Finite Group Databasebeta

← #25129231

Subgroups — #25129231

Order 11008.

1844 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063299C211yesmaximal, characteristic
2C2×#14583213C212yesmaximal, complemented
3C2×#14582759C211yesmaximal, characteristic, complemented
4#19135778C214yesmaximal, complemented
5#19134910C212yesmaximal, complemented
6#19134785C212yesmaximal, complemented
7#19134527C212yesmaximal, complemented
8C43×#1779C211yesmaximal, characteristic, complemented
9C23×D172C2211yescharacteristic
10C23×C22⋊D43—21no—
11C2×#13063299—28no—
12C2×#13063299—21no—
13C2×#13063299—21no—
14C2×#13063299C2212yes—
15C2×#13063299C2212yes—
16C22×C4⋊Dic43C2211yescharacteristic
17C22×C4⋊Dic43C2211yescharacteristic, complemented
18C22×C4⋊Dic43C2211yescharacteristic, complemented
19C22×C22⋊Dic43—21no—
20C22×(C2×D43)⋊C4—21no—
21#14583213—28no—
22C2×#13063259C2214yescomplemented
23C2×#13063247C2214yescomplemented
24C2×#13063247C2214yescomplemented
25#14582759—21no—
26#14582759—21no—
27#14582759C2212yes—
28#14582759C2212yescomplemented
29C2×C43⋊#95—21no—
30#14582662C2214yescomplemented
31C24×C4×C43C2211yescharacteristic
32C2×C22⋊C8×C43C2212yes—
33C2×C22⋊C8×C43C2212yescomplemented
34C2×C22⋊C8×C43C2212yescomplemented
35C24×D43—21no—
36C22×D172—21no—
37C22×D172—24no—
38C22×D172—24no—
39C22×D172—42no—
40C22×D172C2312yes—
41C22×C22⋊D43—28no—
42C22×C22⋊D43—22no—
43C22×C22⋊D43—22no—
44C22×C22⋊D43—44no—
45C22×C22⋊D43—44no—
46C23×Dic43—21no—
47C23×Dic43—21no—
48C23×Dic43—21no—
49#13063299—28no—
50#13063299—28no—
51#13063299—24no—
52#13063299—24no—
53#13063299—24no—
54#13063299D418yes—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43—24no—
57C2×C4⋊Dic43—24no—
58C2×C4⋊Dic43—21no—
59C2×C4⋊Dic43—21no—
60C2×C4⋊Dic43—42no—
61C2×C4⋊Dic43C2312yes—
62C2×C4⋊Dic43C2312yes—
63C2×C4⋊Dic43C2312yes—
64C2×C22⋊Dic43—22no—
65C2×C22⋊Dic43—22no—
66C2×C22⋊Dic43—44no—
67C2×(C2×D43)⋊C4—28no—
68C2×(C2×D43)⋊C4—22no—
69C2×(C2×D43)⋊C4—22no—
70#13063259—28no—
71#13063247—28no—
72#13063247—44no—
73C43⋊#95—22no—
74C43⋊#95—22no—
75C25×C43D411yescharacteristic
76C23×C4×C43—24no—
77C23×C4×C43—24no—
78C23×C4×C43—42no—
79C23×C4×C43D411yescharacteristic
80C23×C4×C43D411yescharacteristic
81C23×C4×C43D411yescharacteristic
82C23×C4×C43C2311yescharacteristic
83C23×C4×C43C2311yescharacteristic
84C23×C4×C43C2311yescharacteristic
85C22×C8×C43C2314yes—
86C22⋊C8×C43—24no—
87C22⋊C8×C43—24no—
88C22⋊C8×C43—42no—
89C24×C43—22no—
90C24×C43—22no—
91C24×C43—22no—
92C24×C43—22no—
93C24×C43—41no—
94C24×C43—41no—
95C24×C43D814yes—
96C24×C43C2×D411yescharacteristic
97C24×C43C2×D411yescharacteristic
98C24×C43C2×D411yescharacteristic
99C23×D43—24no—
100C23×D43—22no—
101C23×D43—42no—
102C23×D43—42no—
103C23×D43—42no—
104C23×D43—42no—
105C23×D43—42no—
106C23×D43—42no—
107C22×C4×C43—22no—
108C22×C4×C43—22no—
109C22×C4×C43—22no—
110C22×C4×C43—22no—
111C22×C4×C43—22no—
112C22×C4×C43—22no—
113C22×C4×C43—22no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—22no—
117C22×C4×C43—22no—
118C22×C4×C43—22no—
119C22×C4×C43—22no—
120C22×C4×C43—41no—
121C22×C4×C43—41no—
122C22×C4×C43—41no—
123C22×C4×C43—416no—
124C22×C4×C43C2×D414yes—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43#4114yes—
129C22×C4×C43C2411yescharacteristic
130C2×C22⋊D43—28no—
131C2×C22⋊D43—28no—
132C2×C22⋊D43—432no—
133C2×C22⋊D43—44no—
134C2×C22⋊D43—44no—
135C2×C22⋊D43—44no—
136C2×C22⋊D43—44no—
137C22×Dic43—24no—
138C22×Dic43—22no—
139C22×Dic43—22no—
140C22×Dic43—22no—
141C22×Dic43—42no—
142C22×Dic43—42no—
143C22×Dic43—42no—
144C22×Dic43—42no—
145C22×Dic43—42no—
146C22×Dic43—42no—
147C22×Dic43—42no—
148C22×Dic43—42no—
149C2×D172—24no—
150C2×D172—22no—
151C2×D172—24no—
152C2×D172—24no—
153C2×D172—416no—
154C2×D172—42no—
155C2×D172—42no—
156C2×D172C2×D414yes—
157C2×C8×C43—24no—
158C2×C8×C43—24no—
159C2×C8×C43—24no—
160C22⋊Dic43—44no—
161C22⋊Dic43—44no—
162(C2×D43)⋊C4—28no—
163(C2×D43)⋊C4—28no—
164C4⋊Dic43—24no—
165C4⋊Dic43—22no—
166C4⋊Dic43—24no—
167C4⋊Dic43—24no—
168C4⋊Dic43—42no—
169C4⋊Dic43—42no—
170C4⋊Dic43C2×D414yes—
171C23×C43—21no—
172C23×C43—21no—
173C23×C43—21no—
174C23×C43—21no—
175C23×C43—21no—
176C23×C43—21no—
177C23×C43—21no—
178C23×C43—21no—
179C23×C43—21no—
180C23×C43—21no—
181C23×C43—21no—
182C23×C43—21no—
183C23×C43—21no—
184C23×C43—21no—
185C23×C43—28no—
186C23×C43—28no—
187C23×C43—44no—
188C23×C43—44no—
189C23×C43—44no—
190C23×C43—44no—
191C23×C43—44no—
192C23×C43#12012yes—
193C23×C43#12312yes—
194C23×C43C2×D812yes—
195C23×C43C2×D812yes—
196C23×C43C2×D812yes—
197C23×C43AGL(1,ℤ/8)12yes—
198C23×C43#13712yes—
199C23×C43C22×D411yescharacteristic
200C22×D43—24no—
201C22×D43—48no—
202C22×D43—48no—
203C22×D43—48no—
204C22×D43—42no—
205C22×D43—42no—
206C22×D43—42no—
207C22×D43—42no—
208C22×D43—42no—
209C22×D43—42no—
210C22×D43—84no—
211C22×D43—84no—
212C22×D43—84no—
213C22×D43—84no—
214C2×C4×C43—21no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—28no—
219C2×C4×C43—28no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—44no—
224C2×C4×C43—44no—
225C2×C4×C43—44no—
226C2×C4×C43—44no—
227C2×C4×C43#12012yes—
228C2×C4×C43#12312yes—
229C2×C4×C43C22×D412yes—
230C2×C4×C43C2×#4112yes—
231C2×C4×C43H(5,𝔽2)12yes—
232C22⋊D43—416no—
233C22⋊D43—416no—
234C2×Dic43—24no—
235C2×Dic43—48no—
236C2×Dic43—42no—
237C2×Dic43—42no—
238C2×Dic43—42no—
239C2×Dic43—42no—
240C2×Dic43—42no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243C2×Dic43—42no—
244C2×Dic43—84no—
245C2×Dic43—84no—
246D172—28no—
247D172—48no—
248D172—44no—
249C8×C43—44no—
250#24002—431nomaximal, Sylow
251C22×C43—22no—
252C22×C43—22no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—22no—
256C22×C43—22no—
257C22×C43—22no—
258C22×C43—22no—
259C22×C43—22no—
260C22×C43—22no—
261C22×C43—22no—
262C22×C43—22no—
263C22×C43—22no—
264C22×C43—22no—
265C22×C43—22no—
266C22×C43—416no—
267C22×C43—41no—
268C22×C43—41no—
269C22×C43—41no—
270C22×C43—41no—
271C22×C43#44714yes—
272C22×C43#48014yescomplemented
273C22×C43C2×#12011yescharacteristic
274C22×C43C2×#12311yescharacteristic
275C22×C43#53411yescharacteristic
276C22×C43C22×D811yescharacteristic
277C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C43C2×#13711yescharacteristic
279C22×C43#57511yescharacteristic
280C2×D43—44no—
281C2×D43—44no—
282C2×D43—44no—
283C2×D43—816no—
284C2×D43—82no—
285C2×D43—82no—
286C2×D43—82no—
287C2×D43—82no—
288C4×C43—22no—
289C4×C43—22no—
290C4×C43—41no—
291C4×C43#54614yes—
292Dic43—44no—
293Dic43—82no—
294Dic43—82no—
295C22×#121—431no—
296C2×#480—432no—
297C2×#380—431no—
298#2033—434no—
299#1874—432no—
300#1852—432no—
301#1786—432no—
302#1779D4311yescharacteristic, complemented
303C2×C43—24no—
304C2×C43—24no—
305C2×C43—42no—
306C2×C43#240511yescharacteristic
307C2×C43C2×#44711yescharacteristic
308C2×C43#300711yescharacteristic
309C2×C43C2×#48011yescharacteristic
310C2×C43#309411yescharacteristic
311C2×C43#318611yescharacteristic
312C2×C43#319311yescharacteristic
313D43—84no—
314C23×D4—431no—
315C23×D4—861no—
316C24×C4C2×D4311yescharacteristic
317C2×#121—432no—
318C2×#121—432no—
319C2×#121—868no—
320C2×#121—861no—
321C2×#121—861no—
322C22×C4⋊C4—431no—
323C22×C4⋊C4—431no—
324C22×C4⋊C4—431no—
325C22×C22⋊C4—861no—
326C22×C22⋊C4—861no—
327#480—868no—
328C2×#107—434no—
329C2×#102—434no—
330C2×#102—434no—
331C2×C22⋊C8C2×D4312yes—
332C2×C22⋊C8C2×D4312yescomplemented
333C2×C22⋊C8C2×D4312yescomplemented
334#380—432no—
335#380—432no—
336#380—861no—
337#380—861no—
338C2×#95—861no—
339#340—434no—
340C43#2400211yescharacteristic, complemented, Sylow
341C25—861no—
342C25D17211yescharacteristic
343C22×D4—432no—
344C22×D4—864no—
345C22×D4—868no—
346C22×D4—862no—
347C22×D4—862no—
348C22×D4—861no—
349C22×D4—864no—
350C22×D4—1722no—
351C22×D4—1724no—
352C22×D4—1724no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—42no—
356C23×C4—861no—
357C23×C4—861no—
358C23×C4—861no—
359C23×C4D17211yescharacteristic
360C23×C4D17211yescharacteristic
361C23×C4D17211yescharacteristic
362C23×C4C22×D4311yescharacteristic
363C23×C4C22×D4311yescharacteristic
364C23×C4C22×D4311yescharacteristic
365C22×C8C22×D4314yes—
366#121—438no—
367#121—864no—
368#121—864no—
369#121—868no—
370#121—868no—
371#121—864no—
372C2×C4⋊C4—432no—
373C2×C4⋊C4—432no—
374C2×C4⋊C4—432no—
375C2×C4⋊C4—864no—
376C2×C4⋊C4—861no—
377C2×C4⋊C4—864no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—1722no—
381C2×C22⋊C4—868no—
382C2×C22⋊C4—862no—
383C2×C22⋊C4—862no—
384C2×C22⋊C4—862no—
385C2×C22⋊C4—862no—
386C2×C22⋊C4—1724no—
387#107—868no—
388#102—868no—
389#102—1724no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—42no—
393#95—862no—
394#95—862no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24—864no—
402C24—862no—
403C24—1722no—
404C24—1722no—
405C24—1722no—
406C24—1722no—
407C24—1722no—
408C24—1722no—
409C24D34414yes—
410C24C2×D17211yescharacteristic
411C24C2×D17211yescharacteristic
412C24C2×D17211yescharacteristic
413C2×D4—434no—
414C2×D4—864no—
415C2×D4—868no—
416C2×D4—868no—
417C2×D4—862no—
418C2×D4—864no—
419C2×D4—864no—
420C2×D4—17216no—
421C2×D4—1722no—
422C2×D4—17232no—
423C2×D4—1724no—
424C2×D4—1724no—
425C2×D4—1724no—
426C2×D4—1724no—
427C2×D4—1722no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—416no—
445C22×C4—864no—
446C22×C4—862no—
447C22×C4—862no—
448C22×C4—862no—
449C22×C4—1722no—
450C22×C4—1722no—
451C22×C4—1722no—
452C22×C4—1722no—
453C22×C4—1722no—
454C22×C4—1722no—
455C22×C4—1722no—
456C22×C4—1722no—
457C22×C4C2×D17211yescharacteristic
458C22×C4C2×D17211yescharacteristic
459C22×C4C2×D17211yescharacteristic
460C22×C4D4×D4314yes—
461C22×C4#1062190014yes—
462C22×C4C23×D4311yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C4⋊C4—434no—
467C4⋊C4—864no—
468C4⋊C4—862no—
469C4⋊C4—864no—
470C4⋊C4—864no—
471C4⋊C4—1722no—
472C4⋊C4—1722no—
473C22⋊C4—868no—
474C22⋊C4—868no—
475C22⋊C4—1724no—
476C22⋊C4—1724no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—28no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—864no—
499C23—3444no—
500C23—3444no—
501C23—3444no—
502C23—3444no—
503C23—1728no—
504C23—1728no—
505C23—1728no—
506C23—1722no—
507C23—1722no—
508C23—1722no—
509C23—1722no—
510C23—1722no—
511C23—1722no—
512C23#1306329512yes—
513C23#1306330912yes—
514C23C2×D34412yes—
515C23C2×D34412yes—
516C23C2×D34412yes—
517C23#1306335312yes—
518C23#1306335812yes—
519C23C22×D17211yescharacteristic
520D4—868no—
521D4—1728no—
522D4—17216no—
523D4—17216no—
524D4—1724no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—864no—
539C2×C4—3444no—
540C2×C4—3444no—
541C2×C4—1722no—
542C2×C4—1722no—
543C2×C4—1722no—
544C2×C4—1722no—
545C2×C4—1722no—
546C2×C4—1722no—
547C2×C4—1728no—
548C2×C4—1722no—
549C2×C4—1722no—
550C2×C4#1306329512yes—
551C2×C4#1306330912yes—
552C2×C4C2×D4×D4312yes—
553C2×C4C2×#1062190012yes—
554C2×C4#1306337012yes—
555C8—44no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—34416no—
577C22—3442no—
578C22—3442no—
579C22—3442no—
580C22—3442no—
581C22—1724no—
582C22—1724no—
583C22—1724no—
584C22#1458303514yes—
585C22#1458321314yescomplemented
586C22C2×#1306329511yescharacteristic
587C22C2×#1306330911yescharacteristic
588C22#1458346111yescharacteristic
589C22C22×D34411yescharacteristic
590C22C2×#1306335311yescharacteristic
591C22C2×#1306335811yescharacteristic
592C22#1458369911yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—3442no—
597C4—3442no—
598C4—1724no—
599C4#1458353614yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—3444no—
604C2#1913792311yescharacteristic
605C2C2×#1458303511yescharacteristic
606C2#1914385711yescharacteristic
607C2C2×#1458321311yescharacteristic
608C2#1914453611yescharacteristic
609C2#1914534611yescharacteristic
610C2#1914540211yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1779C2
socleC23×C43C22×D4
prime radicalC22×#13063299C2
hypercenter#1779D43
hypocenterC43#24002
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#14583213C2semidirect
3. C2×#14582759C2semidirect
4. #19135778C2semidirect
5. #19134910C2semidirect
6. #19134785C2semidirect
7. #19134527C2semidirect
8. C43×#1779C2semidirect
17. C22×C4⋊Dic43C22semidirect
18. C22×C4⋊Dic43C22semidirect
22. C2×#13063259C22semidirect
23. C2×#13063247C22semidirect
24. C2×#13063247C22semidirect
28. #14582759C22semidirect
30. #14582662C22semidirect
33. C2×C22⋊C8×C43C22semidirect
34. C2×C22⋊C8×C43C22semidirect
272. C22×C43#480semidirect
302. #1779D43semidirect
332. C2×C22⋊C8C2×D43semidirect
333. C2×C22⋊C8C2×D43semidirect
340. C43#24002semidirect
585. C22#14583213semidirect

Maximal subgroups

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

subgroupindex
C22×#130632992
C2×#145832132
C2×#145827592
#191357782
#191349102
#191347852
#191345272
C43×#17792
#2400243