Finite Group Databasebeta

← #25159662

Subgroups — #25159662

Order 11008.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583187C211yesmaximal, characteristic
2#19138428C211yesmaximal, characteristic, complemented
3D4×Dic43⋊C4C211yesmaximal, characteristic
4#19136136C211yesmaximal, characteristic, complemented
5#19136074C211yesmaximal, characteristic, complemented
6#19135974C211yesmaximal, characteristic, complemented
7#19135926C211yesmaximal, characteristic, complemented
8#19135323C211yesmaximal, characteristic, complemented
9#19135152C211yesmaximal, characteristic, complemented
10#19134931C211yesmaximal, characteristic
11#19134931C211yesmaximal, characteristic
12#19134863C211yesmaximal, characteristic, complemented
13#19134858C211yesmaximal, characteristic
14#19134856C211yesmaximal, characteristic
15C2×C4×SD16×C43C211yesmaximal, characteristic
16C2×#13063324C2211yescharacteristic
17C2×D4×Dic43C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C22×Dic43⋊C4—21no—
20#14583187—24no—
21C2×#13063257C2211yescharacteristic
22C2×#13063257C2211yescharacteristic
23C2×#13063256C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×Dic43⋊C8C2211yescharacteristic
26C2×#13063249C2211yescharacteristic
27C2×#13063249C2211yescharacteristic
28C2×#13063246C2211yescharacteristic
29C2×#13063246C2211yescharacteristic
30#14582832—21no—
31#14582819—21no—
32#14582806C2211yescharacteristic
33#14582806C2211yescharacteristic
34#14582805—21no—
35#14582798C2211yescharacteristic
36#14582797C2211yescharacteristic
37#14582797C2211yescharacteristic
38#14582795C2211yescharacteristic
39#14582783—21no—
40#14582772—21no—
41#14582771—21no—
42#14582768C2211yescharacteristic
43#14582757—21no—
44C4×Dic43⋊C4C2211yescharacteristic
45#14582662C2211yescharacteristic
46#14582662C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582654C2211yescharacteristic
50#14582653C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53C22×SD16×C43C2211yescharacteristic
54C2×C4×Q8×C43C2211yescharacteristic
55C2×C4×D4×C43C2211yescharacteristic
56C4×SD16×C43—24no—
57C2×#106×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C2×#102×C43C2211yescharacteristic
60C2×C4×C8×C43C2211yescharacteristic
61C22×Dic86—21no—
62C23×Dic43—21no—
63C23×Dic43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66#13063324—22no—
67#13063324—41no—
68D4×Dic43—22no—
69D4×Dic43—22no—
70D4×Dic43—22no—
71D4×Dic43—41no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43C2311yescharacteristic
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—44no—
84C2×Dic43⋊C4D411yescharacteristic
85C2×Dic43⋊C4D411yescharacteristic
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×C4×Dic43C2311yescharacteristic
89C2×C4×Dic43C2311yescharacteristic
90#13063257—22no—
91#13063257—22no—
92#13063256—22no—
93#13063256—41no—
94Dic43⋊C8—22no—
95#13063249—22no—
96#13063249—41no—
97#13063246—22no—
98#13063246—22no—
99C43⋊#95—21no—
100C43⋊#95—21no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95D411yescharacteristic
108C43⋊#95D411yescharacteristic
109C22×Q8×C43C2311yescharacteristic
110C22×D4×C43C2311yescharacteristic
111C23×C4×C43—21no—
112C2×SD16×C43—22no—
113C2×SD16×C43—22no—
114C2×SD16×C43Q814yescomplemented
115C22×C8×C43C2311yescharacteristic
116C22×C8×C43C2311yescharacteristic
117C4×Q8×C43—22no—
118C4×Q8×C43—41no—
119C4×D4×C43—22no—
120C4×D4×C43—41no—
121C2×C4⋊C4×C43—21no—
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C22⋊C4×C43—21no—
125C2×C42×C43—21no—
126C2×C42×C43C2311yescharacteristic
127#106×C43—22no—
128#103×C43—22no—
129#102×C43D414yes—
130C4×C8×C43—22no—
131C24×C43—21no—
132C2×Q8×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43C2×Q812yes—
138C2×D4×C43—21no—
139C2×D4×C43—21no—
140C2×D4×C43—21no—
141C2×D4×C43C2×D412yes—
142C2×D4×C43C2×Q812yes—
143C2×D4×C43#4112yes—
144C22×C4×C43—21no—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—42no—
148C22×C4×C43—42no—
149C22×C4×C43—81no—
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2411yescharacteristic
153C22×Dic43—24no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—21no—
158C22×Dic43—42no—
159C22×Dic43—42no—
160C22×Dic43—42no—
161C22×Dic43—42no—
162C22×Dic43—81no—
163C22×Dic43C2×D411yescharacteristic
164C22×Dic43C2×D411yescharacteristic
165C22×Dic43C2×D411yescharacteristic
166C22×Dic43C2×D411yescharacteristic
167C2×Dic86—42no—
168C2×Dic86—42no—
169C2×Dic86—81no—
170SD16×C43—28no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43C2×D412yes—
176C2×C8×C43C2×Q812yes—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—42no—
181C4⋊C4×C43C2×D412yes—
182C22⋊C4×C43—42no—
183C42×C43—21no—
184C42×C43—21no—
185C42×C43—42no—
186C22⋊Dic43—24no—
187C22⋊Dic43—42no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—81no—
196C4⋊Dic43#4112yes—
197Dic43⋊C4—41no—
198Dic43⋊C4—42no—
199Dic43⋊C4—42no—
200Dic43⋊C4—42no—
201Dic43⋊C4—42no—
202Dic43⋊C4—81no—
203Dic43⋊C4—81no—
204Dic43⋊C4SD1614yescomplemented
205C4×Dic43—21no—
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C23×C43—22no—
216C23×C43—22no—
217C23×C43—22no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43—41no—
221C23×C43—41no—
222C23×C43C22×D411yescharacteristic
223Q8×C43—22no—
224Q8×C43—22no—
225Q8×C43—41no—
226Q8×C43—41no—
227D4×C43—22no—
228D4×C43—22no—
229D4×C43—22no—
230D4×C43C22⋊Q814yes—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—22no—
234C2×C4×C43—21no—
235C2×C4×C43—22no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—84no—
246C2×C4×C43C2×SD1612yes—
247C2×C4×C43C22×D411yescharacteristic
248C2×C4×C43C22×Q811yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43H(5,𝔽2)11yescharacteristic
252C2×C4×C432−1+411yescharacteristic
253C2×Dic43—21no—
254C2×Dic43—21no—
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—21no—
258C2×Dic43—22no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—48no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—84no—
277C2×Dic43C2×SD1612yes—
278C2×Dic43C2×SD1612yes—
279Dic86—84no—
280C8×C43—22no—
281C8×C43—22no—
282C43⋊C8—41no—
283C43⋊C8—41no—
284#26376—431nomaximal, Sylow
285C22×C43—24no—
286C22×C43—42no—
287C22×C43—42no—
288C22×C43—42no—
289C22×C43—42no—
290C22×C43—42no—
291C22×C43—42no—
292C22×C43#52511yescharacteristic
293C22×C43D4211yescharacteristic
294C22×C43D4×Q811yescharacteristic
295C22×C43C22×SD1611yescharacteristic
296C22×C43#57511yescharacteristic
297C22×C43#57711yescharacteristic
298C22×C43#57711yescharacteristic
299C4×C43—22no—
300C4×C43—42no—
301C4×C43—42no—
302C4×C43—41no—
303C4×C43—81no—
304C4×C43C2×C22⋊Q811yescharacteristic
305C4×C43#53911yescharacteristic
306C4×C43#54711yescharacteristic
307C4×C43#55411yescharacteristic
308Dic43—22no—
309Dic43—22no—
310Dic43—42no—
311Dic43—41no—
312Dic43—41no—
313Dic43—81no—
314Dic43—81no—
315Dic43—81no—
316C2×#476—431no—
317C2×C4×SD16D4311yescharacteristic
318#2451—431no—
319D4×C4⋊C4—431no—
320#2096—431no—
321#2084—431no—
322#2066—431no—
323#2058—431no—
324#1941—431no—
325#1916—431no—
326#1877—431no—
327#1877—431no—
328#1865—431no—
329#1864—431no—
330#1864—431no—
331C2×C43—44no—
332C2×C43#244911yescharacteristic
333C2×C43#310111yescharacteristic
334C2×C43#312511yescharacteristic
335C2×C43#326711yescharacteristic
336C2×C43D4×SD1611yescharacteristic
337C2×C43Q8×SD1611yescharacteristic
338C2×C43#338811yescharacteristic
339C22×SD16C2×D4311yescharacteristic
340C2×C4⋊Q8—431no—
341C2×C4×Q8C2×D4311yescharacteristic
342C2×C4×D4—431no—
343C2×C4×D4—431no—
344C2×C4×D4C2×D4311yescharacteristic
345C22×C4⋊C4—861no—
346#476—864no—
347C4×SD16—24no—
348C2×#106—431no—
349C2×#106—431no—
350C2×#106—431no—
351C2×#106—431no—
352C2×#106C2×D4311yescharacteristic
353C2×C4⋊C8—431no—
354C2×#103—431no—
355C2×#103—431no—
356C2×#103C2×D4311yescharacteristic
357C2×#102—431no—
358C2×#102—431no—
359C2×#102C2×D4311yescharacteristic
360C2×C4×C8C2×D4311yescharacteristic
361#398—861no—
362#395—861no—
363#391—431no—
364#391—431no—
365#391—861no—
366#389—431no—
367#389—431no—
368#389—431no—
369#389—431no—
370#387—861no—
371#385—861no—
372#385—861no—
373#384—431no—
374#380—861no—
375C4×C4⋊C4—431no—
376#340—431no—
377#340—431no—
378#340—431no—
379#340—431no—
380#336—431no—
381#336—431no—
382#336—431no—
383#336—431no—
384C43#2637611yescharacteristic, complemented, Sylow
385C22×Q8—861no—
386C22×Q8C22×D4311yescharacteristic
387C22×D4C22×D4311yescharacteristic
388C23×C4—21no—
389C23×C4—861no—
390C23×C4—861no—
391C2×SD16—22no—
392C2×SD16—22no—
393C2×SD16Dic8614yescomplemented
394C22×C8—431no—
395C22×C8—431no—
396C22×C8C22×D4311yescharacteristic
397C22×C8C22×D4311yescharacteristic
398C4⋊Q8—862no—
399C4⋊Q8—1721no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×D4—22no—
403C4×D4—41no—
404C4×D4—862no—
405C4×D4—862no—
406C4×D4—862no—
407C4×D4—1721no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—431no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4—861no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—1724no—
423C2×C4⋊C4C22×D4311yescharacteristic
424C2×C4⋊C4C22×D4311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—861no—
427C2×C22⋊C4—861no—
428C2×C42—21no—
429C2×C42—431no—
430C2×C42—431no—
431C2×C42C22×D4311yescharacteristic
432#106—22no—
433#106—862no—
434#106—862no—
435#106—862no—
436#106—1721no—
437C4⋊C8—862no—
438#103—22no—
439#103—862no—
440#103—1721no—
441#102—862no—
442#102—862no—
443#102C22⋊D4314yes—
444C4×C8—22no—
445#95—431no—
446#95—431no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453#95—861no—
454#95—861no—
455C24—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—3441no—
462C2×Q8—1722no—
463C2×Q8—1722no—
464C2×Q8C2×Dic8612yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic8612yes—
469C2×D4#1062189812yes—
470C2×D4C2×C22⋊D4312yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—81no—
477C22×C4—431no—
478C22×C4—431no—
479C22×C4—431no—
480C22×C4—431no—
481C22×C4—864no—
482C22×C4—861no—
483C22×C4—861no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—3441no—
487C22×C4—1722no—
488C22×C4—1722no—
489C22×C4—1722no—
490C22×C4—1722no—
491C22×C4D4×D4311yescharacteristic
492C22×C4D4×D4311yescharacteristic
493C22×C4C23×D4311yescharacteristic
494SD16—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—861no—
500C2×C8—861no—
501C2×C8—861no—
502C2×C8—861no—
503C2×C8—861no—
504C2×C8—861no—
505C2×C8C2×Dic8612yes—
506C2×C8C2×C22⋊D4312yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—434no—
512C4⋊C4—432no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—3441no—
521C4⋊C4—3441no—
522C4⋊C4—3441no—
523C4⋊C4—1721no—
524C4⋊C4—1722no—
525C4⋊C4—1722no—
526C4⋊C4—1722no—
527C4⋊C4—1722no—
528C4⋊C4C2×C22⋊D4312yes—
529C22⋊C4—42no—
530C22⋊C4—864no—
531C22⋊C4—1722no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—861no—
536C42—861no—
537C42—861no—
538C42—861no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×D4×D4311yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—3444no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#1306330414yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—861no—
574C2×C4—861no—
575C2×C4—861no—
576C2×C4—861no—
577C2×C4—861no—
578C2×C4—862no—
579C2×C4—861no—
580C2×C4—861no—
581C2×C4—861no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—3444no—
585C2×C4—1728no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4SD16×D4312yes—
598C2×C4C22×C22⋊D4311yescharacteristic
599C2×C4C22×Dic8611yescharacteristic
600C2×C4#41×D4311yescharacteristic
601C2×C4C2×#1062189811yescharacteristic
602C2×C4#1306337011yescharacteristic
603C2×C4#1306337211yescharacteristic
604C8—22no—
605C8—22no—
606C8—1721no—
607C8—1721no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1458340811yescharacteristic
616C22D4×C22⋊D4311yescharacteristic
617C22D4×Dic8611yescharacteristic
618C22C2×SD16×D4311yescharacteristic
619C22#1458369611yescharacteristic
620C22#1458370511yescharacteristic
621C22#1458370611yescharacteristic
622C4—22no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—862no—
628C4—862no—
629C4—3441no—
630C4—3441no—
631C4—3441no—
632C4—1721no—
633C4—1722no—
634C4—1721no—
635C4C2×#1306330411yescharacteristic
636C4#1458349111yescharacteristic
637C4#1458354811yescharacteristic
638C4#1458360811yescharacteristic
639C2—44no—
640C2#1913839811yescharacteristic
641C2#1914460111yescharacteristic
642C2#1914486811yescharacteristic
643C2#1914612011yescharacteristic
644C2SD16×C22⋊D4311yescharacteristic
645C2SD16×Dic8611yescharacteristic
646C2#1914753611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C4×SD16×C43C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenterC2×C4×SD16D43
hypocenterC43#26376
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. #19138428C2semidirect
4. #19136136C2semidirect
5. #19136074C2semidirect
6. #19135974C2semidirect
7. #19135926C2semidirect
8. #19135323C2semidirect
9. #19135152C2semidirect
12. #19134863C2semidirect
114. C2×SD16×C43Q8semidirect
204. Dic43⋊C4SD16semidirect
384. C43#26376semidirect
393. C2×SD16Dic86semidirect

Maximal subgroups

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

subgroupindex
C2×#145831872
#191384282
D4×Dic43⋊C42
#191361362
#191360742
#191359742
#191359262
#191353232
#191351522
#191349312
#191348632
#191348582
#191348562
C2×C4×SD16×C432
#2637643