Finite Group Databasebeta

← #25102546

Subgroups — #25102546

Order 11008.

1322 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13063373C211yesmaximal, characteristic
2C2×#14583081C211yesmaximal, characteristic
3C4×#13063356C212yesmaximal, complemented
4#19134989C211yesmaximal, characteristic, complemented
5#19134967C212yesmaximal, complemented
6#19134827C212yesmaximal, complemented
7#19134809C211yesmaximal, characteristic, complemented
8#19134553C211yesmaximal, characteristic
9#19134406C212yesmaximal, complemented
10C4×#13063244C211yesmaximal, characteristic, complemented
11C43×#2327C211yesmaximal, characteristic, complemented
12C2×#13063373C2211yescharacteristic
13C2×#13063356C2212yes—
14#14583381—22no—
15#14583380—22no—
16#14583380C2212yes—
17#14583377C2211yescharacteristic
18#14583375—22no—
19C4×#10621902—22no—
20C4×#10621898—22no—
21C4×#10621898C2212yes—
22C4×Q8×D43—22no—
23#14583081C418yescomplemented
24#14583004—22no—
25#14582997—22no—
26C4×#10621885—22no—
27C4×#10621884—22no—
28#14582939C2212yes—
29C2×#13063253C2212yes—
30C2×#13063253C2212yes—
31#14582908C2212yes—
32#14582898C2212yes—
33C2×#13063244C2211yescharacteristic
34C2×#13063244C2211yescharacteristic
35C4×#10621877C2212yes—
36#14582668C2212yes—
37#14582659C2212yes—
38#14582659C2212yes—
39C2×C4⋊Q8×C43C2211yescharacteristic
40C2×C4×Q8×C43C2211yescharacteristic
41C2×C4×Q8×C43C2211yescharacteristic
42C43×#391C2211yescharacteristic
43C43×#391C2211yescharacteristic
44C43×#387—22no—
45C43×#387—22no—
46C43×#382C2211yescharacteristic
47C43×#376C2211yescharacteristic
48#13063373—22no—
49#13063373—24no—
50#13063373C2×C414yes—
51C2×#10621902—22no—
52C2×#10621898—22no—
53C2×#10621898C2312yes—
54C2×Q8×D43—22no—
55#13063356—24no—
56#13063356C2×C418yescomplemented
57C2×#10621885—22no—
58C2×#10621884—22no—
59C2×#10621877C2312yes—
60C2×#10621877C2312yes—
61Q8×Dic43—22no—
62Q8×Dic43—22no—
63C4×Dic86—22no—
64C4×Dic86—22no—
65C4×Dic86D412yes—
66C4×Dic86D412yes—
67#13063290—24no—
68#13063290—22no—
69C4×D172—22no—
70C4×D172—22no—
71C4×D172D412yes—
72C4×D172D412yes—
73#13063287—24no—
74#13063287—22no—
75C4×C22⋊D43—22no—
76C4×C22⋊D43—24no—
77C4×C22⋊D43—22no—
78#13063284—22no—
79#13063284C2312yes—
80#13063283—24no—
81#13063283—22no—
82#13063282—24no—
83#13063282—22no—
84C4⋊C4×D43—24no—
85C4⋊C4×D43—22no—
86C42×D43—24no—
87C42×D43—22no—
88#13063258—22no—
89#13063256—22no—
90#13063253—24no—
91#13063253C2×C418yescomplemented
92#13063249—22no—
93#13063249—22no—
94#13063248—22no—
95#13063245—22no—
96#13063244—22no—
97#13063244C2×C414yescomplemented
98#13063236—22no—
99C4×C43⋊C8—22no—
100C22×Q8×C43C2311yescharacteristic
101C22×Q8×C43C2311yescharacteristic
102C4⋊Q8×C43—22no—
103C4⋊Q8×C43C2×C414yescomplemented
104C4×Q8×C43—22no—
105C4×Q8×C43—22no—
106C4×Q8×C43—22no—
107C4×Q8×C43—22no—
108C2×C4⋊C4×C43—22no—
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43D412yes—
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C42×C43—21no—
114C2×C42×C43D412yes—
115C2×C42×C43C2311yescharacteristic
116C2×C42×C43C2311yescharacteristic
117C2×C42×C43C2311yescharacteristic
118#95×C43—21no—
119#95×C43—21no—
120#95×C43—22no—
121#95×C43—22no—
122#95×C43—22no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43—21no—
126C2×Q8×C43—22no—
127C2×Q8×C43—22no—
128C2×Q8×C43C22×C412yes—
129C2×Q8×C43C22×C412yes—
130C2×Q8×C43#4112yes—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43C2×D412yes—
136C22×C4×C43C2×D412yes—
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2411yescharacteristic
140C2×C22⋊D43—22no—
141#10621902—28no—
142#10621902—28no—
143Q8×D43—28no—
144Q8×D43—28no—
145#10621898—28no—
146#10621898—28no—
147#10621898—22no—
148#10621898—22no—
149#10621898C22×C414yes—
150#10621898#4114yes—
151C2×D172—22no—
152C2×D172C2×D412yes—
153C2×C4×D43—24no—
154C2×C4×D43—22no—
155C2×C4×D43—22no—
156C2×C4×D43—24no—
157C2×C4×D43—22no—
158C2×Dic86—22no—
159C2×Dic86C2×D412yes—
160C4⋊C4×C43—28no—
161C4⋊C4×C43—24no—
162C4⋊C4×C43—22no—
163C4⋊C4×C43—22no—
164C4⋊C4×C43—41no—
165C4⋊C4×C43—41no—
166C42×C43—22no—
167C42×C43—22no—
168C42×C43—21no—
169C42×C43—22no—
170C42×C43—42no—
171C42×C43C22×C412yes—
172C42×C43C2×D412yes—
173C42×C43C2×D412yes—
174C22⋊Dic43—22no—
175#10621885—28no—
176#10621884—28no—
177(C2×D43)⋊C4—24no—
178(C2×D43)⋊C4—22no—
179(C2×D43)⋊C4—22no—
180C4⋊Dic43—22no—
181C4⋊Dic43—22no—
182C4⋊Dic43C2×D412yes—
183C4⋊Dic43C2×D412yes—
184Dic43⋊C4—24no—
185Dic43⋊C4—22no—
186Dic43⋊C4—22no—
187C4×Dic43—24no—
188C4×Dic43—24no—
189C4×Dic43—22no—
190C4×Dic43—22no—
191C4×Dic43—22no—
192#10621877—22no—
193#10621877—22no—
194#10621877—22no—
195#10621877C22×C414yes—
196C2×C43⋊C8—22no—
197C2×C43⋊C8—22no—
198C23×C43C22×D411yescharacteristic
199C22×D43—22no—
200Q8×C43—24no—
201Q8×C43—24no—
202Q8×C43—24no—
203Q8×C43—42no—
204C2×C4×C43—22no—
205C2×C4×C43—22no—
206C2×C4×C43—22no—
207C2×C4×C43—22no—
208C2×C4×C43—22no—
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×C43—42no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—42no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43C4×D414yes—
222C2×C4×C43#12011yescharacteristic
223C2×C4×C43#12012yes—
224C2×C4×C43#12011yescharacteristic
225C2×C4×C43C23×C411yescharacteristic
226C2×C4×C43C22×D412yes—
227C2×C4×C43C2×#4112yes—
228C2×C4×C43C2×#4111yescharacteristic
229C22⋊D43—24no—
230C22⋊D43—24no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234D172—28no—
235D172—22no—
236D172C4×D414yes—
237C4×D43—216no—
238C4×D43—24no—
239C4×D43—24no—
240C4×D43—44no—
241C4×D43—82no—
242C4×D43—84no—
243Dic86—28no—
244Dic86—22no—
245Dic86C4×D414yes—
246C43⋊C8—24no—
247C43⋊C8—42no—
248#21476—431nomaximal, Sylow
249C22×C43—21no—
250C22×C43—21no—
251C22×C43C2×C4×D411yescharacteristic
252C22×C43C2×#12011yescharacteristic
253C22×C43#52511yescharacteristic
254C2×D43—24no—
255C2×D43—42no—
256C2×D43—42no—
257C4×C43—21no—
258C4×C43—21no—
259C4×C43—22no—
260C4×C43—24no—
261C4×C43—42no—
262C4×C43—42no—
263C4×C43—42no—
264C4×C43—81no—
265C4×C43—81no—
266C4×C43C2×C4×D412yes—
267C4×C43#52512yes—
268Dic43—24no—
269Dic43—42no—
270Dic43—42no—
271C4×2−1+4—431no—
272C2×#456—431no—
273C4×#137—432no—
274#2327D4311yescharacteristic, complemented
275#1886—431no—
276#1882—432no—
277#1860—432no—
278#1857—431no—
279#1793—431no—
280#1759—432no—
281C4×#101—431no—
282C2×C43—21no—
283C2×C43—21no—
284C2×C43C4×#12011yescharacteristic
285C2×C43#302011yescharacteristic
286C2×C43#302211yescharacteristic
287D43—44no—
288C2×2−1+4—431no—
289C2×#137—432no—
290C2×C4⋊Q8C2×D4311yescharacteristic
291#520—432no—
292#520—862no—
293#520—862no—
294#519—431no—
295#519—862no—
296C4×#41—432no—
297C4×#41—862no—
298C4×#41—862no—
299C2×C4×Q8—862no—
300C2×C4×Q8C2×D4311yescharacteristic
301C2×C4×Q8C2×D4311yescharacteristic
302#456—438no—
303#441—862no—
304#440—862no—
305C4×Q16—862no—
306C4×SD16—862no—
307#428—432no—
308C2×#104—432no—
309C2×#104—432no—
310#419—432no—
311#417—432no—
312C2×#101—431no—
313C2×#101—431no—
314C4×M16—432no—
315#391C2×D4311yescharacteristic
316#391C2×D4311yescharacteristic
317#387—22no—
318#387—22no—
319#382C2×D4311yescharacteristic
320#376C2×D4311yescharacteristic
321#343—432no—
322#339—432no—
323#339—432no—
324C43#2147611yescharacteristic, complemented, Sylow
3252−1+4—434no—
3262−1+4—864no—
3272−1+4—862no—
328C2×#41—432no—
329C2×#41—862no—
330C2×#41—862no—
331C22×Q8—862no—
332C22×Q8C22×D4311yescharacteristic
333C22×Q8C22×D4311yescharacteristic
334#137—438no—
335#137—864no—
336C2×Q16—862no—
337C2×SD16—862no—
338C2×M16—432no—
339C2×M16—432no—
340C4⋊Q8—22no—
341C4⋊Q8C4×D4314yescomplemented
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—432no—
347C4×Q8—432no—
348C4×Q8—862no—
349C4×Q8—862no—
350C4×Q8—862no—
351C4×Q8—864no—
352C4×Q8—862no—
353C4×Q8—862no—
354C4×D4—432no—
355C4×D4—432no—
356C4×D4—862no—
357C4×D4—862no—
358C4×D4—862no—
359C4×D4—864no—
360C4×D4—862no—
361C4×D4—862no—
362C4×D4—864no—
363#117—432no—
364#117—864no—
365#117—862no—
366#117—864no—
367#117—862no—
368#117—862no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—862no—
372C2×C4⋊C4—864no—
373C2×C4⋊C4C22⋊D4312yes—
374C2×C4⋊C4C22×D4311yescharacteristic
375C2×C4⋊C4C22×D4311yescharacteristic
376C2×C42—21no—
377C2×C42—864no—
378C2×C42—862no—
379C2×C42C22⋊D4312yes—
380C2×C42C22×D4311yescharacteristic
381C2×C42C22×D4311yescharacteristic
382C2×C42C22×D4311yescharacteristic
383#107—862no—
384#106—862no—
385#104—438no—
386#104—864no—
387#103—862no—
388#103—862no—
389#103—862no—
390#102—862no—
391#101—434no—
392#101—862no—
393#97—862no—
394C4×C8—862no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#41—434no—
401#41—434no—
402#41—868no—
403#41—868no—
404#41—868no—
405#41—868no—
406#41—862no—
407#41—862no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—432no—
414C2×Q8—862no—
415C2×Q8—868no—
416C2×Q8—868no—
417C2×Q8C2×C4×D4312yes—
418C2×Q8C2×C4×D4312yes—
419C2×Q8#1062189812yes—
420C2×D4—432no—
421C2×D4—862no—
422C2×D4—862no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—864no—
428C22×C4—864no—
429C22×C4—862no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4C2×C22⋊D4312yes—
433C22×C4C2×C22⋊D4312yes—
434C22×C4C2×C22⋊D4311yescharacteristic
435C22×C4C2×C22⋊D4311yescharacteristic
436C22×C4C23×D4311yescharacteristic
437Q16—868no—
438SD16—868no—
439M16—434no—
440M16—862no—
441M16—862no—
442M16—862no—
443C2×C8—862no—
444C2×C8—862no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—432no—
452C4⋊C4—432no—
453C4⋊C4—864no—
454C4⋊C4—862no—
455C4⋊C4—862no—
456C4⋊C4—862no—
457C4⋊C4—862no—
458C22⋊C4—864no—
459C22⋊C4—862no—
460C22⋊C4—862no—
461C22⋊C4—862no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—22no—
466C42—42no—
467C42—864no—
468C42—864no—
469C42—862no—
470C42—862no—
471C42—862no—
472C42C2×C4×D4312yes—
473C42D4×D4312yes—
474C42D4×D4312yes—
475C23—862no—
476C23C22×C22⋊D4311yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—24no—
480Q8—42no—
481Q8—434no—
482Q8—868no—
483Q8—862no—
484D4—434no—
485D4—868no—
486D4—864no—
487D4—864no—
488D4—862no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—8616no—
507C2×C4—862no—
508C2×C4—864no—
509C2×C4—864no—
510C2×C4—862no—
511C2×C4—862no—
512C2×C4—3442no—
513C2×C4—3444no—
514C2×C4—1724no—
515C2×C4C4×C22⋊D4314yes—
516C2×C4#1306329311yescharacteristic
517C2×C4#1306329311yescharacteristic
518C2×C4#1306329312yes—
519C2×C4C22×C4×D4311yescharacteristic
520C2×C4C2×D4×D4312yes—
521C2×C4#41×D4312yes—
522C2×C4C2×#1062189811yescharacteristic
523C8—864no—
524C8—1722no—
525C22—21no—
526C22—21no—
527C22—864no—
528C22—1722no—
529C22—1722no—
530C22C2×C4×C22⋊D4311yescharacteristic
531C22C2×#1306329311yescharacteristic
532C22#1458340611yescharacteristic
533C4—21no—
534C4—21no—
535C4—22no—
536C4—24no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—81no—
541C4—81no—
542C4—864no—
543C4—1722no—
544C4—1722no—
545C4C4×D4×D4312yes—
546C4#1458340812yes—
547C2—21no—
548C2—21no—
549C2—1724no—
550C2C4×#1306329311yescharacteristic
551C2#1914395811yescharacteristic
552C2#1914397311yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#2327C2
socleC22×C43C2×#120
prime radicalC2×#13063373C22
cosocleC22×C4×C43C24
hypercenter#2327D43
hypocenterC43#21476
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
3. C4×#13063356C2semidirect
4. #19134989C2semidirect
5. #19134967C2semidirect
6. #19134827C2semidirect
7. #19134809C2semidirect
9. #19134406C2semidirect
10. C4×#13063244C2semidirect
11. C43×#2327C2semidirect
23. #14583081C4semidirect
56. #13063356C2×C4semidirect
91. #13063253C2×C4semidirect
97. #13063244C2×C4semidirect
103. C4⋊Q8×C43C2×C4semidirect
274. #2327D43semidirect
324. C43#21476semidirect
341. C4⋊Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C4×#130633732
C2×#145830812
C4×#130633562
#191349892
#191349672
#191348272
#191348092
#191345532
#191344062
C4×#130632442
C43×#23272
#2147643