×C4×D43C22×C412yes—
163C2×C4×D43C22×C412yes—
164C2×C4×D43C2×D412yes—
165C2×C4×D43C2×D412yes—
166C2×Dic86—22no—
167C2×Dic86—22no—
168C2×Dic86—21no—
169C2×Dic86—24no—
170C2×Dic86—44no—
171C2×Dic86C22×C412yes—
172M16×C43—44no—
173C2×C8×C43—22no—
174C2×C8×C43—22no—
175C2×C8×C43—22no—
176C4⋊C4×C43—24no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—22no—
179C4⋊C4×C43—22no—
180C4⋊C4×C43C22×C412yes—
181C22⋊C4×C43—24no—
182C22⋊C4×C43—42no—
183C22⋊Dic43—24no—
184C22⋊Dic43—42no—
185(C2×D43)⋊C4—24no—
186(C2×D43)⋊C4—24no—
187(C2×D43)⋊C4—44no—
188(C2×D43)⋊C4—44no—
189C4⋊Dic43—21no—
190C4⋊Dic43—24no—
191C4⋊Dic43—22no—
192C4⋊Dic43—22no—
193C4⋊Dic43C22×C412yes—
194Dic43⋊C4—24no—
195Dic43⋊C4—24no—
196Dic43⋊C4—24no—
197C4×Dic43—44no—
198#10621877—44no—
199C2×C43⋊C8—22no—
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C23×C43—22no—
203C23×C43—22no—
204C23×C43—21no—
205C23×C43—21no—
206C23×C43C2×C22⋊C412yes—
207C23×C43C22×D411yescharacteristic
208C22×D43—22no—
209C22×D43—21no—
210C22×D43—21no—
211C22×D43—22no—
212C22×D43—22no—
213C22×D43—21no—
214C22×D43—21no—
215C22×D43—28no—
216C22×D43—44no—
217C22×D43—44no—
218C22×D43—48no—
219C22×D43—48no—
220C22×D43C2×C22⋊C412yes—
221C22×D43C2×C22⋊C412yes—
222C22×D43#13714yes—
223Q8×C43—24no—
224Q8×C43—41no—
225Q8×C43—41no—
226C2×C4×C43—22no—
227C2×C4×C43—24no—
228C2×C4×C43—22no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—42no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43C2×C22⋊C412yes—
239C2×C4×C43C23×C411yescharacteristic
240C2×C4×C43C22×D411yescharacteristic
241C2×C4×C43H(5,𝔽2)12yes—
242C2×C4×C43H(5,𝔽2)12yes—
243C2×Dic43—22no—
244C2×Dic43—22no—
245C2×Dic43—22no—
246C2×Dic43—22no—
247C2×Dic43—44no—
248C2×Dic43—42no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—42no—
252C2×Dic43—42no—
253C2×Dic43—42no—
254C2×Dic43C2×C22⋊C412yes—
255C2×Dic43C2×C22⋊C412yes—
256C4×D43—24no—
257C4×D43—24no—
258C4×D43—22no—
259C4×D43—22no—
260C4×D43—22no—
261C4×D43—22no—
262C4×D43—44no—
263C4×D43—44no—
264C4×D43—48no—
265C4×D43—48no—
266Dic86—24no—
267Dic86—48no—
268Dic86—41no—
269Dic86—41no—
270C8×C43—42no—
271C43⋊C8—42no—
272#21768—431nomaximal, Sylow
273C22×C43—22no—
274C22×C43—22no—
275C22×C43—22no—
276C22×C43—42no—
277C22×C43—42no—
278C22×C43C22×C22⋊C411yescharacteristic
279C22×C43#53412yes—
280C22×C43C2×#13712yes—
281C22×C43#57511yescharacteristic
282C22×C43#57511yescharacteristic
283C2×D43—22no—
284C2×D43—22no—
285C2×D43—24no—
286C2×D43—24no—
287C2×D43—22no—
288C2×D43—22no—
289C2×D43—22no—
290C2×D43—416no—
291C2×D43—42no—
292C2×D43—42no—
293C2×D43#41914yes—
294C4×C43—22no—
295C4×C43—42no—
296C4×C43—42no—
297C4×C43—82no—
298C4×C43#51812yes—
299C4×C43#53411yescharacteristic
300C4×C43#53411yescharacteristic
301Dic43—22no—
302Dic43—22no—
303Dic43—42no—
304Dic43—42no—
305Dic43—82no—
306C22×C22⋊Q8—431no—
307C2×#407—431no—
308#2280—431no—
309#1894—431no—
310#1894—431no—
311#1894—431no—
312#1894D4311yescharacteristic, complemented
313#1873—432no—
314#1873—432no—
315#1824—432no—
316#1787—432no—
317C2×C43—42no—
318C2×C43#230811yescharacteristic
319C2×C43C2×#41911yescharacteristic
320C2×C43#289611yescharacteristic
321C2×C43#319612yes—
322C2×C43#319612yes—
323D43—24no—
324D43—42no—
325C23×Q8—431no—
326C24×C4—431no—
327C22×M16—431no—
328C22×M16C2×D4311yescharacteristic
329C2×C22⋊Q8—431no—
330C2×C22⋊Q8—431no—
331C2×C22⋊Q8—438no—
332C2×C22⋊Q8—431no—
333C2×C22⋊Q8—861no—
334C2×C22⋊Q8—861no—
335C2×C22⋊Q8—864no—
336C2×C22⋊Q8C2×D4311yescharacteristic
337C2×#117—432no—
338C22×C4⋊C4—431no—
339C22×C4⋊C4—861no—
340C22×C22⋊C4—861no—
341C2×#103—432no—
342C2×#103—432no—
343C2×#103—432no—
344C2×#103C2×D4312yescomplemented
345#407—864no—
346C2×C22⋊C8—432no—
347C2×C22⋊C8—432no—
348#381—861no—
349#380—431no—
350#380—432no—
351#380C2×D4311yescharacteristic
352#379—861no—
353C4×C22⋊C4—861no—
354C4×C22⋊C4—861no—
355#340—432no—
356#340—432no—
357#340—432no—
358#340C2×D4312yescomplemented
359C43#2176811yescharacteristic, complemented, Sylow
360C25—431no—
361C22×Q8—434no—
362C22×Q8—431no—
363C22×Q8—861no—
364C22×Q8—862no—
365C22×Q8—1722no—
366C22×Q8—1722no—
367C22×Q8C22×D4311yescharacteristic
368C23×C4—431no—
369C23×C4—431no—
370C23×C4—431no—
371C23×C4—434no—
372C23×C4—431no—
373C23×C4—431no—
374C23×C4—434no—
375C23×C4—861no—
376C23×C4—861no—
377C23×C4—1722no—
378C23×C4—1722no—
379C23×C4C22×D4311yescharacteristic
380C2×M16—22no—
381C2×M16—42no—
382C2×M16—862no—
383C2×M16—1722no—
384C22×C8—432no—
385C22×C8C22×D4312yes—
386C22⋊Q8—22no—
387C22⋊Q8—434no—
388C22⋊Q8—434no—
389C22⋊Q8—434no—
390C22⋊Q8—868no—
391C22⋊Q8—868no—
392C22⋊Q8—862no—
393C22⋊Q8—862no—
394C22⋊Q8—862no—
395C22⋊Q8C4×D4314yes—
396#117—1722no—
397#117—1722no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—431no—
400C2×C4⋊C4—434no—
401C2×C4⋊C4—432no—
402C2×C4⋊C4—862no—
403C2×C4⋊C4—868no—
404C2×C4⋊C4—861no—
405C2×C4⋊C4—861no—
406C2×C4⋊C4—861no—
407C2×C4⋊C4—861no—
408C2×C4⋊C4C22×D4312yes—
409C2×C4⋊C4C22×D4311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—868no—
413C2×C22⋊C4—861no—
414C2×C22⋊C4—861no—
415C2×C22⋊C4—861no—
416C2×C22⋊C4—862no—
417C2×C22⋊C4—862no—
418C2×C22⋊C4—861no—
419C2×C42—862no—
420#103—42no—
421#103—1722no—
422#103—1722no—
423#103—1722no—
424C22⋊C8—864no—
425C22⋊C8—864no—
426#95—21no—
427#95—862no—
428#95—861no—
429C24—431no—
430C24—431no—
431C24—434no—
432C24—864no—
433C24—1722no—
434C24—1721no—
435C24—1721no—
436C24D4×D4311yescharacteristic
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—432no—
441C2×Q8—864no—
442C2×Q8—861no—
443C2×Q8—862no—
444C2×Q8—862no—
445C2×Q8—1724no—
446C2×Q8—1728no—
447C2×Q8—1724no—
448C2×Q8—1724no—
449C2×Q8C2×C4×D4312yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—431no—
456C22×C4—431no—
457C22×C4—434no—
458C22×C4—432no—
459C22×C4—432no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—434no—
463C22×C4—434no—
464C22×C4—864no—
465C22×C4—864no—
466C22×C4—862no—
467C22×C4—862no—
468C22×C4—862no—
469C22×C4—862no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—862no—
473C22×C4—1728no—
474C22×C4—1724no—
475C22×C4—1724no—
476C22×C4—1722no—
477C22×C4—1722no—
478C22×C4—1724no—
479C22×C4—1722no—
480C22×C4—1722no—
481C22×C4—1722no—
482C22×C4C2×C4×D4312yes—
483C22×C4D4×D4311yescharacteristic
484C22×C4D4×D4312yes—
485C22×C4C23×D4311yescharacteristic
486M16—44no—
487M16—1724no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—862no—
492C2×C8—862no—
493C2×C8—862no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—432no—
499C4⋊C4—864no—
500C4⋊C4—864no—
501C4⋊C4—864no—
502C4⋊C4—864no—
503C4⋊C4—861no—
504C4⋊C4—862no—
505C4⋊C4—862no—
506C4⋊C4C2×C4×D4312yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—864no—
510C22⋊C4—864no—
511C22⋊C4—864no—
512C22⋊C4—1724no—
513C22⋊C4—1724no—
514C22⋊C4—1722no—
515C42—1724no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—432no—
521C23—434no—
522C23—432no—
523C23—868no—
524C23—861no—
525C23—861no—
526C23—862no—
527C23—862no—
528C23—861no—
529C23—861no—
530C23—862no—
531C23—1724no—
532C23—1724no—
533C23—1728no—
534C23—1728no—
535C23C22⋊C4×D4312yes—
536C23C2×D4×D4311yescharacteristic
537Q8—24no—
538Q8—41no—
539Q8—41no—
540Q8—864no—
541Q8—1728no—
542Q8—1721no—
543Q8—1721no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—432no—
557C2×C4—432no—
558C2×C4—862no—
559C2×C4—862no—
560C2×C4—864no—
561C2×C4—864no—
562C2×C4—862no—
563C2×C4—862no—
564C2×C4—862no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—862no—
568C2×C4—1724no—
569C2×C4—1724no—
570C2×C4—1728no—
571C2×C4—1728no—
572C2×C4—1724no—
573C2×C4—1722no—
574C2×C4—1722no—
575C2×C4—1722no—
576C2×C4—1721no—
577C2×C4—1721no—
578C2×C4—1722no—
579C2×C4C22⋊C4×D4312yes—
580C2×C4C22×C4×D4311yescharacteristic
581C2×C4C2×D4×D4311yescharacteristic
582C2×C4#1306337012yes—
583C2×C4#1306337112yes—
584C8—42no—
585C8—1722no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—42no—
590C22—42no—
591C22—434no—
592C22—862no—
593C22—862no—
594C22—864no—
595C22—864no—
596C22—862no—
597C22—862no—
598C22—862no—
599C22—17216no—
600C22—1722no—
601C22—1722no—
602C22C2×C22⋊C4×D4311yescharacteristic
603C22#1458345712yes—
604C22#137×D4312yes—
605C22#1458369511yescharacteristic
606C22#1458369811yescharacteristic
607C4—22no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—862no—
612C4—862no—
613C4—3442no—
614C4—1722no—
615C4—1722no—
616C4#1458337112yes—
617C4#1458345911yescharacteristic
618C4#1458346011yescharacteristic
619C2—42no—
620C2—864no—
621C2—1722no—
622C2#1913716911yescharacteristic
623C2#419×D4311yescharacteristic
624C2#1914311111yescharacteristic
625C2#1914542012yes—
626C2#1914542612yes—
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1894C2
socleC23×C43C22×D4
prime radicalC24×D43D4
cosocleC22×C4×C43C24
hypercenter#1894D43
hypocenterC43#21768
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. #19135029C2semidirect
5. #19135028C2semidirect
6. #19135026C2semidirect
7. #19134906C2semidirect
8. #19134904C2semidirect
9. #19134684C2semidirect
10. #19134528C2semidirect
11. C43×#1894C2semidirect
21. C22⋊Q8×D43C4semidirect
26. C2×#13063250C22semidirect
27. C2×#13063249C22semidirect
28. C2×#13063248C22semidirect
38. #14582662C22semidirect
39. #14582661C22semidirect
40. #14582661C22semidirect
43. C2×#103×C43C22semidirect
45. C43×#340C22semidirect
312. #1894D43semidirect
344. C2×#103C2×D43semidirect
358. #340C2×D43semidirect
359. C43#21768semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D432
C2×#145828622
#191369772
#191350292
#191350282
#191350262
#191349062
#191349042
#191346842
#191345282
C43×#18942
#2176843