Finite Group Databasebeta

← #25112248

Subgroups — #25112248

Order 11008.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583552C211yesmaximal, characteristic
2C2×#14582978C211yesmaximal, characteristic, complemented
3C2×#14582899C212yesmaximal, complemented
4#19137495C211yesmaximal, characteristic, complemented
5#19135519C211yesmaximal, characteristic, complemented
6#19135025C212yesmaximal, complemented
7#19134899C212yesmaximal, complemented
8#19134592C211yesmaximal, characteristic, complemented
9#19134534C212yesmaximal, complemented
10C4×#13063247C211yesmaximal, characteristic, complemented
11C43×#1956C211yesmaximal, characteristic, complemented
12C22×#10621902C2212yes—
13C22×#10621872C2212yes—
14#14583552—24no—
15#14583552C418yescomplemented
16C2×#13063310—22no—
17C2×#13063301C2212yes—
18C2×#13063300—21no—
19C2×C4×D172C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×#13063287—21no—
22C2×C4⋊C4×D43—21no—
23C2×C4⋊C4×D43C2212yes—
24#14582978—24no—
25C2×C4⋊(C43⋊C8)C2211yescharacteristic
26#14582899—28no—
27C2×#13063249C2212yes—
28C2×#13063249C2212yes—
29C2×#13063247C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31C2×#13063245C2212yes—
32C2×(C2×D43)⋊C8C2212yes—
33#14582795—21no—
34#14582770—22no—
35#14582756C2212yes—
36C4×C4⋊Dic43C2211yescharacteristic
37#14582662C2211yescharacteristic
38#14582662C2211yescharacteristic
39#14582661C2212yes—
40C2×C4⋊Q8×C43C2211yescharacteristic
41C2×#103×C43C2212yes—
42C2×#103×C43C2212yes—
43C2×C4×C8×C43C2211yescharacteristic
44C43×#384C2211yescharacteristic
45C2×#10621902—24no—
46C2×#10621902—24no—
47C2×#10621902—24no—
48C2×#10621902—42no—
49C2×#10621902—42no—
50C2×#10621902C2×C418yescomplemented
51C22×D172—22no—
52C22×D172D411yescharacteristic
53C22×D172C2311yescharacteristic
54C22×C4×D43—21no—
55C22×C4×D43—22no—
56C22×C4×D43—22no—
57C22×C4×D43C2312yes—
58C2×#10621872—24no—
59C2×#10621872—24no—
60C2×#10621872—42no—
61C22×C43⋊C8C2312yes—
62#13063310—216no—
63#13063301—24no—
64#13063301C2×C418yescomplemented
65#13063300—28no—
66C4×D172—22no—
67C4×D172C2×C414yes—
68C4×D172C2×C414yescomplemented
69C4×D172D414yes—
70#13063287—44no—
71C2×C4⋊Dic43D411yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×Dic43⋊C4—22no—
76C2×Dic43⋊C4—22no—
77C4⋊C4×D43—28no—
78C4⋊C4×D43—24no—
79C4⋊C4×D43—42no—
80C2×(C2×D43)⋊C4—22no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—22no—
83C2×(C2×D43)⋊C4D412yes—
84C2×C4×Dic43—21no—
85C4⋊(C43⋊C8)—22no—
86#13063249—24no—
87#13063249—42no—
88#13063247—22no—
89#13063247D414yescomplemented
90#13063245—24no—
91(C2×D43)⋊C8—24no—
92C43⋊#95—21no—
93C43⋊#95—22no—
94C22×Q8×C43C2312yes—
95C22×C8×C43C2311yescharacteristic
96C22×C8×C43C2311yescharacteristic
97C4⋊Q8×C43—22no—
98C4⋊Q8×C43C2×C414yescomplemented
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43C2312yes—
104C2×C42×C43C2311yescharacteristic
105#103×C43—24no—
106#103×C43—42no—
107C4×C8×C43—22no—
108C23×D43—21no—
109C23×D43C2×D412yes—
110C2×Q8×C43—22no—
111C2×Q8×C43—22no—
112C2×Q8×C43—22no—
113C2×Q8×C43—22no—
114C2×Q8×C43—22no—
115C2×Q8×C43C22×C414yes—
116C22×C4×C43—22no—
117C22×C4×C43—22no—
118C22×C4×C43C2×D411yescharacteristic
119C22×C4×C43C2×D411yescharacteristic
120C22×C4×C43C2411yescharacteristic
121C22×Dic43—21no—
122C22×Dic43—21no—
123C22×Dic43—21no—
124C22×Dic43C2×D412yes—
125#10621902—216no—
126#10621902—48no—
127#10621902—48no—
128#10621902—48no—
129C2×D172—22no—
130C2×D172—22no—
131C2×D172—21no—
132C2×D172—21no—
133C2×D172—21no—
134C2×D172—28no—
135C2×D172—44no—
136C2×D172—44no—
137C2×D172C22⋊C414yes—
138C2×D172C22×C412yes—
139C2×D172C2×D412yes—
140C2×D172#4112yes—
141C2×C4×D43—22no—
142C2×C4×D43—22no—
143C2×C4×D43—22no—
144C2×C4×D43—22no—
145C2×C4×D43—24no—
146C2×C4×D43—28no—
147C2×C4×D43—44no—
148C2×C4×D43—44no—
149C2×C4×D43—44no—
150C2×C4×D43—44no—
151C2×C4×D43—42no—
152C2×C4×D43—42no—
153C2×C4×D43—82no—
154C2×C4×D43C22×C414yes—
155C2×C4×D43C2×D414yes—
156C2×C8×C43—21no—
157C2×C8×C43—21no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43C2×D412yes—
162C4⋊C4×C43—24no—
163C4⋊C4×C43—24no—
164C4⋊C4×C43—22no—
165C4⋊C4×C43—22no—
166C4⋊C4×C43—42no—
167C4⋊C4×C43—81no—
168C42×C43C22×C412yes—
169C42×C43C2×D412yes—
170(C2×D43)⋊C4—24no—
171(C2×D43)⋊C4—28no—
172(C2×D43)⋊C4—44no—
173(C2×D43)⋊C4C22⋊C418yes—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43C22⋊C414yes—
179C4⋊Dic43C22×C412yes—
180C4⋊Dic43C2×D412yes—
181Dic43⋊C4—28no—
182Dic43⋊C4—44no—
183C4×Dic43—42no—
184C2×C43⋊C8—22no—
185C2×C43⋊C8—22no—
186C2×C43⋊C8—22no—
187#10621872—48no—
188C23×C43C22×D411yescharacteristic
189C22×D43—22no—
190C22×D43—22no—
191C22×D43—22no—
192C22×D43—22no—
193C22×D43—22no—
194C22×D43—22no—
195C22×D43—22no—
196C22×D43—22no—
197C22×D43—22no—
198C22×D43—41no—
199C22×D43—41no—
200C22×D43—41no—
201C22×D43—41no—
202C22×D43C2×C22⋊C414yes—
203Q8×C43—24no—
204Q8×C43—24no—
205Q8×C43—42no—
206Q8×C43—42no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—24no—
210C2×C4×C43—42no—
211C2×C4×C43—42no—
212C2×C4×C43—42no—
213C2×C4×C43—42no—
214C2×C4×C43—42no—
215C2×C4×C43C2×C22⋊C412yes—
216C2×C4×C43C2×C22⋊C412yes—
217C2×C4×C43AGL(1,ℤ/8)12yes—
218C2×C4×C43#13712yes—
219C2×C4×C43C23×C411yescharacteristic
220C2×C4×C43C22×D411yescharacteristic
221C2×C4×C43C22×D411yescharacteristic
222C2×C4×C43C2×#4111yescharacteristic
223C2×C4×C43H(5,𝔽2)11yescharacteristic
224C2×C4×C43H(5,𝔽2)11yescharacteristic
225C2×Dic43—22no—
226C2×Dic43—22no—
227C2×Dic43—22no—
228C2×Dic43—22no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43—41no—
232C2×Dic43—41no—
233C2×Dic43—41no—
234C2×Dic43—41no—
235C2×Dic43—41no—
236C2×Dic43C2×C22⋊C414yes—
237D172—22no—
238D172—22no—
239D172—22no—
240D172—28no—
241D172—416no—
242D172C4×D414yescomplemented
243C4×D43—24no—
244C4×D43—24no—
245C4×D43—24no—
246C4×D43—24no—
247C4×D43—416no—
248C4×D43—48no—
249C4×D43—88no—
250C8×C43—22no—
251C8×C43—41no—
252C43⋊C8—42no—
253#22431—431nomaximal, Sylow
254C22×C43C22×C22⋊C411yescharacteristic
255C22×C43D4211yescharacteristic
256C22×C43#54611yescharacteristic
257C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C43C2×#13711yescharacteristic
259C22×C43#57711yescharacteristic
260C22×C43#57811yescharacteristic
261C2×D43—24no—
262C2×D43—24no—
263C2×D43—24no—
264C2×D43—24no—
265C2×D43—42no—
266C2×D43—42no—
267C2×D43—42no—
268C2×D43—42no—
269C2×D43—42no—
270C2×D43—42no—
271C2×D43—42no—
272C2×D43—42no—
273C2×D43—42no—
274C2×D43—42no—
275C4×C43—22no—
276C4×C43—44no—
277C4×C43—82no—
278C4×C43#41714yescomplemented
279C4×C43C2×C4×D411yescharacteristic
280C4×C43#51811yescharacteristic
281C4×C43D4211yescharacteristic
282C4×C43#54611yescharacteristic
283Dic43—24no—
284Dic43—42no—
285Dic43—42no—
286Dic43—81no—
287C2×#547—431no—
288C2×#436—431no—
289C2×#417—432no—
290D4×C4⋊C4—431no—
291#1981—431no—
292#1956D4311yescharacteristic, complemented
293#1894—432no—
294#1872—432no—
295#1802—431no—
296#1788—432no—
297C4×#102—431no—
298C2×C43D4×C22⋊C411yescharacteristic
299C2×C43C2×#41711yescharacteristic
300C2×C43#289811yescharacteristic
301C2×C43#327211yescharacteristic
302C2×C43#327611yescharacteristic
303C2×C43#333011yescharacteristic
304C2×C43#333111yescharacteristic
305D43—44no—
306D43—44no—
307C22×#41—432no—
308C22×M16—432no—
309#547—438no—
310#547—864no—
311C2×C4⋊Q8C2×D4311yescharacteristic
312C2×#123—862no—
313C2×C22⋊Q8—432no—
314C2×#121—861no—
315C2×C4×D4—431no—
316C2×C4×D4—431no—
317C2×C4×D4—861no—
318C22×C4⋊C4—432no—
319C22×C4⋊C4—861no—
320#436—864no—
321C2×C4⋊C8—431no—
322#417—868no—
323C2×#103—432no—
324C2×#103—432no—
325C2×#103C2×D4312yes—
326C2×#103C2×D4312yes—
327C2×#102—431no—
328C2×#102—431no—
329C2×#102—432no—
330C2×C22⋊C8—432no—
331C2×C4×C8C2×D4311yescharacteristic
332#389—861no—
333#385—862no—
334#384C2×D4311yescharacteristic
335#380—432no—
336C4×C4⋊C4—431no—
337#340—431no—
338#340—431no—
339#340—432no—
340C43#2243111yescharacteristic, complemented, Sylow
341C2×#41—438no—
342C2×#41—864no—
343C2×#41—864no—
344C2×#41—864no—
345C2×#41—1722no—
346C2×#41—1722no—
347C22×Q8C22×D4312yes—
348C22×D4—431no—
349C22×D4—431no—
350C22×D4—862no—
351C23×C4—432no—
352C23×C4—862no—
353C23×C4—862no—
354C23×C4—861no—
355C2×M16—864no—
356C2×M16—864no—
357C2×M16—1722no—
358C22×C8—432no—
359C22×C8C22×D4311yescharacteristic
360C22×C8C22×D4311yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8C4×D4314yescomplemented
363#123—8616no—
364C22⋊Q8—438no—
365C22⋊Q8—864no—
366#121—868no—
367C4×D4—434no—
368C4×D4—434no—
369C4×D4—434no—
370C4×D4—862no—
371C4×D4—1724no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—431no—
377C2×C4⋊C4—431no—
378C2×C4⋊C4—431no—
379C2×C4⋊C4—431no—
380C2×C4⋊C4—868no—
381C2×C4⋊C4—864no—
382C2×C4⋊C4—862no—
383C2×C4⋊C4—862no—
384C2×C4⋊C4—1722no—
385C2×C4⋊C4C22×D4312yes—
386C2×C22⋊C4—432no—
387C2×C22⋊C4—862no—
388C2×C22⋊C4—861no—
389C2×C22⋊C4—862no—
390C2×C42—861no—
391C2×C42C22×D4311yescharacteristic
392C4⋊C8—862no—
393#103—24no—
394#103—42no—
395#103—864no—
396#103—1722no—
397#102—434no—
398#102—864no—
399#102—862no—
400C22⋊C8—864no—
401C4×C8—22no—
402#95—862no—
403#95—861no—
404C24—432no—
405C24—861no—
406#41—8616no—
407#41—1728no—
408#41—1728no—
409#41—1728no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C2×C4×D4314yes—
416C2×D4—434no—
417C2×D4—432no—
418C2×D4—432no—
419C2×D4—432no—
420C2×D4—868no—
421C2×D4—862no—
422C2×D4—862no—
423C2×D4—861no—
424C2×D4—861no—
425C2×D4—861no—
426C2×D4—1724no—
427C2×D4—1724no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—432no—
431C22×C4—434no—
432C22×C4—434no—
433C22×C4—868no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—862no—
437C22×C4—862no—
438C22×C4—861no—
439C22×C4—861no—
440C22×C4—864no—
441C22×C4—861no—
442C22×C4—3442no—
443C22×C4—1724no—
444C22×C4—1724no—
445C22×C4—1724no—
446C22×C4—1724no—
447C22×C4—1722no—
448C22×C4—1722no—
449C22×C4D4×D4311yescharacteristic
450C22×C4D4×D4311yescharacteristic
451C22×C4C23×D4311yescharacteristic
452M16—1728no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—862no—
459C2×C8—862no—
460C2×C8—862no—
461C2×C8D4×D4312yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—434no—
469C4⋊C4—432no—
470C4⋊C4—432no—
471C4⋊C4—868no—
472C4⋊C4—861no—
473C4⋊C4—861no—
474C4⋊C4—861no—
475C4⋊C4—861no—
476C4⋊C4—1724no—
477C22⋊C4—438no—
478C22⋊C4—868no—
479C22⋊C4—864no—
480C22⋊C4—1724no—
481C42—1722no—
482C42C2×C4×D4312yes—
483C42D4×D4312yes—
484C23—434no—
485C23—862no—
486C23—862no—
487C23—862no—
488C23—862no—
489C23—862no—
490C23—862no—
491C23—862no—
492C23—862no—
493C23—862no—
494C23—1721no—
495C23—1721no—
496C23—1721no—
497C23—1721no—
498C23C2×D4×D4311yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—434no—
504D4—862no—
505D4—868no—
506D4—862no—
507D4—862no—
508D4—17216no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—434no—
518C2×C4—862no—
519C2×C4—862no—
520C2×C4—864no—
521C2×C4—864no—
522C2×C4—864no—
523C2×C4—864no—
524C2×C4—862no—
525C2×C4—862no—
526C2×C4—3448no—
527C2×C4—17216no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—1721no—
532C2×C4—1721no—
533C2×C4—1721no—
534C2×C4—1721no—
535C2×C4—1728no—
536C2×C4C22⋊C4×D4312yes—
537C2×C4C22⋊C4×D4312yes—
538C2×C4#1306335212yes—
539C2×C4#1306335712yes—
540C2×C4C22×C4×D4311yescharacteristic
541C2×C4C2×D4×D4311yescharacteristic
542C2×C4C2×D4×D4311yescharacteristic
543C2×C4#41×D4311yescharacteristic
544C2×C4#1306337011yescharacteristic
545C2×C4#1306337111yescharacteristic
546C8—22no—
547C8—41no—
548C8—1722no—
549C22—864no—
550C22—864no—
551C22—864no—
552C22—864no—
553C22—1722no—
554C22—1722no—
555C22—1722no—
556C22—1722no—
557C22—1722no—
558C22—1722no—
559C22—1722no—
560C22—1722no—
561C22—1722no—
562C22—1722no—
563C22C2×C22⋊C4×D4311yescharacteristic
564C22#1458353411yescharacteristic
565C22#1458354211yescharacteristic
566C22C2×#1306335211yescharacteristic
567C22C2×#1306335711yescharacteristic
568C22#1458370511yescharacteristic
569C22#1458371211yescharacteristic
570C4—22no—
571C4—44no—
572C4—82no—
573C4—864no—
574C4—3441no—
575C4—1722no—
576C4—1722no—
577C4#1458289914yescomplemented
578C4C4×D4×D4311yescharacteristic
579C4#1458337111yescharacteristic
580C4#1458353311yescharacteristic
581C4#1458354611yescharacteristic
582C2—1724no—
583C2—1724no—
584C2#1913741611yescharacteristic
585C2C2×#1458289911yescharacteristic
586C2#1914312311yescharacteristic
587C2#1914619411yescharacteristic
588C2#1914624011yescharacteristic
589C2#1914689911yescharacteristic
590C2#1914691011yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1956C2
socleC23×C43C22×D4
prime radicalC2×#14583552C2
cosocleC22×C4×C43C24
hypercenter#1956D43
hypocenterC43#22431
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#14582978C2semidirect
3. C2×#14582899C2semidirect
4. #19137495C2semidirect
5. #19135519C2semidirect
6. #19135025C2semidirect
7. #19134899C2semidirect
8. #19134592C2semidirect
9. #19134534C2semidirect
10. C4×#13063247C2semidirect
11. C43×#1956C2semidirect
15. #14583552C4semidirect
50. C2×#10621902C2×C4semidirect
64. #13063301C2×C4semidirect
68. C4×D172C2×C4semidirect
89. #13063247D4semidirect
98. C4⋊Q8×C43C2×C4semidirect
242. D172C4×D4semidirect
278. C4×C43#417semidirect
292. #1956D43semidirect
340. C43#22431semidirect
362. C4⋊Q8C4×D43semidirect
577. C4#14582899semidirect

Maximal subgroups

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

subgroupindex
C2×#145835522
C2×#145829782
C2×#145828992
#191374952
#191355192
#191350252
#191348992
#191345922
#191345342
C4×#130632472
C43×#19562
#2243143