Finite Group Databasebeta

← #25098757

Subgroups — #25098757

Order 11008.

926 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582934C211yesmaximal, characteristic
2#19137076C211yesmaximal, characteristic
3C4×#13063283C211yesmaximal, characteristic
4#19135190C211yesmaximal, characteristic, complemented
5#19135184C211yesmaximal, characteristic, complemented
6#19135182C211yesmaximal, characteristic, complemented
7#19135174C211yesmaximal, characteristic, complemented
8#19135172C211yesmaximal, characteristic, complemented
9#19135166C211yesmaximal, characteristic, complemented
10#19135164C211yesmaximal, characteristic, complemented
11#19134310C211yesmaximal, characteristic
12#19134310C211yesmaximal, characteristic
13#19134310C211yesmaximal, characteristic
14#336×D43C211yesmaximal, characteristic
15C43×#1920C211yesmaximal, characteristic, complemented
16C22×C8×D43C2211yescharacteristic
17C22×C8×D43C2211yescharacteristic
18C2×#13063283C2211yescharacteristic
19C2×#13063283C2211yescharacteristic
20C2×#13063283C2211yescharacteristic
21C2×#13063283C2211yescharacteristic
22C2×C42×D43C2211yescharacteristic
23#14582934C418yescomplemented
24C2×#13063258C2211yescharacteristic
25C2×#13063258C2211yescharacteristic
26C2×#13063257C2211yescharacteristic
27#14582798C2211yescharacteristic
28#14582797C2211yescharacteristic
29#14582795C2211yescharacteristic
30#14582773—21no—
31#14582770—21no—
32C4⋊C4×Dic43—21no—
33C4⋊C4×Dic43C2211yescharacteristic
34C4⋊C4×Dic43C2211yescharacteristic
35C4×C4⋊Dic43C2211yescharacteristic
36#14582750—21no—
37C4×(C2×D43)⋊C4—21no—
38#14582746—21no—
39#14582744—21no—
40C42×Dic43—21no—
41#14582662C2211yescharacteristic
42#14582662C2211yescharacteristic
43#14582662C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582654C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53C2×#106×C43C2211yescharacteristic
54C43×#389C2211yescharacteristic
55C4×C4⋊C4×C43C2211yescharacteristic
56C43×#340C2211yescharacteristic
57C43×#340C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×C4×D43D411yescharacteristic
61C22×C4×D43D411yescharacteristic
62C22×C4×D43C2311yescharacteristic
63C22×C43⋊C8C2311yescharacteristic
64C22×C43⋊C8C2311yescharacteristic
65C2×C8×D43—22no—
66C2×C8×D43—22no—
67C2×C8×D43—22no—
68C2×C8×D43—22no—
69C2×C8×D43—22no—
70C2×C8×D43C2×C414yes—
71#13063283—22no—
72#13063283—22no—
73#13063283—22no—
74#13063283—22no—
75#13063283—22no—
76#13063283C2×C414yes—
77#13063283C2×C414yes—
78#13063283Q814yes—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×C4×Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43—21no—
97C2×C4×Dic43D411yescharacteristic
98C2×C4×Dic43D411yescharacteristic
99C2×C4×Dic43C2311yescharacteristic
100C42×D43—22no—
101C42×D43—22no—
102#13063258C2×C414yescomplemented
103#13063258C2×C414yescomplemented
104#13063257C2×C414yescomplemented
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C22×C8×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113C2×C4⋊C4×C43—21no—
114C2×C4⋊C4×C43C2311yescharacteristic
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C42×C43—21no—
119C2×C42×C43C2311yescharacteristic
120#106×C43C2×C414yescomplemented
121#95×C43—21no—
122#95×C43—21no—
123C23×D43C2×D411yescharacteristic
124C22×C4×C43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2411yescharacteristic
131C22×Dic43—21no—
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43C2×D411yescharacteristic
136C22×Dic43C2×D411yescharacteristic
137C22×Dic43C2×D411yescharacteristic
138C2×C4×D43—22no—
139C2×C4×D43—22no—
140C2×C4×D43—22no—
141C2×C4×D43—22no—
142C2×C4×D43—41no—
143C2×C4×D43—41no—
144C2×C4×D43C22×C412yes—
145C2×C4×D43C2×Q812yes—
146C2×C4×D43#4112yes—
147C2×C4×D43#4112yes—
148C2×C4×D43#4112yes—
149C2×C4×D43#4112yes—
150C2×C8×C43—21no—
151C2×C8×C43—21no—
152C2×C8×C43—21no—
153C2×C8×C43—21no—
154C2×C8×C43—21no—
155C2×C8×C43C22×C412yes—
156C4⋊C4×C43—21no—
157C4⋊C4×C43—21no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—21no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—42no—
162C4⋊C4×C43C22×C412yes—
163C4⋊C4×C43C22×C412yes—
164C4⋊C4×C43C2×Q812yes—
165C42×C43—21no—
166C42×C43—21no—
167C42×C43—42no—
168(C2×D43)⋊C4—24no—
169(C2×D43)⋊C4—24no—
170(C2×D43)⋊C4—24no—
171(C2×D43)⋊C4—42no—
172C4⋊Dic43—21no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—42no—
178C4⋊Dic43C22×C412yes—
179C4⋊Dic43C22×C412yes—
180C4⋊Dic43C2×Q812yes—
181Dic43⋊C4—42no—
182Dic43⋊C4—42no—
183C4×Dic43—21no—
184C4×Dic43—21no—
185C4×Dic43—24no—
186C4×Dic43—24no—
187C4×Dic43—24no—
188C4×Dic43—22no—
189C4×Dic43—22no—
190C4×Dic43—42no—
191C4×Dic43—42no—
192C4×Dic43—42no—
193C4×Dic43—42no—
194C2×C43⋊C8—21no—
195C2×C43⋊C8—21no—
196C2×C43⋊C8—21no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8C22×C412yes—
200C8×D43—28no—
201C8×D43—44no—
202C23×C43C22×D411yescharacteristic
203C22×D43—21no—
204C22×D43—21no—
205C22×D43—21no—
206C22×D43—21no—
207C22×D43C4×D412yes—
208C22×D43C22⋊Q812yes—
209C22×D43#12312yes—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—22no—
213C2×C4×C43—22no—
214C2×C4×C43—22no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43#13512yes—
228C2×C4×C43C23×C411yescharacteristic
229C2×C4×C43C22×Q811yescharacteristic
230C2×C4×C43C2×#4111yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C2×C4×C43C2×#4111yescharacteristic
233C2×C4×C43C2×#4111yescharacteristic
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—22no—
237C2×Dic43—22no—
238C2×Dic43—22no—
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—21no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43C4×D412yes—
252C2×Dic43C22⋊Q812yes—
253C2×Dic43#12312yes—
254C2×Dic43#13512yes—
255C4×D43—44no—
256C4×D43—44no—
257C4×D43#11714yes—
258C4×D43C4×Q814yes—
259C4×D43#12514yes—
260C4×D43#12614yes—
261C8×C43—22no—
262C8×C43—41no—
263C43⋊C8—22no—
264C43⋊C8—41no—
265#21032—431nomaximal, Sylow
266C22×C43C2×C4×D411yescharacteristic
267C22×C43C2×C22⋊Q811yescharacteristic
268C22×C43C2×#12311yescharacteristic
269C22×C43C2×#13511yescharacteristic
270C22×C43C2×#13511yescharacteristic
271C22×C43#57511yescharacteristic
272C22×C43#57511yescharacteristic
273C2×D43—22no—
274C2×D43—22no—
275C2×D43—22no—
276C2×D43—22no—
277C2×D43—22no—
278C2×D43—22no—
279C2×D43#38714yes—
280C4×C43—22no—
281C4×C43—42no—
282C4×C43—41no—
283C4×C43—42no—
284C4×C43—42no—
285C4×C43—81no—
286C4×C43C2×#11711yescharacteristic
287C4×C43C2×C4×Q811yescharacteristic
288C4×C43C2×#12511yescharacteristic
289C4×C43C2×#12611yescharacteristic
290Dic43—22no—
291Dic43—42no—
292Dic43—41no—
293Dic43—42no—
294Dic43—42no—
295Dic43—81no—
296Dic43#38714yes—
297C2×#427—431no—
298#2296—431no—
299C4×#117—431no—
300#1921—431no—
301#1921—431no—
302#1920—431no—
303#1920D4311yescharacteristic, complemented
304#1919—431no—
305#1919—431no—
306#1918—431no—
307#1918—431no—
308#1735—431no—
309#1735—431no—
310#1735—431no—
311C2×#336—431no—
312C2×C43C2×#38711yescharacteristic
313C2×C43C4×#13511yescharacteristic
314C2×C43#294311yescharacteristic
315C2×C43#307611yescharacteristic
316C2×C43#307611yescharacteristic
317C2×C43#309111yescharacteristic
318C2×C43#309111yescharacteristic
319D43—24no—
320C23×C8—431no—
321C23×C8—431no—
322C2×#117—431no—
323C2×#117—431no—
324C2×#117—431no—
325C2×#117—431no—
326C22×C42—431no—
327#427—438no—
328C2×#107—431no—
329C2×#107—431no—
330C2×#106—431no—
331C2×#106C2×D4311yescharacteristic
332#389—431no—
333#389—431no—
334#389—431no—
335#389C2×D4311yescharacteristic
336#385—861no—
337#385—861no—
338C4×C4⋊C4—431no—
339C4×C4⋊C4—431no—
340C4×C4⋊C4—431no—
341C4×C4⋊C4—861no—
342C4×C4⋊C4—861no—
343C4×C4⋊C4C2×D4311yescharacteristic
344C4×C22⋊C4—861no—
345C4×C22⋊C4—861no—
346#376—861no—
347C43—861no—
348#340—431no—
349#340—431no—
350#340—431no—
351#340—431no—
352#340—431no—
353#340—431no—
354#340—431no—
355#340—431no—
356#340—431no—
357#340C2×D4311yescharacteristic
358#340C2×D4311yescharacteristic
359#340C2×D4311yescharacteristic
360#336—431no—
361#336—431no—
362#336—431no—
363#336C2×D4311yescharacteristic
364C43#2103211yescharacteristic, complemented, Sylow
365C23×C4—431no—
366C23×C4—431no—
367C23×C4—431no—
368C22×C8—431no—
369C22×C8—434no—
370C22×C8—431no—
371C22×C8—862no—
372C22×C8—862no—
373C22×C8—862no—
374C22×C8—862no—
375C22×C8—862no—
376C22×C8C22×D4311yescharacteristic
377C22×C8C22×D4311yescharacteristic
378#117—434no—
379#117—434no—
380#117—434no—
381#117—862no—
382#117—862no—
383#117—862no—
384#117—862no—
385#117—862no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4—431no—
390C2×C4⋊C4—431no—
391C2×C4⋊C4—861no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4C22×D4311yescharacteristic
395C2×C4⋊C4C22×D4311yescharacteristic
396C2×C4⋊C4C22×D4311yescharacteristic
397C2×C4⋊C4C22×D4311yescharacteristic
398C2×C22⋊C4—861no—
399C2×C22⋊C4—861no—
400C2×C22⋊C4—861no—
401C2×C22⋊C4—861no—
402C2×C42—21no—
403C2×C42—431no—
404C2×C42—431no—
405C2×C42—431no—
406C2×C42—862no—
407C2×C42—862no—
408C2×C42—861no—
409C2×C42—861no—
410C2×C42—861no—
411C2×C42—861no—
412C2×C42—861no—
413C2×C42—861no—
414C2×C42—861no—
415C2×C42C22×D4311yescharacteristic
416#107—434no—
417#107—434no—
418#106—434no—
419#106C4×D4314yescomplemented
420#95—21no—
421#95—21no—
422#95—861no—
423#95—861no—
424#95—861no—
425#95—861no—
426#95—861no—
427#95—861no—
428C24—431no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—431no—
434C22×C4—432no—
435C22×C4—432no—
436C22×C4—432no—
437C22×C4—432no—
438C22×C4—432no—
439C22×C4—432no—
440C22×C4—431no—
441C22×C4—431no—
442C22×C4—862no—
443C22×C4—862no—
444C22×C4—862no—
445C22×C4—861no—
446C22×C4—861no—
447C22×C4—861no—
448C22×C4—862no—
449C22×C4—861no—
450C22×C4—1721no—
451C22×C4—1721no—
452C22×C4D4×D4311yescharacteristic
453C22×C4D4×D4311yescharacteristic
454C22×C4C23×D4311yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—432no—
461C2×C8—868no—
462C2×C8—861no—
463C2×C8—861no—
464C2×C8—861no—
465C2×C8—861no—
466C2×C8—861no—
467C2×C8—1724no—
468C2×C8C2×C4×D4312yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—432no—
476C4⋊C4—432no—
477C4⋊C4—432no—
478C4⋊C4—861no—
479C4⋊C4—861no—
480C4⋊C4—861no—
481C4⋊C4—861no—
482C4⋊C4—861no—
483C4⋊C4—1722no—
484C4⋊C4—1722no—
485C4⋊C4—1722no—
486C4⋊C4C2×C4×D4312yes—
487C4⋊C4C2×C4×D4312yes—
488C4⋊C4Q8×D4312yes—
489C22⋊C4—864no—
490C22⋊C4—864no—
491C22⋊C4—864no—
492C22⋊C4—1722no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—862no—
497C42—862no—
498C42—864no—
499C42—864no—
500C42—864no—
501C42—861no—
502C42—861no—
503C42—1722no—
504C42—1722no—
505C42—1722no—
506C42—1722no—
507C23—432no—
508C23—432no—
509C23—432no—
510C23—861no—
511C23—861no—
512C23—861no—
513C23—861no—
514C23C2×D4×D4311yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—432no—
533C2×C4—432no—
534C2×C4—432no—
535C2×C4—434no—
536C2×C4—434no—
537C2×C4—434no—
538C2×C4—434no—
539C2×C4—432no—
540C2×C4—861no—
541C2×C4—861no—
542C2×C4—862no—
543C2×C4—862no—
544C2×C4—862no—
545C2×C4—861no—
546C2×C4—861no—
547C2×C4—861no—
548C2×C4—1724no—
549C2×C4—1724no—
550C2×C4—1721no—
551C2×C4—1721no—
552C2×C4—1721no—
553C2×C4—1721no—
554C2×C4—1721no—
555C2×C4—1721no—
556C2×C4—1721no—
557C2×C4—1721no—
558C2×C4—1721no—
559C2×C4#1306334612yes—
560C2×C4C22×C4×D4311yescharacteristic
561C2×C4C2×Q8×D4311yescharacteristic
562C2×C4#41×D4311yescharacteristic
563C2×C4#41×D4311yescharacteristic
564C2×C4#41×D4311yescharacteristic
565C2×C4#41×D4311yescharacteristic
566C8—22no—
567C8—41no—
568C8—862no—
569C8—1721no—
570C22—434no—
571C22—862no—
572C22—862no—
573C22—862no—
574C22—862no—
575C22—862no—
576C22—862no—
577C22C4×D4×D4311yescharacteristic
578C22C22⋊Q8×D4311yescharacteristic
579C22#123×D4311yescharacteristic
580C22#135×D4311yescharacteristic
581C22C2×#1306334611yescharacteristic
582C22#1458369511yescharacteristic
583C22#1458369811yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—434no—
591C4—862no—
592C4—3441no—
593C4—1721no—
594C4—1722no—
595C4—1722no—
596C4—1722no—
597C4#117×D4311yescharacteristic
598C4C4×Q8×D4311yescharacteristic
599C4#125×D4311yescharacteristic
600C4#126×D4311yescharacteristic
601C2—864no—
602C2#387×D4311yescharacteristic
603C2C4×#1306334611yescharacteristic
604C2#1914339011yescharacteristic
605C2#1914438211yescharacteristic
606C2#1914438411yescharacteristic
607C2#1914450711yescharacteristic
608C2#1914450911yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1920C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
cosocleC22×C4×C43C24
hypercenter#1920D43
hypocenterC43#21032
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. #19135190C2semidirect
5. #19135184C2semidirect
6. #19135182C2semidirect
7. #19135174C2semidirect
8. #19135172C2semidirect
9. #19135166C2semidirect
10. #19135164C2semidirect
15. C43×#1920C2semidirect
23. #14582934C4semidirect
102. #13063258C2×C4semidirect
103. #13063258C2×C4semidirect
104. #13063257C2×C4semidirect
120. #106×C43C2×C4semidirect
303. #1920D43semidirect
364. C43#21032semidirect
419. #106C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145829342
#191370762
C4×#130632832
#191351902
#191351842
#191351822
#191351742
#191351722
#191351662
#191351642
#191343102
#336×D432
C43×#19202
#2103243