Finite Group Databasebeta

← #25151818

Subgroups — #25151818

Order 11008.

1550 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583291C211yesmaximal, characteristic, complemented
2C2×#14583121C211yesmaximal, characteristic, complemented
3C2×#14583055C212yesmaximal, complemented
4C2×#14583038C212yesmaximal, complemented
5#19139534C211yesmaximal, characteristic
6#19135603C212yesmaximal, complemented
7#19135596C212yesmaximal, complemented
8#19135519C211yesmaximal, characteristic, complemented
9#19135394C211yesmaximal, characteristic, complemented
10#19134425C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C43C211yesmaximal, characteristic, complemented
12C22×#10621885C2211yescharacteristic
13C22×#10621885C2212yes—
14C22×#10621884C2211yescharacteristic
15C22×#10621884C2212yes—
16C2×#13063313C2211yescharacteristic
17C2×#13063304C2212yes—
18C2×#13063299C2212yes—
19#14583291—24no—
20#14583121—24no—
21#14583055—28no—
22#14583038—28no—
23C2×C4⋊(C43⋊C8)C2211yescharacteristic
24C2×#13063249C2212yes—
25C2×#13063249C2212yes—
26C2×#13063249C2212yes—
27C2×#13063248C2211yescharacteristic
28C2×#13063245C2211yescharacteristic
29C2×C22⋊(C43⋊C8)C2212yes—
30C2×C22⋊(C43⋊C8)C2212yes—
31C2×#13063236C2211yescharacteristic
32#14582834—21no—
33#14582807—21no—
34#14582799—22no—
35C4×C4⋊Dic43C2211yescharacteristic
36#14582661C2212yes—
37C23×Q8×C43C2212yes—
38D4×Q8×C43—24no—
39D4×Q8×C43—24no—
40C2×C4⋊Q8×C43—21no—
41C2×C4⋊Q8×C43C2211yescharacteristic
42C2×C22⋊Q8×C43—22no—
43C2×C22⋊Q8×C43C2212yes—
44C2×C4×Q8×C43C2211yescharacteristic
45C2×C4×D4×C43—21no—
46C2×C4×D4×C43C2211yescharacteristic
47C22×Dic86C2311yescharacteristic
48C22×D172C2311yescharacteristic
49C22×C22⋊D43—22no—
50C2×#10621885—24no—
51C2×#10621885—22no—
52C2×#10621885—24no—
53C2×#10621885—24no—
54C2×#10621885—42no—
55C2×#10621884—22no—
56C2×#10621884—24no—
57C2×#10621884—24no—
58C2×#10621884—24no—
59C2×#10621884—42no—
60C22×C43⋊C8C2311yescharacteristic
61C22×C43⋊C8C2311yescharacteristic
62C22×C43⋊C8C2312yes—
63#13063313—22no—
64#13063313—22no—
65#13063304—24no—
66#13063304—42no—
67#13063299—24no—
68#13063299—42no—
69C2×C4⋊Dic43C2312yes—
70C2×Dic43⋊C4—22no—
71C2×C22⋊Dic43—22no—
72C2×(C2×D43)⋊C4—21no—
73C2×(C2×D43)⋊C4—22no—
74C2×(C2×D43)⋊C4—21no—
75C2×C4×Dic43—21no—
76C4⋊(C43⋊C8)—22no—
77#13063249—24no—
78#13063249—24no—
79#13063249D418yescomplemented
80#13063248—22no—
81#13063245—22no—
82C22⋊(C43⋊C8)—24no—
83C22⋊(C43⋊C8)—24no—
84#13063236—22no—
85C43⋊#95—21no—
86C22×Q8×C43—24no—
87C22×Q8×C43—24no—
88C22×Q8×C43—22no—
89C22×Q8×C43—24no—
90C22×Q8×C43—24no—
91C22×Q8×C43—42no—
92C22×Q8×C43—42no—
93C22×Q8×C43C2312yes—
94C22×Q8×C43C2311yescharacteristic
95C22×D4×C43D411yescharacteristic
96C23×C4×C43—22no—
97C23×C4×C43C2312yes—
98C4⋊Q8×C43—28no—
99C4⋊Q8×C43—22no—
100C4⋊Q8×C43D414yescomplemented
101C22⋊Q8×C43—216no—
102C22⋊Q8×C43—24no—
103C22⋊Q8×C43—24no—
104C4×Q8×C43—22no—
105C4×Q8×C43D414yescomplemented
106C4×D4×C43—28no—
107C4×D4×C43—22no—
108C4×D4×C43—22no—
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43—21no—
111C2×C4⋊C4×C43—21no—
112C2×C4⋊C4×C43—22no—
113C2×C4⋊C4×C43D411yescharacteristic
114C2×C4⋊C4×C43C2311yescharacteristic
115C2×C22⋊C4×C43—22no—
116C2×C22⋊C4×C43D412yes—
117C2×C42×C43—21no—
118C2×C42×C43C2311yescharacteristic
119C24×C43C2×D412yes—
120C23×D43—21no—
121C2×Q8×C43—28no—
122C2×Q8×C43—21no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43—22no—
126C2×Q8×C43—22no—
127C2×Q8×C43—22no—
128C2×Q8×C43—22no—
129C2×Q8×C43—44no—
130C2×Q8×C43—44no—
131C2×Q8×C43—48no—
132C2×Q8×C43—48no—
133C2×Q8×C43—48no—
134C2×Q8×C43—48no—
135C2×Q8×C43C2×D414yes—
136C2×Q8×C43C2×D414yes—
137C2×Q8×C43C2×D412yes—
138C2×D4×C43—22no—
139C2×D4×C43—24no—
140C22×C4×C43—28no—
141C22×C4×C43—22no—
142C22×C4×C43—22no—
143C22×C4×C43—22no—
144C22×C4×C43—22no—
145C22×C4×C43—22no—
146C22×C4×C43—22no—
147C22×C4×C43—21no—
148C22×C4×C43—21no—
149C22×C4×C43—22no—
150C22×C4×C43—44no—
151C22×C4×C43—44no—
152C22×C4×C43C2×D412yes—
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43C2×D411yescharacteristic
155C22×C4×C43C2411yescharacteristic
156C2×C22⋊D43—44no—
157C2×C22⋊D43—44no—
158C2×C22⋊D43—82no—
159C22×Dic43—22no—
160C22×Dic43—21no—
161C2×D172—21no—
162C2×D172—22no—
163C2×D172—21no—
164C2×D172—22no—
165C2×Dic86—21no—
166C2×Dic86—22no—
167C2×Dic86—21no—
168C2×Dic86—22no—
169C4⋊C4×C43—28no—
170C4⋊C4×C43—24no—
171C4⋊C4×C43—24no—
172C4⋊C4×C43—24no—
173C4⋊C4×C43—22no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43C2×D412yes—
176C22⋊C4×C43—28no—
177C22⋊C4×C43—24no—
178C42×C43—24no—
179C42×C43C2×D412yes—
180C42×C43C2×D412yes—
181C22⋊Dic43—44no—
182#10621885—48no—
183#10621885—44no—
184#10621884—48no—
185#10621884—44no—
186(C2×D43)⋊C4—44no—
187(C2×D43)⋊C4—42no—
188(C2×D43)⋊C4—42no—
189C4⋊Dic43—22no—
190C4⋊Dic43C2×D414yes—
191Dic43⋊C4—44no—
192C4×Dic43—81no—
193C2×C43⋊C8—22no—
194C2×C43⋊C8—22no—
195C2×C43⋊C8—22no—
196C2×C43⋊C8—22no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8C2×D414yes—
200C23×C43—22no—
201C23×C43—22no—
202C23×C43—22no—
203C23×C43—22no—
204C23×C43—22no—
205C23×C43—22no—
206C23×C43#13714yes—
207C23×C43C22×D411yescharacteristic
208C22×D43—41no—
209C22×D43—42no—
210C22×D43—42no—
211C22×D43—41no—
212C22×D43—41no—
213Q8×C43—22no—
214Q8×C43—24no—
215Q8×C43—24no—
216Q8×C43—24no—
217Q8×C43—416no—
218Q8×C43—42no—
219Q8×C43—42no—
220Q8×C43#12714yes—
221D4×C43—28no—
222C2×C4×C43—22no—
223C2×C4×C43—24no—
224C2×C4×C43—22no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—22no—
228C2×C4×C43—24no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—416no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43—42no—
238C2×C4×C43—42no—
239C2×C4×C43—42no—
240C2×C4×C43—42no—
241C2×C4×C43#12012yes—
242C2×C4×C43#12012yes—
243C2×C4×C43C22×D412yes—
244C2×C4×C43C22×D411yescharacteristic
245C2×C4×C43C22×D411yescharacteristic
246C2×C4×C43H(5,𝔽2)12yes—
247C22⋊D43—88no—
248C2×Dic43—24no—
249C2×Dic43—42no—
250C2×Dic43—42no—
251C2×Dic43—41no—
252C2×Dic43—42no—
253D172—41no—
254D172—41no—
255D172—42no—
256Dic86—41no—
257Dic86—41no—
258Dic86—42no—
259C43⋊C8—24no—
260C43⋊C8—41no—
261C43⋊C8—41no—
262#25766—431nomaximal, Sylow
263C22×C43—24no—
264C22×C43—24no—
265C22×C43—24no—
266C22×C43—42no—
267C22×C43—42no—
268C22×C43—42no—
269C22×C43—42no—
270C22×C43C2×#12011yescharacteristic
271C22×C43D4212yes—
272C22×C43C2×#13712yes—
273C22×C43#57711yescharacteristic
274C22×C43#57811yescharacteristic
275C2×D43—81no—
276C2×D43—81no—
277C2×D43—82no—
278C2×D43—81no—
279C2×D43—82no—
280C4×C43—24no—
281C4×C43—22no—
282C4×C43—22no—
283C4×C43—42no—
284C4×C43—44no—
285C4×C43—42no—
286C4×C43C2×#12711yescharacteristic
287C4×C43#53511yescharacteristic
288C4×C43D4212yes—
289Dic43—44no—
290Dic43—81no—
291C2×D4×Q8D4311yescharacteristic, complemented
292C2×#497—431no—
293C2×#464—431no—
294C2×#451—432no—
295C2×#448—432no—
296#2540—431no—
297#2001—432no—
298#2000—432no—
299#1981—431no—
300#1955—431no—
301#1763—431no—
302C2×C43—44no—
303C2×C43#243311yescharacteristic
304C2×C43#300411yescharacteristic
305C2×C43#301311yescharacteristic
306C2×C43#326612yes—
307C2×C43#327012yes—
308D43—161no—
309C23×Q8C2×D4312yes—
310C22×Q16—431no—
311C22×Q16—432no—
312C22×SD16—431no—
313C22×SD16—432no—
314D4×Q8—24no—
315D4×Q8—24no—
316C2×C4⋊Q8—21no—
317C2×C4⋊Q8C2×D4311yescharacteristic
318C2×#124—431no—
319C2×C22⋊Q8—22no—
320C2×C22⋊Q8—432no—
321C2×C22⋊Q8C2×D4312yes—
322C2×#121—432no—
323C2×C4×Q8C2×D4311yescharacteristic
324C2×C4×D4—21no—
325C2×C4×D4C2×D4311yescharacteristic
326#497—864no—
327#464—864no—
328#451—868no—
329#448—868no—
330C2×C4⋊C8—431no—
331C2×#103—432no—
332C2×#103—432no—
333C2×#103—432no—
334C2×#103—431no—
335C2×#102—431no—
336C2×C22⋊C8—432no—
337C2×C22⋊C8—432no—
338C2×#97—431no—
339#399—861no—
340#392—861no—
341#390—862no—
342C4×C4⋊C4—431no—
343#340—432no—
344C43#2576611yescharacteristic, complemented, Sylow
345C22×Q8—24no—
346C22×Q8—24no—
347C22×Q8—22no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—42no—
351C22×Q8—42no—
352C22×Q8—431no—
353C22×Q8C22×D4312yes—
354C22×Q8C22×D4311yescharacteristic
355C22×D4—431no—
356C22×D4—862no—
357C22×D4C22⋊D4311yescharacteristic
358C23×C4—22no—
359C23×C4C22×D4312yes—
360C2×Q16—864no—
361C2×Q16—864no—
362C2×Q16—862no—
363C2×Q16—864no—
364C2×Q16—1722no—
365C2×SD16—864no—
366C2×SD16—864no—
367C2×SD16—862no—
368C2×SD16—864no—
369C2×SD16—1722no—
370C22×C8—432no—
371C22×C8—431no—
372C22×C8—431no—
373C4⋊Q8—28no—
374C4⋊Q8—22no—
375C4⋊Q8C22⋊D4314yescomplemented
376#124—862no—
377#124—862no—
378C22⋊Q8—216no—
379C22⋊Q8—24no—
380C22⋊Q8—24no—
381C22⋊Q8—864no—
382C22⋊Q8—1722no—
383#121—864no—
384#121—1722no—
385C4×Q8—22no—
386C4×Q8C22⋊D4314yescomplemented
387C4×D4—28no—
388C4×D4—22no—
389C4×D4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—432no—
395C2×C4⋊C4—862no—
396C2×C4⋊C4C22⋊D4311yescharacteristic
397C2×C4⋊C4C22×D4311yescharacteristic
398C2×C22⋊C4—22no—
399C2×C22⋊C4—862no—
400C2×C22⋊C4—862no—
401C2×C22⋊C4—861no—
402C2×C22⋊C4—861no—
403C2×C22⋊C4C22⋊D4312yes—
404C2×C42—21no—
405C2×C42—861no—
406C2×C42C22×D4311yescharacteristic
407C4⋊C8—862no—
408#103—438no—
409#103—864no—
410#103—864no—
411#103—862no—
412#102—862no—
413C22⋊C8—864no—
414C22⋊C8—864no—
415#97—862no—
416#95—861no—
417C24—861no—
418C24C2×C22⋊D4312yes—
419C2×Q8—28no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—862no—
434C2×Q8—861no—
435C2×Q8—861no—
436C2×Q8—862no—
437C2×Q8D4×D4314yes—
438C2×Q8C2×C22⋊D4314yes—
439C2×Q8C2×C22⋊D4312yes—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—862no—
443C2×D4—861no—
444C2×D4—861no—
445C2×D4—862no—
446C2×D4—3442no—
447C2×D4—1724no—
448C2×D4—1724no—
449C22×C4—21no—
450C22×C4—28no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—862no—
462C22×C4—861no—
463C22×C4C2×C22⋊D4312yes—
464C22×C4C2×C22⋊D4311yescharacteristic
465C22×C4C2×C22⋊D4311yescharacteristic
466C22×C4C23×D4311yescharacteristic
467Q16—1728no—
468Q16—1724no—
469SD16—1728no—
470SD16—1724no—
471C2×C8—434no—
472C2×C8—862no—
473C2×C8—862no—
474C2×C8—862no—
475C2×C8—862no—
476C2×C8—861no—
477C2×C8—861no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—434no—
485C4⋊C4—862no—
486C4⋊C4—1724no—
487C4⋊C4C2×C22⋊D4312yes—
488C22⋊C4—28no—
489C22⋊C4—24no—
490C22⋊C4—1724no—
491C22⋊C4—1724no—
492C22⋊C4—1722no—
493C22⋊C4—1722no—
494C42—24no—
495C42—3441no—
496C42C2×C22⋊D4312yes—
497C42C2×C22⋊D4312yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—1721no—
505C23—1722no—
506C23—1721no—
507C23—1721no—
508C23—1722no—
509C23#1306335614yes—
510C23C22×C22⋊D4311yescharacteristic
511Q8—22no—
512Q8—24no—
513Q8—24no—
514Q8—24no—
515Q8—416no—
516Q8—42no—
517Q8—42no—
518Q8—1721no—
519Q8—1721no—
520Q8—1722no—
521Q8#1306332114yes—
522D4—28no—
523D4—3448no—
524D4—1721no—
525D4—1721no—
526D4—1722no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—416no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—864no—
547C2×C4—1721no—
548C2×C4—1722no—
549C2×C4—1722no—
550C2×C4—1722no—
551C2×C4#1306329412yes—
552C2×C4#1306329412yes—
553C2×C4C2×D4×D4312yes—
554C2×C4C22×C22⋊D4311yescharacteristic
555C2×C4C22×C22⋊D4311yescharacteristic
556C2×C4#1306337012yes—
557C8—864no—
558C8—1721no—
559C8—1721no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—3442no—
568C22—3441no—
569C22—3441no—
570C22—3442no—
571C22—3441no—
572C22C2×#1306329411yescharacteristic
573C22D4×C22⋊D4312yes—
574C22C2×#1306335612yes—
575C22#1458370711yescharacteristic
576C22#1458371411yescharacteristic
577C4—22no—
578C4—24no—
579C4—22no—
580C4—42no—
581C4—44no—
582C4—42no—
583C4—3441no—
584C4—1724no—
585C4C2×#1306332111yescharacteristic
586C4#1458346611yescharacteristic
587C4D4×C22⋊D4312yes—
588C2—44no—
589C2—6881no—
590C2#1913822811yescharacteristic
591C2#1914382911yescharacteristic
592C2#1914391011yescharacteristic
593C2#1914610612yes—
594C2#1914616612yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×D4×Q8×C43C2
socleC23×C43C22×D4
prime radical#19139534C2
hypercenterC2×D4×Q8D43
hypocenterC43#25766
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14583291C2semidirect
2. C2×#14583121C2semidirect
3. C2×#14583055C2semidirect
4. C2×#14583038C2semidirect
6. #19135603C2semidirect
7. #19135596C2semidirect
8. #19135519C2semidirect
9. #19135394C2semidirect
10. #19134425C2semidirect
11. C2×D4×Q8×C43C2semidirect
79. #13063249D4semidirect
100. C4⋊Q8×C43D4semidirect
105. C4×Q8×C43D4semidirect
291. C2×D4×Q8D43semidirect
344. C43#25766semidirect
375. C4⋊Q8C22⋊D43semidirect
386. C4×Q8C22⋊D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832912
C2×#145831212
C2×#145830552
C2×#145830382
#191395342
#191356032
#191355962
#191355192
#191353942
#191344252
C2×D4×Q8×C432
#2576643