Finite Group Databasebeta

← #25129915

Subgroups — #25129915

Order 11008.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583430C211yesmaximal, characteristic
2C2×#14583229C211yesmaximal, characteristic, complemented
3C2×#14582795C211yesmaximal, characteristic, complemented
4#19135931C212yesmaximal, complemented
5#19135834C212yesmaximal, complemented
6#19134913C211yesmaximal, characteristic
7#19134900C211yesmaximal, characteristic, complemented
8#19134854C212yesmaximal, complemented
9#19134846C212yesmaximal, complemented
10#19134661C211yesmaximal, characteristic, complemented
11C2×C43×#480C211yesmaximal, characteristic
12#14583430—24no—
13#14583430—24no—
14C2×#13063318—21no—
15C2×#13063316C2211yescharacteristic
16C2×#13063312C2211yescharacteristic
17C2×#13063306—21no—
18C2×#13063304C2211yescharacteristic
19C2×#13063290C2211yescharacteristic
20C2×D4×Dic43—21no—
21C2×D4×Dic43C2211yescharacteristic
22C2×#13063281—21no—
23C22×C4⋊Dic43C2211yescharacteristic
24C22×Dic43⋊C4—21no—
25C22×C4×Dic43C2211yescharacteristic
26#14583229—24no—
27C2×#13063256C2212yes—
28C2×#13063250C2212yes—
29C2×#13063248C2212yes—
30C2×#13063246C2212yes—
31C2×C22⋊(C43⋊C8)C2211yescharacteristic
32#14582795—21no—
33#14582795—21no—
34#14582795C2212yes—
35#14582795C2212yes—
36C2×C43⋊#95—21no—
37#14582662C2212yes—
38#14582661C2212yes—
39#14582653C2212yes—
40#14582653C2212yes—
41C2×#121×C43C2211yescharacteristic
42C22×C4⋊C4×C43C2211yescharacteristic
43C43×#480—24no—
44C2×#107×C43C2212yes—
45C2×#102×C43C2212yes—
46C2×C22⋊C8×C43C2211yescharacteristic
47C22×Dic86C2311yescharacteristic
48C23×Dic43—21no—
49C23×Dic43—21no—
50C23×Dic43D411yescharacteristic
51C23×Dic43D411yescharacteristic
52C22×C43⋊C8C2312yes—
53#13063318—28no—
54#13063316—22no—
55#13063316—22no—
56#13063312—22no—
57#13063312—22no—
58#13063306—28no—
59#13063304—22no—
60#13063304D414yes—
61#13063290—22no—
62#13063290—22no—
63D4×Dic43—28no—
64D4×Dic43—22no—
65D4×Dic43—22no—
66#13063281—28no—
67C2×C4⋊Dic43—21no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—24no—
70C2×C4⋊Dic43C2311yescharacteristic
71C2×C4⋊Dic43C2312yes—
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—22no—
76C2×Dic43⋊C4—22no—
77C2×Dic43⋊C4—44no—
78C2×C22⋊Dic43—21no—
79C2×C22⋊Dic43—21no—
80C2×C22⋊Dic43—21no—
81C2×C22⋊Dic43—21no—
82C2×C4×Dic43—22no—
83C2×C4×Dic43—22no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43D411yescharacteristic
86C2×C4×Dic43D411yescharacteristic
87C2×C4×Dic43C2311yescharacteristic
88C2×C4×Dic43C2311yescharacteristic
89#13063256—24no—
90#13063250—24no—
91#13063248—24no—
92#13063246—24no—
93C22⋊(C43⋊C8)—22no—
94C43⋊#95—22no—
95C43⋊#95—22no—
96C22×D4×C43—21no—
97C22×D4×C43C2311yescharacteristic
98C23×C4×C43—21no—
99C23×C4×C43C2311yescharacteristic
100C22×C8×C43C2312yes—
101#121×C43—22no—
102#121×C43D414yes—
103C2×C4⋊C4×C43—22no—
104C2×C4⋊C4×C43—22no—
105C2×C4⋊C4×C43—21no—
106C2×C4⋊C4×C43C2312yes—
107C2×C4⋊C4×C43C2311yescharacteristic
108C2×C22⋊C4×C43—21no—
109#107×C43—24no—
110#102×C43—24no—
111C22⋊C8×C43—22no—
112C24×C43—21no—
113C24×C43C2×D411yescharacteristic
114C2×D4×C43—21no—
115C2×D4×C43—24no—
116C2×D4×C43—22no—
117C2×D4×C43—22no—
118C2×D4×C43—42no—
119C2×D4×C43—42no—
120C2×D4×C43C2×D412yes—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—21no—
124C22×C4×C43—22no—
125C22×C4×C43—42no—
126C22×C4×C43—42no—
127C22×C4×C43—42no—
128C22×C4×C43C2×D412yes—
129C22×C4×C43C2×D412yes—
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2411yescharacteristic
132C22×Dic43—24no—
133C22×Dic43—24no—
134C22×Dic43—22no—
135C22×Dic43—22no—
136C22×Dic43—22no—
137C22×Dic43—22no—
138C22×Dic43—22no—
139C22×Dic43—22no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—41no—
143C22×Dic43—42no—
144C22×Dic43—42no—
145C22×Dic43—42no—
146C22×Dic43—42no—
147C22×Dic43C2×D411yescharacteristic
148C22×Dic43C2×D411yescharacteristic
149C22×Dic43C2×D411yescharacteristic
150C22×Dic43C2×D411yescharacteristic
151C2×Dic86—22no—
152C2×Dic86—22no—
153C2×Dic86—21no—
154C2×Dic86C2×D412yes—
155C2×C8×C43—22no—
156C2×C8×C43—22no—
157C2×C8×C43—22no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—22no—
160C4⋊C4×C43—22no—
161C4⋊C4×C43—42no—
162C4⋊C4×C43C2×D412yes—
163C22⋊C4×C43—24no—
164C22⋊Dic43—24no—
165C22⋊Dic43—24no—
166C22⋊Dic43—24no—
167C22⋊Dic43—24no—
168C4⋊Dic43—24no—
169C4⋊Dic43—24no—
170C4⋊Dic43—21no—
171C4⋊Dic43—22no—
172C4⋊Dic43C2×D412yes—
173C4⋊Dic43#4114yes—
174Dic43⋊C4—24no—
175Dic43⋊C4—24no—
176Dic43⋊C4—24no—
177Dic43⋊C4—44no—
178Dic43⋊C4—44no—
179C4×Dic43—24no—
180C4×Dic43—22no—
181C4×Dic43—22no—
182C4×Dic43—21no—
183C4×Dic43—21no—
184C4×Dic43—21no—
185C4×Dic43—21no—
186C2×C43⋊C8—22no—
187C2×C43⋊C8—22no—
188C2×C43⋊C8—22no—
189C23×C43—21no—
190C23×C43—21no—
191C23×C43—21no—
192C23×C43—21no—
193C23×C43—22no—
194C23×C43—41no—
195C23×C43—41no—
196C23×C43—41no—
197C23×C43—41no—
198C23×C43—41no—
199C23×C43—41no—
200C23×C43#12012yes—
201C23×C43#13512yes—
202C23×C43AGL(1,ℤ/8)12yes—
203C23×C43C22×D411yescharacteristic
204D4×C43—24no—
205D4×C43—48no—
206D4×C43—42no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—24no—
210C2×C4×C43—24no—
211C2×C4×C43—22no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—42no—
215C2×C4×C43—42no—
216C2×C4×C43—42no—
217C2×C4×C43—84no—
218C2×C4×C43#12012yes—
219C2×C4×C43C22×D411yescharacteristic
220C2×C4×C43C22×D411yescharacteristic
221C2×C4×C43C2×#4112yes—
222C2×C4×C432−1+412yes—
223C2×Dic43—21no—
224C2×Dic43—21no—
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—21no—
228C2×Dic43—22no—
229C2×Dic43—22no—
230C2×Dic43—24no—
231C2×Dic43—21no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—44no—
238C2×Dic43—48no—
239C2×Dic43—48no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—42no—
243C2×Dic43—41no—
244C2×Dic43—42no—
245C2×Dic43—41no—
246C2×Dic43—44no—
247C2×Dic43#13712yes—
248Dic86—24no—
249Dic86—42no—
250C8×C43—42no—
251C43⋊C8—42no—
252#24066—431nomaximal, Sylow
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—22no—
256C22×C43—22no—
257C22×C43—22no—
258C22×C43—22no—
259C22×C43—42no—
260C22×C43—42no—
261C22×C43—42no—
262C22×C43—81no—
263C22×C43—81no—
264C22×C43—81no—
265C22×C43—81no—
266C22×C43#44914yes—
267C22×C43C2×#12011yescharacteristic
268C22×C43#52511yescharacteristic
269C22×C43#53611yescharacteristic
270C22×C43C2×#13511yescharacteristic
271C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C43C2×#13711yescharacteristic
273C22×C43#57511yescharacteristic
274C4×C43—22no—
275C4×C43—42no—
276C4×C43—82no—
277C4×C43#54712yes—
278C4×C43#54712yes—
279Dic43—22no—
280Dic43—44no—
281Dic43—42no—
282Dic43—41no—
283Dic43—41no—
284Dic43—41no—
285Dic43—42no—
286C2×#529—431no—
287C2×#483—431no—
288C2×#480D4311yescharacteristic
289C2×#389—431no—
290#2059—432no—
291#2042—432no—
292#1874—431no—
293#1872—431no—
294#1864—432no—
295#1863—432no—
296#1817—431no—
297C2×C43—24no—
298C2×C43—82no—
299C2×C43#240611yescharacteristic
300C2×C43C2×#44911yescharacteristic
301C2×C43#300711yescharacteristic
302C2×C43#310811yescharacteristic
303C2×C43#311411yescharacteristic
304C2×C43#323211yescharacteristic
305C2×C43#323811yescharacteristic
306#529—864no—
307#529—864no—
308C2×#126—861no—
309C2×#125—431no—
310C2×#124—431no—
311C2×#123—861no—
312C2×C22⋊Q8—431no—
313C2×#121C2×D4311yescharacteristic
314C2×C4×Q8—431no—
315C2×C4×D4—431no—
316C2×C4×D4—861no—
317C2×#117—861no—
318C22×C4⋊C4—431no—
319C22×C4⋊C4—861no—
320C22×C4⋊C4C2×D4311yescharacteristic
321C22×C42—431no—
322#483—864no—
323#480—24no—
324C2×#107C2×D4312yes—
325C2×#106—432no—
326C2×#103—432no—
327C2×#103—432no—
328C2×#102—432no—
329C2×#102C2×D4312yes—
330C2×C22⋊C8—431no—
331C2×C22⋊C8C2×D4311yescharacteristic
332#389—432no—
333#389—432no—
334#389—861no—
335#389—861no—
336C2×#95—861no—
337#340—432no—
338#340—432no—
339#336—432no—
340#336—432no—
341C43#2406611yescharacteristic, complemented, Sylow
342C22×Q8—431no—
343C22×D4—21no—
344C22×D4C22×D4311yescharacteristic
345C23×C4—21no—
346C23×C4—431no—
347C23×C4—431no—
348C23×C4—861no—
349C23×C4—861no—
350C23×C4C22×D4311yescharacteristic
351C22×C8—432no—
352C22×C8C22×D4312yes—
353#126—868no—
354#125—862no—
355#125—862no—
356#124—862no—
357#124—862no—
358#123—868no—
359C22⋊Q8—434no—
360C22⋊Q8—862no—
361#121—22no—
362#121C22⋊D4314yes—
363C4×Q8—862no—
364C4×Q8—862no—
365C4×D4—862no—
366C4×D4—868no—
367C4×D4—862no—
368#117—868no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—432no—
373C2×C4⋊C4—431no—
374C2×C4⋊C4—864no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—861no—
377C2×C4⋊C4—861no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—862no—
381C2×C4⋊C4—862no—
382C2×C4⋊C4—1724no—
383C2×C4⋊C4C22×D4312yes—
384C2×C4⋊C4C22×D4311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C22⋊C4—861no—
389C2×C22⋊C4—861no—
390C2×C42—431no—
391C2×C42—431no—
392C2×C42—431no—
393C2×C42—431no—
394C2×C42—861no—
395C2×C42—862no—
396C2×C42—862no—
397#107—24no—
398#106—864no—
399#103—864no—
400#103—864no—
401#102—24no—
402#102—864no—
403C22⋊C8—22no—
404C22⋊C8—862no—
405#95—862no—
406#95—862no—
407C24—21no—
408C24D4×D4311yescharacteristic
409C2×Q8—432no—
410C2×Q8—861no—
411C2×Q8—862no—
412C2×Q8—862no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C22⋊D4312yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—431no—
428C22×C4—431no—
429C22×C4—431no—
430C22×C4—431no—
431C22×C4—862no—
432C22×C4—864no—
433C22×C4—864no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—862no—
437C22×C4—862no—
438C22×C4—862no—
439C22×C4—861no—
440C22×C4—861no—
441C22×C4—1722no—
442C22×C4—1721no—
443C22×C4—1722no—
444C22×C4—1722no—
445C22×C4—1722no—
446C22×C4D4×D4311yescharacteristic
447C22×C4D4×D4312yes—
448C22×C4C2×C22⋊D4312yes—
449C22×C4C23×D4311yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—862no—
454C2×C8—862no—
455C2×C8—862no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—432no—
461C4⋊C4—434no—
462C4⋊C4—864no—
463C4⋊C4—864no—
464C4⋊C4—864no—
465C4⋊C4—864no—
466C4⋊C4—864no—
467C4⋊C4—861no—
468C4⋊C4—862no—
469C4⋊C4—1724no—
470C4⋊C4—1724no—
471C4⋊C4C2×C22⋊D4312yes—
472C22⋊C4—24no—
473C22⋊C4—864no—
474C22⋊C4—864no—
475C22⋊C4—864no—
476C22⋊C4—864no—
477C42—864no—
478C42—862no—
479C42—862no—
480C42—861no—
481C42—861no—
482C42—861no—
483C42—861no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#1306329312yes—
496C23#1306334412yes—
497C23#1306334912yes—
498C23C2×D4×D4311yescharacteristic
499Q8—864no—
500Q8—1722no—
501D4—24no—
502D4—48no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—432no—
516C2×C4—861no—
517C2×C4—861no—
518C2×C4—861no—
519C2×C4—861no—
520C2×C4—861no—
521C2×C4—862no—
522C2×C4—862no—
523C2×C4—864no—
524C2×C4—861no—
525C2×C4—861no—
526C2×C4—861no—
527C2×C4—861no—
528C2×C4—861no—
529C2×C4—861no—
530C2×C4—1724no—
531C2×C4—1728no—
532C2×C4—1728no—
533C2×C4—1724no—
534C2×C4—1721no—
535C2×C4—1722no—
536C2×C4—1721no—
537C2×C4—1722no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4#1306329312yes—
541C2×C4C2×D4×D4311yescharacteristic
542C2×C4C22×C22⋊D4311yescharacteristic
543C2×C4#41×D4312yes—
544C2×C4#1306337212yes—
545C8—42no—
546C8—1722no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1458304114yes—
561C22C2×#1306329311yescharacteristic
562C22#1458340711yescharacteristic
563C22#1458347111yescharacteristic
564C22C2×#1306334411yescharacteristic
565C22C2×#1306334911yescharacteristic
566C22#137×D4311yescharacteristic
567C22#1458369811yescharacteristic
568C4—22no—
569C4—42no—
570C4—82no—
571C4—862no—
572C4—1724no—
573C4—1721no—
574C4—1721no—
575C4—1722no—
576C4—1721no—
577C4—1722no—
578C4#1458354712yes—
579C4#1458354812yes—
580C2—24no—
581C2—82no—
582C2#1913792711yescharacteristic
583C2C2×#1458304111yescharacteristic
584C2#1914385111yescharacteristic
585C2#1914467211yescharacteristic
586C2#1914473011yescharacteristic
587C2#1914576011yescharacteristic
588C2#1914581411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#480C2
socleC23×C43C22×D4
prime radicalC2×#121×C43C22
hypercenterC2×#480D43
hypocenterC43#24066
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C2×#14583229C2semidirect
3. C2×#14582795C2semidirect
4. #19135931C2semidirect
5. #19135834C2semidirect
7. #19134900C2semidirect
8. #19134854C2semidirect
9. #19134846C2semidirect
10. #19134661C2semidirect
341. C43#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#145834302
C2×#145832292
C2×#145827952
#191359312
#191358342
#191349132
#191349002
#191348542
#191348462
#191346612
C2×C43×#4802
#2406643