Finite Group Databasebeta

← #25123898

Subgroups — #25123898

Order 11008.

1498 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×C4⋊Q8×D43C211yesmaximal, characteristic, complemented
2C2×#14582899C212yesmaximal
3#19137084C211yesmaximal, characteristic
4#19135736C211yesmaximal, characteristic, complemented
5#19135734C212yesmaximal, complemented
6#19135732C211yesmaximal, characteristic, complemented
7#19135730C212yesmaximal, complemented
8#19135728C211yesmaximal, characteristic, complemented
9#19134313C212yesmaximal, complemented
10#19134287C211yesmaximal, characteristic, complemented
11C43×#2024C211yesmaximal, characteristic, complemented
12C22×#10621902C2211yescharacteristic
13C22×Q8×D43—21no—
14C22×#10621872C2212yes—
15C2×#13063325C2211yescharacteristic
16C2×#13063324—21no—
17C2×#13063324C2212yes—
18C2×#13063323—21no—
19C4⋊Q8×D43—28no—
20C2×#13063283C2211yescharacteristic
21C2×C4⋊C4×D43—21no—
22C2×C4⋊C4×D43C2212yes—
23C2×C42×D43C2211yescharacteristic
24#14582899—28no—
25C2×#13063249C2212yes—
26C2×#13063247C2212yes—
27C2×#13063245C2212yes—
28#14582806C2211yescharacteristic
29#14582801C2211yescharacteristic
30#14582800C2211yescharacteristic
31#14582777—21no—
32#14582777—21no—
33#14582746—21no—
34#14582746—21no—
35#14582662C2212yes—
36#14582661C2212yes—
37#14582661C2212yes—
38#14582654C2211yescharacteristic
39#14582653C2212yes—
40C2×C4⋊Q8×C43C2211yescharacteristic
41C2×#103×C43C2212yes—
42C43×#391C2211yescharacteristic
43C43×#340C2212yes—
44C43×#336C2211yescharacteristic
45C2×#10621902—22no—
46C2×#10621902—24no—
47C2×#10621902—24no—
48C2×#10621902Q814yescomplemented
49C22×Dic86—21no—
50C22×Dic86—22no—
51C2×Q8×D43—28no—
52C2×Q8×D43—44no—
53C2×Q8×D43—44no—
54C22×D172—21no—
55C22×D172C2311yescharacteristic
56C22×C4×D43—22no—
57C22×C4×D43—21no—
58C22×C4×D43D411yescharacteristic
59C22×C4×D43D411yescharacteristic
60C22×C4×D43C2311yescharacteristic
61C2×#10621872—24no—
62C2×#10621872—24no—
63C2×#10621872—24no—
64C22×C43⋊C8C2312yes—
65#13063325—24no—
66#13063324—28no—
67#13063324—24no—
68#13063324—24no—
69#13063323—28no—
70#13063283—24no—
71C2×C4⋊Dic43—21no—
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2312yes—
74C2×Dic43⋊C4—22no—
75C2×Dic43⋊C4—22no—
76C4⋊C4×D43—28no—
77C4⋊C4×D43—24no—
78C4⋊C4×D43D418yes—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88C42×D43—24no—
89#13063249—24no—
90#13063247—24no—
91#13063245—24no—
92C43⋊#95—21no—
93C43⋊#95—21no—
94C22×Q8×C43—21no—
95C22×Q8×C43C2311yescharacteristic
96C22×C8×C43C2312yes—
97C4⋊Q8×C43—24no—
98C2×C4⋊C4×C43—21no—
99C2×C4⋊C4×C43C2311yescharacteristic
100C2×C4⋊C4×C43C2312yes—
101C2×C42×C43C2311yescharacteristic
102#103×C43—24no—
103#95×C43—21no—
104#95×C43—21no—
105C23×D43—21no—
106C23×D43C2×D411yescharacteristic
107C2×Q8×C43—24no—
108C2×Q8×C43—21no—
109C2×Q8×C43—22no—
110C2×Q8×C43—22no—
111C2×Q8×C43—42no—
112C2×Q8×C43—42no—
113C2×Q8×C43C2×Q812yescomplemented
114C22×C4×C43—21no—
115C22×C4×C43—22no—
116C22×C4×C43—21no—
117C22×C4×C43C2×D411yescharacteristic
118C22×C4×C43C2×D411yescharacteristic
119C22×C4×C43C2411yescharacteristic
120C22×Dic43—22no—
121C22×Dic43—21no—
122C22×Dic43C2×D411yescharacteristic
123C22×Dic43C2×D411yescharacteristic
124C22×Dic43C2×D411yescharacteristic
125#10621902—216no—
126#10621902—48no—
127Q8×D43—432no—
128C2×D172—21no—
129C2×D172—24no—
130C2×D172—22no—
131C2×D172—22no—
132C2×D172—42no—
133C2×D172—42no—
134C2×D172C2×Q812yescomplemented
135C2×C4×D43—22no—
136C2×C4×D43—22no—
137C2×C4×D43—22no—
138C2×C4×D43—22no—
139C2×C4×D43—24no—
140C2×C4×D43—22no—
141C2×C4×D43—22no—
142C2×C4×D43—28no—
143C2×C4×D43—44no—
144C2×C4×D43—44no—
145C2×C4×D43—42no—
146C2×C4×D43—42no—
147C2×C4×D43C2×D414yes—
148C2×C4×D43C2×Q812yes—
149C2×C4×D43#4112yes—
150C2×C4×D43#4114yes—
151C2×Dic86—24no—
152C2×Dic86—28no—
153C2×Dic86—44no—
154C2×Dic86—42no—
155C2×Dic86—42no—
156C2×Dic86—44no—
157C2×C8×C43—22no—
158C2×C8×C43—22no—
159C2×C8×C43—22no—
160C4⋊C4×C43—24no—
161C4⋊C4×C43—22no—
162C4⋊C4×C43—22no—
163C4⋊C4×C43C2×D414yes—
164C42×C43—22no—
165(C2×D43)⋊C4—42no—
166(C2×D43)⋊C4—42no—
167(C2×D43)⋊C4—42no—
168(C2×D43)⋊C4—42no—
169(C2×D43)⋊C4—42no—
170C4⋊Dic43—24no—
171C4⋊Dic43—22no—
172C4⋊Dic43—22no—
173C4⋊Dic43C2×D414yes—
174Dic43⋊C4—28no—
175Dic43⋊C4—28no—
176C4×Dic43—22no—
177C4×Dic43—22no—
178C4×Dic43—22no—
179C4×Dic43—42no—
180C2×C43⋊C8—22no—
181C2×C43⋊C8—22no—
182C2×C43⋊C8—22no—
183#10621872—48no—
184C23×C43C22×D411yescharacteristic
185C22×D43—21no—
186C22×D43—21no—
187C22×D43—21no—
188C22×D43—21no—
189C22×D43—22no—
190C22×D43—41no—
191C22×D43—41no—
192C22×D43—41no—
193C22×D43—41no—
194C22×D43—41no—
195C22×D43—41no—
196C22×D43#12012yes—
197C22×D43C22⋊Q812yes—
198C22×D43#12412yes—
199Q8×C43—24no—
200Q8×C43—48no—
201Q8×C43—42no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—21no—
205C2×C4×C43—21no—
206C2×C4×C43—24no—
207C2×C4×C43—22no—
208C2×C4×C43—41no—
209C2×C4×C43—41no—
210C2×C4×C43—42no—
211C2×C4×C43—42no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43AGL(1,ℤ/8)12yes—
216C2×C4×C43#13712yes—
217C2×C4×C43C22×D412yes—
218C2×C4×C43C22×Q811yescharacteristic
219C2×C4×C43C2×#4112yes—
220C2×C4×C43C2×#4111yescharacteristic
221C2×Dic43—21no—
222C2×Dic43—21no—
223C2×Dic43—21no—
224C2×Dic43—21no—
225C2×Dic43—24no—
226C2×Dic43—42no—
227C2×Dic43—42no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43#12012yes—
232C2×Dic43C22⋊Q812yes—
233C2×Dic43#12412yes—
234C2×Dic43AGL(1,ℤ/8)12yes—
235C2×Dic43#13712yes—
236D172—24no—
237D172—48no—
238D172—42no—
239C4×D43—28no—
240C4×D43—28no—
241C4×D43—416no—
242C4×D43—48no—
243C4×D43C22⋊Q818yes—
244C4×D43#12418yes—
245Dic86—416no—
246Dic86—48no—
247C8×C43—42no—
248C43⋊C8—42no—
249#23452—431nomaximal, Sylow
250C22×C43C2×#12011yescharacteristic
251C22×C43C2×C22⋊Q811yescharacteristic
252C22×C43C2×#12411yescharacteristic
253C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C43C2×#13711yescharacteristic
256C22×C43C2×#13711yescharacteristic
257C2×D43—22no—
258C2×D43—22no—
259C2×D43—22no—
260C2×D43—22no—
261C2×D43—22no—
262C2×D43—22no—
263C2×D43—42no—
264C2×D43—42no—
265C2×D43—42no—
266C2×D43—81no—
267C2×D43—81no—
268C2×D43—81no—
269C2×D43—81no—
270C2×D43#39314yes—
271C4×C43—22no—
272C4×C43—22no—
273C4×C43—42no—
274C4×C43—44no—
275C4×C43—81no—
276C4×C43C2×C22⋊Q812yes—
277C4×C43C2×#12412yes—
278Dic43—22no—
279Dic43—22no—
280Dic43—44no—
281Dic43—81no—
282Dic43#39314yes—
283C22×C4⋊Q8—431no—
284C2×#417—432no—
285#2297—431no—
286#2024—432no—
287#2024—431no—
288#2024D4311yescharacteristic, complemented
289#2023—432no—
290#2023—431no—
291#2023—431no—
292#1736—432no—
293#1728—431no—
294C2×C43C2×#39311yescharacteristic
295C2×C43#300311yescharacteristic
296C2×C43#300411yescharacteristic
297C2×C43#307911yescharacteristic
298C2×C43#307911yescharacteristic
299C2×C43#314111yescharacteristic
300C2×C43#314111yescharacteristic
301D43—24no—
302D43—82no—
303C22×#41—431no—
304C23×Q8—861no—
305C22×M16—432no—
306C2×C4⋊Q8—432no—
307C2×C4⋊Q8—431no—
308C2×C4⋊Q8—861no—
309C2×C4⋊Q8—861no—
310C2×C4⋊Q8—868no—
311C2×C4⋊Q8C2×D4311yescharacteristic
312C2×#117—431no—
313C22×C4⋊C4—432no—
314C22×C4⋊C4—861no—
315C22×C42—431no—
316#417—868no—
317C2×#103—432no—
318C2×#103C2×D4312yes—
319C2×#102—432no—
320C2×#102—432no—
321#391—431no—
322#391C2×D4311yescharacteristic
323#390—431no—
324#390—431no—
325#386—861no—
326#386—861no—
327C4×C22⋊C4—861no—
328C4×C22⋊C4—861no—
329#340—432no—
330#340—432no—
331#340—432no—
332#340C2×D4312yes—
333#336—432no—
334#336—431no—
335#336C2×D4311yescharacteristic
336C43#2345211yescharacteristic, complemented, Sylow
337C2×#41—434no—
338C2×#41—862no—
339C2×#41—864no—
340C2×#41—864no—
341C22×Q8—21no—
342C22×Q8—862no—
343C22×Q8—868no—
344C22×Q8—861no—
345C22×Q8—1724no—
346C22×Q8—1724no—
347C22×Q8C22×D4311yescharacteristic
348C22×D4—431no—
349C22×D4—861no—
350C23×C4—431no—
351C23×C4—431no—
352C23×C4—431no—
353C23×C4—861no—
354C23×C4—862no—
355C2×M16—864no—
356C2×M16—864no—
357C2×M16—864no—
358C22×C8—432no—
359C22×C8C22×D4312yes—
360C4⋊Q8—24no—
361C4⋊Q8—864no—
362C4⋊Q8—864no—
363C4⋊Q8—868no—
364C4⋊Q8—868no—
365C4⋊Q8—864no—
366#117—864no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—432no—
369C2×C4⋊C4—438no—
370C2×C4⋊C4—431no—
371C2×C4⋊C4—864no—
372C2×C4⋊C4—868no—
373C2×C4⋊C4—862no—
374C2×C4⋊C4—862no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4C22×D4311yescharacteristic
377C2×C4⋊C4C22×D4312yes—
378C2×C22⋊C4—861no—
379C2×C22⋊C4—861no—
380C2×C22⋊C4—861no—
381C2×C22⋊C4—861no—
382C2×C22⋊C4—861no—
383C2×C42—431no—
384C2×C42—431no—
385C2×C42—431no—
386C2×C42—864no—
387C2×C42—861no—
388C2×C42C22×D4311yescharacteristic
389#103—24no—
390#103—864no—
391#102—864no—
392#102—864no—
393#95—21no—
394#95—21no—
395#95—861no—
396#95—861no—
397C24—431no—
398C24—861no—
399#41—8616no—
400#41—1728no—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—864no—
408C2×Q8—868no—
409C2×Q8—1722no—
410C2×Q8—1724no—
411C2×Q8—1724no—
412C2×Q8—17232no—
413C2×Q8—1722no—
414C2×Q8Q8×D4312yescomplemented
415C2×D4—432no—
416C2×D4—864no—
417C2×D4—862no—
418C2×D4—862no—
419C2×D4—861no—
420C2×D4—1722no—
421C2×D4—1722no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—431no—
426C22×C4—434no—
427C22×C4—432no—
428C22×C4—434no—
429C22×C4—432no—
430C22×C4—431no—
431C22×C4—431no—
432C22×C4—862no—
433C22×C4—862no—
434C22×C4—864no—
435C22×C4—862no—
436C22×C4—862no—
437C22×C4—868no—
438C22×C4—862no—
439C22×C4—862no—
440C22×C4—862no—
441C22×C4—861no—
442C22×C4—1722no—
443C22×C4—1722no—
444C22×C4—1724no—
445C22×C4—1724no—
446C22×C4D4×D4311yescharacteristic
447C22×C4D4×D4311yescharacteristic
448C22×C4C23×D4311yescharacteristic
449M16—1728no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—862no—
454C2×C8—862no—
455C2×C8—862no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—434no—
460C4⋊C4—868no—
461C4⋊C4—868no—
462C4⋊C4—864no—
463C4⋊C4—862no—
464C4⋊C4—862no—
465C4⋊C4D4×D4314yes—
466C22⋊C4—1722no—
467C22⋊C4—1722no—
468C22⋊C4—1722no—
469C22⋊C4—1722no—
470C22⋊C4—1722no—
471C42—22no—
472C42—862no—
473C42—862no—
474C42—862no—
475C42—1722no—
476C23—432no—
477C23—432no—
478C23—432no—
479C23—861no—
480C23—861no—
481C23—861no—
482C23—861no—
483C23—862no—
484C23—1721no—
485C23—1721no—
486C23—1721no—
487C23—1721no—
488C23—1721no—
489C23—1721no—
490C23C2×D4×D4311yescharacteristic
491Q8—24no—
492Q8—48no—
493Q8—42no—
494Q8—1728no—
495Q8—17216no—
496D4—864no—
497D4—1728no—
498D4—1722no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—432no—
513C2×C4—432no—
514C2×C4—432no—
515C2×C4—438no—
516C2×C4—438no—
517C2×C4—432no—
518C2×C4—432no—
519C2×C4—868no—
520C2×C4—868no—
521C2×C4—861no—
522C2×C4—861no—
523C2×C4—861no—
524C2×C4—861no—
525C2×C4—864no—
526C2×C4—1728no—
527C2×C4—17216no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—1722no—
532C2×C4—1722no—
533C2×C4#1306335212yes—
534C2×C4#1306335712yes—
535C2×C4C2×D4×D4312yes—
536C2×C4C2×Q8×D4311yescharacteristic
537C2×C4#41×D4312yes—
538C2×C4#41×D4311yescharacteristic
539C8—42no—
540C8—1722no—
541C22—434no—
542C22—862no—
543C22—862no—
544C22—862no—
545C22—862no—
546C22—862no—
547C22—862no—
548C22—3441no—
549C22—3441no—
550C22—3441no—
551C22—3441no—
552C22—1722no—
553C22—1722no—
554C22—1722no—
555C22#120×D4311yescharacteristic
556C22C22⋊Q8×D4311yescharacteristic
557C22#124×D4311yescharacteristic
558C22AGL(1,ℤ/8)×D4311yescharacteristic
559C22C2×#1306335211yescharacteristic
560C22#137×D4311yescharacteristic
561C22C2×#1306335711yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—44no—
566C4—81no—
567C4—434no—
568C4—862no—
569C4—862no—
570C4—3441no—
571C4—1724no—
572C4C22⋊Q8×D4312yes—
573C4#124×D4312yes—
574C2—864no—
575C2—3442no—
576C2#393×D4311yescharacteristic
577C2#1914382311yescharacteristic
578C2#1914383011yescharacteristic
579C2#1914440511yescharacteristic
580C2#1914440911yescharacteristic
581C2#1914498711yescharacteristic
582C2#1914499011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2024C2
socleC23×C43C22×D4
prime radicalC22×#10621902C22
hypercenter#2024D43
hypocenterC43#23452
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D43C2semidirect
4. #19135736C2semidirect
5. #19135734C2semidirect
6. #19135732C2semidirect
7. #19135730C2semidirect
8. #19135728C2semidirect
9. #19134313C2semidirect
10. #19134287C2semidirect
11. C43×#2024C2semidirect
48. C2×#10621902Q8semidirect
113. C2×Q8×C43C2×Q8semidirect
134. C2×D172C2×Q8semidirect
288. #2024D43semidirect
336. C43#23452semidirect
414. C2×Q8Q8×D43semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D432
C2×#145828992
#191370842
#191357362
#191357342
#191357322
#191357302
#191357282
#191343132
#191342872
C43×#20242
#2345243