Finite Group Databasebeta

← #25025606

Subgroups — #25025606

Order 11008.

1918 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063313C211yesmaximal, characteristic
2C2×#14582803C212yesmaximal, complemented
3#19132839C212yesmaximal, complemented
4#19132839C212yesmaximal, complemented
5#19132835C212yesmaximal
6#19132831C212yesmaximal, complemented
7#19132831C212yesmaximal, complemented
8#19132822C211yesmaximal, characteristic, complemented
9C43×#1434C211yesmaximal, characteristic, complemented
10C23×Dic86C2211yescharacteristic
11C23×D172C2211yescharacteristic
12C2×#13063313—28no—
13C2×#13063313C418yescomplemented
14C22×Dic43⋊C4C2212yes—
15C22×(C2×D43)⋊C4C2212yes—
16C22×(C2×D43)⋊C4C2212yes—
17C22×(C2×D43)⋊C4C2212yes—
18C22×(C2×D43)⋊C4C2212yes—
19C22×(C2×D43)⋊C4C2212yes—
20C22×(C2×D43)⋊C4C2212yes—
21C22×C4×Dic43C2212yes—
22#14582803—28no—
23C2×C43⋊#95C2212yes—
24C2×C43⋊#95C2212yes—
25C2×C43⋊#95C2212yes—
26C2×C43⋊#95C2212yes—
27C2×C43⋊#95C2211yescharacteristic
28C2×C43⋊#95C2211yescharacteristic
29C22×C42×C43C2211yescharacteristic
30C2×#95×C43C2212yes—
31C2×#95×C43C2212yes—
32C2×#95×C43C2211yescharacteristic
33C2×#95×C43C2211yescharacteristic
34C24×D43C2312yes—
35C22×Dic86—41no—
36C22×Dic86—42no—
37C22×Dic86—41no—
38C22×Dic86C2×C414yes—
39C22×Dic86D418yes—
40C22×D172—24no—
41C22×D172—42no—
42C22×D172—41no—
43C22×D172—41no—
44C22×D172C2×C414yes—
45C23×Dic43C2312yes—
46C23×Dic43C2312yes—
47C23×Dic43C2312yes—
48#13063313—232no—
49C2×Dic43⋊C4—24no—
50C2×Dic43⋊C4—42no—
51C2×Dic43⋊C4—42no—
52C2×(C2×D43)⋊C4—24no—
53C2×(C2×D43)⋊C4—24no—
54C2×(C2×D43)⋊C4—24no—
55C2×(C2×D43)⋊C4—24no—
56C2×(C2×D43)⋊C4—24no—
57C2×(C2×D43)⋊C4—24no—
58C2×(C2×D43)⋊C4—24no—
59C2×(C2×D43)⋊C4—24no—
60C2×(C2×D43)⋊C4—42no—
61C2×(C2×D43)⋊C4—42no—
62C2×(C2×D43)⋊C4—42no—
63C2×(C2×D43)⋊C4—42no—
64C2×(C2×D43)⋊C4—42no—
65C2×(C2×D43)⋊C4—42no—
66C2×(C2×D43)⋊C4—42no—
67C2×(C2×D43)⋊C4—42no—
68C2×(C2×D43)⋊C4C2×C418yes—
69C2×(C2×D43)⋊C4C2×C418yes—
70C2×C4×Dic43—24no—
71C2×C4×Dic43—42no—
72C2×C4×Dic43—42no—
73C43⋊#95—24no—
74C43⋊#95—24no—
75C43⋊#95—24no—
76C43⋊#95—24no—
77C43⋊#95—41no—
78C43⋊#95—41no—
79C23×C4×C43C2312yes—
80C23×C4×C43C2311yescharacteristic
81C23×C4×C43C2311yescharacteristic
82C23×C4×C43C2312yes—
83C23×C4×C43C2311yescharacteristic
84C2×C42×C43—24no—
85C2×C42×C43C2×C414yes—
86#95×C43—24no—
87#95×C43—24no—
88#95×C43—41no—
89#95×C43—41no—
90C24×C43C2411yescharacteristic
91C23×D43—22no—
92C23×D43—22no—
93C23×D43—22no—
94C23×D43—22no—
95C23×D43—22no—
96C23×D43—22no—
97C23×D43—22no—
98C23×D43—22no—
99C23×D43—22no—
100C23×D43—22no—
101C23×D43—22no—
102C23×D43—22no—
103C23×D43—22no—
104C23×D43—22no—
105C23×D43C22×C414yes—
106C22×C4×C43—22no—
107C22×C4×C43—22no—
108C22×C4×C43—22no—
109C22×C4×C43—22no—
110C22×C4×C43—22no—
111C22×C4×C43—22no—
112C22×C4×C43—22no—
113C22×C4×C43—22no—
114C22×C4×C43—22no—
115C22×C4×C43—22no—
116C22×C4×C43—21no—
117C22×C4×C43—21no—
118C22×C4×C43—21no—
119C22×C4×C43—22no—
120C22×C4×C43—21no—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—22no—
126C22×C4×C43—22no—
127C22×C4×C43—21no—
128C22×C4×C43—22no—
129C22×C4×C43—21no—
130C22×C4×C43C22×C412yes—
131C22×C4×C43C22×C412yes—
132C22×C4×C43C22×C412yes—
133C22×C4×C43C2×D414yes—
134C22×C4×C43#4114yes—
135C22×Dic43—22no—
136C22×Dic43—22no—
137C22×Dic43—22no—
138C22×Dic43—22no—
139C22×Dic43—22no—
140C22×Dic43—22no—
141C22×Dic43—22no—
142C22×Dic43—22no—
143C22×Dic43—22no—
144C22×Dic43—22no—
145C22×Dic43—22no—
146C22×Dic43—22no—
147C22×Dic43—22no—
148C22×Dic43—22no—
149C22×Dic43—22no—
150C22×Dic43—22no—
151C22×Dic43—22no—
152C22×Dic43C22×C414yes—
153C22×Dic43C2×D414yes—
154C22×Dic43C2×D414yes—
155C22×Dic43#4114yes—
156C2×D172—28no—
157C2×D172—44no—
158C2×D172—44no—
159C2×D172—48no—
160C2×D172—48no—
161C2×Dic86—44no—
162C2×Dic86—44no—
163C2×Dic86—48no—
164C2×Dic86—48no—
165C2×Dic86C22⋊C4116yescomplemented
166C42×C43—28no—
167(C2×D43)⋊C4—216no—
168(C2×D43)⋊C4—216no—
169(C2×D43)⋊C4—48no—
170(C2×D43)⋊C4—48no—
171(C2×D43)⋊C4—48no—
172(C2×D43)⋊C4—48no—
173Dic43⋊C4—48no—
174C4×Dic43—48no—
175C23×C43C23×C411yescharacteristic
176C23×C43C22×D412yes—
177C23×C43C2×#4112yes—
178C23×C43C2×#4112yes—
179C23×C43H(5,𝔽2)11yescharacteristic
180C23×C43H(5,𝔽2)12yes—
181C23×C43H(5,𝔽2)11yescharacteristic
182C23×C43H(5,𝔽2)11yescharacteristic
183C23×C43H(5,𝔽2)11yescharacteristic
184C23×C432−1+411yescharacteristic
185C23×C432−1+411yescharacteristic
186C22×D43—24no—
187C22×D43—24no—
188C22×D43—24no—
189C22×D43—24no—
190C22×D43—24no—
191C22×D43—24no—
192C22×D43—24no—
193C22×D43—42no—
194C22×D43—42no—
195C22×D43—42no—
196C22×D43—42no—
197C22×D43—42no—
198C22×D43—42no—
199C22×D43—42no—
200C22×D43—42no—
201C22×D43—42no—
202C22×D43—42no—
203C22×D43—42no—
204C22×D43—42no—
205C22×D43—42no—
206C22×D43—42no—
207C22×D43—42no—
208C22×D43—42no—
209C22×D43—42no—
210C22×D43—42no—
211C22×D43—42no—
212C22×D43—42no—
213C22×D43—42no—
214C22×D43—42no—
215C22×D43—42no—
216C22×D43—42no—
217C22×D43—42no—
218C22×D43—42no—
219C22×D43—42no—
220C22×D43—42no—
221C2×C4×C43—22no—
222C2×C4×C43—22no—
223C2×C4×C43—24no—
224C2×C4×C43—24no—
225C2×C4×C43—22no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—22no—
229C2×C4×C43—22no—
230C2×C4×C43—22no—
231C2×C4×C43—24no—
232C2×C4×C43—24no—
233C2×C4×C43—24no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43—42no—
237C2×C4×C43—42no—
238C2×C4×C43—42no—
239C2×C4×C43—42no—
240C2×C4×C43—42no—
241C2×C4×C43—42no—
242C2×C4×C43—42no—
243C2×C4×C43—42no—
244C2×C4×C43—42no—
245C2×C4×C43—42no—
246C2×C4×C43—42no—
247C2×C4×C43—42no—
248C2×C4×C43—42no—
249C2×C4×C43C2×C22⋊C414yes—
250C2×Dic43—24no—
251C2×Dic43—24no—
252C2×Dic43—24no—
253C2×Dic43—24no—
254C2×Dic43—24no—
255C2×Dic43—24no—
256C2×Dic43—24no—
257C2×Dic43—24no—
258C2×Dic43—24no—
259C2×Dic43—42no—
260C2×Dic43—42no—
261C2×Dic43—42no—
262C2×Dic43—42no—
263C2×Dic43—42no—
264C2×Dic43—42no—
265C2×Dic43—42no—
266C2×Dic43—42no—
267C2×Dic43—42no—
268C2×Dic43—42no—
269C2×Dic43—42no—
270C2×Dic43C2×C22⋊C418yes—
271D172—416no—
272Dic86—416no—
273#14492—431nomaximal, Sylow
274C22×C43C22×C22⋊C411yescharacteristic
275C22×C43C4×#4111yescharacteristic
276C22×C43C4×#4111yescharacteristic
277C22×C43#51811yescharacteristic
278C22×C43#51811yescharacteristic
279C22×C43#52011yescharacteristic
280C22×C43#52011yescharacteristic
281C22×C43#53412yes—
282C22×C43#53412yes—
283C22×C43#53812yes—
284C22×C43#53812yes—
285C22×C43#54212yes—
286C22×C43#54212yes—
287C22×C43#54812yes—
288C22×C43#54812yes—
289C22×C43#55012yes—
290C22×C43#55012yes—
291C22×C43#55112yes—
292C22×C43#55512yes—
293C22×C43H(3,𝔽4)11yescharacteristic
294C22×C43#56211yescharacteristic
295C22×C43#56211yescharacteristic
296C22×C43#56311yescharacteristic
297C2×D43—44no—
298C2×D43—44no—
299C2×D43—44no—
300C2×D43—44no—
301C2×D43—44no—
302C2×D43—44no—
303C2×D43—44no—
304C2×D43—82no—
305C2×D43—82no—
306C2×D43—82no—
307C2×D43—82no—
308C2×D43—82no—
309C2×D43—82no—
310C2×D43—82no—
311C2×D43—82no—
312C4×C43—24no—
313C4×C43—42no—
314C4×C43—42no—
315C4×C43—44no—
316C4×C43—82no—
317Dic43—28no—
318Dic43—44no—
319Dic43—82no—
320C22×#124—431no—
321C2×#391—432no—
322#1439—432no—
323#1439—432no—
324#1438—432no—
325#1437—432no—
326#1437—432no—
327#1434—431no—
328#1434D4311yescharacteristic, complemented
329C2×C43#230811yescharacteristic
330C2×C43#233711yescharacteristic
331C2×C43#233711yescharacteristic
332C2×C43#234111yescharacteristic
333C2×C43#234111yescharacteristic
334C2×C43#237211yescharacteristic
335C2×C43#237911yescharacteristic
336C2×C43#273012yes—
337C2×C43#273012yes—
338C2×C43#273112yes—
339C2×C43#273312yes—
340D43—84no—
341C23×Q8—431no—
342C23×D4—431no—
343C2×#124—438no—
344C2×#124—868no—
345C22×C4⋊C4—432no—
346C22×C22⋊C4—432no—
347C22×C22⋊C4—432no—
348C22×C22⋊C4—432no—
349C22×C22⋊C4—432no—
350C22×C22⋊C4—432no—
351C22×C22⋊C4—432no—
352C22×C42—432no—
353C22×C42C2×D4311yescharacteristic
354#391—868no—
355C2×#95—432no—
356C2×#95—432no—
357C2×#95—432no—
358C2×#95—432no—
359C2×#95—431no—
360C2×#95—431no—
361C2×#95C2×D4312yes—
362C2×#95C2×D4312yes—
363C2×#95C2×D4311yescharacteristic
364C2×#95C2×D4311yescharacteristic
365C43#1449211yescharacteristic, complemented, Sylow
366C25—432no—
367C22×Q8—434no—
368C22×Q8—438no—
369C22×Q8—1721no—
370C22×Q8—1722no—
371C22×Q8—1721no—
372C22×D4—434no—
373C22×D4—864no—
374C22×D4—1721no—
375C22×D4—1721no—
376C22×D4—1722no—
377C23×C4—432no—
378C23×C4—432no—
379C23×C4—432no—
380C23×C4C22×D4312yes—
381C23×C4C22×D4311yescharacteristic
382C23×C4C22×D4311yescharacteristic
383C23×C4C22×D4312yes—
384C23×C4C22×D4311yescharacteristic
385#124—8632no—
386C2×C4⋊C4—864no—
387C2×C4⋊C4—1722no—
388C2×C4⋊C4—1722no—
389C2×C22⋊C4—438no—
390C2×C22⋊C4—438no—
391C2×C22⋊C4—864no—
392C2×C22⋊C4—864no—
393C2×C22⋊C4—864no—
394C2×C22⋊C4—864no—
395C2×C22⋊C4—864no—
396C2×C22⋊C4—864no—
397C2×C22⋊C4—864no—
398C2×C22⋊C4—864no—
399C2×C22⋊C4—1722no—
400C2×C22⋊C4—1722no—
401C2×C22⋊C4—1722no—
402C2×C22⋊C4—1722no—
403C2×C22⋊C4—1722no—
404C2×C22⋊C4—1722no—
405C2×C22⋊C4—1722no—
406C2×C22⋊C4—1722no—
407C2×C42—24no—
408C2×C42—864no—
409C2×C42—1722no—
410C2×C42—1722no—
411C2×C42C4×D4314yes—
412#95—24no—
413#95—24no—
414#95—41no—
415#95—41no—
416#95—864no—
417#95—864no—
418#95—864no—
419#95—864no—
420#95—1721no—
421#95—1721no—
422C24—434no—
423C24—862no—
424C24—862no—
425C24—862no—
426C24—862no—
427C24—862no—
428C24—862no—
429C24—862no—
430C24—862no—
431C24—862no—
432C24—862no—
433C24—862no—
434C24—862no—
435C24—862no—
436C24—862no—
437C24C23×D4311yescharacteristic
438C2×Q8—4316no—
439C2×Q8—1724no—
440C2×Q8—1724no—
441C2×Q8—1728no—
442C2×Q8—1728no—
443C2×D4—868no—
444C2×D4—1724no—
445C2×D4—1724no—
446C2×D4—1728no—
447C2×D4—1728no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—434no—
473C22×C4—434no—
474C22×C4—434no—
475C22×C4—434no—
476C22×C4—862no—
477C22×C4—862no—
478C22×C4—862no—
479C22×C4—862no—
480C22×C4—862no—
481C22×C4—862no—
482C22×C4—862no—
483C22×C4—862no—
484C22×C4—862no—
485C22×C4—862no—
486C22×C4—862no—
487C22×C4—862no—
488C22×C4—862no—
489C22×C4—862no—
490C22×C4—862no—
491C22×C4—862no—
492C22×C4—862no—
493C22×C4C2×C4×D4312yes—
494C22×C4C2×C4×D4312yes—
495C22×C4C2×C4×D4312yes—
496C22×C4#1062189814yes—
497C22×C4D4×D4314yes—
498C4⋊C4—1728no—
499C22⋊C4—8616no—
500C22⋊C4—8616no—
501C22⋊C4—1728no—
502C22⋊C4—1728no—
503C22⋊C4—1728no—
504C22⋊C4—1728no—
505C42—28no—
506C42—1728no—
507C23—864no—
508C23—864no—
509C23—864no—
510C23—864no—
511C23—864no—
512C23—864no—
513C23—864no—
514C23—1722no—
515C23—1722no—
516C23—1722no—
517C23—1722no—
518C23—1722no—
519C23—1722no—
520C23—1722no—
521C23—1722no—
522C23—1722no—
523C23—1722no—
524C23—1722no—
525C23—1722no—
526C23—1722no—
527C23—1722no—
528C23—1722no—
529C23—1722no—
530C23—1722no—
531C23—1722no—
532C23—1722no—
533C23—1722no—
534C23—1722no—
535C23—1722no—
536C23—1722no—
537C23—1722no—
538C23—1722no—
539C23—1722no—
540C23—1722no—
541C23—1722no—
542C23C22×C4×D4311yescharacteristic
543C23C2×D4×D4312yes—
544C23#41×D4312yes—
545C23C2×#1062189812yes—
546C23#1306337011yescharacteristic
547C23#1306337111yescharacteristic
548C23#1306337111yescharacteristic
549C23#1306337111yescharacteristic
550C23#1306337112yes—
551C23#1306337211yescharacteristic
552C23#1306337311yescharacteristic
553Q8—17216no—
554D4—17216no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—438no—
584C2×C4—864no—
585C2×C4—864no—
586C2×C4—864no—
587C2×C4—864no—
588C2×C4—864no—
589C2×C4—864no—
590C2×C4—864no—
591C2×C4—864no—
592C2×C4—864no—
593C2×C4—1722no—
594C2×C4—1722no—
595C2×C4—1722no—
596C2×C4—1722no—
597C2×C4—1722no—
598C2×C4—1722no—
599C2×C4—1722no—
600C2×C4—1722no—
601C2×C4—1722no—
602C2×C4—1722no—
603C2×C4—1722no—
604C2×C4C22⋊C4×D4314yes—
605C22—3442no—
606C22—3442no—
607C22—3442no—
608C22—3442no—
609C22—3442no—
610C22—3442no—
611C22—3442no—
612C22—3442no—
613C22—1724no—
614C22—1724no—
615C22—1724no—
616C22—1724no—
617C22—1724no—
618C22—1724no—
619C22—1724no—
620C22C2×C22⋊C4×D4311yescharacteristic
621C22#1458336711yescharacteristic
622C22C4×#1062189811yescharacteristic
623C22#1458337311yescharacteristic
624C22#1458337311yescharacteristic
625C22#1458337911yescharacteristic
626C22#1458338011yescharacteristic
627C22#1458346012yes—
628C22#1458346012yes—
629C22#1458348212yes—
630C22#1458348812yes—
631C22#1458351512yes—
632C22#1458352112yes—
633C22#1458355812yes—
634C22#1458355912yes—
635C22#1458356712yes—
636C22#1458357012yes—
637C22#1458357312yes—
638C22#1458361212yes—
639C22#1458364111yescharacteristic
640C22#1458364311yescharacteristic
641C22#1458364711yescharacteristic
642C22#1458365311yescharacteristic
643C4—24no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—82no—
648C4—868no—
649C4—3442no—
650C4—1724no—
651C2—3444no—
652C2#1913717211yescharacteristic
653C2#1913740511yescharacteristic
654C2#1913740811yescharacteristic
655C2#1913743211yescharacteristic
656C2#1913743511yescharacteristic
657C2#1913770511yescharacteristic
658C2#1913776611yescharacteristic
659C2#1914159212yes—
660C2#1914159612yes—
661C2#1914160512yes—
662C2#1914162212yes—
subgroupG/H
center, FrattiniC24C23×D43
commutatorC23×C43C23×C4
FittingC43×#1434C2
socle, cosocleC24×C43C24
prime radicalC23×D172C22
hypercenter#1434D43
hypocenterC43#14492
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#14582803C2semidirect
3. #19132839C2semidirect
4. #19132839C2semidirect
6. #19132831C2semidirect
7. #19132831C2semidirect
8. #19132822C2semidirect
9. C43×#1434C2semidirect
13. C2×#13063313C4semidirect
165. C2×Dic86C22⋊C4semidirect
328. #1434D43semidirect
365. C43#14492semidirect

Maximal subgroups

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

subgroupindex
C22×#130633132
C2×#145828032
#191328392
#191328352
#191328312
#191328222
C43×#14342
#1449243