Finite Group Databasebeta

← #25153716

Subgroups — #25153716

Order 11008.

994 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583177C211yesmaximal, characteristic, complemented
2C2×#14582895C211yesmaximal, characteristic
3#19138287C211yesmaximal, characteristic, complemented
4#19137425C211yesmaximal, characteristic
5#19136145C211yesmaximal, characteristic, complemented
6#19135988C211yesmaximal, characteristic, complemented
7#19135986C211yesmaximal, characteristic, complemented
8#19135734C211yesmaximal, characteristic, complemented
9#19134858C211yesmaximal, characteristic, complemented
10#19134840C211yesmaximal, characteristic, complemented
11#19134796C211yesmaximal, characteristic, complemented
12#19134772C211yesmaximal, characteristic, complemented
13#19134622C211yesmaximal, characteristic, complemented
14#19134603C211yesmaximal, characteristic, complemented
15C43×#1788C211yesmaximal, characteristic
16C23×C43⋊C8C2211yescharacteristic
17C2×#13063324C2211yescharacteristic
18C2×D4×Dic43C2211yescharacteristic
19C2×D4×Dic43C2211yescharacteristic
20C2×#13063284C2211yescharacteristic
21#14583177—24no—
22C2×#13063259C2211yescharacteristic
23C2×#13063258C2211yescharacteristic
24C2×#13063258C2211yescharacteristic
25C2×#13063257C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27C2×Dic43⋊C8C2211yescharacteristic
28#14582895—24no—
29C2×#13063249C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063246C2211yescharacteristic
32C2×#13063246C2211yescharacteristic
33#14582822—21no—
34#14582811—21no—
35#14582806C2211yescharacteristic
36#14582806C2211yescharacteristic
37#14582803—21no—
38#14582798C2211yescharacteristic
39#14582795C2211yescharacteristic
40#14582782—21no—
41#14582779—21no—
42#14582779—21no—
43#14582764C2211yescharacteristic
44#14582762—21no—
45#14582759C2211yescharacteristic
46C4×C22⋊Dic43C2211yescharacteristic
47C2×C43⋊#95—21no—
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C22×#41×C43C2211yescharacteristic
55C2×#103×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C2×#102×C43C2211yescharacteristic
59C2×C22⋊C8×C43C2211yescharacteristic
60C43×#380C2211yescharacteristic
61C22×Dic86—21no—
62C23×Dic43—21no—
63C23×Dic43—21no—
64C22×C43⋊C8—22no—
65C22×C43⋊C8—22no—
66C22×C43⋊C8—41no—
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69#13063324—22no—
70#13063324—41no—
71D4×Dic43—22no—
72D4×Dic43—22no—
73D4×Dic43—41no—
74D4×Dic43—41no—
75#13063284—22no—
76#13063284—41no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×C22⋊Dic43—21no—
85C2×C22⋊Dic43—21no—
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×C22⋊Dic43D411yescharacteristic
89C2×C22⋊Dic43D411yescharacteristic
90C2×C4×Dic43—21no—
91C2×C4×Dic43C2311yescharacteristic
92C2×C4×Dic43C2311yescharacteristic
93#13063259—22no—
94#13063258—22no—
95#13063258—41no—
96#13063257—41no—
97Dic43⋊C8—22no—
98Dic43⋊C8—41no—
99#13063249—22no—
100#13063249—22no—
101#13063246—22no—
102#13063246D414yes—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95D411yescharacteristic
113C43⋊#95D411yescharacteristic
114C2×#41×C43—22no—
115C2×#41×C43—22no—
116C2×#41×C43—22no—
117C2×#41×C43—41no—
118C2×#41×C43—41no—
119C2×#41×C43Q814yescomplemented
120C22×Q8×C43C2311yescharacteristic
121C22×D4×C43—21no—
122C22×D4×C43C2311yescharacteristic
123C23×C4×C43—21no—
124C23×C4×C43C2311yescharacteristic
125C22×C8×C43C2311yescharacteristic
126C22×C8×C43C2311yescharacteristic
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C4⋊C4×C43C2311yescharacteristic
129C2×C22⋊C4×C43—21no—
130#103×C43—41no—
131#103×C43—41no—
132#102×C43—22no—
133#102×C43—22no—
134C22⋊C8×C43—22no—
135#95×C43—21no—
136C24×C43—21no—
137C24×C43C2×D411yescharacteristic
138#41×C43—28no—
139#41×C43—44no—
140#41×C43—44no—
141#41×C43—44no—
142C2×Q8×C43—21no—
143C2×Q8×C43—21no—
144C2×Q8×C43—21no—
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43C2×Q812yes—
148C2×D4×C43—21no—
149C2×D4×C43—21no—
150C2×D4×C43—21no—
151C2×D4×C43—24no—
152C2×D4×C43—42no—
153C2×D4×C43—42no—
154C2×D4×C43C2×D412yes—
155C2×D4×C43C2×Q812yes—
156C2×D4×C43#4112yes—
157C22×C4×C43—21no—
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43—21no—
161C22×C4×C43—24no—
162C22×C4×C43—21no—
163C22×C4×C43—21no—
164C22×C4×C43—21no—
165C22×C4×C43—42no—
166C22×C4×C43—42no—
167C22×C4×C43C2×D411yescharacteristic
168C22×C4×C43C2×Q812yes—
169C22×C4×C43#4112yes—
170C22×C4×C43C2411yescharacteristic
171C22×Dic43—21no—
172C22×Dic43—21no—
173C22×Dic43—21no—
174C22×Dic43—21no—
175C22×Dic43—42no—
176C22×Dic43—42no—
177C22×Dic43—42no—
178C22×Dic43—42no—
179C22×Dic43—81no—
180C22×Dic43—81no—
181C22×Dic43C2×D411yescharacteristic
182C22×Dic43C2×D411yescharacteristic
183C22×Dic43C2×D411yescharacteristic
184C22×Dic43C2×D411yescharacteristic
185C2×Dic86—42no—
186C2×Dic86—42no—
187C2×Dic86—81no—
188C2×C8×C43—21no—
189C2×C8×C43—21no—
190C2×C8×C43—21no—
191C2×C8×C43—21no—
192C2×C8×C43—21no—
193C2×C8×C43—21no—
194C4⋊C4×C43—21no—
195C4⋊C4×C43—21no—
196C4⋊C4×C43—21no—
197C4⋊C4×C43—21no—
198C22⋊C4×C43—81no—
199C22⋊Dic43—22no—
200C22⋊Dic43—41no—
201C22⋊Dic43—42no—
202C22⋊Dic43—42no—
203C22⋊Dic43—42no—
204C22⋊Dic43—81no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—42no—
213C4⋊Dic43C2×D412yes—
214Dic43⋊C4—42no—
215Dic43⋊C4—81no—
216C4×Dic43—21no—
217C4×Dic43—21no—
218C4×Dic43—21no—
219C4×Dic43—21no—
220C4×Dic43—42no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8—44no—
227C2×C43⋊C8C2×D412yes—
228C23×C43—22no—
229C23×C43—21no—
230C23×C43—22no—
231C23×C43—22no—
232C23×C43—21no—
233C23×C43—21no—
234C23×C43—21no—
235C23×C43—21no—
236C23×C43—21no—
237C23×C43—41no—
238C23×C43—41no—
239C23×C43—41no—
240C23×C43—41no—
241C23×C43C22⋊Q812yes—
242C23×C43C22×D411yescharacteristic
243Q8×C43—22no—
244Q8×C43—22no—
245Q8×C43—41no—
246Q8×C43—41no—
247D4×C43—22no—
248D4×C43—22no—
249D4×C43—22no—
250D4×C43—48no—
251D4×C43C22⋊Q814yes—
252C2×C4×C43—21no—
253C2×C4×C43—21no—
254C2×C4×C43—22no—
255C2×C4×C43—22no—
256C2×C4×C43—22no—
257C2×C4×C43—22no—
258C2×C4×C43—22no—
259C2×C4×C43—48no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—41no—
263C2×C4×C43—41no—
264C2×C4×C43—41no—
265C2×C4×C43—41no—
266C2×C4×C43—41no—
267C2×C4×C43—41no—
268C2×C4×C43C22⋊Q812yes—
269C2×C4×C43C22×D411yescharacteristic
270C2×C4×C43C22×Q811yescharacteristic
271C2×C4×C43C2×#4111yescharacteristic
272C2×C4×C43C2×#4111yescharacteristic
273C2×C4×C43H(5,𝔽2)11yescharacteristic
274C2×C4×C43H(5,𝔽2)11yescharacteristic
275C2×Dic43—21no—
276C2×Dic43—21no—
277C2×Dic43—21no—
278C2×Dic43—22no—
279C2×Dic43—21no—
280C2×Dic43—21no—
281C2×Dic43—21no—
282C2×Dic43—21no—
283C2×Dic43—21no—
284C2×Dic43—21no—
285C2×Dic43—21no—
286C2×Dic43—21no—
287C2×Dic43—41no—
288C2×Dic43—41no—
289C2×Dic43—41no—
290C2×Dic43—41no—
291C2×Dic43—41no—
292C2×Dic43—41no—
293C2×Dic43—41no—
294C2×Dic43—41no—
295C2×Dic43—41no—
296C2×Dic43—41no—
297C2×Dic43—41no—
298C2×Dic43—84no—
299C2×Dic43—84no—
300C2×Dic43#13512yes—
301Dic86—84no—
302C8×C43—41no—
303C8×C43—41no—
304C43⋊C8—22no—
305C43⋊C8—41no—
306#25954—431nomaximal, Sylow
307C22×C43—22no—
308C22×C43—22no—
309C22×C43—24no—
310C22×C43—22no—
311C22×C43—42no—
312C22×C43—42no—
313C22×C43—41no—
314C22×C43—42no—
315C22×C43—41no—
316C22×C43—42no—
317C22×C43—41no—
318C22×C43—42no—
319C22×C43—41no—
320C22×C43—42no—
321C22×C43C2×C22⋊Q811yescharacteristic
322C22×C43D4211yescharacteristic
323C22×C43#54611yescharacteristic
324C22×C43C2×#13511yescharacteristic
325C22×C43#57511yescharacteristic
326C22×C43#57611yescharacteristic
327C22×C43#57811yescharacteristic
328C4×C43—22no—
329C4×C43—42no—
330C4×C43—81no—
331C4×C43—81no—
332C4×C43C2×C22⋊Q811yescharacteristic
333C4×C43#54311yescharacteristic
334C4×C43#54611yescharacteristic
335C4×C43#55111yescharacteristic
336Dic43—22no—
337Dic43—41no—
338Dic43—42no—
339Dic43—41no—
340Dic43—41no—
341Dic43—81no—
342Dic43—81no—
343Dic43—81no—
344C2×#474—431no—
345C2×#416—431no—
346#2439—431no—
347#2339—431no—
348#2097—431no—
349#2068—431no—
350#2068—431no—
351#2024—431no—
352#1864—431no—
353#1862—431no—
354#1854—431no—
355#1850—431no—
356#1808—431no—
357#1804—431no—
358#1788D4311yescharacteristic
359C2×C43—42no—
360C2×C43—44no—
361C2×C43#243711yescharacteristic
362C2×C43#307611yescharacteristic
363C2×C43#308311yescharacteristic
364C2×C43#326911yescharacteristic
365C2×C43#328011yescharacteristic
366C2×C43#330211yescharacteristic
367C2×C43#331911yescharacteristic
368C22×#41C2×D4311yescharacteristic
369C23×C8—431no—
370C2×C4⋊Q8—431no—
371C2×C4×D4—431no—
372C2×C4×D4—431no—
373C2×#117—431no—
374#474—864no—
375C2×#107—431no—
376C2×#107—431no—
377C2×#107—431no—
378C2×#106—431no—
379C2×C4⋊C8—431no—
380C2×C4⋊C8—431no—
381#416—864no—
382C2×#103—431no—
383C2×#103—431no—
384C2×#103C2×D4311yescharacteristic
385C2×#103C2×D4311yescharacteristic
386C2×#102—431no—
387C2×#102—431no—
388C2×#102C2×D4311yescharacteristic
389C2×#102C2×D4311yescharacteristic
390C2×C22⋊C8C2×D4311yescharacteristic
391#396—861no—
392#393—861no—
393#391—431no—
394#391—431no—
395#391—861no—
396#389—431no—
397#389—431no—
398#387—861no—
399#386—861no—
400#386—861no—
401#382—431no—
402#381—861no—
403#380—431no—
404#380C2×D4311yescharacteristic
405C4×C22⋊C4—431no—
406C2×#95—861no—
407#340—431no—
408#340—431no—
409#340—431no—
410#340—431no—
411#336—431no—
412#336—431no—
413C43#2595411yescharacteristic, complemented, Sylow
414C2×#41—22no—
415C2×#41—22no—
416C2×#41—22no—
417C2×#41—41no—
418C2×#41—41no—
419C2×#41Dic8614yescomplemented
420C22×Q8—861no—
421C22×Q8C22×D4311yescharacteristic
422C22×D4—21no—
423C22×D4C22×D4311yescharacteristic
424C23×C4—21no—
425C23×C4—861no—
426C23×C4—861no—
427C23×C4C22×D4311yescharacteristic
428C22×C8—431no—
429C22×C8—431no—
430C22×C8—862no—
431C22×C8—862no—
432C22×C8—1721no—
433C22×C8C22×D4311yescharacteristic
434C22×C8C22×D4311yescharacteristic
435C4⋊Q8—862no—
436C4⋊Q8—1721no—
437C4×D4—862no—
438C4×D4—862no—
439C4×D4—1721no—
440C4×D4—1721no—
441#117—862no—
442#117—1721no—
443C2×C4⋊C4—431no—
444C2×C4⋊C4—431no—
445C2×C4⋊C4—431no—
446C2×C4⋊C4—431no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4—861no—
449C2×C4⋊C4—861no—
450C2×C4⋊C4C22×D4311yescharacteristic
451C2×C4⋊C4C22×D4311yescharacteristic
452C2×C22⋊C4—21no—
453C2×C22⋊C4—431no—
454C2×C22⋊C4—431no—
455C2×C22⋊C4—861no—
456C2×C22⋊C4—861no—
457C2×C22⋊C4—861no—
458C2×C22⋊C4—861no—
459C2×C42—431no—
460C2×C42—431no—
461C2×C42—861no—
462#107—862no—
463#107—862no—
464#107—1721no—
465#106—1721no—
466C4⋊C8—862no—
467C4⋊C8—1721no—
468#103—41no—
469#103—41no—
470#103—862no—
471#103—862no—
472#102—22no—
473#102—22no—
474#102—434no—
475#102—862no—
476C22⋊C8—22no—
477#95—21no—
478#95—431no—
479#95—431no—
480#95—861no—
481#95—861no—
482#95—861no—
483#95—861no—
484#95—861no—
485#95—861no—
486#95—861no—
487#95—861no—
488#95—861no—
489C24—21no—
490C24D4×D4311yescharacteristic
491#41—28no—
492#41—44no—
493#41—44no—
494#41—44no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—3441no—
501C2×Q8—1722no—
502C2×Q8—1722no—
503C2×Q8C2×Dic8612yes—
504C2×D4—24no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×Dic8612yes—
511C2×D4D4×D4312yes—
512C2×D4#1062190012yes—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—24no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—431no—
524C22×C4—431no—
525C22×C4—431no—
526C22×C4—431no—
527C22×C4—861no—
528C22×C4—861no—
529C22×C4—861no—
530C22×C4—861no—
531C22×C4—3441no—
532C22×C4—3441no—
533C22×C4—1722no—
534C22×C4—1722no—
535C22×C4—1722no—
536C22×C4—1722no—
537C22×C4C2×Dic8612yes—
538C22×C4D4×D4311yescharacteristic
539C22×C4#1062190012yes—
540C22×C4C23×D4311yescharacteristic
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—432no—
548C2×C8—861no—
549C2×C8—861no—
550C2×C8—861no—
551C2×C8—861no—
552C2×C8—861no—
553C2×C8—1724no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—432no—
559C4⋊C4—861no—
560C4⋊C4—861no—
561C4⋊C4—861no—
562C4⋊C4—861no—
563C4⋊C4—861no—
564C4⋊C4—861no—
565C4⋊C4—861no—
566C4⋊C4—3441no—
567C4⋊C4—1722no—
568C4⋊C4—1722no—
569C22⋊C4—81no—
570C22⋊C4—862no—
571C22⋊C4—3441no—
572C22⋊C4—1721no—
573C22⋊C4—1722no—
574C22⋊C4—1722no—
575C22⋊C4—1722no—
576C42—861no—
577C42—861no—
578C42—861no—
579C42—861no—
580C42—1722no—
581C23—22no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—22no—
589C23—21no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23#1306330212yes—
595C23C2×D4×D4311yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600Q8—3444no—
601D4—22no—
602D4—22no—
603D4—22no—
604D4—48no—
605D4#1306330214yes—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—432no—
623C2×C4—861no—
624C2×C4—861no—
625C2×C4—861no—
626C2×C4—861no—
627C2×C4—861no—
628C2×C4—861no—
629C2×C4—861no—
630C2×C4—861no—
631C2×C4—861no—
632C2×C4—862no—
633C2×C4—861no—
634C2×C4—861no—
635C2×C4—3444no—
636C2×C4—3444no—
637C2×C4—1721no—
638C2×C4—1721no—
639C2×C4—1721no—
640C2×C4—1721no—
641C2×C4—1721no—
642C2×C4—1721no—
643C2×C4—1721no—
644C2×C4—1721no—
645C2×C4—1721no—
646C2×C4—1721no—
647C2×C4—1721no—
648C2×C4#1306330212yes—
649C2×C4C2×D4×D4311yescharacteristic
650C2×C4C22×Dic8611yescharacteristic
651C2×C4C2×#1062190011yescharacteristic
652C2×C4C2×#1062190011yescharacteristic
653C2×C4#1306337011yescharacteristic
654C2×C4#1306337011yescharacteristic
655C8—41no—
656C8—41no—
657C8—862no—
658C8—1721no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—42no—
664C22—42no—
665C22—41no—
666C22—41no—
667C22—42no—
668C22—42no—
669C22—41no—
670C22—41no—
671C22—42no—
672C22—42no—
673C22C2×#1306330211yescharacteristic
674C22#1458353111yescharacteristic
675C22#1458353611yescharacteristic
676C22#135×D4311yescharacteristic
677C22#1458369611yescharacteristic
678C22#1458370311yescharacteristic
679C22#1458371511yescharacteristic
680C4—22no—
681C4—42no—
682C4—81no—
683C4—81no—
684C4—862no—
685C4—3441no—
686C4—3441no—
687C4—3441no—
688C4—1721no—
689C4—1721no—
690C4—1722no—
691C4—1721no—
692C4C2×#1306330211yescharacteristic
693C4#1458352511yescharacteristic
694C4#1458353611yescharacteristic
695C4#1458358011yescharacteristic
696C2—42no—
697C2—44no—
698C2#1913826111yescharacteristic
699C2#1914437911yescharacteristic
700C2#1914445011yescharacteristic
701C2#1914615411yescharacteristic
702C2#1914627811yescharacteristic
703C2#1914650511yescharacteristic
704C2#1914675511yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1788C2
socleC23×C43C22×D4
prime radicalC22×#41×C43C22
hypercenter#1788D43
hypocenterC43#25954
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
1. C2×#14583177C2semidirect
3. #19138287C2semidirect
5. #19136145C2semidirect
6. #19135988C2semidirect
7. #19135986C2semidirect
8. #19135734C2semidirect
9. #19134858C2semidirect
10. #19134840C2semidirect
11. #19134796C2semidirect
12. #19134772C2semidirect
13. #19134622C2semidirect
14. #19134603C2semidirect
119. C2×#41×C43Q8semidirect
413. C43#25954semidirect
419. C2×#41Dic86semidirect

Maximal subgroups

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

subgroupindex
C2×#145831772
C2×#145828952
#191382872
#191374252
#191361452
#191359882
#191359862
#191357342
#191348582
#191348402
#191347962
#191347722
#191346222
#191346032
C43×#17882
#2595443