Finite Group Databasebeta

← #25133170

Subgroups — #25133170

Order 11008.

1064 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583608C211yesmaximal, characteristic
2C2×#14583183C211yesmaximal, characteristic, complemented
3C2×#14582996C211yesmaximal, characteristic
4#19139963C211yesmaximal, characteristic, complemented
5#19135935C211yesmaximal, characteristic, complemented
6#19135933C211yesmaximal, characteristic, complemented
7#19135747C211yesmaximal, characteristic, complemented
8#19135745C211yesmaximal, characteristic, complemented
9#19135566C211yesmaximal, characteristic, complemented
10#19135533C211yesmaximal, characteristic, complemented
11#19134929C211yesmaximal, characteristic
12#19134927C211yesmaximal, characteristic
13#19134854C211yesmaximal, characteristic
14#19134603C211yesmaximal, characteristic
15C43×#1864C211yesmaximal, characteristic
16C22×#10621883C2211yescharacteristic
17#14583608—24no—
18#14583608—24no—
19C2×#13063324C2211yescharacteristic
20C2×#13063317C2211yescharacteristic
21C2×#13063316C2211yescharacteristic
22C2×#13063304—21no—
23C2×#13063302—21no—
24C2×#13063302—21no—
25C2×C4×Dic86C2211yescharacteristic
26C2×#13063290C2211yescharacteristic
27C2×D4×Dic43C2211yescharacteristic
28C2×D4×Dic43C2211yescharacteristic
29C22×Dic43⋊C4—21no—
30#14583183—24no—
31#14582996—24no—
32C2×#13063258C2211yescharacteristic
33C2×#13063256C2211yescharacteristic
34C2×Dic43⋊C8C2211yescharacteristic
35C2×#13063250C2211yescharacteristic
36C2×#13063250C2211yescharacteristic
37C2×#13063248C2211yescharacteristic
38C2×#13063246C2211yescharacteristic
39C2×#13063235C2211yescharacteristic
40#14582840—21no—
41#14582819—21no—
42#14582805C2211yescharacteristic
43#14582804C2211yescharacteristic
44#14582795C2211yescharacteristic
45#14582785—21no—
46#14582781—21no—
47#14582764C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582654C2211yescharacteristic
54#14582653C2211yescharacteristic
55#14582653C2211yescharacteristic
56C2×C4×D4×C43C2211yescharacteristic
57C2×#106×C43C2211yescharacteristic
58C2×#102×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C43×#389C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C43×#336C2211yescharacteristic
63C22×Dic86—21no—
64C22×Dic86D411yescharacteristic
65C22×Dic86C2311yescharacteristic
66C23×Dic43—21no—
67C23×Dic43—21no—
68C2×#10621883—22no—
69C2×#10621883—22no—
70C2×#10621883Q814yescomplemented
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73#13063324—22no—
74#13063324D414yes—
75#13063317—22no—
76#13063317—22no—
77#13063316—22no—
78#13063316—22no—
79#13063304—28no—
80#13063302—28no—
81#13063302—28no—
82C4×Dic86—22no—
83C4×Dic86—22no—
84#13063290—22no—
85#13063290—22no—
86D4×Dic43—22no—
87D4×Dic43—22no—
88D4×Dic43—22no—
89D4×Dic43D414yes—
90C2×C4⋊Dic43—21no—
91C2×C4⋊Dic43D411yescharacteristic
92C2×C4⋊Dic43C2311yescharacteristic
93C2×C4⋊Dic43C2311yescharacteristic
94C2×C4⋊Dic43C2311yescharacteristic
95C2×Dic43⋊C4—28no—
96C2×Dic43⋊C4—21no—
97C2×Dic43⋊C4—21no—
98C2×Dic43⋊C4—21no—
99C2×Dic43⋊C4—21no—
100C2×Dic43⋊C4—21no—
101C2×Dic43⋊C4—21no—
102C2×Dic43⋊C4D411yescharacteristic
103C2×Dic43⋊C4D411yescharacteristic
104C2×C22⋊Dic43—21no—
105C2×C22⋊Dic43—21no—
106C2×C4×Dic43—21no—
107C2×C4×Dic43C2311yescharacteristic
108C2×C4×Dic43C2311yescharacteristic
109#13063258—22no—
110#13063256—22no—
111Dic43⋊C8—22no—
112#13063250—22no—
113#13063250—22no—
114#13063248—22no—
115#13063246—22no—
116#13063235—22no—
117C43⋊#95—21no—
118C43⋊#95—21no—
119C43⋊#95—21no—
120C43⋊#95—21no—
121C43⋊#95—21no—
122C43⋊#95—21no—
123C43⋊#95—21no—
124C22×D4×C43C2311yescharacteristic
125C23×C4×C43—21no—
126C22×C8×C43C2311yescharacteristic
127C22×C8×C43C2311yescharacteristic
128C4×D4×C43—22no—
129C4×D4×C43—22no—
130C2×C4⋊C4×C43—21no—
131C2×C4⋊C4×C43C2311yescharacteristic
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C4⋊C4×C43C2311yescharacteristic
134C2×C22⋊C4×C43—21no—
135C2×C42×C43C2311yescharacteristic
136#106×C43—22no—
137#102×C43—22no—
138#102×C43—22no—
139#95×C43—21no—
140C24×C43—21no—
141C2×D4×C43—21no—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43C2×D412yes—
145C2×D4×C43C2×Q812yes—
146C2×D4×C43#4112yes—
147C22×C4×C43—24no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—81no—
152C22×C4×C43—81no—
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2411yescharacteristic
156C22×Dic43—24no—
157C22×Dic43—24no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C22×Dic43—42no—
163C22×Dic43—42no—
164C22×Dic43—81no—
165C22×Dic43—81no—
166C22×Dic43C2×D411yescharacteristic
167C22×Dic43C2×D411yescharacteristic
168C22×Dic43C2×D411yescharacteristic
169C22×Dic43C2×D411yescharacteristic
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86—22no—
176C2×Dic86—24no—
177C2×Dic86—41no—
178C2×Dic86—41no—
179C2×Dic86—81no—
180C2×Dic86—81no—
181C2×Dic86C2×Q812yes—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43—21no—
186C2×C8×C43—21no—
187C2×C8×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43—21no—
191C4⋊C4×C43—21no—
192C4⋊C4×C43—21no—
193C4⋊C4×C43—42no—
194C4⋊C4×C43C2×D412yes—
195C22⋊C4×C43—24no—
196C42×C43—21no—
197C42×C43—21no—
198C22⋊Dic43—24no—
199C22⋊Dic43—24no—
200#10621883—28no—
201C4⋊Dic43—24no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43—22no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43C2×D412yes—
208C4⋊Dic43C2×D412yes—
209Dic43⋊C4—24no—
210Dic43⋊C4—24no—
211Dic43⋊C4—24no—
212Dic43⋊C4—22no—
213Dic43⋊C4—24no—
214Dic43⋊C4—22no—
215Dic43⋊C4—42no—
216Dic43⋊C4—42no—
217C4×Dic43—21no—
218C4×Dic43—21no—
219C4×Dic43—42no—
220C4×Dic43C2×D412yes—
221C4×Dic43C2×D412yes—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227C2×C43⋊C8C2×Q812yes—
228C23×C43—22no—
229C23×C43—22no—
230C23×C43—22no—
231C23×C43—41no—
232C23×C43—41no—
233C23×C43—41no—
234C23×C43—41no—
235C23×C43C22×D411yescharacteristic
236D4×C43—22no—
237D4×C43—22no—
238D4×C43—22no—
239D4×C43C22⋊Q814yes—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—22no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—84no—
256C2×C4×C43C22×D411yescharacteristic
257C2×C4×C43C22×D411yescharacteristic
258C2×C4×C43C22×Q811yescharacteristic
259C2×C4×C43C2×#4111yescharacteristic
260C2×C4×C432−1+411yescharacteristic
261C2×C4×C432−1+411yescharacteristic
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—22no—
269C2×Dic43—22no—
270C2×Dic43—22no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—21no—
274C2×Dic43—21no—
275C2×Dic43—48no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—84no—
286C2×Dic43#12012yes—
287C2×Dic43#12012yes—
288Dic86—22no—
289Dic86—22no—
290Dic86—44no—
291Dic86—41no—
292Dic86—41no—
293Dic86—84no—
294C8×C43—41no—
295C8×C43—41no—
296C43⋊C8—22no—
297C43⋊C8—41no—
298#24343—431nomaximal, Sylow
299C22×C43—24no—
300C22×C43—42no—
301C22×C43—42no—
302C22×C43—42no—
303C22×C43—42no—
304C22×C43—42no—
305C22×C43—42no—
306C22×C43C2×#12011yescharacteristic
307C22×C43#54811yescharacteristic
308C22×C43D4×Q811yescharacteristic
309C22×C43#57511yescharacteristic
310C22×C43#57511yescharacteristic
311C22×C43#57711yescharacteristic
312C22×C43#57711yescharacteristic
313C4×C43—41no—
314C4×C43—42no—
315C4×C43—41no—
316C4×C43—81no—
317C4×C43—81no—
318C4×C43C2×C22⋊Q811yescharacteristic
319C4×C43#53611yescharacteristic
320C4×C43#54811yescharacteristic
321C4×C43D4×Q811yescharacteristic
322Dic43—22no—
323Dic43—22no—
324Dic43—42no—
325Dic43—41no—
326Dic43—42no—
327Dic43—42no—
328Dic43—41no—
329Dic43—81no—
330C2×#554—431no—
331C2×#475—431no—
332C2×#440—431no—
333#2581—431no—
334#2059—431no—
335#2059—431no—
336#2026—431no—
337#2026—431no—
338#1993—431no—
339#1985—431no—
340#1877—431no—
341#1877—431no—
342#1864—431no—
343#1864D4311yescharacteristic
344#1804—431no—
345C2×C43—44no—
346C2×C43#240911yescharacteristic
347C2×C43#300511yescharacteristic
348C2×C43#300911yescharacteristic
349C2×C43#337611yescharacteristic
350C2×C43#337611yescharacteristic
351C2×C43#338811yescharacteristic
352C2×C43#338811yescharacteristic
353C22×SD16—431no—
354#554—864no—
355#554—864no—
356C2×C4⋊Q8—431no—
357C2×#125—431no—
358C2×#125—431no—
359C2×C22⋊Q8—861no—
360C2×C22⋊Q8—861no—
361C2×C22⋊Q8—861no—
362C2×C4×Q8—431no—
363C2×C4×Q8—431no—
364C2×C4×D4—431no—
365C2×C4×D4—431no—
366C2×C4×D4C2×D4311yescharacteristic
367C22×C4⋊C4—861no—
368#475—864no—
369#440—864no—
370C2×#107—431no—
371C2×#106—431no—
372C2×#106C2×D4311yescharacteristic
373C2×C4⋊C8—431no—
374C2×#103—431no—
375C2×#103—431no—
376C2×#103—431no—
377C2×#102—431no—
378C2×#102C2×D4311yescharacteristic
379C2×#102C2×D4311yescharacteristic
380C2×#97—431no—
381#400—861no—
382#395—861no—
383#391—431no—
384#391—431no—
385#389—431no—
386#389C2×D4311yescharacteristic
387#387—861no—
388#387—861no—
389#382—431no—
390#340—431no—
391#340—431no—
392#340—431no—
393#340—431no—
394#340—431no—
395#340C2×D4311yescharacteristic
396#336—431no—
397#336—431no—
398#336—431no—
399#336C2×D4311yescharacteristic
400C43#2434311yescharacteristic, complemented, Sylow
401C22×Q8—431no—
402C22×Q8—431no—
403C22×Q8—861no—
404C22×D4C22×D4311yescharacteristic
405C23×C4—21no—
406C23×C4—861no—
407C23×C4—861no—
408C2×SD16—434no—
409C2×SD16—862no—
410C2×SD16—862no—
411C22×C8—431no—
412C22×C8—431no—
413C22×C8C22×D4311yescharacteristic
414C22×C8C22×D4311yescharacteristic
415C4⋊Q8—434no—
416C4⋊Q8—862no—
417#125—862no—
418#125—862no—
419#125—862no—
420#125—862no—
421C22⋊Q8—868no—
422C22⋊Q8—868no—
423C22⋊Q8—868no—
424C4×Q8—862no—
425C4×Q8—862no—
426C4×Q8—862no—
427C4×Q8—862no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—434no—
431C4×D4—862no—
432C4×D4—862no—
433C4×D4—862no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—431no—
436C2×C4⋊C4—431no—
437C2×C4⋊C4—431no—
438C2×C4⋊C4—431no—
439C2×C4⋊C4—431no—
440C2×C4⋊C4—431no—
441C2×C4⋊C4—861no—
442C2×C4⋊C4—861no—
443C2×C4⋊C4—861no—
444C2×C4⋊C4—868no—
445C2×C4⋊C4—861no—
446C2×C4⋊C4—861no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4—861no—
449C2×C4⋊C4C22×D4311yescharacteristic
450C2×C4⋊C4C22×D4311yescharacteristic
451C2×C4⋊C4C22×D4311yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—861no—
454C2×C22⋊C4—861no—
455C2×C42—431no—
456C2×C42—431no—
457C2×C42—861no—
458C2×C42C22×D4311yescharacteristic
459#107—862no—
460#106—22no—
461#106—862no—
462C4⋊C8—862no—
463#103—862no—
464#103—862no—
465#103—862no—
466#102—22no—
467#102—22no—
468#102—862no—
469#97—862no—
470#95—21no—
471#95—861no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476#95—861no—
477#95—861no—
478C24—21no—
479C2×Q8—432no—
480C2×Q8—864no—
481C2×Q8—862no—
482C2×Q8—861no—
483C2×Q8—861no—
484C2×Q8—861no—
485C2×Q8—861no—
486C2×Q8—861no—
487C2×Q8—3441no—
488C2×Q8—3441no—
489C2×Q8—1721no—
490C2×Q8—1721no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4#1062189812yes—
495C2×D4D4×D4312yes—
496C2×D4Q8×D4312yes—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—431no—
504C22×C4—431no—
505C22×C4—431no—
506C22×C4—431no—
507C22×C4—864no—
508C22×C4—864no—
509C22×C4—861no—
510C22×C4—861no—
511C22×C4—861no—
512C22×C4—861no—
513C22×C4—3441no—
514C22×C4—3441no—
515C22×C4—1722no—
516C22×C4—1722no—
517C22×C4D4×D4311yescharacteristic
518C22×C4D4×D4311yescharacteristic
519C22×C4C23×D4311yescharacteristic
520SD16—868no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—432no—
528C2×C8—861no—
529C2×C8—861no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8—861no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—432no—
540C4⋊C4—432no—
541C4⋊C4—862no—
542C4⋊C4—862no—
543C4⋊C4—862no—
544C4⋊C4—864no—
545C4⋊C4—864no—
546C4⋊C4—864no—
547C4⋊C4—864no—
548C4⋊C4—864no—
549C4⋊C4—861no—
550C4⋊C4—861no—
551C4⋊C4—861no—
552C4⋊C4—861no—
553C4⋊C4—1722no—
554C4⋊C4—1722no—
555C4⋊C4D4×D4312yes—
556C22⋊C4—24no—
557C22⋊C4—864no—
558C22⋊C4—864no—
559C42—21no—
560C42—21no—
561C42—432no—
562C42—432no—
563C42—861no—
564C42—861no—
565C42—1722no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×D4×D4311yescharacteristic
574Q8—862no—
575Q8—862no—
576Q8—3444no—
577Q8—1724no—
578Q8—1721no—
579Q8—1721no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4#1306330314yes—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—432no—
601C2×C4—432no—
602C2×C4—861no—
603C2×C4—861no—
604C2×C4—861no—
605C2×C4—861no—
606C2×C4—862no—
607C2×C4—861no—
608C2×C4—861no—
609C2×C4—862no—
610C2×C4—862no—
611C2×C4—861no—
612C2×C4—861no—
613C2×C4—861no—
614C2×C4—861no—
615C2×C4—3444no—
616C2×C4—1728no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4—1721no—
620C2×C4—1721no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4C2×D4×D4311yescharacteristic
627C2×C4C2×D4×D4311yescharacteristic
628C2×C4C2×Q8×D4311yescharacteristic
629C2×C4C2×#1062189811yescharacteristic
630C2×C4#1306337211yescharacteristic
631C2×C4#1306337211yescharacteristic
632C8—41no—
633C8—41no—
634C8—862no—
635C8—1721no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22#120×D4311yescharacteristic
644C22#1458355811yescharacteristic
645C22#1458356211yescharacteristic
646C22#1458369511yescharacteristic
647C22#1458369611yescharacteristic
648C22#1458370511yescharacteristic
649C22#1458370911yescharacteristic
650C4—41no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—862no—
656C4—862no—
657C4—3441no—
658C4—1721no—
659C4—1722no—
660C4—1722no—
661C4—1721no—
662C4—1722no—
663C4C2×#1306330311yescharacteristic
664C4#1458347111yescharacteristic
665C4#1458355811yescharacteristic
666C4#1458356211yescharacteristic
667C2—44no—
668C2#1913795911yescharacteristic
669C2#1914383311yescharacteristic
670C2#1914386711yescharacteristic
671C2#1914741811yescharacteristic
672C2#1914742311yescharacteristic
673C2#1914753311yescharacteristic
674C2#1914754411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1864C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#1864D43
hypocenterC43#24343
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×#14583183C2semidirect
4. #19139963C2semidirect
5. #19135935C2semidirect
6. #19135933C2semidirect
7. #19135747C2semidirect
8. #19135745C2semidirect
9. #19135566C2semidirect
10. #19135533C2semidirect
70. C2×#10621883Q8semidirect
400. C43#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#145836082
C2×#145831832
C2×#145829962
#191399632
#191359352
#191359332
#191357472
#191357452
#191355662
#191355332
#191349292
#191349272
#191348542
#191346032
C43×#18642
#2434343