Finite Group Databasebeta

← #25169359

Subgroups — #25169359

Order 11008.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19138989C211yesmaximal, characteristic
2#19137596C211yesmaximal, characteristic
3#19136016C211yesmaximal, characteristic, complemented
4#19135974C211yesmaximal, characteristic, complemented
5#19135796C211yesmaximal, characteristic
6#19135790C211yesmaximal, characteristic
7#19135362C211yesmaximal, characteristic, complemented
8#19135323C211yesmaximal, characteristic, complemented
9#19135201C211yesmaximal, characteristic
10#19135187C211yesmaximal, characteristic, complemented
11#19135178C211yesmaximal, characteristic, complemented
12#19134863C211yesmaximal, characteristic, complemented
13#19134789C211yesmaximal, characteristic
14#19134680C211yesmaximal, characteristic
15C4×#103×C43C211yesmaximal, characteristic, complemented
16C2×#13063301C2211yescharacteristic
17C2×C4⋊C4×D43C2211yescharacteristic
18C2×#13063258C2211yescharacteristic
19C2×#13063257C2211yescharacteristic
20C2×#13063257C2211yescharacteristic
21C2×#13063256C2211yescharacteristic
22C2×Dic43⋊C8C2211yescharacteristic
23C2×#13063249C2211yescharacteristic
24C2×#13063249C2211yescharacteristic
25C2×(C2×D43)⋊C8C2211yescharacteristic
26C2×(C2×D43)⋊C8C2211yescharacteristic
27C43⋊#401—21no—
28#14582828—21no—
29#14582806C2211yescharacteristic
30#14582797C2211yescharacteristic
31#14582797C2211yescharacteristic
32#14582795C2211yescharacteristic
33#14582794—21no—
34#14582790—21no—
35#14582783—21no—
36#14582783—21no—
37#14582770—21no—
38#14582764C2211yescharacteristic
39#14582756C2211yescharacteristic
40#14582756C2211yescharacteristic
41C4⋊C4×Dic43C2211yescharacteristic
42C4×(C2×D43)⋊C4—21no—
43C4×(C2×D43)⋊C4C2211yescharacteristic
44#14582662C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582654C2211yescharacteristic
51#14582653C2211yescharacteristic
52C2×C4×Q8×C43C2211yescharacteristic
53C2×#103×C43C2211yescharacteristic
54C2×#103×C43C2211yescharacteristic
55C2×C4×C8×C43C2211yescharacteristic
56C4×C4⋊C4×C43C2211yescharacteristic
57C43×#340C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C22×C4×D43—21no—
60C22×C4×D43C2311yescharacteristic
61C22×C43⋊C8C2311yescharacteristic
62C22×C43⋊C8C2311yescharacteristic
63#13063301—22no—
64#13063301—22no—
65C2×C4⋊Dic43C2311yescharacteristic
66C2×C4⋊Dic43C2311yescharacteristic
67C2×C4⋊Dic43C2311yescharacteristic
68C2×C4⋊Dic43C2311yescharacteristic
69C2×Dic43⋊C4—21no—
70C2×Dic43⋊C4—21no—
71C2×Dic43⋊C4—21no—
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4—21no—
74C4⋊C4×D43—22no—
75C4⋊C4×D43—41no—
76C2×(C2×D43)⋊C4—21no—
77C2×(C2×D43)⋊C4—21no—
78C2×(C2×D43)⋊C4—21no—
79C2×(C2×D43)⋊C4D411yescharacteristic
80C2×(C2×D43)⋊C4D411yescharacteristic
81C2×C4×Dic43—21no—
82C2×C4×Dic43C2311yescharacteristic
83#13063258—22no—
84#13063257—22no—
85#13063257—41no—
86#13063256—22no—
87Dic43⋊C8—22no—
88#13063249—22no—
89#13063249—41no—
90(C2×D43)⋊C8—22no—
91(C2×D43)⋊C8—41no—
92C43⋊#95—21no—
93C43⋊#95—21no—
94C43⋊#95—21no—
95C43⋊#95—21no—
96C43⋊#95—21no—
97C43⋊#95—21no—
98C43⋊#95—21no—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95D411yescharacteristic
103C43⋊#95D411yescharacteristic
104C22×Q8×C43C2311yescharacteristic
105C22×C8×C43C2311yescharacteristic
106C22×C8×C43C2311yescharacteristic
107C4×Q8×C43—22no—
108C4×Q8×C43—41no—
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43C2311yescharacteristic
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C42×C43—21no—
114C2×C42×C43—21no—
115C2×C42×C43C2311yescharacteristic
116#103×C43—22no—
117#103×C43—22no—
118C4×C8×C43—41no—
119#95×C43—21no—
120C23×D43C2×D411yescharacteristic
121C2×Q8×C43—21no—
122C2×Q8×C43—21no—
123C2×Q8×C43—21no—
124C2×Q8×C43—21no—
125C2×Q8×C43—21no—
126C2×Q8×C43#4112yes—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2411yescharacteristic
134C22×Dic43—21no—
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43C2×D411yescharacteristic
139C22×Dic43C2×D411yescharacteristic
140C22×Dic43C2×D411yescharacteristic
141C2×C4×D43—21no—
142C2×C4×D43—21no—
143C2×C4×D43—21no—
144C2×C4×D43—21no—
145C2×C4×D43—21no—
146C2×C4×D43—21no—
147C2×C4×D43—42no—
148C2×C4×D43—42no—
149C2×C4×D43—81no—
150C2×C8×C43—21no—
151C2×C8×C43—21no—
152C2×C8×C43—21no—
153C2×C8×C43—21no—
154C2×C8×C43—21no—
155C2×C8×C43#4112yes—
156C4⋊C4×C43—21no—
157C4⋊C4×C43—21no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—21no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—42no—
162C4⋊C4×C43#4112yes—
163C42×C43—21no—
164C42×C43—21no—
165C42×C43—42no—
166C42×C43—81no—
167(C2×D43)⋊C4—24no—
168(C2×D43)⋊C4—22no—
169(C2×D43)⋊C4—22no—
170(C2×D43)⋊C4—42no—
171(C2×D43)⋊C4—81no—
172C4⋊Dic43—21no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43#4112yes—
180Dic43⋊C4—24no—
181Dic43⋊C4—42no—
182Dic43⋊C4—42no—
183Dic43⋊C4—81no—
184Dic43⋊C4—81no—
185C4×Dic43—21no—
186C4×Dic43—21no—
187C4×Dic43—81no—
188C2×C43⋊C8—21no—
189C2×C43⋊C8—21no—
190C2×C43⋊C8—21no—
191C2×C43⋊C8—21no—
192C2×C43⋊C8—21no—
193C2×C43⋊C8—21no—
194C23×C43C22×D411yescharacteristic
195C22×D43—21no—
196C22×D43—21no—
197C22×D43—21no—
198C22×D43—21no—
199C22×D43—21no—
200C22×D43—21no—
201C22×D43#13512yes—
202Q8×C43—22no—
203Q8×C43—22no—
204Q8×C43—41no—
205Q8×C43—41no—
206C2×C4×C43—21no—
207C2×C4×C43—21no—
208C2×C4×C43—22no—
209C2×C4×C43—22no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43#13712yes—
224C2×C4×C43C2×#4111yescharacteristic
225C2×C4×C43C2×#4111yescharacteristic
226C2×C4×C43C2×#4111yescharacteristic
227C2×C4×C43C2×#4111yescharacteristic
228C2×C4×C43H(5,𝔽2)11yescharacteristic
229C2×C4×C432−1+411yescharacteristic
230C2×Dic43—21no—
231C2×Dic43—22no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—21no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C4×D43—41no—
252C4×D43—41no—
253C4×D43—41no—
254C4×D43—41no—
255C4×D43—84no—
256C8×C43—22no—
257C8×C43—41no—
258C43⋊C8—41no—
259C43⋊C8—41no—
260#27037—431nomaximal, Sylow
261C22×C43#52511yescharacteristic
262C22×C43#54611yescharacteristic
263C22×C43#54711yescharacteristic
264C22×C43C2×#13511yescharacteristic
265C22×C43C2×#13711yescharacteristic
266C22×C43#57711yescharacteristic
267C22×C43#57711yescharacteristic
268C2×D43—22no—
269C2×D43—22no—
270C2×D43—22no—
271C2×D43—41no—
272C2×D43—41no—
273C2×D43—41no—
274C2×D43—41no—
275C4×C43—22no—
276C4×C43—42no—
277C4×C43—42no—
278C4×C43—41no—
279C4×C43—81no—
280C4×C43—81no—
281C4×C43#52911yescharacteristic
282C4×C43#53911yescharacteristic
283C4×C43#55311yescharacteristic
284C4×C43#55611yescharacteristic
285Dic43—41no—
286Dic43—41no—
287Dic43—42no—
288Dic43—41no—
289Dic43—81no—
290Dic43—81no—
291Dic43—81no—
292#2494—431no—
293#2360—431no—
294#2074—431no—
295#2066—431no—
296#2036—431no—
297#2035—431no—
298#1949—431no—
299#1941—431no—
300#1923—431no—
301#1921—431no—
302#1920—431no—
303#1865—431no—
304#1853—431no—
305#1823—431no—
306C4×#103D4311yescharacteristic, complemented
307C2×C43#246611yescharacteristic
308C2×C43#310811yescharacteristic
309C2×C43#312511yescharacteristic
310C2×C43#329611yescharacteristic
311C2×C43#332011yescharacteristic
312C2×C43#333711yescharacteristic
313C2×C43#335611yescharacteristic
314D43—42no—
315C2×C22⋊Q8—431no—
316C2×C4×Q8C2×D4311yescharacteristic
317C22×C4⋊C4—431no—
318C2×#107—431no—
319C2×#106—431no—
320C2×#106—431no—
321C2×#106—431no—
322C2×C4⋊C8—431no—
323C2×#103—431no—
324C2×#103—431no—
325C2×#103C2×D4311yescharacteristic
326C2×#103C2×D4311yescharacteristic
327C2×C22⋊C8—431no—
328C2×C22⋊C8—431no—
329C2×C4×C8C2×D4311yescharacteristic
330#401—861no—
331#397—861no—
332#391—431no—
333#389—431no—
334#389—431no—
335#389—431no—
336#389—861no—
337#388—861no—
338#387—861no—
339#387—861no—
340#385—861no—
341#382—431no—
342#380—431no—
343#380—431no—
344C4×C4⋊C4—431no—
345C4×C4⋊C4C2×D4311yescharacteristic
346C4×C22⋊C4—431no—
347C4×C22⋊C4—861no—
348#340—431no—
349#340—431no—
350#340—431no—
351#340—431no—
352#340—431no—
353#340—431no—
354#340C2×D4311yescharacteristic
355#340C2×D4311yescharacteristic
356#336—431no—
357#336—431no—
358C43#2703711yescharacteristic, complemented, Sylow
359C22×Q8C22×D4311yescharacteristic
360C23×C4—431no—
361C23×C4—861no—
362C22×C8—431no—
363C22×C8—431no—
364C22×C8C22×D4311yescharacteristic
365C22×C8C22×D4311yescharacteristic
366C22⋊Q8—862no—
367C22⋊Q8—862no—
368C4×Q8—22no—
369C4×Q8—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—431no—
372C2×C4⋊C4—431no—
373C2×C4⋊C4—431no—
374C2×C4⋊C4—431no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—862no—
377C2×C4⋊C4—861no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—861no—
381C2×C4⋊C4—1721no—
382C2×C4⋊C4C22×D4311yescharacteristic
383C2×C4⋊C4C22×D4311yescharacteristic
384C2×C4⋊C4C22×D4311yescharacteristic
385C2×C22⋊C4—431no—
386C2×C22⋊C4—431no—
387C2×C22⋊C4—861no—
388C2×C22⋊C4—861no—
389C2×C22⋊C4—861no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—431no—
393C2×C42—861no—
394C2×C42C22×D4311yescharacteristic
395#107—862no—
396#106—862no—
397#106—862no—
398#106—1721no—
399C4⋊C8—862no—
400#103—22no—
401#103—22no—
402#103—862no—
403#103—1721no—
404C22⋊C8—862no—
405C22⋊C8—1721no—
406C4×C8—41no—
407#95—21no—
408#95—431no—
409#95—431no—
410#95—861no—
411#95—861no—
412#95—861no—
413#95—861no—
414#95—861no—
415#95—861no—
416#95—861no—
417#95—861no—
418#95—861no—
419#95—861no—
420C24—431no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1062189812yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—431no—
432C22×C4—431no—
433C22×C4—431no—
434C22×C4—861no—
435C22×C4—861no—
436C22×C4—861no—
437C22×C4—861no—
438C22×C4—861no—
439C22×C4—861no—
440C22×C4—861no—
441C22×C4—861no—
442C22×C4—861no—
443C22×C4—861no—
444C22×C4—3441no—
445C22×C4—1722no—
446C22×C4—1722no—
447C22×C4D4×D4311yescharacteristic
448C22×C4D4×D4311yescharacteristic
449C22×C4C23×D4311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—861no—
456C2×C8—861no—
457C2×C8—861no—
458C2×C8—861no—
459C2×C8—861no—
460C2×C8—861no—
461C2×C8#1062189812yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—432no—
469C4⋊C4—864no—
470C4⋊C4—861no—
471C4⋊C4—861no—
472C4⋊C4—861no—
473C4⋊C4—861no—
474C4⋊C4—861no—
475C4⋊C4—861no—
476C4⋊C4—861no—
477C4⋊C4—3441no—
478C4⋊C4—3441no—
479C4⋊C4—1722no—
480C4⋊C4—1722no—
481C4⋊C4#1062189812yes—
482C22⋊C4—862no—
483C22⋊C4—864no—
484C22⋊C4—862no—
485C22⋊C4—3441no—
486C22⋊C4—1722no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—81no—
491C42—861no—
492C42—861no—
493C42—3441no—
494C23—432no—
495C23—861no—
496C23—861no—
497C23—861no—
498C23—861no—
499C23—861no—
500C23—861no—
501C23C2×D4×D4311yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—861no—
524C2×C4—861no—
525C2×C4—861no—
526C2×C4—862no—
527C2×C4—861no—
528C2×C4—861no—
529C2×C4—861no—
530C2×C4—861no—
531C2×C4—861no—
532C2×C4—861no—
533C2×C4—3444no—
534C2×C4—1721no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4—1721no—
543C2×C4—1721no—
544C2×C4—1721no—
545C2×C4—1721no—
546C2×C4—1721no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4#1306335712yes—
550C2×C4#41×D4311yescharacteristic
551C2×C4C2×#1062189811yescharacteristic
552C2×C4C2×#1062189811yescharacteristic
553C2×C4C2×#1062189811yescharacteristic
554C2×C4#1306337011yescharacteristic
555C2×C4#1306337211yescharacteristic
556C8—22no—
557C8—41no—
558C8—1721no—
559C8—1721no—
560C22—862no—
561C22—862no—
562C22—862no—
563C22—1721no—
564C22—1721no—
565C22—1721no—
566C22—1721no—
567C22#1458340811yescharacteristic
568C22#1458353911yescharacteristic
569C22#1458355411yescharacteristic
570C22#135×D4311yescharacteristic
571C22C2×#1306335711yescharacteristic
572C22#1458370511yescharacteristic
573C22#1458370611yescharacteristic
574C4—22no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—3441no—
581C4—3441no—
582C4—3441no—
583C4—1721no—
584C4—1722no—
585C4—1721no—
586C4—1721no—
587C4#1458343311yescharacteristic
588C4#1458349111yescharacteristic
589C4#1458359811yescharacteristic
590C4#1458361711yescharacteristic
591C2—1722no—
592C2#1913863111yescharacteristic
593C2#1914467911yescharacteristic
594C2#1914487811yescharacteristic
595C2#1914643111yescharacteristic
596C2#1914676211yescharacteristic
597C2#1914697911yescharacteristic
598C2#1914721211yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC4×#103×C43C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenterC4×#103D43
hypocenterC43#27037
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. #19136016C2semidirect
4. #19135974C2semidirect
7. #19135362C2semidirect
8. #19135323C2semidirect
10. #19135187C2semidirect
11. #19135178C2semidirect
12. #19134863C2semidirect
15. C4×#103×C43C2semidirect
306. C4×#103D43semidirect
358. C43#27037semidirect

Maximal subgroups

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

subgroupindex
#191389892
#191375962
#191360162
#191359742
#191357962
#191357902
#191353622
#191353232
#191352012
#191351872
#191351782
#191348632
#191347892
#191346802
C4×#103×C432
#2703743