Finite Group Databasebeta

← #25099301

Subgroups — #25099301

Order 11008.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582901C211yesmaximal, characteristic
2C2×#14582898C211yesmaximal, characteristic
3C2×M16×Dic43C211yesmaximal, characteristic, complemented
4#19137083C211yesmaximal, characteristic
5C2×#14582795C211yesmaximal, characteristic, complemented
6#19135290C212yesmaximal, complemented
7#19135283C212yesmaximal, complemented
8#19135276C212yesmaximal, complemented
9#19135269C212yesmaximal, complemented
10#19134316C211yesmaximal, characteristic, complemented
11C43×#1736C211yesmaximal, characteristic, complemented
12C22×#10621898C2211yescharacteristic
13C22×#10621877C2211yescharacteristic
14C22×C4⋊Dic43C2211yescharacteristic
15C22×Dic43⋊C4—21no—
16C22×C4×Dic43C2211yescharacteristic
17#14582901C418yescomplemented
18#14582898—24no—
19C2×#13063250C2212yes—
20C2×#13063248C2212yes—
21C2×#13063247C2212yes—
22C2×#13063245C2212yes—
23M16×Dic43—24no—
24C2×#13063235C2212yes—
25C2×C8×Dic43C2212yes—
26#14582804C2211yescharacteristic
27#14582804C2211yescharacteristic
28#14582800C2211yescharacteristic
29#14582800C2211yescharacteristic
30#14582795—21no—
31#14582795—21no—
32#14582795C2212yes—
33#14582795C2212yes—
34#14582776—21no—
35#14582776—21no—
36C4×(C2×D43)⋊C4—21no—
37C4×(C2×D43)⋊C4—21no—
38C2×C43⋊#95—21no—
39#14582661C2212yes—
40#14582661C2212yes—
41C22×M16×C43C2211yescharacteristic
42C2×#117×C43C2211yescharacteristic
43C22×C4⋊C4×C43C2211yescharacteristic
44C43×#340C2212yes—
45C43×#340C2212yes—
46C2×#10621898—22no—
47C2×#10621898—22no—
48C2×#10621898—22no—
49C2×#10621898—22no—
50C2×#10621898—22no—
51C2×#10621898C2×C414yes—
52C22×Dic86C2311yescharacteristic
53C22×D172C2311yescharacteristic
54C22×C22⋊D43—21no—
55C23×Dic43—21no—
56C23×Dic43D411yescharacteristic
57C23×Dic43D411yescharacteristic
58C22×C4×D43—21no—
59C2×#10621877—22no—
60C2×#10621877—22no—
61C2×#10621877D414yes—
62C22×C43⋊C8C2312yes—
63C2×C4⋊Dic43—21no—
64C2×C4⋊Dic43—22no—
65C2×C4⋊Dic43C2×C414yes—
66C2×C4⋊Dic43C2312yes—
67C2×Dic43⋊C4—22no—
68C2×Dic43⋊C4—22no—
69C2×Dic43⋊C4—44no—
70C2×(C2×D43)⋊C4—21no—
71C2×(C2×D43)⋊C4—21no—
72C2×(C2×D43)⋊C4—21no—
73C2×(C2×D43)⋊C4—21no—
74C2×C4×Dic43—22no—
75C2×C4×Dic43—22no—
76C2×C4×Dic43D411yescharacteristic
77C2×C4×Dic43D411yescharacteristic
78C2×C4×Dic43C2311yescharacteristic
79C2×C4×Dic43C2311yescharacteristic
80#13063250C2×C418yescomplemented
81#13063248—24no—
82#13063247C2×C418yescomplemented
83#13063245—24no—
84#13063235—24no—
85C8×Dic43—24no—
86C43⋊#95—22no—
87C43⋊#95—22no—
88C43⋊#95—21no—
89C43⋊#95—21no—
90C43⋊#95—21no—
91C43⋊#95—21no—
92C23×C4×C43—21no—
93C23×C4×C43C2311yescharacteristic
94C2×M16×C43—22no—
95C2×M16×C43—22no—
96C2×M16×C43C2×C414yes—
97C22×C8×C43C2312yes—
98#117×C43—22no—
99#117×C43—22no—
100C2×C4⋊C4×C43—22no—
101C2×C4⋊C4×C43—22no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43C2312yes—
104C2×C4⋊C4×C43C2311yescharacteristic
105C2×C4⋊C4×C43C2311yescharacteristic
106C2×C22⋊C4×C43—21no—
107C2×C42×C43—21no—
108C24×C43C2×D411yescharacteristic
109C23×D43—21no—
110C22×C4×C43—21no—
111C22×C4×C43—21no—
112C22×C4×C43—22no—
113C22×C4×C43—42no—
114C22×C4×C43—42no—
115C22×C4×C43—42no—
116C22×C4×C43C22×C412yes—
117C22×C4×C43C2×D412yes—
118C22×C4×C43C2×D411yescharacteristic
119C22×C4×C43#4112yes—
120C22×C4×C43#4112yes—
121C22×C4×C43#4112yes—
122C22×C4×C43#4112yes—
123C22×C4×C43C2411yescharacteristic
124C2×C22⋊D43—24no—
125C2×C22⋊D43—42no—
126C2×C22⋊D43—42no—
127C22×Dic43—22no—
128C22×Dic43—22no—
129C22×Dic43—24no—
130C22×Dic43—22no—
131C22×Dic43—22no—
132C22×Dic43—21no—
133C22×Dic43—22no—
134C22×Dic43—41no—
135C22×Dic43—41no—
136C22×Dic43—42no—
137C22×Dic43—42no—
138C22×Dic43C2×D411yescharacteristic
139C22×Dic43C2×D411yescharacteristic
140C22×Dic43C2×D411yescharacteristic
141C22×Dic43C2×D411yescharacteristic
142#10621898—28no—
143#10621898—28no—
144#10621898—44no—
145#10621898—44no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×D172—21no—
151C2×D172C22×C412yes—
152C2×C4×D43—24no—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×Dic86—21no—
156C2×Dic86—21no—
157C2×Dic86—21no—
158C2×Dic86—21no—
159C2×Dic86—21no—
160C2×Dic86C22×C412yes—
161M16×C43—28no—
162C2×C8×C43—22no—
163C2×C8×C43—22no—
164C2×C8×C43C22×C414yes—
165C4⋊C4×C43—22no—
166C4⋊C4×C43—22no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—21no—
171C4⋊C4×C43—42no—
172C22⋊C4×C43—42no—
173C42×C43—42no—
174(C2×D43)⋊C4—42no—
175(C2×D43)⋊C4—42no—
176(C2×D43)⋊C4—42no—
177(C2×D43)⋊C4—42no—
178C4⋊Dic43—24no—
179C4⋊Dic43—22no—
180C4⋊Dic43C22×C414yes—
181Dic43⋊C4—44no—
182Dic43⋊C4—44no—
183C4×Dic43—22no—
184C4×Dic43—22no—
185C4×Dic43—21no—
186C4×Dic43—21no—
187C4×Dic43—21no—
188C4×Dic43—21no—
189#10621877—28no—
190C2×C43⋊C8—22no—
191C2×C43⋊C8—22no—
192C2×C43⋊C8C2×D414yes—
193C23×C43—21no—
194C23×C43—21no—
195C23×C43—21no—
196C23×C43—21no—
197C23×C43C4×D412yes—
198C23×C43#12112yes—
199C23×C43#12412yes—
200C23×C43C22×D411yescharacteristic
201C22×D43—22no—
202C22×D43—41no—
203C22×D43—41no—
204C22×D43—41no—
205C22×D43—41no—
206C22×D43—41no—
207C22×D43—41no—
208C2×C4×C43—42no—
209C2×C4×C43—41no—
210C2×C4×C43—41no—
211C2×C4×C43—42no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—42no—
217C2×C4×C43—84no—
218C2×C4×C43#11714yes—
219C2×C4×C43C4×D414yes—
220C2×C4×C43C4×D412yes—
221C2×C4×C43#12112yes—
222C2×C4×C43#12314yes—
223C2×C4×C43#12412yes—
224C2×C4×C43#12614yes—
225C2×C4×C43C23×C411yescharacteristic
226C2×C4×C43C22×D411yescharacteristic
227C2×C4×C43C2×#4111yescharacteristic
228C2×C4×C43C2×#4111yescharacteristic
229C2×C4×C43C2×#4111yescharacteristic
230C2×C4×C43C2×#4111yescharacteristic
231C22⋊D43—48no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—24no—
237C2×Dic43—22no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—21no—
242C2×Dic43—21no—
243C2×Dic43—21no—
244C2×Dic43—44no—
245C2×Dic43—44no—
246C2×Dic43—48no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—42no—
250C2×Dic43—42no—
251C2×Dic43AGL(1,ℤ/8)12yes—
252C2×Dic43#13712yes—
253D172—22no—
254D172—22no—
255D172—41no—
256D172—41no—
257C4×D43—48no—
258Dic86—22no—
259Dic86—22no—
260Dic86—41no—
261Dic86—41no—
262C8×C43—24no—
263C43⋊C8—24no—
264#21093—431nomaximal, Sylow
265C22×C43—22no—
266C22×C43—22no—
267C22×C43—22no—
268C22×C43—22no—
269C22×C43—22no—
270C22×C43—22no—
271C22×C43#38814yes—
272C22×C43C2×C4×D411yescharacteristic
273C22×C43C2×#12111yescharacteristic
274C22×C43C2×#12411yescharacteristic
275C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C43C2×#13711yescharacteristic
277C22×C43#57511yescharacteristic
278C22×C43#57511yescharacteristic
279C2×D43—42no—
280C2×D43—42no—
281C2×D43—42no—
282C2×D43—81no—
283C2×D43—81no—
284C2×D43—81no—
285C2×D43—81no—
286C4×C43—81no—
287C4×C43—81no—
288C4×C43—82no—
289C4×C43#38814yes—
290C4×C43C2×#11711yescharacteristic
291C4×C43C2×C4×D411yescharacteristic
292C4×C43C2×#12311yescharacteristic
293C4×C43C2×#12611yescharacteristic
294Dic43—22no—
295Dic43—22no—
296Dic43—44no—
297Dic43—41no—
298Dic43—41no—
299Dic43—42no—
300C2×#417—431no—
301C2×#417—431no—
302C2×C4×M16—431no—
303#2297—431no—
304C2×#389—431no—
305#1936—432no—
306#1935—432no—
307#1934—432no—
308#1933—432no—
309#1736—431no—
310#1736D4311yescharacteristic, complemented
311C2×C43—24no—
312C2×C43C2×#38811yescharacteristic
313C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C43C4×#13711yescharacteristic
315C2×C43#305511yescharacteristic
316C2×C43#305611yescharacteristic
317C2×C43#314111yescharacteristic
318C2×C43#314211yescharacteristic
319D43—82no—
320C22×#41—431no—
321C22×M16—431no—
322C22×M16C2×D4311yescharacteristic
323C2×#117C2×D4311yescharacteristic
324C22×C4⋊C4—431no—
325C22×C4⋊C4—861no—
326C22×C4⋊C4C2×D4311yescharacteristic
327C22×C42—431no—
328#417—438no—
329#417—864no—
330C2×#103—432no—
331C2×#103—432no—
332C2×#102—432no—
333C2×#102—432no—
334C4×M16—864no—
335C2×#97—432no—
336C2×C4×C8—432no—
337#391—431no—
338#391—431no—
339#390—431no—
340#390—431no—
341#389—432no—
342#389—432no—
343#389—861no—
344#389—861no—
345#386—861no—
346#386—861no—
347C4×C22⋊C4—861no—
348C4×C22⋊C4—861no—
349C2×#95—861no—
350#340—432no—
351#340—432no—
352#340C2×D4312yes—
353#340C2×D4312yes—
354C43#2109311yescharacteristic, complemented, Sylow
355C2×#41—434no—
356C2×#41—862no—
357C2×#41—862no—
358C2×#41—862no—
359C2×#41—862no—
360C2×#41—862no—
361C22×Q8—431no—
362C22×D4—431no—
363C22×D4—861no—
364C23×C4—21no—
365C23×C4—431no—
366C23×C4—431no—
367C23×C4—861no—
368C23×C4—861no—
369C23×C4C22×D4311yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—434no—
373C2×M16—862no—
374C2×M16—862no—
375C2×M16C4×D4314yes—
376C22×C8—432no—
377C22×C8C22×D4312yes—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—434no—
384C2×C4⋊C4—432no—
385C2×C4⋊C4—861no—
386C2×C4⋊C4—862no—
387C2×C4⋊C4—862no—
388C2×C4⋊C4—862no—
389C2×C4⋊C4—1724no—
390C2×C4⋊C4C22×D4312yes—
391C2×C4⋊C4C22×D4311yescharacteristic
392C2×C4⋊C4C22×D4311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—861no—
395C2×C22⋊C4—861no—
396C2×C22⋊C4—861no—
397C2×C22⋊C4—861no—
398C2×C42—21no—
399C2×C42—431no—
400C2×C42—431no—
401C2×C42—431no—
402C2×C42—431no—
403C2×C42—862no—
404C2×C42—862no—
405#103—438no—
406#103—864no—
407#102—438no—
408#102—864no—
409#97—864no—
410C4×C8—864no—
411#95—862no—
412#95—862no—
413#95—861no—
414#95—861no—
415#95—861no—
416#95—861no—
417C24—861no—
418C24D4×D4311yescharacteristic
419#41—868no—
420#41—868no—
421#41—1724no—
422#41—1724no—
423C2×Q8—432no—
424C2×Q8—861no—
425C2×Q8—861no—
426C2×Q8—861no—
427C2×Q8—861no—
428C2×Q8—861no—
429C2×D4—432no—
430C2×D4—864no—
431C2×D4—861no—
432C2×D4—861no—
433C2×D4—861no—
434C2×D4—861no—
435C2×D4—861no—
436C2×D4—1722no—
437C2×D4—1722no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—431no—
445C22×C4—431no—
446C22×C4—431no—
447C22×C4—431no—
448C22×C4—862no—
449C22×C4—862no—
450C22×C4—864no—
451C22×C4—862no—
452C22×C4—862no—
453C22×C4—861no—
454C22×C4—862no—
455C22×C4—864no—
456C22×C4—1721no—
457C22×C4—1721no—
458C22×C4—1722no—
459C22×C4—1722no—
460C22×C4—1722no—
461C22×C4—1722no—
462C22×C4C2×C4×D4312yes—
463C22×C4#1062189812yes—
464C22×C4#1062189812yes—
465C22×C4D4×D4312yes—
466C22×C4D4×D4311yescharacteristic
467C22×C4#1062190012yes—
468C22×C4#1062190212yes—
469C22×C4C23×D4311yescharacteristic
470M16—28no—
471M16—868no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—434no—
475C2×C8—862no—
476C2×C8—862no—
477C2×C8C2×C4×D4314yes—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—434no—
486C4⋊C4—864no—
487C4⋊C4—862no—
488C4⋊C4—1724no—
489C4⋊C4—1724no—
490C22⋊C4—42no—
491C22⋊C4—1722no—
492C22⋊C4—1722no—
493C22⋊C4—1722no—
494C22⋊C4—1722no—
495C42—42no—
496C42—862no—
497C42—862no—
498C42—861no—
499C42—861no—
500C42—861no—
501C42—861no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—862no—
507C23—1721no—
508C23—1721no—
509C23—1721no—
510C23—1721no—
511C23—1721no—
512C23—1721no—
513C23#1306328712yes—
514C23#1306329812yes—
515C23#1306331112yes—
516C23C2×D4×D4311yescharacteristic
517Q8—862no—
518Q8—862no—
519Q8—1721no—
520Q8—1721no—
521D4—862no—
522D4—862no—
523D4—1728no—
524D4—1721no—
525D4—1721no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—432no—
537C2×C4—432no—
538C2×C4—861no—
539C2×C4—861no—
540C2×C4—861no—
541C2×C4—861no—
542C2×C4—864no—
543C2×C4—862no—
544C2×C4—861no—
545C2×C4—861no—
546C2×C4—861no—
547C2×C4—861no—
548C2×C4—861no—
549C2×C4—861no—
550C2×C4—1724no—
551C2×C4—1724no—
552C2×C4—1728no—
553C2×C4—1721no—
554C2×C4—1721no—
555C2×C4—1722no—
556C2×C4—1722no—
557C2×C4—1728no—
558C2×C4#1306328214yes—
559C2×C4(C22⋊D43)⋊C414yes—
560C2×C4#1306328712yes—
561C2×C4#1306329812yes—
562C2×C4#1306331014yes—
563C2×C4#1306331112yes—
564C2×C4#1306332014yes—
565C2×C4C22×C4×D4311yescharacteristic
566C2×C4C2×D4×D4311yescharacteristic
567C2×C4C2×#1062190011yescharacteristic
568C2×C4C2×#1062189811yescharacteristic
569C2×C4C2×#1062189811yescharacteristic
570C2×C4C2×#1062190211yescharacteristic
571C8—24no—
572C8—864no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—3441no—
580C22—3441no—
581C22—3441no—
582C22—3441no—
583C22—1722no—
584C22—1722no—
585C22—1722no—
586C22#1458278814yes—
587C22C2×#1306328711yescharacteristic
588C22C2×#1306329811yescharacteristic
589C22C2×#1306331111yescharacteristic
590C22AGL(1,ℤ/8)×D4311yescharacteristic
591C22#137×D4311yescharacteristic
592C22#1458369511yescharacteristic
593C22#1458369811yescharacteristic
594C4—82no—
595C4—81no—
596C4—81no—
597C4—862no—
598C4—862no—
599C4—1722no—
600C4—1721no—
601C4—1721no—
602C4—1724no—
603C4#1458278814yes—
604C4C2×#1306328211yescharacteristic
605C4C2×(C22⋊D43)⋊C411yescharacteristic
606C4C2×#1306331011yescharacteristic
607C4C2×#1306332011yescharacteristic
608C2—24no—
609C2—3442no—
610C2C2×#1458278811yescharacteristic
611C2#1914340211yescharacteristic
612C2#1914341311yescharacteristic
613C2#1914421211yescharacteristic
614C2#1914422311yescharacteristic
615C2#1914498611yescharacteristic
616C2#1914499311yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1736C2
socleC23×C43C22×D4
prime radicalC22×#10621898C22
cosocleC22×C4×C43C24
hypercenter#1736D43
hypocenterC43#21093
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×M16×Dic43C2semidirect
5. C2×#14582795C2semidirect
6. #19135290C2semidirect
7. #19135283C2semidirect
8. #19135276C2semidirect
9. #19135269C2semidirect
10. #19134316C2semidirect
11. C43×#1736C2semidirect
17. #14582901C4semidirect
80. #13063250C2×C4semidirect
82. #13063247C2×C4semidirect
310. #1736D43semidirect
354. C43#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#145829012
C2×#145828982
C2×M16×Dic432
#191370832
C2×#145827952
#191352902
#191352832
#191352762
#191352692
#191343162
C43×#17362
#2109343