Finite Group Databasebeta

← #25130514

Subgroups — #25130514

Order 11008.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583610C211yesmaximal, characteristic, complemented
2C2×C4×#10621883C211yesmaximal, characteristic, complemented
3#19139508C211yesmaximal, characteristic
4#19135793C212yesmaximal, complemented
5#19135599C212yesmaximal, complemented
6#19135557C211yesmaximal, characteristic, complemented
7#19135388C211yesmaximal, characteristic, complemented
8#19134909C212yesmaximal, complemented
9#19134786C212yesmaximal, complemented
10#19134601C211yesmaximal, characteristic, complemented
11C2×C43×#476C211yesmaximal, characteristic, complemented
12C22×#10621883C2211yescharacteristic
13#14583610—24no—
14#14583610—24no—
15C2×#13063316—21no—
16C2×#13063305C2212yes—
17C2×#13063301—22no—
18C2×#13063299—21no—
19C2×#13063296C2212yes—
20C2×C4×Dic86C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×#13063287—21no—
23C2×C4⋊C4×D43C2212yes—
24C4×#10621883—24no—
25C2×#13063256C2211yescharacteristic
26C2×#13063250C2212yes—
27C2×#13063248C2211yescharacteristic
28C2×#13063246C2211yescharacteristic
29C2×(C2×D43)⋊C8C2212yes—
30C2×(C2×D43)⋊C8C2212yes—
31C2×C4×C43⋊C8C2211yescharacteristic
32#14582815—22no—
33#14582769C2211yescharacteristic
34#14582759—21no—
35#14582756C2212yes—
36#14582662C2212yes—
37#14582661C2212yes—
38#14582661C2212yes—
39C2×C4⋊Q8×C43C2211yescharacteristic
40C2×C4×D4×C43C2211yescharacteristic
41C43×#476—24no—
42C2×#106×C43C2212yes—
43C2×C4⋊C8×C43C2211yescharacteristic
44C2×#102×C43C2212yes—
45C22×Dic86C2311yescharacteristic
46C22×D172D411yescharacteristic
47C22×C22⋊D43—22no—
48C22×C4×D43—22no—
49C22×C4×D43C2312yes—
50C2×#10621883—22no—
51C2×#10621883—22no—
52C2×#10621883—22no—
53C22×C43⋊C8C2311yescharacteristic
54C22×C43⋊C8C2311yescharacteristic
55#13063316—28no—
56#13063305—24no—
57#13063305—24no—
58#13063301—216no—
59#13063299—44no—
60#13063296—24no—
61#13063296—42no—
62C4×Dic86—22no—
63C4×Dic86D414yescomplemented
64C4×D172—22no—
65C4×D172—22no—
66#13063287—28no—
67C2×C4⋊Dic43—22no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43D411yescharacteristic
71C2×C4⋊Dic43C2312yes—
72C2×C4⋊Dic43C2311yescharacteristic
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—22no—
75C4⋊C4×D43—24no—
76C4⋊C4×D43—24no—
77C2×C22⋊Dic43—22no—
78C2×(C2×D43)⋊C4—22no—
79C2×(C2×D43)⋊C4—22no—
80C2×(C2×D43)⋊C4D412yes—
81C2×C4×Dic43—21no—
82#13063256—22no—
83#13063250—42no—
84#13063248—22no—
85#13063246—22no—
86(C2×D43)⋊C8—24no—
87(C2×D43)⋊C8—42no—
88C4×C43⋊C8—22no—
89C43⋊#95—22no—
90C22×Q8×C43—21no—
91C22×D4×C43C2311yescharacteristic
92C23×C4×C43—21no—
93C22×C8×C43C2312yes—
94C4⋊Q8×C43—22no—
95C4⋊Q8×C43D414yescomplemented
96C4×D4×C43—22no—
97C4×D4×C43—22no—
98C2×C4⋊C4×C43—21no—
99C2×C4⋊C4×C43C2311yescharacteristic
100C2×C4⋊C4×C43C2312yes—
101C2×C22⋊C4×C43—21no—
102C2×C42×C43C2311yescharacteristic
103#106×C43—24no—
104C4⋊C8×C43—22no—
105#102×C43—24no—
106C24×C43—21no—
107C23×D43C2×D412yes—
108C2×Q8×C43—24no—
109C2×Q8×C43—81no—
110C2×Q8×C43—81no—
111C2×D4×C43—21no—
112C2×D4×C43—21no—
113C2×D4×C43—21no—
114C2×D4×C43—21no—
115C2×D4×C43—21no—
116C2×D4×C43#4112yes—
117C22×C4×C43—22no—
118C22×C4×C43—21no—
119C22×C4×C43—42no—
120C22×C4×C43—42no—
121C22×C4×C43—42no—
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2411yescharacteristic
125C2×C22⋊D43—44no—
126C2×C22⋊D43—44no—
127C2×C22⋊D43—82no—
128C22×Dic43—21no—
129C22×Dic43—21no—
130C22×Dic43—22no—
131C22×Dic43C2×D412yes—
132C2×D172—22no—
133C2×D172—22no—
134C2×D172SD1614yescomplemented
135C2×C4×D43—22no—
136C2×C4×D43—22no—
137C2×C4×D43—22no—
138C2×C4×D43—22no—
139C2×C4×D43—22no—
140C2×C4×D43—22no—
141C2×C4×D43—28no—
142C2×C4×D43—82no—
143C2×C4×D43—82no—
144C2×Dic86—21no—
145C2×Dic86—21no—
146C2×Dic86—21no—
147C2×Dic86—21no—
148C2×Dic86—21no—
149C2×Dic86C2×D412yes—
150C2×C8×C43—22no—
151C2×C8×C43—22no—
152C2×C8×C43—22no—
153C4⋊C4×C43—24no—
154C4⋊C4×C43—22no—
155C4⋊C4×C43—21no—
156C4⋊C4×C43—21no—
157C4⋊C4×C43C2×D414yes—
158C22⋊C4×C43—42no—
159C42×C43C2×D412yes—
160C42×C43C2×D412yes—
161C22⋊Dic43—44no—
162#10621883—28no—
163(C2×D43)⋊C4—28no—
164(C2×D43)⋊C4—24no—
165(C2×D43)⋊C4—44no—
166C4⋊Dic43—28no—
167C4⋊Dic43—22no—
168C4⋊Dic43—21no—
169C4⋊Dic43—22no—
170C4⋊Dic43—22no—
171C4⋊Dic43—42no—
172C4⋊Dic43—81no—
173C4⋊Dic43C2×D412yes—
174Dic43⋊C4—28no—
175Dic43⋊C4—24no—
176C4×Dic43—24no—
177C2×C43⋊C8—21no—
178C2×C43⋊C8—21no—
179C2×C43⋊C8—21no—
180C2×C43⋊C8—21no—
181C2×C43⋊C8—21no—
182C2×C43⋊C8#4112yes—
183C23×C43—22no—
184C23×C43—41no—
185C23×C43—41no—
186C23×C43—41no—
187C23×C43—41no—
188C23×C43—41no—
189C23×C43—41no—
190C23×C43C22×D411yescharacteristic
191C22×D43—22no—
192C22×D43—22no—
193C22×D43—22no—
194C22×D43—22no—
195C22×D43—22no—
196C22×D43—22no—
197C22×D43C2×SD1614yes—
198Q8×C43—84no—
199D4×C43—22no—
200D4×C43—22no—
201D4×C43—41no—
202D4×C43—41no—
203C2×C4×C43—21no—
204C2×C4×C43—21no—
205C2×C4×C43—24no—
206C2×C4×C43—42no—
207C2×C4×C43—42no—
208C2×C4×C43—41no—
209C2×C4×C43—41no—
210C2×C4×C43—41no—
211C2×C4×C43—84no—
212C2×C4×C43#12012yes—
213C2×C4×C43#12012yes—
214C2×C4×C43C2×SD1612yes—
215C2×C4×C43#13712yes—
216C2×C4×C43C22×D411yescharacteristic
217C2×C4×C43C22×D411yescharacteristic
218C2×C4×C43C2×#4111yescharacteristic
219C2×C4×C43C2×#4111yescharacteristic
220C2×C4×C43H(5,𝔽2)11yescharacteristic
221C2×C4×C432−1+411yescharacteristic
222C22⋊D43—88no—
223C2×Dic43—22no—
224C2×Dic43—22no—
225C2×Dic43—22no—
226C2×Dic43—22no—
227C2×Dic43—22no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43—41no—
232C2×Dic43—42no—
233C2×Dic43—42no—
234C2×Dic43—42no—
235D172—28no—
236C4×D43—42no—
237C4×D43—42no—
238C4×D43—42no—
239C4×D43—42no—
240C4×D43—88no—
241Dic86—22no—
242Dic86—22no—
243Dic86—41no—
244Dic86—41no—
245C8×C43—42no—
246C43⋊C8—22no—
247C43⋊C8—41no—
248#24118—431nomaximal, Sylow
249C22×C43—42no—
250C22×C43—42no—
251C22×C43—42no—
252C22×C43—81no—
253C22×C43—81no—
254C22×C43—81no—
255C22×C43—81no—
256C22×C43C2×#12011yescharacteristic
257C22×C43#54611yescharacteristic
258C22×C43#54711yescharacteristic
259C22×C43C22×SD1611yescharacteristic
260C22×C43C2×#13711yescharacteristic
261C22×C43#57611yescharacteristic
262C22×C43#57711yescharacteristic
263C2×D43—24no—
264C2×D43—24no—
265C2×D43—24no—
266C2×D43—42no—
267C2×D43—42no—
268C2×D43—42no—
269C2×D43—42no—
270C4×C43—22no—
271C4×C43—44no—
272C4×C43—81no—
273C4×C43#44814yes—
274C4×C43#52511yescharacteristic
275C4×C43#53711yescharacteristic
276C4×C43#54811yescharacteristic
277C4×C43#55011yescharacteristic
278Dic43—42no—
279Dic43—42no—
280Dic43—42no—
281Dic43—82no—
282C2×#554—431no—
283C2×#476D4311yescharacteristic, complemented
284C2×C4×SD16—431no—
285#2537—431no—
286#2035—432no—
287#2000—432no—
288#1990—431no—
289#1954—431no—
290#1874—432no—
291#1853—432no—
292#1803—431no—
293C2×C43—82no—
294C2×C43#240711yescharacteristic
295C2×C43C2×#44811yescharacteristic
296C2×C43#301211yescharacteristic
297C2×C43#329511yescharacteristic
298C2×C43#333111yescharacteristic
299C2×C43#333511yescharacteristic
300C2×C43#335711yescharacteristic
301D43—44no—
302C22×SD16—431no—
303#554—864no—
304#554—864no—
305C2×C4⋊Q8C2×D4311yescharacteristic
306C2×#125—861no—
307C2×C22⋊Q8—432no—
308C2×C22⋊Q8—862no—
309C2×#121—432no—
310C2×#121—861no—
311C2×C4×Q8—431no—
312C2×C4×D4—431no—
313C2×C4×D4—861no—
314C2×C4×D4C2×D4311yescharacteristic
315C22×C4⋊C4—432no—
316#476—24no—
317C4×SD16—864no—
318C2×#106—431no—
319C2×#106C2×D4312yes—
320C2×C4⋊C8C2×D4311yescharacteristic
321C2×#103—431no—
322C2×#103—432no—
323C2×#102—431no—
324C2×#102C2×D4312yes—
325C2×C22⋊C8—432no—
326C2×C22⋊C8—432no—
327C2×C4×C8—431no—
328#394—862no—
329#384—431no—
330#380—432no—
331#380—861no—
332#340—432no—
333#340—432no—
334#340—432no—
335C43#2411811yescharacteristic, complemented, Sylow
336C22×Q8—21no—
337C22×Q8—431no—
338C22×D4—431no—
339C22×D4—862no—
340C22×D4C22×D4311yescharacteristic
341C23×C4—21no—
342C23×C4—432no—
343C23×C4—862no—
344C2×SD16—862no—
345C2×SD16—862no—
346C2×SD16—862no—
347C22×C8—431no—
348C22×C8—431no—
349C22×C8C22×D4312yes—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D4314yescomplemented
352#125—868no—
353C22⋊Q8—864no—
354C22⋊Q8—8616no—
355C22⋊Q8—864no—
356#121—864no—
357#121—1722no—
358#121—1724no—
359C4×Q8—434no—
360C4×Q8—862no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—868no—
364C4×D4—862no—
365C4×D4—862no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—431no—
368C2×C4⋊C4—431no—
369C2×C4⋊C4—432no—
370C2×C4⋊C4—864no—
371C2×C4⋊C4—862no—
372C2×C4⋊C4—862no—
373C2×C4⋊C4—861no—
374C2×C4⋊C4—864no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—861no—
377C2×C4⋊C4C22×D4311yescharacteristic
378C2×C4⋊C4C22×D4312yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—432no—
381C2×C22⋊C4—862no—
382C2×C22⋊C4—862no—
383C2×C22⋊C4—862no—
384C2×C42—861no—
385C2×C42C22×D4311yescharacteristic
386#106—24no—
387#106—862no—
388C4⋊C8—22no—
389#103—862no—
390#103—1722no—
391#102—24no—
392#102—862no—
393C22⋊C8—864no—
394C22⋊C8—1722no—
395C4×C8—862no—
396#95—862no—
397C24—21no—
398C24—432no—
399C2×Q8—24no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—432no—
403C2×Q8—861no—
404C2×Q8—861no—
405C2×Q8—861no—
406C2×Q8—861no—
407C2×Q8—861no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—434no—
414C2×D4—862no—
415C2×D4—862no—
416C2×D4—3442no—
417C2×D4—1724no—
418C2×D4—1724no—
419C2×D4#1062190212yes—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—432no—
426C22×C4—868no—
427C22×C4—861no—
428C22×C4—861no—
429C22×C4—862no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—862no—
433C22×C4—862no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—3442no—
437C22×C4—3442no—
438C22×C4D4×D4311yescharacteristic
439C22×C4D4×D4311yescharacteristic
440C22×C4C23×D4311yescharacteristic
441SD16—868no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—432no—
446C2×C8—861no—
447C2×C8—861no—
448C2×C8—861no—
449C2×C8—861no—
450C2×C8—861no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—432no—
456C4⋊C4—868no—
457C4⋊C4—868no—
458C4⋊C4—862no—
459C4⋊C4—864no—
460C4⋊C4—861no—
461C4⋊C4—862no—
462C4⋊C4—862no—
463C4⋊C4—3441no—
464C4⋊C4—1722no—
465C4⋊C4C2×C22⋊D4314yes—
466C22⋊C4—42no—
467C22⋊C4—868no—
468C22⋊C4—864no—
469C22⋊C4—1724no—
470C22⋊C4—1724no—
471C42—864no—
472C42D4×D4312yes—
473C42C2×C22⋊D4312yes—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—434no—
482C23—862no—
483C23—862no—
484C23—862no—
485C23—862no—
486C23—862no—
487C23—862no—
488C23C2×D4×D4311yescharacteristic
489Q8—84no—
490Q8—862no—
491Q8—862no—
492Q8—1721no—
493Q8—1721no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498D4—868no—
499D4—3448no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—862no—
510C2×C4—862no—
511C2×C4—862no—
512C2×C4—862no—
513C2×C4—862no—
514C2×C4—3448no—
515C2×C4—1722no—
516C2×C4—1722no—
517C2×C4—1722no—
518C2×C4—1721no—
519C2×C4—1721no—
520C2×C4—1721no—
521C2×C4—1721no—
522C2×C4—1722no—
523C2×C4—1722no—
524C2×C4—1722no—
525C2×C4—1722no—
526C2×C4#1306329312yes—
527C2×C4#1306329312yes—
528C2×C4SD16×D4312yes—
529C2×C4#1306335412yes—
530C2×C4C2×D4×D4311yescharacteristic
531C2×C4C22×C22⋊D4311yescharacteristic
532C2×C4#41×D4311yescharacteristic
533C2×C4C2×#1062190211yescharacteristic
534C2×C4#1306337111yescharacteristic
535C2×C4#1306337311yescharacteristic
536C8—42no—
537C8—862no—
538C8—1721no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—864no—
547C22—864no—
548C22—864no—
549C22—1722no—
550C22—1722no—
551C22—1722no—
552C22—1722no—
553C22C2×#1306329311yescharacteristic
554C22#1458354011yescharacteristic
555C22#1458355111yescharacteristic
556C22C2×SD16×D4311yescharacteristic
557C22C2×#1306335411yescharacteristic
558C22#1458370311yescharacteristic
559C22#1458370911yescharacteristic
560C4—22no—
561C4—44no—
562C4—81no—
563C4—3442no—
564C4—1722no—
565C4—1722no—
566C4—1722no—
567C4#1458303614yes—
568C4#1458341211yescharacteristic
569C4#1458347911yescharacteristic
570C4#1458356011yescharacteristic
571C4#1458356811yescharacteristic
572C2—82no—
573C2—1724no—
574C2#1913794211yescharacteristic
575C2C2×#1458303611yescharacteristic
576C2#1914389811yescharacteristic
577C2#1914641611yescharacteristic
578C2#1914690811yescharacteristic
579C2#1914694911yescharacteristic
580C2#1914722411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#476C2
socleC23×C43C22×D4
prime radical#19139508C2
hypercenterC2×#476D43
hypocenterC43#24118
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#14583610C2semidirect
2. C2×C4×#10621883C2semidirect
4. #19135793C2semidirect
5. #19135599C2semidirect
6. #19135557C2semidirect
7. #19135388C2semidirect
8. #19134909C2semidirect
9. #19134786C2semidirect
10. #19134601C2semidirect
11. C2×C43×#476C2semidirect
63. C4×Dic86D4semidirect
95. C4⋊Q8×C43D4semidirect
134. C2×D172SD16semidirect
283. C2×#476D43semidirect
335. C43#24118semidirect
351. C4⋊Q8C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145836102
C2×C4×#106218832
#191395082
#191357932
#191355992
#191355572
#191353882
#191349092
#191347862
#191346012
C2×C43×#4762
#2411843