Finite Group Databasebeta

← #25153712

Subgroups — #25153712

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×#14582892C211yesmaximal, characteristic
3#19138284C211yesmaximal, characteristic, complemented
4#19137420C211yesmaximal, characteristic
5#19136142C211yesmaximal, characteristic, complemented
6#19135987C211yesmaximal, characteristic, complemented
7#19135985C211yesmaximal, characteristic, complemented
8#19135735C211yesmaximal, characteristic, complemented
9#19134858C211yesmaximal, characteristic, complemented
10#19134795C211yesmaximal, characteristic, complemented
11#19134774C211yesmaximal, characteristic, complemented
12#19134621C211yesmaximal, characteristic, complemented
13#19134605C211yesmaximal, characteristic, complemented
14#19134532C211yesmaximal, characteristic
15C43×#1862C211yesmaximal, characteristic, complemented
16C22×#10621900C2211yescharacteristic
17C22×C8×D43C2211yescharacteristic
18C2×#13063324C2211yescharacteristic
19C2×D4×Dic43C2211yescharacteristic
20C2×#13063283C2211yescharacteristic
21#14583177—24no—
22C2×#13063259C2211yescharacteristic
23C2×#13063258C2211yescharacteristic
24C2×#13063257C2211yescharacteristic
25C2×C4⋊(C43⋊C8)C2211yescharacteristic
26C2×Dic43⋊C8C2211yescharacteristic
27#14582892—24no—
28C2×#13063250C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063248C2211yescharacteristic
31C2×#13063248C2211yescharacteristic
32C2×#13063246C2211yescharacteristic
33C2×#13063246C2211yescharacteristic
34C2×(C2×D43)⋊C8C2211yescharacteristic
35#14582821—21no—
36#14582812—21no—
37#14582805C2211yescharacteristic
38#14582804—21no—
39#14582804C2211yescharacteristic
40#14582798C2211yescharacteristic
41#14582784—21no—
42#14582778—21no—
43#14582776—21no—
44#14582765C2211yescharacteristic
45#14582760—21no—
46#14582756C2211yescharacteristic
47#14582756C2211yescharacteristic
48C4×(C2×D43)⋊C4C2211yescharacteristic
49C2×C43⋊#95—21no—
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582661C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×C4×D4×C43C2211yescharacteristic
55C2×#107×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C43×#389C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C2×#10621900—22no—
62C2×#10621900—22no—
63C2×#10621900—22no—
64C2×#10621900—41no—
65C2×#10621900—41no—
66C2×#10621900Q814yescomplemented
67C22×Dic86—21no—
68C22×Dic86C2311yescharacteristic
69C22×C22⋊D43—21no—
70C23×Dic43—21no—
71C23×Dic43—21no—
72C22×C4×D43C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74C22×C43⋊C8C2311yescharacteristic
75C2×C8×D43—22no—
76C2×C8×D43—22no—
77C2×C8×D43—41no—
78#13063324—22no—
79#13063324—41no—
80D4×Dic43—22no—
81D4×Dic43—41no—
82#13063283—22no—
83#13063283—41no—
84C2×C4⋊Dic43—21no—
85C2×C4⋊Dic43C2311yescharacteristic
86C2×C4⋊Dic43C2311yescharacteristic
87C2×C4⋊Dic43C2311yescharacteristic
88C2×Dic43⋊C4—21no—
89C2×C22⋊Dic43—21no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4D411yescharacteristic
94C2×(C2×D43)⋊C4D411yescharacteristic
95C2×C4×Dic43—21no—
96C2×C4×Dic43C2311yescharacteristic
97#13063259—22no—
98#13063258—22no—
99#13063257—41no—
100C4⋊(C43⋊C8)—41no—
101Dic43⋊C8—22no—
102#13063250—22no—
103#13063250—41no—
104#13063248—22no—
105#13063248—41no—
106#13063246—22no—
107#13063246—22no—
108(C2×D43)⋊C8—22no—
109C43⋊#95—21no—
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⋊#95D411yescharacteristic
119C43⋊#95D411yescharacteristic
120C22×D4×C43C2311yescharacteristic
121C23×C4×C43—21no—
122C22×C8×C43C2311yescharacteristic
123C22×C8×C43C2311yescharacteristic
124C4×D4×C43—22no—
125C4×D4×C43—41no—
126C2×C4⋊C4×C43—21no—
127C2×C4⋊C4×C43C2311yescharacteristic
128C2×C4⋊C4×C43C2311yescharacteristic
129C2×C4⋊C4×C43C2311yescharacteristic
130C2×C22⋊C4×C43—21no—
131C2×C42×C43C2311yescharacteristic
132#107×C43—41no—
133#102×C43—22no—
134#102×C43D414yes—
135#95×C43—21no—
136C24×C43—21no—
137C23×D43C2×D411yescharacteristic
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
153C2×C22⋊D43—24no—
154C2×C22⋊D43—42no—
155C2×C22⋊D43—42no—
156C22×Dic43—24no—
157C22×Dic43—21no—
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—42no—
162C22×Dic43—42no—
163C22×Dic43—42no—
164C22×Dic43—42no—
165C22×Dic43—81no—
166C22×Dic43C2×D411yescharacteristic
167C22×Dic43C2×D411yescharacteristic
168C22×Dic43C2×D411yescharacteristic
169#10621900—28no—
170#10621900—44no—
171#10621900—44no—
172#10621900—44no—
173C2×C4×D43—21no—
174C2×C4×D43—21no—
175C2×C4×D43—21no—
176C2×C4×D43—21no—
177C2×C4×D43C2×Q812yes—
178C2×C4×D43#4112yes—
179C2×Dic86—21no—
180C2×Dic86—21no—
181C2×Dic86—21no—
182C2×Dic86—21no—
183C2×Dic86—21no—
184C2×Dic86—42no—
185C2×Dic86—42no—
186C2×Dic86—81no—
187C2×Dic86C2×Q812yes—
188C2×C8×C43—21no—
189C2×C8×C43—21no—
190C2×C8×C43—21no—
191C2×C8×C43—21no—
192C2×C8×C43—21no—
193C2×C8×C43C2×D412yes—
194C4⋊C4×C43—21no—
195C4⋊C4×C43—21no—
196C4⋊C4×C43—21no—
197C4⋊C4×C43—21no—
198C4⋊C4×C43—21no—
199C4⋊C4×C43—81no—
200C4⋊C4×C43C2×D412yes—
201C22⋊C4×C43—42no—
202C42×C43—21no—
203C42×C43—21no—
204C22⋊Dic43—42no—
205(C2×D43)⋊C4—22no—
206(C2×D43)⋊C4—41no—
207(C2×D43)⋊C4—42no—
208(C2×D43)⋊C4—81no—
209(C2×D43)⋊C4—81no—
210C4⋊Dic43—21no—
211C4⋊Dic43—21no—
212C4⋊Dic43—21no—
213C4⋊Dic43—21no—
214C4⋊Dic43—21no—
215C4⋊Dic43—21no—
216C4⋊Dic43—42no—
217Dic43⋊C4—42no—
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—21no—
227C8×D43—44no—
228C23×C43—22no—
229C23×C43—22no—
230C23×C43—22no—
231C23×C43—41no—
232C23×C43—41no—
233C23×C43—41no—
234C23×C43—41no—
235C23×C43C22×D411yescharacteristic
236C22×D43—21no—
237C22×D43—21no—
238C22×D43—21no—
239C22×D43—21no—
240C22×D43—21no—
241C22×D43—21no—
242C22×D43C22⋊Q812yes—
243D4×C43—22no—
244D4×C43—22no—
245D4×C43—22no—
246D4×C43C22⋊Q814yes—
247C2×C4×C43—22no—
248C2×C4×C43—21no—
249C2×C4×C43—21no—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—21no—
253C2×C4×C43—21no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—41no—
257C2×C4×C43—41no—
258C2×C4×C43—41no—
259C2×C4×C43—41no—
260C2×C4×C43—41no—
261C2×C4×C43—41no—
262C2×C4×C43—84no—
263C2×C4×C43C22×D411yescharacteristic
264C2×C4×C43C22×Q811yescharacteristic
265C2×C4×C43C2×#4111yescharacteristic
266C2×C4×C43C2×#4111yescharacteristic
267C2×C4×C43H(5,𝔽2)11yescharacteristic
268C2×C4×C43H(5,𝔽2)11yescharacteristic
269C22⋊D43—48no—
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—21no—
274C2×Dic43—22no—
275C2×Dic43—21no—
276C2×Dic43—21no—
277C2×Dic43—21no—
278C2×Dic43—48no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282C2×Dic43—41no—
283C2×Dic43—41no—
284C2×Dic43—41no—
285C2×Dic43—41no—
286C2×Dic43—41no—
287C2×Dic43—41no—
288C2×Dic43—41no—
289C2×Dic43—41no—
290C2×Dic43—84no—
291C2×Dic43C22⋊Q812yes—
292C2×Dic43#13512yes—
293C4×D43—22no—
294C4×D43—22no—
295C4×D43—22no—
296C4×D43—22no—
297Dic86—22no—
298Dic86—22no—
299Dic86—41no—
300Dic86—41no—
301Dic86—84no—
302C8×C43—22no—
303C8×C43—41no—
304C43⋊C8—41no—
305C43⋊C8—41no—
306#25954—431nomaximal, Sylow
307C22×C43—24no—
308C22×C43—42no—
309C22×C43—42no—
310C22×C43—42no—
311C22×C43—42no—
312C22×C43—42no—
313C22×C43—42no—
314C22×C43C2×C22⋊Q811yescharacteristic
315C22×C43D4211yescharacteristic
316C22×C43#54611yescharacteristic
317C22×C43C2×#13511yescharacteristic
318C22×C43#57511yescharacteristic
319C22×C43#57611yescharacteristic
320C22×C43#57811yescharacteristic
321C2×D43—22no—
322C2×D43—22no—
323C2×D43—22no—
324C2×D43—41no—
325C2×D43—41no—
326C2×D43—41no—
327C2×D43—41no—
328C4×C43—42no—
329C4×C43—41no—
330C4×C43—41no—
331C4×C43—81no—
332C4×C43—81no—
333C4×C43C2×C22⋊Q811yescharacteristic
334C4×C43#54311yescharacteristic
335C4×C43#54611yescharacteristic
336C4×C43#55111yescharacteristic
337Dic43—22no—
338Dic43—22no—
339Dic43—42no—
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#1862D4311yescharacteristic, complemented
354#1854—431no—
355#1850—431no—
356#1808—431no—
357#1804—431no—
358#1788—431no—
359C2×C43—44no—
360C2×C43#243711yescharacteristic
361C2×C43#307611yescharacteristic
362C2×C43#308311yescharacteristic
363C2×C43#326911yescharacteristic
364C2×C43#328011yescharacteristic
365C2×C43#330211yescharacteristic
366C2×C43#331911yescharacteristic
367D43—42no—
368C22×#41—431no—
369C23×C8—431no—
370C2×C4⋊Q8—431no—
371C2×C4×D4—431no—
372C2×C4×D4C2×D4311yescharacteristic
373C2×#117—431no—
374#474—864no—
375C2×#107—431no—
376C2×#107—431no—
377C2×#107C2×D4311yescharacteristic
378C2×#106—431no—
379C2×C4⋊C8—431no—
380C2×C4⋊C8—431no—
381#416—864no—
382C2×#103—431no—
383C2×#103—431no—
384C2×#103—431no—
385C2×#103—431no—
386C2×#102—431no—
387C2×#102—431no—
388C2×#102C2×D4311yescharacteristic
389C2×#102C2×D4311yescharacteristic
390C2×C22⋊C8—431no—
391#396—861no—
392#393—861no—
393#391—431no—
394#391—431no—
395#391—861no—
396#389—431no—
397#389C2×D4311yescharacteristic
398#387—861no—
399#386—861no—
400#386—861no—
401#382—431no—
402#381—861no—
403#380—431no—
404#380—431no—
405C4×C22⋊C4—431no—
406C2×#95—861no—
407#340—431no—
408#340—431no—
409#340—431no—
410#340C2×D4311yescharacteristic
411#336—431no—
412#336C2×D4311yescharacteristic
413C43#2595411yescharacteristic, complemented, Sylow
414C2×#41—434no—
415C2×#41—862no—
416C2×#41—862no—
417C2×#41—862no—
418C2×#41—1721no—
419C2×#41—1721no—
420C22×Q8—431no—
421C22×Q8—861no—
422C22×D4—861no—
423C22×D4C22×D4311yescharacteristic
424C23×C4—21no—
425C23×C4—431no—
426C23×C4—861no—
427C23×C4—861no—
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—22no—
438C4×D4—41no—
439C4×D4—862no—
440C4×D4—1721no—
441#117—862no—
442#117—1721no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—431no—
445C2×C4⋊C4—431no—
446C2×C4⋊C4—431no—
447C2×C4⋊C4—861no—
448C2×C4⋊C4—861no—
449C2×C4⋊C4C22×D4311yescharacteristic
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—861no—
461C2×C42C22×D4311yescharacteristic
462#107—41no—
463#107—862no—
464#107—862no—
465#106—1721no—
466C4⋊C8—862no—
467C4⋊C8—1721no—
468#103—862no—
469#103—862no—
470#103—1721no—
471#103—1721no—
472#102—22no—
473#102—862no—
474#102—862no—
475#102D17214yes—
476C22⋊C8—862no—
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—
490C24—431no—
491#41—868no—
492#41—1724no—
493#41—1724no—
494#41—1724no—
495C2×Q8—432no—
496C2×Q8—861no—
497C2×Q8—861no—
498C2×Q8—861no—
499C2×Q8—861no—
500C2×Q8—861no—
501C2×Q8—3441no—
502C2×Q8—1722no—
503C2×Q8—1722no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—864no—
508C2×D4—1722no—
509C2×D4—1722no—
510C2×D4C2×D17212yes—
511C2×D4#1062190012yes—
512C2×D4Q8×D4312yes—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—81no—
519C22×C4—431no—
520C22×C4—431no—
521C22×C4—432no—
522C22×C4—432no—
523C22×C4—431no—
524C22×C4—864no—
525C22×C4—861no—
526C22×C4—861no—
527C22×C4—861no—
528C22×C4—861no—
529C22×C4—861no—
530C22×C4—861no—
531C22×C4—861no—
532C22×C4—861no—
533C22×C4—3441no—
534C22×C4—1722no—
535C22×C4—1722no—
536C22×C4—1722no—
537C22×C4—1722no—
538C22×C4D4×D4311yescharacteristic
539C22×C4D4×D4311yescharacteristic
540C22×C4C23×D4311yescharacteristic
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—861no—
547C2×C8—861no—
548C2×C8—861no—
549C2×C8—861no—
550C2×C8—861no—
551C2×C8—861no—
552C2×C8—1724no—
553C2×C8C2×D17212yes—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—81no—
560C4⋊C4—861no—
561C4⋊C4—861no—
562C4⋊C4—861no—
563C4⋊C4—861no—
564C4⋊C4—861no—
565C4⋊C4—861no—
566C4⋊C4—1722no—
567C4⋊C4—1722no—
568C4⋊C4C2×D17212yes—
569C22⋊C4—42no—
570C22⋊C4—862no—
571C22⋊C4—3441no—
572C22⋊C4—3441no—
573C22⋊C4—1721no—
574C22⋊C4—1722no—
575C22⋊C4—1722no—
576C42—21no—
577C42—21no—
578C42—861no—
579C42—861no—
580C42—1722no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—432no—
589C23—861no—
590C23—861no—
591C23—861no—
592C23—861no—
593C23—861no—
594C23—861no—
595C23C2×D4×D4311yescharacteristic
596Q8—862no—
597Q8—862no—
598Q8—3444no—
599Q8—1721no—
600Q8—1721no—
601D4—22no—
602D4—22no—
603D4—22no—
604D4—1728no—
605D4#1306330514yes—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—432no—
623C2×C4—432no—
624C2×C4—861no—
625C2×C4—861no—
626C2×C4—861no—
627C2×C4—861no—
628C2×C4—862no—
629C2×C4—862no—
630C2×C4—862no—
631C2×C4—862no—
632C2×C4—862no—
633C2×C4—861no—
634C2×C4—861no—
635C2×C4—861no—
636C2×C4—3444no—
637C2×C4—1728no—
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—1721no—
649C2×C4C22×D17211yescharacteristic
650C2×C4C2×Q8×D4311yescharacteristic
651C2×C4#41×D4311yescharacteristic
652C2×C4C2×#1062190011yescharacteristic
653C2×C4#1306337011yescharacteristic
654C2×C4#1306337111yescharacteristic
655C8—22no—
656C8—41no—
657C8—1721no—
658C8—1721no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—862no—
667C22—862no—
668C22—862no—
669C22—1721no—
670C22—1721no—
671C22—1721no—
672C22—1721no—
673C22C22⋊Q8×D4311yescharacteristic
674C22D4×D17211yescharacteristic
675C22#1458353611yescharacteristic
676C22#135×D4311yescharacteristic
677C22#1458369511yescharacteristic
678C22#1458370011yescharacteristic
679C22#1458371511yescharacteristic
680C4—41no—
681C4—42no—
682C4—41no—
683C4—81no—
684C4—81no—
685C4—862no—
686C4—862no—
687C4—3441no—
688C4—3441no—
689C4—3441no—
690C4—1721no—
691C4—1722no—
692C4C2×#1306330511yescharacteristic
693C4#1458352411yescharacteristic
694C4#1458354411yescharacteristic
695C4#1458357211yescharacteristic
696C2—44no—
697C2—1722no—
698C2#1913825911yescharacteristic
699C2#1914438011yescharacteristic
700C2#1914444111yescharacteristic
701C2#1914615211yescharacteristic
702C2#1914628111yescharacteristic
703C2#1914650511yescharacteristic
704C2#1914674411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1862C2
socleC23×C43C22×D4
prime radicalC22×#10621900C22
hypercenter#1862D43
hypocenterC43#25954
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
1. C2×#14583177C2semidirect
3. #19138284C2semidirect
5. #19136142C2semidirect
6. #19135987C2semidirect
7. #19135985C2semidirect
8. #19135735C2semidirect
9. #19134858C2semidirect
10. #19134795C2semidirect
11. #19134774C2semidirect
12. #19134621C2semidirect
13. #19134605C2semidirect
15. C43×#1862C2semidirect
66. C2×#10621900Q8semidirect
353. #1862D43semidirect
413. C43#25954semidirect

Maximal subgroups

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

subgroupindex
C2×#145831772
C2×#145828922
#191382842
#191374202
#191361422
#191359872
#191359852
#191357352
#191348582
#191347952
#191347742
#191346212
#191346052
#191345322
C43×#18622
#2595443