Finite Group Databasebeta

← #25096129

Subgroups — #25096129

Order 11008.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583380C211yesmaximal, characteristic
2C2×#14582925C211yesmaximal, characteristic, complemented
3C2×#14582898C211yesmaximal, characteristic
4C2×#14582896C211yesmaximal, characteristic
5#19137219C211yesmaximal, characteristic, complemented
6#19134609C212yesmaximal, complemented
7#19134604C212yesmaximal, complemented
8#19134599C212yesmaximal, complemented
9#19134594C212yesmaximal, complemented
10#19134314C211yesmaximal, characteristic, complemented
11C2×C43×#340C211yesmaximal, characteristic, complemented
12C22×#10621898C2211yescharacteristic
13C22×#10621877C2211yescharacteristic
14#14583380—24no—
15#14583380C418yescomplemented
16C2×#13063290C2212yes—
17C2×#13063287C2212yes—
18C2×C4×C22⋊D43—22no—
19C2×#13063284C2211yescharacteristic
20C2×#13063284C2211yescharacteristic
21C2×#13063283—21no—
22C2×C4⋊C4×D43—21no—
23#14582925—24no—
24C2×Dic43⋊C8C2212yes—
25C2×Dic43⋊C8C2212yes—
26#14582898—24no—
27#14582896—24no—
28C2×#13063250C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063248C2212yes—
31C2×#13063247C2211yescharacteristic
32C2×#13063247C2211yescharacteristic
33C2×#13063245C2212yes—
34#14582795—21no—
35#14582795—21no—
36#14582772—22no—
37#14582768C2212yes—
38#14582764C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C23×C8×C43C2211yescharacteristic
42C22×C4⋊C4×C43C2211yescharacteristic
43C22×C4⋊C4×C43C2211yescharacteristic
44C43×#340C2212yes—
45C43×#340C2212yes—
46C2×#10621898—22no—
47C2×#10621898—22no—
48C2×#10621898—22no—
49C2×#10621898C2×C414yes—
50C2×#10621898D414yes—
51C2×#10621898Q814yes—
52C22×Dic86C2311yescharacteristic
53C22×D172C2311yescharacteristic
54C22×C22⋊D43—21no—
55C22×C4×D43—21no—
56C22×C4×D43—22no—
57C2×#10621877—22no—
58C2×#10621877—22no—
59C2×#10621877—22no—
60C22×C43⋊C8C2312yes—
61#13063290—24no—
62#13063290C2×C418yescomplemented
63#13063287—24no—
64#13063287C2×C418yescomplemented
65C4×C22⋊D43—216no—
66#13063284—22no—
67#13063284—22no—
68#13063284C2×C414yes—
69#13063284D414yes—
70#13063283—28no—
71C2×C4⋊Dic43D411yescharacteristic
72C2×C4⋊Dic43D411yescharacteristic
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×Dic43⋊C4—22no—
76C2×Dic43⋊C4—22no—
77C2×Dic43⋊C4—22no—
78C4⋊C4×D43—28no—
79C2×C22⋊Dic43—21no—
80C2×C22⋊Dic43D412yes—
81C2×(C2×D43)⋊C4—22no—
82C2×C4×Dic43—22no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43C2312yes—
85Dic43⋊C8—24no—
86Dic43⋊C8—24no—
87#13063250—22no—
88#13063250—22no—
89#13063248—24no—
90#13063247—22no—
91#13063247—22no—
92#13063245—24no—
93C43⋊#95—22no—
94C43⋊#95—22no—
95C23×C4×C43—21no—
96C23×C4×C43—21no—
97C23×C4×C43C2311yescharacteristic
98C22×C8×C43—22no—
99C22×C8×C43—22no—
100C22×C8×C43—22no—
101C22×C8×C43C2311yescharacteristic
102C22×C8×C43C2311yescharacteristic
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—22no—
105C2×C4⋊C4×C43—22no—
106C2×C4⋊C4×C43—22no—
107C2×C4⋊C4×C43—21no—
108C2×C4⋊C4×C43C2×C414yes—
109C2×C4⋊C4×C43C2312yes—
110C2×C4⋊C4×C43C2312yes—
111C24×C43C2×D411yescharacteristic
112C23×D43—21no—
113C22×C4×C43—24no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—42no—
117C22×C4×C43—81no—
118C22×C4×C43—81no—
119C22×C4×C43—81no—
120C22×C4×C43—81no—
121C22×C4×C43C22×C412yes—
122C22×C4×C43C2×D412yes—
123C22×C4×C43C2×D412yes—
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2×Q812yes—
126C22×C4×C43#4112yes—
127C22×C4×C43#4112yes—
128C22×C4×C43C2411yescharacteristic
129C2×C22⋊D43—24no—
130C2×C22⋊D43—24no—
131C2×C22⋊D43—24no—
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43C2×D412yes—
136C22×Dic43C2×D412yes—
137#10621898—28no—
138#10621898—28no—
139#10621898—28no—
140#10621898C4⋊C4116yescomplemented
141C2×D172—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172C22×C412yes—
145C2×D172C2×D412yes—
146C2×D172C2×Q812yes—
147C2×C4×D43—24no—
148C2×C4×D43—24no—
149C2×C4×D43—24no—
150C2×C4×D43—28no—
151C2×C4×D43—82no—
152C2×C4×D43—82no—
153C2×Dic86—21no—
154C2×Dic86—21no—
155C2×Dic86—21no—
156C2×Dic86C22×C412yes—
157C2×Dic86C2×D412yes—
158C2×Dic86C2×Q812yes—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43—21no—
165C2×C8×C43—44no—
166C4⋊C4×C43—24no—
167C4⋊C4×C43—22no—
168C4⋊C4×C43—22no—
169C4⋊C4×C43—22no—
170C4⋊C4×C43—42no—
171C4⋊C4×C43C22×C414yes—
172C22⋊Dic43—42no—
173C22⋊Dic43C22⋊C418yes—
174(C2×D43)⋊C4—28no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43C22⋊C414yes—
180C4⋊Dic43C22⋊C414yes—
181Dic43⋊C4—28no—
182Dic43⋊C4—82no—
183Dic43⋊C4—82no—
184C4×Dic43—28no—
185C4×Dic43—42no—
186C4×Dic43C22×C414yes—
187C4×Dic43C2×D414yes—
188#10621877—44no—
189C2×C43⋊C8—22no—
190C2×C43⋊C8—22no—
191C2×C43⋊C8—22no—
192C23×C43—21no—
193C23×C43—21no—
194C23×C43—21no—
195C23×C43—21no—
196C23×C43C2×C22⋊C412yes—
197C23×C43#12112yes—
198C23×C43C22⋊Q812yes—
199C23×C43C22×D411yescharacteristic
200C22×D43—22no—
201C22×D43—22no—
202C22×D43—22no—
203C22×D43—41no—
204C22×D43—41no—
205C22×D43—41no—
206C22×D43—41no—
207C2×C4×C43—24no—
208C2×C4×C43—42no—
209C2×C4×C43—42no—
210C2×C4×C43—42no—
211C2×C4×C43—42no—
212C2×C4×C43—42no—
213C2×C4×C43—84no—
214C2×C4×C43—84no—
215C2×C4×C43C2×C22⋊C412yes—
216C2×C4×C43C2×C4⋊C414yes—
217C2×C4×C43#11714yes—
218C2×C4×C43#12114yes—
219C2×C4×C43#12112yes—
220C2×C4×C43C22⋊Q814yes—
221C2×C4×C43C22⋊Q812yes—
222C2×C4×C43C23×C411yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43C22×D411yescharacteristic
225C2×C4×C43C22×Q811yescharacteristic
226C2×C4×C43C2×#4111yescharacteristic
227C2×C4×C43C2×#4111yescharacteristic
228C22⋊D43—216no—
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—22no—
235C2×Dic43—22no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43C2×C22⋊C414yes—
243C2×Dic43C2×C22⋊C414yes—
244D172—22no—
245D172—22no—
246D172—22no—
247D172C2×C4⋊C414yes—
248C4×D43—216no—
249C4×D43—88no—
250Dic86—22no—
251Dic86—22no—
252Dic86—22no—
253Dic86C2×C4⋊C414yes—
254C8×C43—41no—
255C8×C43—41no—
256C43⋊C8—42no—
257#20652—431nomaximal, Sylow
258C22×C43—22no—
259C22×C43—22no—
260C22×C43—22no—
261C22×C43—22no—
262C22×C43—22no—
263C22×C43—22no—
264C22×C43#38014yes—
265C22×C43C22×C22⋊C411yescharacteristic
266C22×C43C2×#12111yescharacteristic
267C22×C43C2×C22⋊Q811yescharacteristic
268C22×C43#57611yescharacteristic
269C22×C43#57711yescharacteristic
270C22×C43#57711yescharacteristic
271C22×C43#57811yescharacteristic
272C2×D43—24no—
273C2×D43—42no—
274C2×D43—42no—
275C2×D43—42no—
276C2×D43—42no—
277C2×D43—42no—
278C2×D43—42no—
279C4×C43—44no—
280C4×C43—82no—
281C4×C43#38014yes—
282C4×C43C22×C4⋊C411yescharacteristic
283C4×C43C2×#11711yescharacteristic
284C4×C43C2×#12111yescharacteristic
285C4×C43C2×C22⋊Q811yescharacteristic
286Dic43—24no—
287Dic43—24no—
288Dic43—24no—
289Dic43—81no—
290Dic43—81no—
291C2×#520—431no—
292C2×#424—431no—
293C2×#417—431no—
294C2×#416—431no—
295#2317—431no—
296#1805—432no—
297#1804—432no—
298#1803—432no—
299#1802—432no—
300#1736—431no—
301C2×#340D4311yescharacteristic, complemented
302C2×C43—24no—
303C2×C43C2×#38011yescharacteristic
304C2×C43#289711yescharacteristic
305C2×C43#289811yescharacteristic
306C2×C43#304511yescharacteristic
307C2×C43#304611yescharacteristic
308C2×C43#308311yescharacteristic
309C2×C43#308411yescharacteristic
310D43—44no—
311C22×#41—431no—
312C22×M16—431no—
313C23×C8C2×D4311yescharacteristic
314#520—438no—
315#520—864no—
316C2×C4×Q8—432no—
317C2×C4×D4—432no—
318C2×C4×D4—862no—
319C2×#117—431no—
320C2×#117—431no—
321C2×#117—861no—
322C22×C4⋊C4—861no—
323C22×C4⋊C4C2×D4311yescharacteristic
324C22×C4⋊C4C2×D4311yescharacteristic
325#424—864no—
326C2×C4⋊C8—432no—
327C2×C4⋊C8—432no—
328#417—864no—
329#416—864no—
330C2×#103—431no—
331C2×#103—431no—
332C2×#103—432no—
333C2×#102—431no—
334C2×#102—431no—
335C2×#102—432no—
336#389—861no—
337#389—861no—
338#385—862no—
339#384—432no—
340#382—432no—
341#340—432no—
342#340—432no—
343#340C2×D4312yes—
344#340C2×D4312yes—
345C43#2065211yescharacteristic, complemented, Sylow
346C2×#41—434no—
347C2×#41—434no—
348C2×#41—434no—
349C2×#41—862no—
350C2×#41—862no—
351C2×#41—862no—
352C22×Q8—431no—
353C22×D4—431no—
354C22×D4—861no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—862no—
358C23×C4—861no—
359C23×C4C22×D4311yescharacteristic
360C2×M16—862no—
361C2×M16—862no—
362C2×M16—862no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—432no—
367C22×C8C22×D4311yescharacteristic
368C22×C8C22×D4311yescharacteristic
369C4×Q8—438no—
370C4×Q8—864no—
371C4×D4—438no—
372C4×D4—864no—
373C4×D4—8616no—
374#117—434no—
375#117—434no—
376#117—868no—
377#117—862no—
378#117—862no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—431no—
385C2×C4⋊C4—431no—
386C2×C4⋊C4—431no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—868no—
389C2×C4⋊C4—862no—
390C2×C4⋊C4—862no—
391C2×C4⋊C4—862no—
392C2×C4⋊C4C4×D4314yes—
393C2×C4⋊C4C22×D4312yes—
394C2×C4⋊C4C22×D4312yes—
395C2×C22⋊C4—432no—
396C2×C22⋊C4—862no—
397C2×C22⋊C4—861no—
398C2×C42—432no—
399C2×C42—862no—
400C2×C42—861no—
401C4⋊C8—864no—
402C4⋊C8—864no—
403#103—864no—
404#103—862no—
405#103—862no—
406#102—864no—
407#102—862no—
408#102—862no—
409#95—862no—
410#95—862no—
411C24—861no—
412C24D4×D4311yescharacteristic
413#41—4316no—
414#41—868no—
415#41—868no—
416#41—868no—
417C2×Q8—432no—
418C2×Q8—432no—
419C2×Q8—432no—
420C2×Q8—861no—
421C2×Q8—861no—
422C2×Q8—861no—
423C2×D4—432no—
424C2×D4—432no—
425C2×D4—432no—
426C2×D4—864no—
427C2×D4—864no—
428C2×D4—864no—
429C2×D4—861no—
430C2×D4—861no—
431C2×D4—861no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—432no—
441C22×C4—432no—
442C22×C4—868no—
443C22×C4—861no—
444C22×C4—864no—
445C22×C4—864no—
446C22×C4—864no—
447C22×C4—861no—
448C22×C4—861no—
449C22×C4—3442no—
450C22×C4—3442no—
451C22×C4C2×C4×D4312yes—
452C22×C4#1062189812yes—
453C22×C4#1062189812yes—
454C22×C4D4×D4311yescharacteristic
455C22×C4D4×D4312yes—
456C22×C4D4×D4312yes—
457C22×C4Q8×D4312yes—
458C22×C4C23×D4311yescharacteristic
459M16—1724no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—862no—
468C2×C8—862no—
469C2×C8—862no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—434no—
476C4⋊C4—434no—
477C4⋊C4—868no—
478C4⋊C4—861no—
479C4⋊C4—861no—
480C4⋊C4—861no—
481C4⋊C4—861no—
482C4⋊C4—3442no—
483C4⋊C4—3442no—
484C4⋊C4C2×C4×D4314yes—
485C22⋊C4—438no—
486C22⋊C4—868no—
487C22⋊C4—1722no—
488C42—434no—
489C42—434no—
490C42—868no—
491C42—1722no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—862no—
497C23—862no—
498C23—862no—
499C23—1721no—
500C23—1721no—
501C23—1721no—
502C23—1721no—
503C23C22⋊C4×D4312yes—
504C23#1306329812yes—
505C23#1306330312yes—
506C23C2×D4×D4311yescharacteristic
507Q8—434no—
508Q8—862no—
509Q8—862no—
510Q8—862no—
511D4—434no—
512D4—8616no—
513D4—862no—
514D4—862no—
515D4—862no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—434no—
525C2×C4—434no—
526C2×C4—862no—
527C2×C4—862no—
528C2×C4—862no—
529C2×C4—862no—
530C2×C4—862no—
531C2×C4—862no—
532C2×C4—862no—
533C2×C4—8616no—
534C2×C4—3448no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4—1721no—
541C2×C4C22⋊C4×D4312yes—
542C2×C4C4⋊C4×D4314yes—
543C2×C4#1306328214yes—
544C2×C4#1306329814yes—
545C2×C4#1306329812yes—
546C2×C4#1306330314yes—
547C2×C4#1306330312yes—
548C2×C4C22×C4×D4311yescharacteristic
549C2×C4C2×D4×D4311yescharacteristic
550C2×C4C2×D4×D4311yescharacteristic
551C2×C4C2×Q8×D4311yescharacteristic
552C2×C4C2×#1062189811yescharacteristic
553C2×C4C2×#1062189811yescharacteristic
554C8—41no—
555C8—41no—
556C8—1722no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—864no—
564C22—1722no—
565C22—1722no—
566C22—1722no—
567C22—1722no—
568C22—1722no—
569C22—1722no—
570C22#1458275814yes—
571C22C2×C22⋊C4×D4311yescharacteristic
572C22C2×#1306329811yescharacteristic
573C22C2×#1306330311yescharacteristic
574C22#1458370011yescharacteristic
575C22#1458370511yescharacteristic
576C22#1458370511yescharacteristic
577C22#1458371211yescharacteristic
578C4—44no—
579C4—82no—
580C4—864no—
581C4—864no—
582C4—864no—
583C4—3441no—
584C4—3441no—
585C4#1458275814yes—
586C4C2×C4⋊C4×D4311yescharacteristic
587C4C2×#1306328211yescharacteristic
588C4C2×#1306329811yescharacteristic
589C4C2×#1306330311yescharacteristic
590C2—24no—
591C2—1724no—
592C2C2×#1458275811yescharacteristic
593C2#1914311611yescharacteristic
594C2#1914312311yescharacteristic
595C2#1914413911yescharacteristic
596C2#1914415011yescharacteristic
597C2#1914445111yescharacteristic
598C2#1914446211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#340C2
socleC23×C43C22×D4
prime radicalC22×#10621898C22
cosocleC22×C4×C43C24
hypercenterC2×#340D43
hypocenterC43#20652
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#14582925C2semidirect
5. #19137219C2semidirect
6. #19134609C2semidirect
7. #19134604C2semidirect
8. #19134599C2semidirect
9. #19134594C2semidirect
10. #19134314C2semidirect
11. C2×C43×#340C2semidirect
15. #14583380C4semidirect
62. #13063290C2×C4semidirect
64. #13063287C2×C4semidirect
140. #10621898C4⋊C4semidirect
301. C2×#340D43semidirect
345. C43#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#145833802
C2×#145829252
C2×#145828982
C2×#145828962
#191372192
#191346092
#191346042
#191345992
#191345942
#191343142
C2×C43×#3402
#2065243