Finite Group Databasebeta

← #25196802

Subgroups — #25196802

Order 11008.

784 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19140416C211yesmaximal, characteristic, complemented
2#19136163C211yesmaximal, characteristic, complemented
3#19136116C211yesmaximal, characteristic, complemented
4#19135928C211yesmaximal, characteristic, complemented
5#19135731C211yesmaximal, characteristic, complemented
6#19135173C211yesmaximal, characteristic, complemented
7#19135166C211yesmaximal, characteristic, complemented
8#19135034C211yesmaximal, characteristic
9#19135024C211yesmaximal, characteristic
10#19134856C211yesmaximal, characteristic, complemented
11#19134856C211yesmaximal, characteristic, complemented
12#19134626C211yesmaximal, characteristic, complemented
13#19134618C211yesmaximal, characteristic, complemented
14#19134292C211yesmaximal, characteristic, complemented
15C43×#2324C211yesmaximal, characteristic
16C22×#10621877C2211yescharacteristic
17C23×C43⋊C8C2211yescharacteristic
18C2×#13063325C2211yescharacteristic
19C2×#13063258C2211yescharacteristic
20C2×#13063256C2211yescharacteristic
21C2×#13063256C2211yescharacteristic
22C2×#13063256C2211yescharacteristic
23C2×#13063246C2211yescharacteristic
24C2×#13063246C2211yescharacteristic
25C2×#13063246C2211yescharacteristic
26C2×#13063246C2211yescharacteristic
27#14582838—21no—
28#14582835—21no—
29#14582819—21no—
30#14582811—21no—
31#14582797C2211yescharacteristic
32#14582797C2211yescharacteristic
33#14582765C2211yescharacteristic
34#14582759C2211yescharacteristic
35#14582759C2211yescharacteristic
36#14582661C2211yescharacteristic
37#14582661C2211yescharacteristic
38#14582661C2211yescharacteristic
39#14582661C2211yescharacteristic
40#14582661C2211yescharacteristic
41#14582661C2211yescharacteristic
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582654C2211yescharacteristic
45#14582654C2211yescharacteristic
46#14582654C2211yescharacteristic
47#14582654C2211yescharacteristic
48C2×#121×C43C2211yescharacteristic
49C2×C4×D4×C43C2211yescharacteristic
50C2×#117×C43C2211yescharacteristic
51C43×#390—21no—
52C43×#390C2211yescharacteristic
53C43×#389C2211yescharacteristic
54C43×#388—21no—
55C43×#386—21no—
56C43×#385—21no—
57C43×#380C2211yescharacteristic
58C4×C4⋊C4×C43C2211yescharacteristic
59C22×Dic86—21no—
60C2×#10621877—22no—
61C2×#10621877—22no—
62C2×#10621877—41no—
63C22×C43⋊C8—22no—
64C22×C43⋊C8—22no—
65C22×C43⋊C8—41no—
66C22×C43⋊C8C2311yescharacteristic
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2311yescharacteristic
70#13063325—41no—
71#13063325—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×C22⋊Dic43—21no—
80C2×C22⋊Dic43—21no—
81#13063258—22no—
82#13063256—22no—
83#13063256—41no—
84#13063256—41no—
85#13063246—22no—
86#13063246—22no—
87#13063246—41no—
88#13063246—41no—
89C43⋊#95—21no—
90C43⋊#95—21no—
91C43⋊#95—21no—
92C43⋊#95—21no—
93C43⋊#95—21no—
94C43⋊#95—21no—
95C22×D4×C43—21no—
96C22×D4×C43C2311yescharacteristic
97C23×C4×C43—21no—
98C23×C4×C43C2311yescharacteristic
99#121×C43—22no—
100#121×C43Q814yescomplemented
101C4×D4×C43—41no—
102C4×D4×C43—41no—
103#117×C43—41no—
104#117×C43—41no—
105C2×C4⋊C4×C43—21no—
106C2×C4⋊C4×C43D411yescharacteristic
107C2×C4⋊C4×C43D411yescharacteristic
108C2×C4⋊C4×C43C2311yescharacteristic
109C2×C4⋊C4×C43C2311yescharacteristic
110C2×C4⋊C4×C43C2311yescharacteristic
111C2×C22⋊C4×C43—21no—
112C2×C22⋊C4×C43—21no—
113C2×C22⋊C4×C43—21no—
114C2×C22⋊C4×C43—21no—
115C2×C22⋊C4×C43—21no—
116C2×C22⋊C4×C43D411yescharacteristic
117C2×C22⋊C4×C43D411yescharacteristic
118C2×C42×C43—21no—
119C2×C42×C43C2311yescharacteristic
120C2×C42×C43C2311yescharacteristic
121#95×C43—21no—
122#95×C43—21no—
123#95×C43—21no—
124C24×C43—21no—
125C24×C43C2×D411yescharacteristic
126C2×D4×C43—21no—
127C2×D4×C43—24no—
128C2×D4×C43—21no—
129C2×D4×C43—21no—
130C2×D4×C43—21no—
131C2×D4×C43—21no—
132C2×D4×C43—81no—
133C2×D4×C43—81no—
134C2×D4×C43C2×Q812yes—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—42no—
143C22×C4×C43—81no—
144C22×C4×C43—81no—
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2×D411yescharacteristic
148C22×C4×C43C2×D411yescharacteristic
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43C2×Q812yes—
151C22×C4×C43#4112yes—
152C22×C4×C43C2411yescharacteristic
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C2×Dic86—42no—
158C2×Dic86—81no—
159C2×Dic86—81no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—41no—
166C4⋊C4×C43—41no—
167C4⋊C4×C43—81no—
168C4⋊C4×C43C2×Q812yes—
169C22⋊C4×C43—24no—
170C22⋊C4×C43—41no—
171C22⋊C4×C43—41no—
172C22⋊C4×C43—42no—
173C22⋊C4×C43—81no—
174C22⋊C4×C43—81no—
175C22⋊C4×C43—81no—
176C42×C43—21no—
177C42×C43—21no—
178C42×C43—21no—
179C42×C43—21no—
180C42×C43—81no—
181C22⋊Dic43—81no—
182C22⋊Dic43—81no—
183C4⋊Dic43—21no—
184C4⋊Dic43—21no—
185C4⋊Dic43—21no—
186C4⋊Dic43—21no—
187C4⋊Dic43—21no—
188C4⋊Dic43—21no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—42no—
192Dic43⋊C4—81no—
193Dic43⋊C4—81no—
194#10621877—44no—
195C2×C43⋊C8—21no—
196C2×C43⋊C8—21no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—44no—
206C2×C43⋊C8#4112yes—
207C2×C43⋊C8#4112yes—
208C23×C43—21no—
209C23×C43—21no—
210C23×C43—21no—
211C23×C43—21no—
212C23×C43—21no—
213C23×C43—21no—
214C23×C43—22no—
215C23×C43—41no—
216C23×C43—41no—
217C23×C43—41no—
218C23×C43—41no—
219C23×C43—41no—
220C23×C43—41no—
221C23×C43C22⋊Q812yes—
222C23×C43C22×D411yescharacteristic
223D4×C43—22no—
224D4×C43—22no—
225D4×C43—41no—
226D4×C43—41no—
227D4×C43—84no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—22no—
231C2×C4×C43—22no—
232C2×C4×C43—22no—
233C2×C4×C43—22no—
234C2×C4×C43—22no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
245C2×C4×C43—21no—
246C2×C4×C43—21no—
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×C43—41no—
254C2×C4×C43—41no—
255C2×C4×C43—84no—
256C2×C4×C43C22⋊Q812yes—
257C2×C4×C43C22×Q811yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×C4×C43C2×#4111yescharacteristic
260C2×C4×C43C2×#4111yescharacteristic
261C2×C4×C43H(5,𝔽2)11yescharacteristic
262C2×C4×C43H(5,𝔽2)11yescharacteristic
263C2×Dic43—41no—
264C2×Dic43—41no—
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—
275Dic86—84no—
276C43⋊C8—22no—
277C43⋊C8—22no—
278C43⋊C8—41no—
279C43⋊C8—41no—
280#29174—431nomaximal, Sylow
281C22×C43—22no—
282C22×C43—22no—
283C22×C43—22no—
284C22×C43—41no—
285C22×C43—41no—
286C22×C43—41no—
287C22×C43—41no—
288C22×C43—42no—
289C22×C43—42no—
290C22×C43—42no—
291C22×C43—81no—
292C22×C43—81no—
293C22×C43—81no—
294C22×C43—81no—
295C22×C43C2×C22⋊Q811yescharacteristic
296C22×C43#53211yescharacteristic
297C22×C43#53411yescharacteristic
298C22×C43#57611yescharacteristic
299C22×C43#57711yescharacteristic
300C22×C43#57711yescharacteristic
301C22×C43#57711yescharacteristic
302C4×C43—22no—
303C4×C43—42no—
304C4×C43—41no—
305C4×C43—41no—
306C4×C43—41no—
307C4×C43—41no—
308C4×C43—81no—
309C4×C43—81no—
310C4×C43#55111yescharacteristic
311C4×C43#55111yescharacteristic
312C4×C43#55411yescharacteristic
313C4×C43#55411yescharacteristic
314Dic43—81no—
315Dic43—81no—
316Dic43—81no—
317Dic43—81no—
318#2628—431no—
319#2324D4311yescharacteristic
320#2101—431no—
321#2092—431no—
322#2058—431no—
323#2023—431no—
324#1919—431no—
325#1918—431no—
326#1895—431no—
327#1893—431no—
328#1864—431no—
329#1864—431no—
330#1809—431no—
331#1807—431no—
332#1729—431no—
333C2×C43—42no—
334C2×C43—82no—
335C2×C43#256411yescharacteristic
336C2×C43#308111yescharacteristic
337C2×C43#308411yescharacteristic
338C2×C43#317311yescharacteristic
339C2×C43#317911yescharacteristic
340C2×C43#319811yescharacteristic
341C2×C43#320011yescharacteristic
342C22×M16—431no—
343C23×C8—431no—
344C2×C4⋊Q8—431no—
345C2×#121C2×D4311yescharacteristic
346C2×C4×D4C2×D4311yescharacteristic
347C2×#117C2×D4311yescharacteristic
348C2×#107—431no—
349C2×#106—431no—
350C2×#106—431no—
351C2×#106—431no—
352C2×#102—431no—
353C2×#102—431no—
354C2×#102—431no—
355C2×#102—431no—
356#400—861no—
357#399—861no—
358#395—861no—
359#393—861no—
360#390—21no—
361#390C2×D4311yescharacteristic
362#389—431no—
363#389—431no—
364#389C2×D4311yescharacteristic
365#388—21no—
366#386—21no—
367#385—21no—
368#382—431no—
369#380—431no—
370#380—431no—
371#380C2×D4311yescharacteristic
372C4×C4⋊C4C2×D4311yescharacteristic
373#340—431no—
374#340—431no—
375#340—431no—
376#340—431no—
377#340—431no—
378#340—431no—
379#340—431no—
380#340—431no—
381#336—431no—
382#336—431no—
383#336—431no—
384#336—431no—
385C43#2917411yescharacteristic, complemented, Sylow
386C22×Q8—861no—
387C22×D4—21no—
388C22×D4C22×D4311yescharacteristic
389C23×C4—21no—
390C23×C4C22×D4311yescharacteristic
391C2×M16—862no—
392C2×M16—862no—
393C2×M16—1721no—
394C22×C8—431no—
395C22×C8—431no—
396C22×C8—431no—
397C22×C8—431no—
398C22×C8—862no—
399C22×C8—862no—
400C22×C8—1721no—
401C4⋊Q8—1721no—
402C4⋊Q8—1721no—
403#121—22no—
404#121Dic8614yescomplemented
405C4×D4—41no—
406C4×D4—41no—
407#117—41no—
408#117—41no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4C22⋊D4311yescharacteristic
418C2×C4⋊C4C22⋊D4311yescharacteristic
419C2×C4⋊C4C22×D4311yescharacteristic
420C2×C4⋊C4C22×D4311yescharacteristic
421C2×C4⋊C4C22×D4311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—861no—
428C2×C22⋊C4—861no—
429C2×C22⋊C4C22⋊D4311yescharacteristic
430C2×C22⋊C4C22⋊D4311yescharacteristic
431C2×C42—21no—
432C2×C42C22×D4311yescharacteristic
433C2×C42C22×D4311yescharacteristic
434#107—862no—
435#106—862no—
436#106—1721no—
437#106—1721no—
438#102—862no—
439#102—862no—
440#102—1721no—
441#102—1721no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—861no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451C24—21no—
452C24C2×C22⋊D4311yescharacteristic
453C2×Q8—3441no—
454C2×Q8—3441no—
455C2×Q8—1722no—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—81no—
463C2×D4—81no—
464C2×D4C2×Dic8612yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—861no—
479C22×C4C2×Dic8612yes—
480C22×C4#1062189812yes—
481C22×C4C2×C22⋊D4311yescharacteristic
482C22×C4C2×C22⋊D4311yescharacteristic
483C22×C4C2×C22⋊D4311yescharacteristic
484C22×C4C2×C22⋊D4311yescharacteristic
485C22×C4C2×C22⋊D4311yescharacteristic
486C22×C4C23×D4311yescharacteristic
487M16—1724no—
488C2×C8—432no—
489C2×C8—432no—
490C2×C8—861no—
491C2×C8—861no—
492C2×C8—861no—
493C2×C8—861no—
494C2×C8—861no—
495C2×C8—861no—
496C2×C8—861no—
497C2×C8—861no—
498C2×C8—861no—
499C2×C8—861no—
500C2×C8—1724no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—81no—
509C4⋊C4—861no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—3441no—
518C4⋊C4—3441no—
519C4⋊C4—1722no—
520C4⋊C4C2×Dic8612yes—
521C22⋊C4—24no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C22⋊C4—3441no—
529C22⋊C4—3441no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—81no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#1306330412yes—
549C23C22×C22⋊D4311yescharacteristic
550Q8—3444no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—1721no—
585C2×C4—1721no—
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#1306330412yes—
597C2×C4C22×Dic8611yescharacteristic
598C2×C4#41×D4311yescharacteristic
599C2×C4#41×D4311yescharacteristic
600C2×C4C2×#1062189811yescharacteristic
601C2×C4#1306337011yescharacteristic
602C2×C4#1306337011yescharacteristic
603C8—862no—
604C8—862no—
605C8—1721no—
606C8—1721no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#1306330411yescharacteristic
622C22#1458344511yescharacteristic
623C22#1458345811yescharacteristic
624C22#1458370211yescharacteristic
625C22#1458370711yescharacteristic
626C22#1458371011yescharacteristic
627C22#1458371011yescharacteristic
628C4—22no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—3441no—
637C4—3441no—
638C4—3441no—
639C4—3441no—
640C4#1458357811yescharacteristic
641C4#1458357811yescharacteristic
642C4#1458360811yescharacteristic
643C4#1458360811yescharacteristic
644C2—42no—
645C2—82no—
646C2#1913977111yescharacteristic
647C2#1914442611yescharacteristic
648C2#1914445711yescharacteristic
649C2#1914520611yescharacteristic
650C2#1914528011yescharacteristic
651C2#1914544111yescharacteristic
652C2#1914546711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2324C2
socleC23×C43C22×D4
prime radicalC2×#121×C43C22
hypercenter#2324D43
hypocenterC43#29174
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. #19140416C2semidirect
2. #19136163C2semidirect
3. #19136116C2semidirect
4. #19135928C2semidirect
5. #19135731C2semidirect
6. #19135173C2semidirect
7. #19135166C2semidirect
10. #19134856C2semidirect
11. #19134856C2semidirect
12. #19134626C2semidirect
13. #19134618C2semidirect
14. #19134292C2semidirect
100. #121×C43Q8semidirect
385. C43#29174semidirect
404. #121Dic86semidirect

Maximal subgroups

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

subgroupindex
#191404162
#191361632
#191361162
#191359282
#191357312
#191351732
#191351662
#191350342
#191350242
#191348562
#191346262
#191346182
#191342922
C43×#23242
#2917443