Finite Group Databasebeta

← #25133050

Subgroups — #25133050

Order 11008.

1042 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583196C211yesmaximal, characteristic
2C2×C4×#10621884C211yesmaximal, characteristic
3#19139953C211yesmaximal, characteristic
4#19135930C211yesmaximal, characteristic, complemented
5#19135925C211yesmaximal, characteristic, complemented
6#19135738C211yesmaximal, characteristic
7#19135728C211yesmaximal, characteristic
8#19135561C211yesmaximal, characteristic, complemented
9#19135532C211yesmaximal, characteristic, complemented
10#19134950C211yesmaximal, characteristic
11#19134950C211yesmaximal, characteristic
12#19134863C211yesmaximal, characteristic, complemented
13#19134849C211yesmaximal, characteristic, complemented
14#19134610C211yesmaximal, characteristic, complemented
15C2×C43×#557C211yesmaximal, characteristic, complemented
16C22×#10621884C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18#14583196—24no—
19C4×#10621884—24no—
20C2×#13063258C2211yescharacteristic
21C2×#13063256C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×C4⋊(C43⋊C8)C2211yescharacteristic
24C2×#13063249C2211yescharacteristic
25C2×#13063249C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063245C2211yescharacteristic
28C2×#13063245C2211yescharacteristic
29C2×#13063245C2211yescharacteristic
30C2×C4×C43⋊C8C2211yescharacteristic
31#14582834—21no—
32#14582810—21no—
33#14582801C2211yescharacteristic
34#14582801C2211yescharacteristic
35#14582797C2211yescharacteristic
36#14582797C2211yescharacteristic
37#14582770—21no—
38#14582770—21no—
39#14582765C2211yescharacteristic
40#14582661C2211yescharacteristic
41#14582661C2211yescharacteristic
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582654C2211yescharacteristic
47#14582654C2211yescharacteristic
48#14582654C2211yescharacteristic
49#14582654C2211yescharacteristic
50C43×#557—24no—
51C43×#557—24no—
52C2×C4⋊Q8×C43—21no—
53C2×C4⋊Q8×C43C2211yescharacteristic
54C2×#125×C43—21no—
55C2×#125×C43—21no—
56C2×#125×C43C2211yescharacteristic
57C2×#125×C43C2211yescharacteristic
58C2×C4×Q8×C43—21no—
59C2×C4×Q8×C43C2211yescharacteristic
60C2×C4×Q8×C43C2211yescharacteristic
61C2×C4×Q8×C43C2211yescharacteristic
62C2×C4×Q8×C43C2211yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C2×#10621884—22no—
66C2×#10621884—22no—
67C2×#10621884Q814yescomplemented
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2311yescharacteristic
70C22×C43⋊C8C2311yescharacteristic
71C22×C43⋊C8C2311yescharacteristic
72C4×D172—22no—
73C4×D172—22no—
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—21no—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84#13063258—22no—
85#13063256—22no—
86#13063256—22no—
87C4⋊(C43⋊C8)—22no—
88#13063249—22no—
89#13063249—22no—
90#13063249—22no—
91#13063245—22no—
92#13063245—22no—
93#13063245—22no—
94C4×C43⋊C8—22no—
95C43⋊#95—21no—
96C43⋊#95—21no—
97C43⋊#95—21no—
98C43⋊#95—21no—
99C22×Q8×C43—21no—
100C22×Q8×C43D411yescharacteristic
101C22×Q8×C43C2311yescharacteristic
102C4⋊Q8×C43—28no—
103C4⋊Q8×C43—22no—
104C4⋊Q8×C43—22no—
105#125×C43—28no—
106#125×C43—28no—
107#125×C43—22no—
108#125×C43—22no—
109#125×C43—22no—
110#125×C43D414yes—
111C4×Q8×C43—28no—
112C4×Q8×C43—22no—
113C4×Q8×C43—22no—
114C4×Q8×C43—22no—
115C4×Q8×C43—22no—
116C4×Q8×C43—22no—
117C4×Q8×C43—22no—
118C4×Q8×C43—22no—
119C4×Q8×C43D414yes—
120C2×C4⋊C4×C43—21no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43—21no—
123C2×C4⋊C4×C43—21no—
124C2×C4⋊C4×C43—21no—
125C2×C4⋊C4×C43—21no—
126C2×C4⋊C4×C43—21no—
127C2×C4⋊C4×C43—21no—
128C2×C4⋊C4×C43D411yescharacteristic
129C2×C4⋊C4×C43D411yescharacteristic
130C2×C4⋊C4×C43D411yescharacteristic
131C2×C4⋊C4×C43C2311yescharacteristic
132C2×C4⋊C4×C43C2311yescharacteristic
133C2×C4⋊C4×C43C2311yescharacteristic
134C2×C42×C43—21no—
135C2×C42×C43—21no—
136C2×C42×C43—21no—
137C2×C42×C43C2311yescharacteristic
138C2×C42×C43C2311yescharacteristic
139C2×C42×C43C2311yescharacteristic
140C23×D43—21no—
141C2×Q8×C43—24no—
142C2×Q8×C43—22no—
143C2×Q8×C43—22no—
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—41no—
148C2×Q8×C43—42no—
149C2×Q8×C43—42no—
150C2×Q8×C43C2×D412yes—
151C2×Q8×C43C2×Q812yes—
152C2×Q8×C43#4112yes—
153C22×C4×C43—21no—
154C22×C4×C43—21no—
155C22×C4×C43—21no—
156C22×C4×C43—21no—
157C22×C4×C43C2×D411yescharacteristic
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2×D411yescharacteristic
161C22×C4×C43C2×D411yescharacteristic
162C22×C4×C43C2×D411yescharacteristic
163C22×C4×C43C2411yescharacteristic
164C22×Dic43—21no—
165C22×Dic43—21no—
166C22×Dic43—21no—
167C2×D172—21no—
168C2×D172—21no—
169C2×D172—21no—
170C2×D172—21no—
171C2×D172—21no—
172C2×D172C2×Q812yes—
173C2×C4×D43—42no—
174C2×C4×D43—81no—
175C2×C4×D43—81no—
176C4⋊C4×C43—24no—
177C4⋊C4×C43—24no—
178C4⋊C4×C43—24no—
179C4⋊C4×C43—24no—
180C4⋊C4×C43—24no—
181C4⋊C4×C43—24no—
182C4⋊C4×C43—24no—
183C4⋊C4×C43—24no—
184C4⋊C4×C43—22no—
185C4⋊C4×C43—22no—
186C4⋊C4×C43—22no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C4⋊C4×C43C2×D412yes—
191C4⋊C4×C43C2×D412yes—
192C4⋊C4×C43C2×D412yes—
193C42×C43—24no—
194C42×C43—24no—
195C42×C43—24no—
196C42×C43—21no—
197C42×C43—21no—
198C42×C43—21no—
199C42×C43—21no—
200C42×C43C2×D412yes—
201C42×C43C2×D412yes—
202#10621884—28no—
203(C2×D43)⋊C4—42no—
204(C2×D43)⋊C4—42no—
205(C2×D43)⋊C4—42no—
206(C2×D43)⋊C4—81no—
207(C2×D43)⋊C4—81no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213C4⋊Dic43—21no—
214Dic43⋊C4—81no—
215Dic43⋊C4—81no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—21no—
227C2×C43⋊C8C2×Q812yes—
228C23×C43C22×D411yescharacteristic
229C22×D43—22no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—41no—
233C22×D43—41no—
234C22×D43—41no—
235C22×D43—41no—
236Q8×C43—22no—
237Q8×C43—22no—
238Q8×C43—22no—
239Q8×C43—44no—
240Q8×C43—48no—
241Q8×C43C22⋊Q814yes—
242C2×C4×C43—22no—
243C2×C4×C43—21no—
244C2×C4×C43—22no—
245C2×C4×C43—22no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—22no—
254C2×C4×C43—21no—
255C2×C4×C43—22no—
256C2×C4×C43—22no—
257C2×C4×C43—21no—
258C2×C4×C43—21no—
259C2×C4×C43—21no—
260C2×C4×C43—21no—
261C2×C4×C43—21no—
262C2×C4×C43—21no—
263C2×C4×C43—41no—
264C2×C4×C43—41no—
265C2×C4×C43—41no—
266C2×C4×C43—41no—
267C2×C4×C43—41no—
268C2×C4×C43—41no—
269C2×C4×C43#12012yes—
270C2×C4×C43#12012yes—
271C2×C4×C43#13512yes—
272C2×C4×C43C22×D411yescharacteristic
273C2×C4×C43C22×D411yescharacteristic
274C2×C4×C43C22×Q811yescharacteristic
275C2×C4×C43C2×#4111yescharacteristic
276C2×C4×C432−1+411yescharacteristic
277C2×C4×C432−1+411yescharacteristic
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—41no—
287D172—22no—
288D172—22no—
289D172—41no—
290D172—41no—
291C4×D43—84no—
292C43⋊C8—22no—
293C43⋊C8—41no—
294C43⋊C8—41no—
295C43⋊C8—41no—
296#24334—431nomaximal, Sylow
297C22×C43C2×#12011yescharacteristic
298C22×C43#54811yescharacteristic
299C22×C43D4×Q811yescharacteristic
300C22×C43C2×#13511yescharacteristic
301C22×C43#57511yescharacteristic
302C22×C43#57711yescharacteristic
303C22×C43#57711yescharacteristic
304C2×D43—42no—
305C2×D43—42no—
306C2×D43—42no—
307C2×D43—81no—
308C2×D43—81no—
309C2×D43—81no—
310C2×D43—81no—
311C4×C43—24no—
312C4×C43—22no—
313C4×C43—22no—
314C4×C43—22no—
315C4×C43—41no—
316C4×C43—42no—
317C4×C43—42no—
318C4×C43—41no—
319C4×C43—41no—
320C4×C43—42no—
321C4×C43C2×C22⋊Q811yescharacteristic
322C4×C43#53611yescharacteristic
323C4×C43#54811yescharacteristic
324C4×C43D4×Q811yescharacteristic
325Dic43—81no—
326Dic43—81no—
327Dic43—81no—
328C2×#557D4311yescharacteristic, complemented
329C2×#478—431no—
330C2×C4×SD16—431no—
331#2580—431no—
332#2059—431no—
333#2058—431no—
334#2025—431no—
335#2023—431no—
336#1991—431no—
337#1984—431no—
338#1880—431no—
339#1880—431no—
340#1865—431no—
341#1863—431no—
342#1805—431no—
343C2×C43#240911yescharacteristic
344C2×C43#300211yescharacteristic
345C2×C43#301011yescharacteristic
346C2×C43#336511yescharacteristic
347C2×C43#337111yescharacteristic
348C2×C43#338111yescharacteristic
349C2×C43#338811yescharacteristic
350D43—82no—
351C22×SD16—431no—
352#557—24no—
353#557—24no—
354C2×C4⋊Q8—21no—
355C2×C4⋊Q8C2×D4311yescharacteristic
356C2×#125—21no—
357C2×#125—21no—
358C2×#125C2×D4311yescharacteristic
359C2×#125C2×D4311yescharacteristic
360C2×C4×Q8—21no—
361C2×C4×Q8C2×D4311yescharacteristic
362C2×C4×Q8C2×D4311yescharacteristic
363C2×C4×Q8C2×D4311yescharacteristic
364C2×C4×Q8C2×D4311yescharacteristic
365C2×C4×D4—431no—
366#478—864no—
367C4×SD16—864no—
368C2×#107—431no—
369C2×#106—431no—
370C2×#106—431no—
371C2×C4⋊C8—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#103—431no—
375C2×#102—431no—
376C2×#102—431no—
377C2×#102—431no—
378C2×C4×C8—431no—
379#399—861no—
380#393—861no—
381#390—431no—
382#390—431no—
383#389—431no—
384#389—431no—
385#385—861no—
386#385—861no—
387#382—431no—
388#340—431no—
389#340—431no—
390#340—431no—
391#340—431no—
392#340—431no—
393#340—431no—
394#336—431no—
395#336—431no—
396#336—431no—
397#336—431no—
398C43#2433411yescharacteristic, complemented, Sylow
399C22×Q8—21no—
400C22×Q8C22⋊D4311yescharacteristic
401C22×Q8C22×D4311yescharacteristic
402C22×D4—431no—
403C23×C4—861no—
404C2×SD16—434no—
405C2×SD16—862no—
406C2×SD16—862no—
407C22×C8—431no—
408C22×C8—431no—
409C22×C8—431no—
410C22×C8—431no—
411C4⋊Q8—28no—
412C4⋊Q8—22no—
413C4⋊Q8—22no—
414#125—28no—
415#125—28no—
416#125—22no—
417#125—22no—
418#125—22no—
419#125C22⋊D4314yes—
420C4×Q8—28no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×Q8—22no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8C22⋊D4314yes—
429C4×D4—862no—
430C4×D4—862no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—431no—
440C2×C4⋊C4—431no—
441C2×C4⋊C4—431no—
442C2×C4⋊C4—861no—
443C2×C4⋊C4—861no—
444C2×C4⋊C4C22⋊D4311yescharacteristic
445C2×C4⋊C4C22⋊D4311yescharacteristic
446C2×C4⋊C4C22⋊D4311yescharacteristic
447C2×C4⋊C4C22×D4311yescharacteristic
448C2×C4⋊C4C22×D4311yescharacteristic
449C2×C4⋊C4C22×D4311yescharacteristic
450C2×C22⋊C4—861no—
451C2×C22⋊C4—861no—
452C2×C22⋊C4—861no—
453C2×C22⋊C4—861no—
454C2×C22⋊C4—861no—
455C2×C42—21no—
456C2×C42—21no—
457C2×C42—21no—
458C2×C42C22×D4311yescharacteristic
459C2×C42C22×D4311yescharacteristic
460C2×C42C22×D4311yescharacteristic
461#107—862no—
462#106—862no—
463#106—862no—
464C4⋊C8—862no—
465#103—862no—
466#103—862no—
467#103—862no—
468#102—862no—
469#102—862no—
470#102—862no—
471C4×C8—862no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476C24—861no—
477C2×Q8—24no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—41no—
484C2×Q8—42no—
485C2×Q8—42no—
486C2×Q8Q8×D4312yes—
487C2×Q8#1062190212yes—
488C2×Q8C2×C22⋊D4312yes—
489C2×D4—432no—
490C2×D4—861no—
491C2×D4—861no—
492C2×D4—861no—
493C2×D4—861no—
494C2×D4—861no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—861no—
500C22×C4—861no—
501C22×C4—861no—
502C22×C4—3441no—
503C22×C4—3441no—
504C22×C4—1722no—
505C22×C4C2×C22⋊D4311yescharacteristic
506C22×C4C2×C22⋊D4311yescharacteristic
507C22×C4C2×C22⋊D4311yescharacteristic
508C22×C4C2×C22⋊D4311yescharacteristic
509C22×C4C2×C22⋊D4311yescharacteristic
510C22×C4C2×C22⋊D4311yescharacteristic
511C22×C4C23×D4311yescharacteristic
512SD16—868no—
513C2×C8—432no—
514C2×C8—861no—
515C2×C8—861no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C2×C8—861no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8—861no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—861no—
540C4⋊C4—861no—
541C4⋊C4—861no—
542C4⋊C4—861no—
543C4⋊C4—861no—
544C4⋊C4—861no—
545C4⋊C4—3441no—
546C4⋊C4—3441no—
547C4⋊C4C2×C22⋊D4312yes—
548C4⋊C4C2×C22⋊D4312yes—
549C4⋊C4C2×C22⋊D4312yes—
550C22⋊C4—3441no—
551C22⋊C4—3441no—
552C22⋊C4—1722no—
553C22⋊C4—1722no—
554C22⋊C4—1722no—
555C42—24no—
556C42—24no—
557C42—24no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42C2×C22⋊D4312yes—
563C42C2×C22⋊D4312yes—
564C23—862no—
565C23—1721no—
566C23—1721no—
567C23—1721no—
568C23—1721no—
569C23—1721no—
570C23—1721no—
571C23C22×C22⋊D4311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—44no—
576Q8—48no—
577Q8#1306330114yes—
578D4—862no—
579D4—862no—
580D4—1721no—
581D4—1721no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—3444no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1721no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4—1721no—
618C2×C4—1721no—
619C2×C4#1306329412yes—
620C2×C4#1306329412yes—
621C2×C4#1306334512yes—
622C2×C4C22×C22⋊D4311yescharacteristic
623C2×C4C22×C22⋊D4311yescharacteristic
624C2×C4C2×Q8×D4311yescharacteristic
625C2×C4C2×#1062190211yescharacteristic
626C2×C4#1306337311yescharacteristic
627C2×C4#1306337311yescharacteristic
628C8—862no—
629C8—1721no—
630C8—1721no—
631C8—1721no—
632C22—3441no—
633C22—3441no—
634C22—3441no—
635C22—3441no—
636C22—1722no—
637C22—1722no—
638C22—1722no—
639C22C2×#1306329411yescharacteristic
640C22#1458356011yescharacteristic
641C22Q8×C22⋊D4311yescharacteristic
642C22C2×#1306334511yescharacteristic
643C22#1458369711yescharacteristic
644C22#1458370711yescharacteristic
645C22#1458370711yescharacteristic
646C4—22no—
647C4—22no—
648C4—24no—
649C4—22no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—3441no—
657C4—3441no—
658C4—3441no—
659C4C2×#1306330111yescharacteristic
660C4#1458347311yescharacteristic
661C4#1458356011yescharacteristic
662C4Q8×C22⋊D4311yescharacteristic
663C2—3442no—
664C2#1913795411yescharacteristic
665C2#1914381311yescharacteristic
666C2#1914388111yescharacteristic
667C2#1914729111yescharacteristic
668C2#1914736311yescharacteristic
669C2#1914746611yescharacteristic
670C2#1914753411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#557C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenterC2×#557D43
hypocenterC43#24334
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #19135930C2semidirect
5. #19135925C2semidirect
8. #19135561C2semidirect
9. #19135532C2semidirect
12. #19134863C2semidirect
13. #19134849C2semidirect
14. #19134610C2semidirect
15. C2×C43×#557C2semidirect
67. C2×#10621884Q8semidirect
328. C2×#557D43semidirect
398. C43#24334semidirect

Maximal subgroups

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

subgroupindex
C2×#145831962
C2×C4×#106218842
#191399532
#191359302
#191359252
#191357382
#191357282
#191355612
#191355322
#191349502
#191348632
#191348492
#191346102
C2×C43×#5572
#2433443