Finite Group Databasebeta

← #25107836

Subgroups — #25107836

Order 11008.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583050C211yesmaximal, characteristic
2#19137123C211yesmaximal, characteristic, complemented
3C2×#14582802C211yesmaximal, characteristic
4#19135477C212yesmaximal, complemented
5#19135289C212yesmaximal, complemented
6#19135218C212yesmaximal, complemented
7#19134947C212yesmaximal, complemented
8#19134900C211yesmaximal, characteristic, complemented
9#19134782C211yesmaximal, characteristic, complemented
10#19134359C211yesmaximal, characteristic, complemented
11C43×#1786C211yesmaximal, characteristic
12C22×#10621883C2212yes—
13C2×#13063304C2211yescharacteristic
14C2×#13063290C2211yescharacteristic, complemented
15C22×C4⋊Dic43C2211yescharacteristic
16C22×C22⋊Dic43—21no—
17C22×C22⋊Dic43—21no—
18C22×C4×Dic43C2211yescharacteristic
19#14583050C418yescomplemented
20C2×#13063258C2212yescomplemented
21C2×#13063250C2212yescomplemented
22C2×#13063246C2212yes—
23C2×C22⋊(C43⋊C8)C2211yescharacteristic
24C2×#13063235C2212yescomplemented
25#14582804C2211yescharacteristic, complemented
26#14582802—21no—
27#14582802—21no—
28#14582802C2212yes—
29#14582802C2212yes—
30#14582795C2211yescharacteristic, complemented
31#14582781—21no—
32#14582771—21no—
33C4⋊C4×Dic43C2211yescharacteristic, complemented
34#14582750—21no—
35C22⋊C4×Dic43—21no—
36#14582662C2212yescomplemented
37#14582661C2212yescomplemented
38#14582653C2212yescomplemented
39C23×D4×C43C2211yescharacteristic
40C2×#102×C43C2212yes—
41C2×#102×C43C2212yes—
42C2×C22⋊C8×C43C2211yescharacteristic
43C43×#380C2211yescharacteristic
44C22×Dic86C2311yescharacteristic
45C23×Dic43—21no—
46C23×Dic43D411yescharacteristic
47C23×Dic43D411yescharacteristic
48C2×#10621883—42no—
49C2×#10621883—42no—
50C2×#10621883C2×C418yescomplemented
51C22×C43⋊C8C2312yes—
52#13063304—22no—
53#13063304C2×C414yes—
54#13063290—42no—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43—42no—
57C2×C4⋊Dic43C2312yes—
58C2×C4⋊Dic43C2311yescharacteristic
59C2×Dic43⋊C4—21no—
60C2×Dic43⋊C4—21no—
61C2×Dic43⋊C4—21no—
62C2×C22⋊Dic43—21no—
63C2×C22⋊Dic43—22no—
64C2×C22⋊Dic43—22no—
65C2×C22⋊Dic43—22no—
66C2×C22⋊Dic43—22no—
67C2×C22⋊Dic43—44no—
68C2×C22⋊Dic43—44no—
69C2×C4×Dic43—21no—
70C2×C4×Dic43—21no—
71C2×C4×Dic43—42no—
72C2×C4×Dic43D411yescharacteristic
73C2×C4×Dic43D411yescharacteristic
74C2×C4×Dic43C2311yescharacteristic
75C2×C4×Dic43C2311yescharacteristic
76#13063258—42no—
77#13063250—42no—
78#13063246C2×C418yescomplemented
79C22⋊(C43⋊C8)C2×C414yes—
80#13063235—42no—
81C43⋊#95—21no—
82C43⋊#95—21no—
83C43⋊#95—21no—
84C43⋊#95—21no—
85C25×C43—21no—
86C22×D4×C43—21no—
87C22×D4×C43—22no—
88C22×D4×C43—42no—
89C22×D4×C43—42no—
90C22×D4×C43C2×C414yes—
91C22×D4×C43C2312yes—
92C23×C4×C43C2311yescharacteristic
93C22×C8×C43C2312yes—
94C2×C4⋊C4×C43C2311yescharacteristic
95C2×C4⋊C4×C43C2311yescharacteristic
96C2×C22⋊C4×C43—21no—
97#102×C43—42no—
98#102×C43—42no—
99C22⋊C8×C43—22no—
100#95×C43—21no—
101C24×C43—24no—
102C24×C43—22no—
103C24×C43—42no—
104C24×C43—42no—
105C24×C43—42no—
106C24×C43—42no—
107C24×C43—42no—
108C24×C43—42no—
109C24×C43C2×D411yescharacteristic
110C2×D4×C43—24no—
111C2×D4×C43—22no—
112C2×D4×C43—22no—
113C2×D4×C43—22no—
114C2×D4×C43—22no—
115C2×D4×C43—22no—
116C2×D4×C43—48no—
117C2×D4×C43—48no—
118C2×D4×C43—42no—
119C2×D4×C43—42no—
120C2×D4×C43C22×C414yes—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—21no—
124C22×C4×C43—21no—
125C22×C4×C43C22×C412yes—
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43#4112yes—
128C22×C4×C43C2411yescharacteristic
129C22×Dic43—22no—
130C22×Dic43—22no—
131C22×Dic43—22no—
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—22no—
135C22×Dic43—41no—
136C22×Dic43—41no—
137C22×Dic43—41no—
138C22×Dic43—42no—
139C22×Dic43—42no—
140C22×Dic43—42no—
141C22×Dic43C2×D411yescharacteristic
142C22×Dic43C2×D411yescharacteristic
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C2×Dic86—22no—
146C2×Dic86—22no—
147C2×Dic86—21no—
148C2×Dic86C22×C412yes—
149C2×C8×C43—22no—
150C2×C8×C43—22no—
151C2×C8×C43—22no—
152C4⋊C4×C43—22no—
153C4⋊C4×C43—22no—
154C22⋊C4×C43—42no—
155C22⋊Dic43—24no—
156C22⋊Dic43—44no—
157C22⋊Dic43—44no—
158C22⋊Dic43—44no—
159C22⋊Dic43—44no—
160#10621883—48no—
161C4⋊Dic43—21no—
162C4⋊Dic43—22no—
163C4⋊Dic43—22no—
164C4⋊Dic43—42no—
165C4⋊Dic43C22×C412yes—
166Dic43⋊C4—24no—
167Dic43⋊C4—42no—
168Dic43⋊C4—42no—
169C4×Dic43—22no—
170C4×Dic43—22no—
171C4×Dic43—41no—
172C4×Dic43—41no—
173C4×Dic43—42no—
174C4×Dic43—42no—
175C2×C43⋊C8—22no—
176C2×C43⋊C8—22no—
177C2×C43⋊C8C22×C414yes—
178C23×C43—21no—
179C23×C43—21no—
180C23×C43—21no—
181C23×C43—21no—
182C23×C43—24no—
183C23×C43—48no—
184C23×C43—48no—
185C23×C43—42no—
186C23×C43—42no—
187C23×C43—42no—
188C23×C43—42no—
189C23×C43—42no—
190C23×C43—42no—
191C23×C43—48no—
192C23×C43—84no—
193C23×C43—84no—
194C23×C43—84no—
195C23×C43—84no—
196C23×C43C4×D412yes—
197C23×C43AGL(1,ℤ/8)12yes—
198C23×C43#13712yes—
199C23×C43C22×D411yescharacteristic
200D4×C43—24no—
201D4×C43—24no—
202D4×C43—48no—
203D4×C43—42no—
204D4×C43—42no—
205C2×C4×C43—21no—
206C2×C4×C43—21no—
207C2×C4×C43—24no—
208C2×C4×C43—24no—
209C2×C4×C43—41no—
210C2×C4×C43—41no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43C4×D412yes—
216C2×C4×C43C23×C411yescharacteristic
217C2×C4×C43C2×#4111yescharacteristic
218C2×C4×C43H(5,𝔽2)12yes—
219C2×C4×C43H(5,𝔽2)12yes—
220C2×Dic43—21no—
221C2×Dic43—21no—
222C2×Dic43—22no—
223C2×Dic43—22no—
224C2×Dic43—21no—
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—21no—
228C2×Dic43—21no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43—21no—
233C2×Dic43—44no—
234C2×Dic43—44no—
235C2×Dic43—41no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—42no—
240C2×Dic43—42no—
241C2×Dic43—42no—
242C2×Dic43—84no—
243C2×Dic43AGL(1,ℤ/8)12yes—
244Dic86—24no—
245Dic86—42no—
246C8×C43—42no—
247C43⋊C8—24no—
248#22049—431nomaximal, Sylow
249C22×C43—22no—
250C22×C43—22no—
251C22×C43—22no—
252C22×C43—22no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—44no—
256C22×C43—44no—
257C22×C43—44no—
258C22×C43—816no—
259C22×C43—82no—
260C22×C43—82no—
261C22×C43—82no—
262C22×C43—82no—
263C22×C43#44014yes—
264C22×C43C2×C4×D411yescharacteristic
265C22×C43#53411yescharacteristic
266C22×C43#53511yescharacteristic
267C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C43C2×#13711yescharacteristic
270C22×C43#57511yescharacteristic
271C4×C43—22no—
272C4×C43—81no—
273C4×C43—81no—
274C4×C43#51812yes—
275C4×C43#53812yes—
276Dic43—22no—
277Dic43—42no—
278Dic43—41no—
279Dic43—41no—
280Dic43—41no—
281Dic43—42no—
282Dic43—82no—
283C2×#450—431no—
284C4×C22⋊Q8—431no—
285C2×#390—431no—
286#1973—432no—
287#1936—432no—
288#1925—432no—
289#1880—432no—
290#1872—431no—
291#1852—431no—
292#1786D4311yescharacteristic
293#1749—431no—
294C2×C43—24no—
295C2×C43—84no—
296C2×C43#232111yescharacteristic
297C2×C43C2×#44011yescharacteristic
298C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
299C2×C43#319311yescharacteristic
300C2×C43#319411yescharacteristic
301C2×C43#321511yescharacteristic
302C2×C43#321711yescharacteristic
303C23×D4C2×D4311yescharacteristic
304C22×SD16—432no—
305C2×C22⋊Q8—431no—
306C2×C4×Q8—431no—
307C22×C4⋊C4—431no—
308C22×C22⋊C4—861no—
309C22×C22⋊C4—861no—
310C22×C42—431no—
311#450—438no—
312C2×#107—432no—
313C2×#103—432no—
314C2×#102—432no—
315C2×#102C2×D4312yes—
316C2×#102C2×D4312yes—
317C2×C22⋊C8—431no—
318C2×C22⋊C8C2×D4311yescharacteristic
319C2×#97—432no—
320#391—431no—
321#390—432no—
322#390—432no—
323#390—861no—
324#390—861no—
325#389—431no—
326#387—861no—
327#385—861no—
328#380C2×D4311yescharacteristic
329C4×C4⋊C4—431no—
330C4×C4⋊C4—861no—
331C4×C22⋊C4—861no—
332#340—432no—
333#340—432no—
334#336—432no—
335C43#2204911yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8—431no—
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C4×D4314yes—
343C22×D4C22×D4312yes—
344C23×C4—431no—
345C23×C4—431no—
346C23×C4—861no—
347C23×C4C22×D4311yescharacteristic
348C2×SD16—438no—
349C2×SD16—1722no—
350C2×SD16—1722no—
351C22×C8—432no—
352C22×C8C22×D4312yes—
353C22⋊Q8—434no—
354C22⋊Q8—862no—
355C4×Q8—1722no—
356C2×C4⋊C4—431no—
357C2×C4⋊C4—432no—
358C2×C4⋊C4—861no—
359C2×C4⋊C4—861no—
360C2×C4⋊C4—861no—
361C2×C4⋊C4—861no—
362C2×C4⋊C4—1722no—
363C2×C4⋊C4C22×D4311yescharacteristic
364C2×C4⋊C4C22×D4311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—861no—
367C2×C22⋊C4—862no—
368C2×C22⋊C4—862no—
369C2×C22⋊C4—862no—
370C2×C22⋊C4—862no—
371C2×C22⋊C4—1724no—
372C2×C22⋊C4—1724no—
373C2×C42—431no—
374C2×C42—431no—
375C2×C42—431no—
376C2×C42—431no—
377C2×C42—861no—
378C2×C42—861no—
379C2×C42—1722no—
380#107—1722no—
381#103—1722no—
382#102—42no—
383#102—42no—
384#102—438no—
385C22⋊C8—22no—
386C22⋊C8—434no—
387#97—1722no—
388#95—21no—
389#95—861no—
390#95—861no—
391#95—861no—
392#95—861no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24D4×D4311yescharacteristic
402C2×Q8—432no—
403C2×Q8—861no—
404C2×Q8—862no—
405C2×Q8—862no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C2×C4×D4314yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—431no—
422C22×C4—431no—
423C22×C4—431no—
424C22×C4—431no—
425C22×C4—862no—
426C22×C4—862no—
427C22×C4—862no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—862no—
431C22×C4—1721no—
432C22×C4—1721no—
433C22×C4—1721no—
434C22×C4—1722no—
435C22×C4—1722no—
436C22×C4—1722no—
437C22×C4C2×C4×D4312yes—
438C22×C4D4×D4311yescharacteristic
439C22×C4#1062190012yes—
440C22×C4C23×D4311yescharacteristic
441SD16—1728no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—434no—
446C2×C8—862no—
447C2×C8—862no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—432no—
451C4⋊C4—864no—
452C4⋊C4—861no—
453C4⋊C4—862no—
454C4⋊C4—862no—
455C4⋊C4—1722no—
456C4⋊C4—1722no—
457C4⋊C4—1722no—
458C22⋊C4—42no—
459C22⋊C4—864no—
460C22⋊C4—1724no—
461C22⋊C4—1724no—
462C22⋊C4—1724no—
463C22⋊C4—1724no—
464C42—862no—
465C42—862no—
466C42—1721no—
467C42—1721no—
468C42—1722no—
469C42—1722no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23(C22⋊D43)⋊C412yes—
489C23#1306334912yes—
490C23#1306335412yes—
491C23C2×D4×D4311yescharacteristic
492Q8—864no—
493Q8—1722no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—432no—
510C2×C4—861no—
511C2×C4—861no—
512C2×C4—862no—
513C2×C4—862no—
514C2×C4—861no—
515C2×C4—861no—
516C2×C4—861no—
517C2×C4—861no—
518C2×C4—861no—
519C2×C4—861no—
520C2×C4—861no—
521C2×C4—861no—
522C2×C4—861no—
523C2×C4—3444no—
524C2×C4—1724no—
525C2×C4—1722no—
526C2×C4—1722no—
527C2×C4—1722no—
528C2×C4—1724no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—1721no—
532C2×C4—1721no—
533C2×C4(C22⋊D43)⋊C412yes—
534C2×C4C22×C4×D4311yescharacteristic
535C2×C4C2×#1062190011yescharacteristic
536C2×C4#1306337012yes—
537C2×C4#1306337012yes—
538C8—42no—
539C8—864no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#1458299614yes—
555C22C2×(C22⋊D43)⋊C411yescharacteristic
556C22#1458345911yescharacteristic
557C22#1458346411yescharacteristic
558C22AGL(1,ℤ/8)×D4311yescharacteristic
559C22C2×#1306334911yescharacteristic
560C22C2×#1306335411yescharacteristic
561C22#1458369511yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—862no—
566C4—3442no—
567C4—1721no—
568C4—1721no—
569C4—1721no—
570C4—1722no—
571C4—1722no—
572C4#1458337212yes—
573C4#1458348412yes—
574C2—24no—
575C2—84no—
576C2#1913725011yescharacteristic
577C2C2×#1458299611yescharacteristic
578C2#1914339911yescharacteristic
579C2#1914539811yescharacteristic
580C2#1914540611yescharacteristic
581C2#1914559511yescharacteristic
582C2#1914561111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1786C2
socleC23×C43C22×D4
prime radicalC23×D4×C43C22
cosocleC22×C4×C43C24
hypercenter#1786D43
hypocenterC43#22049
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #19137123C2semidirect
4. #19135477C2semidirect
5. #19135289C2semidirect
6. #19135218C2semidirect
7. #19134947C2semidirect
8. #19134900C2semidirect
9. #19134782C2semidirect
10. #19134359C2semidirect
14. C2×#13063290C22semidirect
19. #14583050C4semidirect
20. C2×#13063258C22semidirect
21. C2×#13063250C22semidirect
24. C2×#13063235C22semidirect
25. #14582804C22semidirect
30. #14582795C22semidirect
33. C4⋊C4×Dic43C22semidirect
36. #14582662C22semidirect
37. #14582661C22semidirect
38. #14582653C22semidirect
50. C2×#10621883C2×C4semidirect
78. #13063246C2×C4semidirect
335. C43#22049semidirect

Maximal subgroups

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

subgroupindex
C2×#145830502
#191371232
C2×#145828022
#191354772
#191352892
#191352182
#191349472
#191349002
#191347822
#191343592
C43×#17862
#2204943