Finite Group Databasebeta

← #25003873

Subgroups — #25003873

Order 11008.

868 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19143184C211yesmaximal, characteristic, complemented
2C2×#14582872C211yesmaximal, characteristic
3C2×#14582872C211yesmaximal, characteristic
4#19143039C211yesmaximal, characteristic
5#19134306C211yesmaximal, characteristic, complemented
6C2×#14582655C211yesmaximal, characteristic, complemented
7C43×#1733C211yesmaximal, characteristic, complemented
8C22×#10621898C2211yescharacteristic
9C22×#10621877C2211yescharacteristic
10C2×#13063284C2211yescharacteristic
11C2×#13063284C2211yescharacteristic
12#14582942—22no—
13#14582939—22no—
14#14582935—22no—
15#14582872—22no—
16#14582872—22no—
17#14582872—22no—
18#14582872—22no—
19C2×#13063239—22no—
20C2×#13063239—22no—
21#14582867—22no—
22#14582865—22no—
23#14582862C414yescomplemented
24#14582660—22no—
25#14582655—22no—
26C22×M16×C43C2211yescharacteristic
27C2×#117×C43C2211yescharacteristic
28C2×#117×C43C2211yescharacteristic
29C43×#339—22no—
30C2×#10621898—21no—
31C2×#10621898—21no—
32C2×#10621898—21no—
33C2×#10621898—21no—
34C2×#10621898—21no—
35C2×#10621898—21no—
36C2×#10621898—21no—
37C2×#10621898—21no—
38C2×#10621898—21no—
39C2×#10621898—21no—
40C2×#10621898—21no—
41C2×#10621898—21no—
42C22×Dic86Q811yescharacteristic
43C22×D172Q811yescharacteristic
44C22×C22⋊D43D412yes—
45C22×C4×D43C2×C412yes—
46C2×#10621877—21no—
47C2×#10621877—21no—
48C2×#10621877—21no—
49C2×#10621877—21no—
50C2×#10621877C2×C412yes—
51C2×#10621877D412yes—
52C22×C43⋊C8D412yes—
53#13063284—21no—
54#13063284—21no—
55#13063284—21no—
56#13063284—21no—
57#13063284—21no—
58#13063284—21no—
59#13063284D412yes—
60#13063284D412yes—
61C2×C4⋊Dic43—21no—
62C2×C4⋊Dic43—21no—
63C2×C22⋊Dic43—21no—
64C2×C22⋊Dic43—21no—
65C2×C4×Dic43—21no—
66C2×C4×Dic43—21no—
67#13063258—24no—
68#13063256—24no—
69#13063239—28no—
70#13063239—28no—
71(C2×D43)⋊C8—28no—
72C23×C4×C43C2311yescharacteristic
73C2×M16×C43—21no—
74C2×M16×C43—21no—
75C2×M16×C43—21no—
76C2×M16×C43—21no—
77C2×M16×C43—21no—
78C2×M16×C43C2×C412yes—
79C22×C8×C43—21no—
80#117×C43—21no—
81#117×C43—21no—
82#117×C43—21no—
83#117×C43—21no—
84#117×C43—21no—
85#117×C43—21no—
86#117×C43D412yes—
87#117×C43D412yes—
88C2×C4⋊C4×C43—21no—
89C2×C4⋊C4×C43—21no—
90C2×C22⋊C4×C43—21no—
91C2×C22⋊C4×C43—21no—
92C2×C42×C43—21no—
93C2×C42×C43—21no—
94C24×C43C2×D411yescharacteristic
95C23×D43C4⋊C412yes—
96C22×C4×C43—41no—
97C22×C4×C43—41no—
98C22×C4×C43C22×C411yescharacteristic
99C22×C4×C43C2×D411yescharacteristic
100C22×C4×C43C2×D411yescharacteristic
101C22×C4×C43C2×D411yescharacteristic
102C22×C4×C43C2×D411yescharacteristic
103C22×C4×C43C2×D411yescharacteristic
104C22×C4×C43C2×D411yescharacteristic
105C22×C4×C43C2×Q811yescharacteristic
106C22×C4×C43#4111yescharacteristic
107C22×C4×C43#4111yescharacteristic
108C22×C4×C43#4111yescharacteristic
109C22×C4×C43#4111yescharacteristic
110C22×C4×C43#4111yescharacteristic
111C22×C4×C43#4111yescharacteristic
112C2×C22⋊D43—22no—
113C2×C22⋊D43—22no—
114C2×C22⋊D43—22no—
115C2×C22⋊D43—22no—
116C2×C22⋊D43—22no—
117C2×C22⋊D43—22no—
118C22×Dic43—41no—
119C22×Dic43—41no—
120C22×Dic43C4⋊C412yes—
121#10621898—41no—
122#10621898—41no—
123#10621898—41no—
124#10621898—41no—
125#10621898—41no—
126#10621898—41no—
127#10621898—41no—
128#10621898—41no—
129#10621898—41no—
130#10621898—41no—
131#10621898—41no—
132#10621898—41no—
133#10621898—41no—
134#10621898—41no—
135#10621898—41no—
136#10621898—41no—
137C2×D172—21no—
138C2×D172—21no—
139C2×D172—21no—
140C2×D172—21no—
141C2×D172—21no—
142C2×D172—21no—
143C2×C4×D43—22no—
144C2×C4×D43—22no—
145C2×C4×D43—22no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—22no—
149C2×Dic86—21no—
150C2×Dic86—21no—
151C2×Dic86—21no—
152C2×Dic86—21no—
153C2×Dic86—21no—
154C2×Dic86—21no—
155M16×C43—22no—
156M16×C43—22no—
157M16×C43—41no—
158M16×C43—41no—
159C2×C8×C43—22no—
160C2×C8×C43—41no—
161C2×C8×C43—41no—
162C4⋊C4×C43—22no—
163C4⋊C4×C43—22no—
164C4⋊C4×C43—41no—
165C4⋊C4×C43—41no—
166C22⋊C4×C43—22no—
167C22⋊C4×C43—22no—
168C22⋊C4×C43—41no—
169C22⋊C4×C43—41no—
170C42×C43—22no—
171C42×C43—22no—
172C42×C43—41no—
173C42×C43—41no—
174C22⋊Dic43—22no—
175C22⋊Dic43—22no—
176C22⋊Dic43—41no—
177C22⋊Dic43—41no—
178C4⋊Dic43—22no—
179C4⋊Dic43—22no—
180C4⋊Dic43—41no—
181C4⋊Dic43—41no—
182C4×Dic43—22no—
183C4×Dic43—22no—
184C4×Dic43—41no—
185C4×Dic43—41no—
186#10621877—22no—
187#10621877—22no—
188#10621877—22no—
189#10621877—22no—
190C2×C43⋊C8—22no—
191C2×C43⋊C8—22no—
192C2×C43⋊C8C22⋊C414yes—
193C23×C43—81no—
194C23×C43C2×C4⋊C411yescharacteristic
195C23×C43#12011yescharacteristic
196C23×C43#12011yescharacteristic
197C23×C43#12111yescharacteristic
198C23×C43#12111yescharacteristic
199C23×C43#12311yescharacteristic
200C23×C43#12311yescharacteristic
201C22×D43—22no—
202C22×D43—22no—
203C22×D43—22no—
204C22×D43—82no—
205C2×C4×C43—42no—
206C2×C4×C43—42no—
207C2×C4×C43—82no—
208C2×C4×C43—82no—
209C2×C4×C43C2×C22⋊C411yescharacteristic
210C2×C4×C43#11711yescharacteristic
211C2×C4×C43C4×D411yescharacteristic
212C2×C4×C43C4×D411yescharacteristic
213C2×C4×C43C4×D411yescharacteristic
214C2×C4×C43C4×D411yescharacteristic
215C2×C4×C43#12011yescharacteristic
216C2×C4×C43#12111yescharacteristic
217C2×C4×C43#12111yescharacteristic
218C2×C4×C43#12111yescharacteristic
219C2×C4×C43#12111yescharacteristic
220C2×C4×C43#12111yescharacteristic
221C2×C4×C43#12111yescharacteristic
222C2×C4×C43C22⋊Q811yescharacteristic
223C2×C4×C43C22⋊Q811yescharacteristic
224C2×C4×C43C22⋊Q811yescharacteristic
225C2×C4×C43C22⋊Q811yescharacteristic
226C2×C4×C43C22⋊Q811yescharacteristic
227C2×C4×C43C22⋊Q811yescharacteristic
228C2×C4×C43#12311yescharacteristic
229C2×C4×C43#12311yescharacteristic
230C2×C4×C43#12311yescharacteristic
231C2×C4×C43#12411yescharacteristic
232C2×C4×C43#12411yescharacteristic
233C2×C4×C43#12411yescharacteristic
234C2×C4×C43#12611yescharacteristic
235C2×C4×C43#12611yescharacteristic
236C2×C4×C43#12711yescharacteristic
237C22⋊D43—42no—
238C22⋊D43—42no—
239C22⋊D43—42no—
240C22⋊D43—42no—
241C2×Dic43—22no—
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—42no—
245C2×Dic43—42no—
246C2×Dic43—82no—
247C2×Dic43—82no—
248D172—41no—
249D172—41no—
250D172—41no—
251D172—41no—
252C4×D43—42no—
253C4×D43—42no—
254C4×D43—42no—
255C4×D43—42no—
256Dic86—41no—
257Dic86—41no—
258Dic86—41no—
259Dic86—41no—
260C8×C43—42no—
261C43⋊C8—24no—
262#11102—431nomaximal, Sylow
263C22×C43—41no—
264C22×C43—41no—
265C22×C43—41no—
266C22×C43—41no—
267C22×C43—41no—
268C22×C43—41no—
269C22×C43—41no—
270C22×C43#38011yescharacteristic
271C22×C43#38511yescharacteristic
272C22×C43#38511yescharacteristic
273C22×C43#39211yescharacteristic
274C22×C43#39411yescharacteristic
275C22×C43#39411yescharacteristic
276C22×C43#39711yescharacteristic
277C2×D43—44no—
278C2×D43—42no—
279C2×D43—82no—
280C2×D43—82no—
281C4×C43—82no—
282C4×C43—82no—
283C4×C43#38611yescharacteristic
284C4×C43#38811yescharacteristic
285C4×C43#38811yescharacteristic
286C4×C43#39011yescharacteristic
287C4×C43#39311yescharacteristic
288C4×C43#39611yescharacteristic
289C4×C43#39611yescharacteristic
290C4×C43#39911yescharacteristic
291Dic43—42no—
292Dic43—82no—
293Dic43—82no—
294#2911—431no—
295C2×#410—431no—
296C2×#410—431no—
297#2879—431no—
298#1733—431no—
299#1733D4311yescharacteristic, complemented
300C2×H(3,ℤ/4)—431no—
301C2×C43—21no—
302C2×C43—21no—
303C2×C43—21no—
304C2×C43—21no—
305C2×C43—21no—
306C2×C43—21no—
307C2×C43—21no—
308C2×C43#143911yescharacteristic
309D43—82no—
310C22×#41—431no—
311C22×M16—431no—
312C22×M16C2×D4311yescharacteristic
313C2×#117—431no—
314C2×#117—431no—
315C2×#117C2×D4311yescharacteristic
316C2×#117C2×D4311yescharacteristic
317#428—862no—
318#428—862no—
319#427—862no—
320#410—862no—
321#410—862no—
322#410—862no—
323#410—862no—
324C2×#99—862no—
325C2×#99—862no—
326#408—862no—
327#408—862no—
328#407—434no—
329#339—22no—
330#339—862no—
331H(3,ℤ/4)—862no—
332C43#1110211yescharacteristic, complemented, Sylow
333C2×#41—861no—
334C2×#41—861no—
335C2×#41—861no—
336C2×#41—861no—
337C2×#41—861no—
338C2×#41—861no—
339C2×#41—861no—
340C2×#41—861no—
341C2×#41—861no—
342C2×#41—861no—
343C2×#41—861no—
344C2×#41—861no—
345C22×Q8—431no—
346C22×D4—432no—
347C22×D4—431no—
348C23×C4—432no—
349C23×C4C22×D4311yescharacteristic
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—432no—
356C2×M16—432no—
357C2×M16—861no—
358C2×M16—861no—
359C2×M16—861no—
360C2×M16—861no—
361C2×M16C4×D4312yes—
362C22×C8—21no—
363C22×C8—432no—
364#117—21no—
365#117—21no—
366#117—21no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—432no—
371#117—432no—
372#117—861no—
373#117—861no—
374#117—861no—
375#117—861no—
376#117—861no—
377#117—861no—
378#117D17212yes—
379#117C22⋊D4312yes—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—861no—
383C2×C4⋊C4—861no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—861no—
391C2×C42—861no—
392#107—864no—
393#106—864no—
394#99—868no—
395#99—868no—
396C22⋊C8—868no—
397C24—432no—
398C24D4×D4311yescharacteristic
399#41—1721no—
400#41—1721no—
401#41—1721no—
402#41—1721no—
403#41—1721no—
404#41—1721no—
405#41—1721no—
406#41—1721no—
407#41—1721no—
408#41—1721no—
409#41—1721no—
410#41—1721no—
411#41—1721no—
412#41—1721no—
413#41—1721no—
414#41—1721no—
415C2×Q8—861no—
416C2×Q8—861no—
417C2×Q8—861no—
418C2×Q8—861no—
419C2×Q8—861no—
420C2×Q8—861no—
421C2×D4—862no—
422C2×D4—862no—
423C2×D4—862no—
424C2×D4—862no—
425C2×D4—862no—
426C2×D4—862no—
427C2×D4—861no—
428C2×D4—861no—
429C2×D4—861no—
430C2×D4—861no—
431C2×D4—861no—
432C2×D4—861no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—432no—
436C22×C4—862no—
437C22×C4—862no—
438C22×C4—862no—
439C22×C4—862no—
440C22×C4—862no—
441C22×C4—862no—
442C22×C4—1721no—
443C22×C4—1721no—
444C22×C4C2×C4×D4311yescharacteristic
445C22×C4C2×D17211yescharacteristic
446C22×C4#1062189811yescharacteristic
447C22×C4#1062189811yescharacteristic
448C22×C4#1062189811yescharacteristic
449C22×C4#1062189811yescharacteristic
450C22×C4D4×D4311yescharacteristic
451C22×C4D4×D4311yescharacteristic
452C22×C4D4×D4311yescharacteristic
453C22×C4D4×D4311yescharacteristic
454C22×C4#1062190011yescharacteristic
455C22×C4Q8×D4311yescharacteristic
456C22×C4#1062190211yescharacteristic
457C22×C4C2×C22⋊D4311yescharacteristic
458M16—22no—
459M16—22no—
460M16—41no—
461M16—41no—
462M16—862no—
463M16—862no—
464M16—862no—
465M16—862no—
466C2×C8—22no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—434no—
470C2×C8—862no—
471C2×C8—862no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—862no—
477C4⋊C4—862no—
478C4⋊C4—1721no—
479C4⋊C4—1721no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—862no—
485C22⋊C4—862no—
486C22⋊C4—1721no—
487C22⋊C4—1721no—
488C42—22no—
489C42—22no—
490C42—41no—
491C42—41no—
492C42—862no—
493C42—862no—
494C42—1721no—
495C42—1721no—
496C23—81no—
497C23—862no—
498C23—862no—
499C23—862no—
500C23—3442no—
501C23C4⋊C4×D4311yescharacteristic
502C23#1306329311yescharacteristic
503C23#1306329511yescharacteristic
504C23#1306329811yescharacteristic
505C23#1306329811yescharacteristic
506C23#1306330711yescharacteristic
507C23#1306331011yescharacteristic
508Q8—1721no—
509Q8—1721no—
510Q8—1721no—
511Q8—1721no—
512D4—1722no—
513D4—1722no—
514D4—1722no—
515D4—1722no—
516D4—1721no—
517D4—1721no—
518D4—1721no—
519D4—1721no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—862no—
525C2×C4—862no—
526C2×C4—862no—
527C2×C4—3442no—
528C2×C4—3442no—
529C2×C4—1722no—
530C2×C4—1722no—
531C2×C4—1722no—
532C2×C4—1722no—
533C2×C4—1722no—
534C2×C4—1722no—
535C2×C4C22⋊C4×D4311yescharacteristic
536C2×C4#1306328211yescharacteristic
537C2×C4(C22⋊D43)⋊C411yescharacteristic
538C2×C4C4×C22⋊D4311yescharacteristic
539C2×C4#1306328711yescharacteristic
540C2×C4C4×D17211yescharacteristic
541C2×C4#1306329511yescharacteristic
542C2×C4#1306329711yescharacteristic
543C2×C4#1306329811yescharacteristic
544C2×C4#1306329811yescharacteristic
545C2×C4#1306329811yescharacteristic
546C2×C4#1306329911yescharacteristic
547C2×C4#1306330011yescharacteristic
548C2×C4#1306330111yescharacteristic
549C2×C4#1306330311yescharacteristic
550C2×C4#1306330311yescharacteristic
551C2×C4#1306330311yescharacteristic
552C2×C4#1306330311yescharacteristic
553C2×C4#1306330511yescharacteristic
554C2×C4#1306330811yescharacteristic
555C2×C4#1306331011yescharacteristic
556C2×C4#1306331011yescharacteristic
557C2×C4#1306331111yescharacteristic
558C2×C4#1306331111yescharacteristic
559C2×C4#1306331311yescharacteristic
560C2×C4#1306331911yescharacteristic
561C2×C4#1306332011yescharacteristic
562C2×C4#1306332111yescharacteristic
563C8—42no—
564C8—864no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3442no—
573C22—3442no—
574C22—1724no—
575C22—1722no—
576C22#1458275811yescharacteristic
577C22#1458277011yescharacteristic
578C22#1458277311yescharacteristic
579C22#1458280811yescharacteristic
580C22#1458281411yescharacteristic
581C22#1458281711yescharacteristic
582C22#1458282811yescharacteristic
583C4—82no—
584C4—82no—
585C4—3442no—
586C4—3442no—
587C4—1722no—
588C4#1458277711yescharacteristic
589C4#1458278811yescharacteristic
590C4#1458279011yescharacteristic
591C4#1458279911yescharacteristic
592C4#1458281211yescharacteristic
593C4#1458282111yescharacteristic
594C4#1458282311yescharacteristic
595C4#1458283411yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—3442no—
604C2#1913283911yescharacteristic
subgroupG/H
centerC4#14582799
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1733C2
socleC2×C43#1439
prime radicalC22×#10621898C22
cosocleC23×C4×C43C23
hypercenter#1733D43
hypocenterC43#11102
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. #19143184C2semidirect
5. #19134306C2semidirect
6. C2×#14582655C2semidirect
7. C43×#1733C2semidirect
23. #14582862C4semidirect
299. #1733D43semidirect
332. C43#11102semidirect

Maximal subgroups

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

subgroupindex
#191431842
C2×#145828722
#191430392
#191343062
C2×#145826552
C43×#17332
#1110243