Finite Group Databasebeta

← #25103249

Subgroups — #25103249

Order 11008.

1002 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583155C211yesmaximal, characteristic
2C2×#14582991C211yesmaximal, characteristic, complemented
3C2×M16×Dic43C211yesmaximal, characteristic
4C22⋊Q8×Dic43C211yesmaximal, characteristic
5#19137123C211yesmaximal, characteristic
6#19135463C211yesmaximal, characteristic, complemented
7#19135337C211yesmaximal, characteristic, complemented
8#19135286C211yesmaximal, characteristic, complemented
9#19135282C211yesmaximal, characteristic, complemented
10#19135028C211yesmaximal, characteristic
11#19134626C211yesmaximal, characteristic
12#19134483C211yesmaximal, characteristic, complemented
13#103×Dic43C211yesmaximal, characteristic, complemented
14#19134414C211yesmaximal, characteristic, complemented
15C43×#1894C211yesmaximal, characteristic
16C22×#10621885C2211yescharacteristic
17C22×#10621877C2211yescharacteristic
18C2×#13063304C2211yescharacteristic
19C2×Q8×Dic43C2211yescharacteristic
20C2×#13063290C2211yescharacteristic
21C22×C4×Dic43C2211yescharacteristic
22#14583155C418yescomplemented
23#14582991—24no—
24C2×#13063258C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×#13063256C2211yescharacteristic
27C2×#13063250C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063249C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31M16×Dic43—24no—
32C2×#13063235C2211yescharacteristic
33C2×#13063235C2211yescharacteristic
34C2×C8×Dic43C2211yescharacteristic
35C2×C8×Dic43C2211yescharacteristic
36#14582806C2211yescharacteristic
37#14582804C2211yescharacteristic
38#14582798C2211yescharacteristic
39#14582795C2211yescharacteristic
40#14582785—21no—
41#14582781—21no—
42#14582774—21no—
43#14582771—21no—
44#14582759C2211yescharacteristic
45C4⋊C4×Dic43—21no—
46C4⋊C4×Dic43C2211yescharacteristic
47C4⋊C4×Dic43C2211yescharacteristic
48#14582750—21no—
49C22⋊C4×Dic43—21no—
50C22⋊C4×Dic43—21no—
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53#14582661C2211yescharacteristic
54#14582661C2211yescharacteristic
55C22×M16×C43C2211yescharacteristic
56C2×C22⋊Q8×C43C2211yescharacteristic
57C2×#103×C43C2211yescharacteristic
58C2×#103×C43C2211yescharacteristic
59C43×#380C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C43×#340C2211yescharacteristic
62C22×Dic86C2311yescharacteristic
63C23×Dic43D411yescharacteristic
64C23×Dic43D411yescharacteristic
65C2×#10621885—22no—
66C2×#10621885—22no—
67C2×#10621885C2×C414yescomplemented
68C2×#10621877—22no—
69C2×#10621877—22no—
70C2×#10621877C2×C414yes—
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73#13063304—22no—
74#13063304C2×C414yes—
75Q8×Dic43—22no—
76Q8×Dic43—22no—
77#13063290—22no—
78#13063290—22no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4—21no—
87C2×C22⋊Dic43—21no—
88C2×C22⋊Dic43—21no—
89C2×C4×Dic43—22no—
90C2×C4×Dic43—22no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43D411yescharacteristic
96C2×C4×Dic43D411yescharacteristic
97C2×C4×Dic43C2311yescharacteristic
98C2×C4×Dic43C2311yescharacteristic
99#13063258—22no—
100#13063256—22no—
101#13063256C2×C414yescomplemented
102#13063250—22no—
103#13063250—22no—
104#13063249C2×C414yescomplemented
105#13063248C2×C414yescomplemented
106#13063235—22no—
107#13063235—22no—
108C8×Dic43—22no—
109C8×Dic43D414yes—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C43⋊#95—21no—
119C22×Q8×C43C2311yescharacteristic
120C23×C4×C43C2311yescharacteristic
121C2×M16×C43—22no—
122C2×M16×C43—22no—
123C2×M16×C43—22no—
124C22×C8×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C22⋊Q8×C43—22no—
127C22⋊Q8×C43C2×C414yes—
128C2×C4⋊C4×C43—21no—
129C2×C4⋊C4×C43C2311yescharacteristic
130C2×C4⋊C4×C43C2311yescharacteristic
131C2×C4⋊C4×C43C2311yescharacteristic
132C2×C22⋊C4×C43—21no—
133C2×C22⋊C4×C43—21no—
134#103×C43—22no—
135#103×C43—22no—
136#95×C43—21no—
137C24×C43C2×D411yescharacteristic
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43—21no—
141C2×Q8×C43—21no—
142C2×Q8×C43—21no—
143C2×Q8×C43C22×C412yes—
144C22×C4×C43—21no—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43—21no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43C22×C412yes—
153C22×C4×C43C2×D411yescharacteristic
154C22×C4×C43#4112yes—
155C22×C4×C43C2411yescharacteristic
156C22×Dic43—22no—
157C22×Dic43—22no—
158C22×Dic43—22no—
159C22×Dic43—22no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C22×Dic43—21no—
163C22×Dic43—21no—
164C22×Dic43—41no—
165C22×Dic43—41no—
166C22×Dic43C2×D411yescharacteristic
167C22×Dic43C2×D411yescharacteristic
168C22×Dic43C2×D411yescharacteristic
169C22×Dic43C2×D411yescharacteristic
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86C22×C412yes—
176M16×C43—28no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43—21no—
181C2×C8×C43C2×D412yes—
182C2×C8×C43#4112yes—
183C4⋊C4×C43—24no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—21no—
189C4⋊C4×C43C22×C412yes—
190C22⋊C4×C43—24no—
191C22⋊C4×C43—42no—
192C22⋊Dic43—24no—
193C22⋊Dic43—42no—
194#10621885—28no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—42no—
201C4⋊Dic43—42no—
202C4⋊Dic43C22×C412yes—
203Dic43⋊C4—24no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—22no—
209C4×Dic43—22no—
210C4×Dic43—42no—
211C4×Dic43—42no—
212C4×Dic43—42no—
213C4×Dic43—42no—
214C4×Dic43C2×D412yes—
215C4×Dic43#4112yes—
216#10621877—28no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8C22×C412yes—
220C2×C43⋊C8C22×C412yes—
221C2×C43⋊C8C2×D412yes—
222C2×C43⋊C8#4112yes—
223C23×C43—21no—
224C23×C43—21no—
225C23×C43—21no—
226C23×C43—21no—
227C23×C43—21no—
228C23×C43—21no—
229C23×C43C4×D412yes—
230C23×C43C22×D411yescharacteristic
231Q8×C43—22no—
232Q8×C43—22no—
233Q8×C43—41no—
234Q8×C43—41no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—22no—
238C2×C4×C43—22no—
239C2×C4×C43—22no—
240C2×C4×C43—22no—
241C2×C4×C43—22no—
242C2×C4×C43—22no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43—41no—
253C2×C4×C43C4×D412yes—
254C2×C4×C43C23×C411yescharacteristic
255C2×C4×C43C22×D411yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×C4×C43C2×#4111yescharacteristic
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—22no—
263C2×Dic43—22no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—44no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—44no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43#12112yes—
283C2×Dic43#12112yes—
284C2×Dic43#13712yes—
285C2×Dic43#13712yes—
286Dic86—22no—
287Dic86—22no—
288Dic86—41no—
289Dic86—41no—
290C8×C43—22no—
291C8×C43—22no—
292C43⋊C8—22no—
293C43⋊C8C4×D414yes—
294C4×#470—431nomaximal, Sylow
295C22×C43—22no—
296C22×C43—22no—
297C22×C43—22no—
298C22×C43—41no—
299C22×C43—41no—
300C22×C43—41no—
301C22×C43—41no—
302C22×C43C2×C4×D411yescharacteristic
303C22×C43C2×#12111yescharacteristic
304C22×C43#53211yescharacteristic
305C22×C43C2×#13711yescharacteristic
306C22×C43C2×#13711yescharacteristic
307C22×C43#57511yescharacteristic
308C22×C43#57511yescharacteristic
309C4×C43—22no—
310C4×C43—42no—
311C4×C43—42no—
312C4×C43—81no—
313C4×C43—81no—
314C4×C43C2×C4×D411yescharacteristic
315C4×C43C4×#4111yescharacteristic
316C4×C43#52511yescharacteristic
317C4×C43#52911yescharacteristic
318Dic43—22no—
319Dic43—42no—
320Dic43—41no—
321Dic43—41no—
322Dic43—42no—
323Dic43—81no—
324Dic43—81no—
325Dic43#47014yescomplemented
326C2×#470—431no—
327C2×C4×Q16—431no—
328C2×C4×M16—431no—
329C4×C22⋊Q8—431no—
330C4×C22⋊Q8—431no—
331#1971—431no—
332#1944—431no—
333#1935—431no—
334#1935—431no—
335#1894—431no—
336#1894D4311yescharacteristic
337#1809—431no—
338C4×#106—431no—
339C4×#103—431no—
340C4×#103—431no—
341C2×C43—42no—
342C2×C43C4×#12111yescharacteristic
343C2×C43C4×#13711yescharacteristic
344C2×C43C4×#13711yescharacteristic
345C2×C43C2×#47011yescharacteristic
346C2×C43#305411yescharacteristic
347C2×C43#317211yescharacteristic
348C2×C43#317211yescharacteristic
349C22×Q16—431no—
350C22×M16—431no—
351C22×M16C2×D4311yescharacteristic
352C2×C22⋊Q8—431no—
353C2×C22⋊Q8C2×D4311yescharacteristic
354C2×C4×Q8—431no—
355C2×C4×Q8—431no—
356C22×C42—431no—
357#470—438no—
358C4×Q16—864no—
359C2×#107—431no—
360C2×#106—431no—
361C2×#106—431no—
362C2×#103—431no—
363C2×#103—431no—
364C2×#103—431no—
365C2×#103—431no—
366C2×#103C2×D4311yescharacteristic
367C2×#103C2×D4311yescharacteristic
368C4×M16—864no—
369C2×#97—431no—
370C2×#97—431no—
371C2×C4×C8—431no—
372C2×C4×C8—431no—
373#391—431no—
374#391—431no—
375#389—431no—
376#389—431no—
377#387—861no—
378#387—861no—
379#385—861no—
380#385—861no—
381#380—431no—
382#380C2×D4311yescharacteristic
383C4×C4⋊C4—431no—
384C4×C4⋊C4—431no—
385C4×C4⋊C4—861no—
386C4×C4⋊C4—861no—
387C4×C22⋊C4—861no—
388C4×C22⋊C4—861no—
389#340—431no—
390#340—431no—
391#340—431no—
392#340—431no—
393#340C2×D4311yescharacteristic
394#340C2×D4311yescharacteristic
395C43C4×#47011yescharacteristic, complemented, Sylow
396C22×Q8—431no—
397C22×Q8C22×D4311yescharacteristic
398C23×C4—431no—
399C23×C4—431no—
400C23×C4C22×D4311yescharacteristic
401C2×Q16—434no—
402C2×Q16—862no—
403C2×Q16—862no—
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—22no—
407C2×M16—434no—
408C2×M16—862no—
409C2×M16—862no—
410C22×C8—431no—
411C22×C8—431no—
412C22×C8C22×D4311yescharacteristic
413C22×C8C22×D4311yescharacteristic
414C22⋊Q8—22no—
415C22⋊Q8—434no—
416C22⋊Q8—862no—
417C22⋊Q8C4×D4314yes—
418C4×Q8—862no—
419C4×Q8—862no—
420C4×Q8—862no—
421C4×Q8—862no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—431no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4—861no—
431C2×C4⋊C4C22×D4311yescharacteristic
432C2×C4⋊C4C22×D4311yescharacteristic
433C2×C4⋊C4C22×D4311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—861no—
437C2×C22⋊C4—861no—
438C2×C42—431no—
439C2×C42—431no—
440C2×C42—431no—
441C2×C42—431no—
442C2×C42—862no—
443C2×C42—862no—
444C2×C42—861no—
445C2×C42—861no—
446C2×C42—861no—
447C2×C42—861no—
448#107—862no—
449#106—434no—
450#106—862no—
451#103—22no—
452#103—22no—
453#103—434no—
454#103—434no—
455#103—862no—
456#103—862no—
457#97—862no—
458#97—862no—
459C4×C8—434no—
460C4×C8—862no—
461#95—21no—
462#95—861no—
463#95—861no—
464#95—861no—
465#95—861no—
466#95—861no—
467#95—861no—
468#95—861no—
469#95—861no—
470#95—861no—
471C24D4×D4311yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—432no—
478C2×Q8—861no—
479C2×Q8—861no—
480C2×Q8—861no—
481C2×Q8—861no—
482C2×Q8—861no—
483C2×Q8C2×C4×D4312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—431no—
493C22×C4—431no—
494C22×C4—431no—
495C22×C4—431no—
496C22×C4—862no—
497C22×C4—862no—
498C22×C4—862no—
499C22×C4—862no—
500C22×C4—861no—
501C22×C4—861no—
502C22×C4—861no—
503C22×C4—861no—
504C22×C4—1721no—
505C22×C4—1721no—
506C22×C4C2×C4×D4312yes—
507C22×C4D4×D4311yescharacteristic
508C22×C4#1062190012yes—
509C22×C4C23×D4311yescharacteristic
510Q16—868no—
511M16—28no—
512M16—868no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—432no—
518C2×C8—432no—
519C2×C8—432no—
520C2×C8—432no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8D4×D4312yes—
524C2×C8#1062190012yes—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—432no—
532C4⋊C4—864no—
533C4⋊C4—861no—
534C4⋊C4—861no—
535C4⋊C4—861no—
536C4⋊C4—861no—
537C4⋊C4—861no—
538C4⋊C4—1722no—
539C4⋊C4—1722no—
540C4⋊C4—1722no—
541C4⋊C4—1722no—
542C4⋊C4C2×C4×D4312yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—864no—
546C22⋊C4—1722no—
547C42—432no—
548C42—432no—
549C42—862no—
550C42—862no—
551C42—861no—
552C42—861no—
553C42—1722no—
554C42—1722no—
555C42—1722no—
556C42—1722no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23(C22⋊D43)⋊C412yes—
564C23C2×D4×D4311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—862no—
570Q8—862no—
571Q8—1721no—
572Q8—1721no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—432no—
592C2×C4—432no—
593C2×C4—432no—
594C2×C4—432no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—862no—
598C2×C4—862no—
599C2×C4—861no—
600C2×C4—861no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—861no—
604C2×C4—861no—
605C2×C4—1724no—
606C2×C4—1724no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1721no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4—1721no—
616C2×C4—1721no—
617C2×C4(C22⋊D43)⋊C412yes—
618C2×C4C22×C4×D4311yescharacteristic
619C2×C4C2×D4×D4311yescharacteristic
620C2×C4#41×D4311yescharacteristic
621C2×C4#41×D4311yescharacteristic
622C2×C4C2×#1062190011yescharacteristic
623C2×C4C2×#1062190011yescharacteristic
624C8—22no—
625C8—22no—
626C8—434no—
627C8—862no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×(C22⋊D43)⋊C411yescharacteristic
636C22#121×D4311yescharacteristic
637C22#1458344411yescharacteristic
638C22#137×D4311yescharacteristic
639C22#137×D4311yescharacteristic
640C22#1458369511yescharacteristic
641C22#1458369811yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—434no—
648C4—862no—
649C4—3441no—
650C4—3441no—
651C4—1721no—
652C4—1721no—
653C4—1722no—
654C4—1722no—
655C4C4×D4×D4311yescharacteristic
656C4C4×#1062190011yescharacteristic
657C4#1458340511yescharacteristic
658C4#1458342611yescharacteristic
659C2—42no—
660C2#1913710511yescharacteristic
661C2#1914341011yescharacteristic
662C2#1914341911yescharacteristic
663C2#470×D4311yescharacteristic
664C2#1914419911yescharacteristic
665C2#1914519411yescharacteristic
666C2#1914519611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1894C2
socleC23×C43C22×D4
prime radicalC24×C43C2×D4
cosocleC22×C4×C43C24
hypercenter#1894D43
hypocenterC43C4×#470
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
2. C2×#14582991C2semidirect
6. #19135463C2semidirect
7. #19135337C2semidirect
8. #19135286C2semidirect
9. #19135282C2semidirect
12. #19134483C2semidirect
13. #103×Dic43C2semidirect
14. #19134414C2semidirect
22. #14583155C4semidirect
67. C2×#10621885C2×C4semidirect
101. #13063256C2×C4semidirect
104. #13063249C2×C4semidirect
105. #13063248C2×C4semidirect
325. Dic43#470semidirect
395. C43C4×#470semidirect

Maximal subgroups

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

subgroupindex
C2×#145831552
C2×#145829912
C2×M16×Dic432
C22⋊Q8×Dic432
#191371232
#191354632
#191353372
#191352862
#191352822
#191350282
#191346262
#191344832
#103×Dic432
#191344142
C43×#18942
C4×#47043