Finite Group Databasebeta

← #25101502

Subgroups — #25101502

Order 11008.

826 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×#14582961C211yesmaximal, characteristic
2#19137200C211yesmaximal, characteristic
3#19136949C211yesmaximal, characteristic
4#19135362C211yesmaximal, characteristic, complemented
5#19135360C211yesmaximal, characteristic, complemented
6#19135355C211yesmaximal, characteristic, complemented
7#19135352C211yesmaximal, characteristic, complemented
8#19134797C211yesmaximal, characteristic
9#19134793C211yesmaximal, characteristic
10#19134789C211yesmaximal, characteristic
11#19134781C211yesmaximal, characteristic
12#19134492C211yesmaximal, characteristic, complemented
13C4×#13063258C211yesmaximal, characteristic, complemented
14C4×#13063256C211yesmaximal, characteristic, complemented
15C43×#1776C211yesmaximal, characteristic, complemented
16C2×#13063283C2211yescharacteristic
17C2×#13063282C2211yescharacteristic
18C2×C4⋊C4×D43C2211yescharacteristic
19#14582961C418yescomplemented
20C2×#13063259C2211yescharacteristic
21C2×#13063258C2211yescharacteristic
22C2×#13063258C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×#13063256C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×Dic43⋊C8C2211yescharacteristic
28C2×(C2×D43)⋊C8C2211yescharacteristic
29C2×(C2×D43)⋊C8C2211yescharacteristic
30C2×C4×C43⋊C8C2211yescharacteristic
31#14582790—21no—
32#14582788—21no—
33#14582783—21no—
34#14582781—21no—
35#14582768C2211yescharacteristic
36#14582764C2211yescharacteristic
37#14582756C2211yescharacteristic
38#14582756C2211yescharacteristic
39C4⋊C4×Dic43C2211yescharacteristic
40C4⋊C4×Dic43C2211yescharacteristic
41C4×Dic43⋊C4—21no—
42C4×C4⋊Dic43C2211yescharacteristic
43C4×C4⋊Dic43C2211yescharacteristic
44#14582750—21no—
45C4×(C2×D43)⋊C4—21no—
46#14582746—21no—
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C2×#106×C43C2211yescharacteristic
55C2×#97×C43C2211yescharacteristic
56C4×C4⋊C4×C43C2211yescharacteristic
57C4×C4⋊C4×C43C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C22×C4×D43—21no—
61C22×C4×D43C2311yescharacteristic
62C22×C43⋊C8C2311yescharacteristic
63C22×C43⋊C8C2311yescharacteristic
64#13063283—22no—
65#13063283—22no—
66#13063282C2×C414yes—
67#13063282Q814yes—
68C2×C4⋊Dic43C2311yescharacteristic
69C2×C4⋊Dic43C2311yescharacteristic
70C2×C4⋊Dic43C2311yescharacteristic
71C2×C4⋊Dic43C2311yescharacteristic
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4D411yescharacteristic
76C2×Dic43⋊C4D411yescharacteristic
77C4⋊C4×D43—22no—
78C4⋊C4×D43—22no—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4D411yescharacteristic
83C2×(C2×D43)⋊C4D411yescharacteristic
84C2×C4×Dic43—21no—
85C2×C4×Dic43—21no—
86C2×C4×Dic43—21no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89C2×C4×Dic43C2311yescharacteristic
90#13063259—22no—
91#13063258—22no—
92#13063258—22no—
93#13063257—22no—
94#13063256—22no—
95#13063256—22no—
96Dic43⋊C8C2×C414yescomplemented
97Dic43⋊C8C2×C414yescomplemented
98(C2×D43)⋊C8C2×C414yes—
99(C2×D43)⋊C8C2×C414yes—
100C4×C43⋊C8C2×C414yescomplemented
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C22×C8×C43C2311yescharacteristic
110C22×C8×C43C2311yescharacteristic
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C4⋊C4×C43C2311yescharacteristic
114C2×C4⋊C4×C43C2311yescharacteristic
115C2×C42×C43—21no—
116C2×C42×C43—21no—
117C2×C42×C43C2311yescharacteristic
118#107×C43—22no—
119#106×C43—22no—
120#97×C43C2×C414yescomplemented
121#95×C43—21no—
122#95×C43—21no—
123C23×D43C2×D411yescharacteristic
124C22×C4×C43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2411yescharacteristic
131C22×Dic43—21no—
132C22×Dic43—21no—
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43C2×D411yescharacteristic
136C22×Dic43C2×D411yescharacteristic
137C22×Dic43C2×D411yescharacteristic
138C2×C4×D43—21no—
139C2×C4×D43—21no—
140C2×C4×D43—21no—
141C2×C4×D43—21no—
142C2×C4×D43—24no—
143C2×C4×D43—81no—
144C2×C4×D43—81no—
145C2×C4×D43C22×C412yes—
146C2×C4×D43C2×Q812yes—
147C2×C8×C43—21no—
148C2×C8×C43—21no—
149C2×C8×C43—21no—
150C2×C8×C43—21no—
151C2×C8×C43C22×C412yes—
152C2×C8×C43C22×C412yes—
153C4⋊C4×C43—21no—
154C4⋊C4×C43—21no—
155C4⋊C4×C43—21no—
156C4⋊C4×C43—21no—
157C4⋊C4×C43—21no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43#4112yes—
160C4⋊C4×C43#4112yes—
161C42×C43—81no—
162C42×C43—81no—
163C42×C43C22×C412yes—
164C42×C43C2×Q812yes—
165(C2×D43)⋊C4—24no—
166(C2×D43)⋊C4—81no—
167(C2×D43)⋊C4—81no—
168(C2×D43)⋊C4C4⋊C414yes—
169(C2×D43)⋊C4C4⋊C414yes—
170C4⋊Dic43—21no—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43—21no—
176C4⋊Dic43#4112yes—
177C4⋊Dic43#4112yes—
178Dic43⋊C4—24no—
179Dic43⋊C4—81no—
180Dic43⋊C4—81no—
181Dic43⋊C4C4⋊C414yes—
182Dic43⋊C4C4⋊C414yes—
183C4×Dic43—24no—
184C4×Dic43—81no—
185C4×Dic43—81no—
186C4×Dic43—81no—
187C4×Dic43—81no—
188C4×Dic43C22×C412yes—
189C4×Dic43C2×Q812yes—
190C2×C43⋊C8—21no—
191C2×C43⋊C8—21no—
192C2×C43⋊C8—21no—
193C2×C43⋊C8—21no—
194C2×C43⋊C8C22×C412yes—
195C2×C43⋊C8C22×C412yes—
196C23×C43C22×D411yescharacteristic
197C22×D43—21no—
198C22×D43—21no—
199C22×D43—21no—
200C22×D43—21no—
201C22×D43—21no—
202C22×D43—21no—
203C22×D43C2×C4⋊C412yes—
204C2×C4×C43—21no—
205C2×C4×C43—21no—
206C2×C4×C43—21no—
207C2×C4×C43—21no—
208C2×C4×C43—22no—
209C2×C4×C43—22no—
210C2×C4×C43—41no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43C2×C4⋊C412yes—
221C2×C4×C43C2×C4⋊C412yes—
222C2×C4×C43C23×C411yescharacteristic
223C2×C4×C43C22×Q811yescharacteristic
224C2×C4×C43C2×#4111yescharacteristic
225C2×C4×C43C2×#4111yescharacteristic
226C2×C4×C43C2×#4111yescharacteristic
227C2×C4×C43C2×#4111yescharacteristic
228C2×Dic43—21no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—22no—
235C2×Dic43—22no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245C2×Dic43—41no—
246C2×Dic43C2×C4⋊C412yes—
247C2×Dic43C2×C4⋊C412yes—
248C2×Dic43C2×C4⋊C412yes—
249C4×D43—22no—
250C4×D43—22no—
251C4×D43—22no—
252C4×D43—22no—
253C4×D43—84no—
254C8×C43—22no—
255C8×C43—22no—
256C43⋊C8—22no—
257C43⋊C8—22no—
258#21367—431nomaximal, Sylow
259C22×C43C22×C4⋊C411yescharacteristic
260C22×C43#52511yescharacteristic
261C22×C43#52511yescharacteristic
262C22×C43#57611yescharacteristic
263C22×C43#57711yescharacteristic
264C22×C43#57711yescharacteristic
265C22×C43#57811yescharacteristic
266C2×D43—22no—
267C2×D43—22no—
268C2×D43—22no—
269C2×D43—41no—
270C2×D43—41no—
271C2×D43—41no—
272C2×D43—41no—
273C4×C43—22no—
274C4×C43—22no—
275C4×C43—42no—
276C4×C43—42no—
277C4×C43—81no—
278C4×C43—81no—
279C4×C43C4×#4111yescharacteristic
280C4×C43C4×#4111yescharacteristic
281C4×C43#53311yescharacteristic
282C4×C43#53311yescharacteristic
283Dic43—22no—
284Dic43—22no—
285Dic43—22no—
286Dic43—42no—
287Dic43—42no—
288Dic43—81no—
289Dic43—81no—
290C2×#433—431no—
291#2314—431no—
292#2274—431no—
293#1949—431no—
294#1949—431no—
295#1948—431no—
296#1947—431no—
297#1854—431no—
298#1854—431no—
299#1853—431no—
300#1852—431no—
301#1776—431no—
302#1776D4311yescharacteristic, complemented
303C4×#107—431no—
304C4×#106—431no—
305C2×C43#229611yescharacteristic
306C2×C43#291211yescharacteristic
307C2×C43#291311yescharacteristic
308C2×C43#311811yescharacteristic
309C2×C43#311911yescharacteristic
310C2×C43#312011yescharacteristic
311C2×C43#312111yescharacteristic
312D43—42no—
313C2×#117—431no—
314C2×#117—431no—
315C22×C4⋊C4—431no—
316#433—438no—
317C2×#107—431no—
318C2×#107—431no—
319C2×#107—431no—
320C2×#107C2×D4311yescharacteristic
321C2×#106—431no—
322C2×#106—431no—
323C2×#106—431no—
324C2×#106C2×D4311yescharacteristic
325C2×C4⋊C8—431no—
326C2×C4⋊C8—431no—
327C2×C22⋊C8—431no—
328C2×C22⋊C8—431no—
329C2×#97C2×D4311yescharacteristic
330C2×C4×C8—431no—
331#388—861no—
332#388—861no—
333#387—861no—
334#387—861no—
335#384—431no—
336#382—431no—
337#380—431no—
338#380—431no—
339C4×C4⋊C4—431no—
340C4×C4⋊C4—431no—
341C4×C4⋊C4—431no—
342C4×C4⋊C4—431no—
343C4×C4⋊C4—861no—
344C4×C4⋊C4—861no—
345C4×C4⋊C4C2×D4311yescharacteristic
346C4×C4⋊C4C2×D4311yescharacteristic
347C4×C22⋊C4—861no—
348C4×C22⋊C4—861no—
349#340—431no—
350#340—431no—
351#340—431no—
352#340—431no—
353#340—431no—
354#340—431no—
355#340C2×D4311yescharacteristic
356#340C2×D4311yescharacteristic
357C43#2136711yescharacteristic, complemented, Sylow
358C23×C4—431no—
359C23×C4—861no—
360C22×C8—431no—
361C22×C8—431no—
362C22×C8C22×D4311yescharacteristic
363C22×C8C22×D4311yescharacteristic
364#117—434no—
365#117—434no—
366#117—862no—
367#117—862no—
368C2×C4⋊C4—431no—
369C2×C4⋊C4—431no—
370C2×C4⋊C4—431no—
371C2×C4⋊C4—431no—
372C2×C4⋊C4—431no—
373C2×C4⋊C4—431no—
374C2×C4⋊C4—862no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—862no—
377C2×C4⋊C4—861no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4C22×D4311yescharacteristic
380C2×C4⋊C4C22×D4311yescharacteristic
381C2×C4⋊C4C22×D4311yescharacteristic
382C2×C4⋊C4C22×D4311yescharacteristic
383C2×C22⋊C4—431no—
384C2×C22⋊C4—431no—
385C2×C22⋊C4—861no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—431no—
391C2×C42—861no—
392C2×C42—861no—
393C2×C42—861no—
394C2×C42—861no—
395C2×C42—861no—
396C2×C42C22×D4311yescharacteristic
397#107—22no—
398#107—862no—
399#107—862no—
400#107—862no—
401#106—22no—
402#106—862no—
403#106—862no—
404#106—862no—
405C4⋊C8—434no—
406C4⋊C8—434no—
407C22⋊C8—434no—
408C22⋊C8—434no—
409#97C4×D4314yescomplemented
410C4×C8—434no—
411#95—21no—
412#95—21no—
413#95—861no—
414#95—861no—
415#95—861no—
416#95—861no—
417#95—861no—
418#95—861no—
419#95—861no—
420#95—861no—
421C24—431no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—431no—
427C22×C4—431no—
428C22×C4—431no—
429C22×C4—432no—
430C22×C4—432no—
431C22×C4—864no—
432C22×C4—861no—
433C22×C4—861no—
434C22×C4—861no—
435C22×C4—861no—
436C22×C4—861no—
437C22×C4—861no—
438C22×C4—861no—
439C22×C4—861no—
440C22×C4—3441no—
441C22×C4—3441no—
442C22×C4D4×D4311yescharacteristic
443C22×C4D4×D4311yescharacteristic
444C22×C4C23×D4311yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—432no—
450C2×C8—432no—
451C2×C8—861no—
452C2×C8—861no—
453C2×C8—861no—
454C2×C8—861no—
455C2×C8C2×C4×D4312yes—
456C2×C8C2×C4×D4312yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—434no—
464C4⋊C4—434no—
465C4⋊C4—432no—
466C4⋊C4—432no—
467C4⋊C4—864no—
468C4⋊C4—861no—
469C4⋊C4—861no—
470C4⋊C4—861no—
471C4⋊C4—861no—
472C4⋊C4—861no—
473C4⋊C4—861no—
474C4⋊C4—3441no—
475C4⋊C4—3441no—
476C4⋊C4#1062189812yes—
477C4⋊C4#1062189812yes—
478C22⋊C4—434no—
479C22⋊C4—434no—
480C22⋊C4—864no—
481C22⋊C4—3441no—
482C22⋊C4—3441no—
483C42—81no—
484C42—81no—
485C42—432no—
486C42—432no—
487C42—864no—
488C42—3441no—
489C42—3441no—
490C42—3441no—
491C42—3441no—
492C42C2×C4×D4312yes—
493C42Q8×D4312yes—
494C23—432no—
495C23—861no—
496C23—861no—
497C23—861no—
498C23—861no—
499C23—861no—
500C23—861no—
501C23C2×D4×D4311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—432no—
519C2×C4—432no—
520C2×C4—432no—
521C2×C4—861no—
522C2×C4—861no—
523C2×C4—861no—
524C2×C4—861no—
525C2×C4—861no—
526C2×C4—861no—
527C2×C4—862no—
528C2×C4—862no—
529C2×C4—862no—
530C2×C4—862no—
531C2×C4—862no—
532C2×C4—862no—
533C2×C4—3444no—
534C2×C4—1721no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4—1721no—
543C2×C4—1721no—
544C2×C4C4⋊C4×D4312yes—
545C2×C4C4⋊C4×D4312yes—
546C2×C4C22×C4×D4311yescharacteristic
547C2×C4C2×Q8×D4311yescharacteristic
548C2×C4#41×D4311yescharacteristic
549C2×C4#41×D4311yescharacteristic
550C2×C4C2×#1062189811yescharacteristic
551C2×C4C2×#1062189811yescharacteristic
552C8—22no—
553C8—22no—
554C8—862no—
555C8—862no—
556C22—862no—
557C22—862no—
558C22—862no—
559C22—1721no—
560C22—1721no—
561C22—1721no—
562C22—1721no—
563C22C2×C4⋊C4×D4311yescharacteristic
564C22#1458340811yescharacteristic
565C22#1458340811yescharacteristic
566C22#1458370311yescharacteristic
567C22#1458370811yescharacteristic
568C22#1458370911yescharacteristic
569C22#1458371511yescharacteristic
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—81no—
576C4—862no—
577C4—862no—
578C4—862no—
579C4—3441no—
580C4—3441no—
581C4—1722no—
582C4—1722no—
583C4#1458336711yescharacteristic
584C4C4×#1062189811yescharacteristic
585C4#1458345311yescharacteristic
586C4#1458345311yescharacteristic
587C2—1722no—
588C2#1913707511yescharacteristic
589C2#1914318911yescharacteristic
590C2#1914319411yescharacteristic
591C2#1914477711yescharacteristic
592C2#1914479211yescharacteristic
593C2#1914481311yescharacteristic
594C2#1914482811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1776C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
cosocleC22×C4×C43C24
hypercenter#1776D43
hypocenterC43#21367
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #19135362C2semidirect
5. #19135360C2semidirect
6. #19135355C2semidirect
7. #19135352C2semidirect
12. #19134492C2semidirect
13. C4×#13063258C2semidirect
14. C4×#13063256C2semidirect
15. C43×#1776C2semidirect
19. #14582961C4semidirect
96. Dic43⋊C8C2×C4semidirect
97. Dic43⋊C8C2×C4semidirect
100. C4×C43⋊C8C2×C4semidirect
120. #97×C43C2×C4semidirect
302. #1776D43semidirect
357. C43#21367semidirect
409. #97C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145829612
#191372002
#191369492
#191353622
#191353602
#191353552
#191353522
#191347972
#191347932
#191347892
#191347812
#191344922
C4×#130632582
C4×#130632562
C43×#17762
#2136743