Finite Group Databasebeta

← #25027284

Subgroups — #25027284

Order 11008.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582810C211yesmaximal, characteristic
2C2×#14582806C211yesmaximal, characteristic, complemented
3C2×#14582788C211yesmaximal, characteristic
4#19132859C211yesmaximal, characteristic, complemented
5#19132853C212yesmaximal, complemented
6#19132850C212yesmaximal, complemented
7#19132843C212yesmaximal
8#19132841C211yesmaximal, characteristic
9#19132839C212yesmaximal
10#19132820C211yesmaximal, characteristic, complemented
11C43×#1446C211yesmaximal, characteristic, complemented
12C22×C4⋊Dic43C2211yescharacteristic
13C22×Dic43⋊C4C2211yescharacteristic
14C22×Dic43⋊C4C2212yes—
15C22×(C2×D43)⋊C4C2211yescharacteristic
16C22×(C2×D43)⋊C4C2211yescharacteristic
17C22×(C2×D43)⋊C4C2211yescharacteristic
18C22×(C2×D43)⋊C4C2212yes—
19C22×(C2×D43)⋊C4C2212yes—
20C22×C4×Dic43C2211yescharacteristic
21#14582810C418yescomplemented
22#14582806—24no—
23#14582788—24no—
24C2×C43⋊#95C2211yescharacteristic
25C2×C43⋊#95C2212yes—
26C2×C43⋊#95C2212yes—
27C2×C43⋊#95C2212yes—
28C2×C43⋊#95C2212yes—
29C2×C43⋊#95C2212yes—
30C2×C43⋊#95C2212yes—
31C2×C43⋊#95C2211yescharacteristic
32C2×C43⋊#95C2212yes—
33C23×Q8×C43C2211yescharacteristic
34C2×#95×C43C2211yescharacteristic
35C2×#95×C43C2211yescharacteristic
36C2×#95×C43C2212yes—
37C2×#95×C43C2212yes—
38C24×D43C2311yescharacteristic
39C23×Dic43C2312yes—
40C23×Dic43C2311yescharacteristic
41C23×Dic43C2312yes—
42C23×Dic43C2311yescharacteristic
43C23×Dic43C2311yescharacteristic
44C2×C4⋊Dic43—22no—
45C2×C4⋊Dic43—42no—
46C2×Dic43⋊C4—22no—
47C2×Dic43⋊C4—42no—
48C2×Dic43⋊C4—42no—
49C2×Dic43⋊C4—42no—
50C2×Dic43⋊C4—42no—
51C2×(C2×D43)⋊C4—22no—
52C2×(C2×D43)⋊C4—22no—
53C2×(C2×D43)⋊C4—22no—
54C2×(C2×D43)⋊C4—22no—
55C2×(C2×D43)⋊C4—42no—
56C2×(C2×D43)⋊C4—42no—
57C2×(C2×D43)⋊C4—42no—
58C2×(C2×D43)⋊C4—42no—
59C2×(C2×D43)⋊C4—42no—
60C2×(C2×D43)⋊C4—42no—
61C2×(C2×D43)⋊C4—42no—
62C2×(C2×D43)⋊C4C2×C414yes—
63C2×(C2×D43)⋊C4C2×C418yes—
64C2×C4×Dic43—42no—
65C2×C4×Dic43D414yes—
66C43⋊#95—22no—
67C43⋊#95—24no—
68C43⋊#95—24no—
69C43⋊#95—42no—
70C43⋊#95—42no—
71C43⋊#95—42no—
72C43⋊#95—42no—
73C43⋊#95C2×C418yes—
74C43⋊#95C2×C414yes—
75C22×Q8×C43—22no—
76C22×Q8×C43—22no—
77C22×Q8×C43—42no—
78C22×Q8×C43—42no—
79C22×Q8×C43C2×C414yes—
80C23×C4×C43C2311yescharacteristic
81C23×C4×C43C2311yescharacteristic
82C23×C4×C43C2312yes—
83C23×C4×C43C2312yes—
84C23×C4×C43C2311yescharacteristic
85#95×C43—22no—
86#95×C43—22no—
87#95×C43—42no—
88#95×C43—42no—
89C24×C43C2411yescharacteristic
90C23×D43—21no—
91C23×D43—21no—
92C23×D43—22no—
93C23×D43—22no—
94C23×D43—22no—
95C23×D43—21no—
96C23×D43—21no—
97C23×D43—21no—
98C23×D43—21no—
99C23×D43—22no—
100C23×D43C22×C412yes—
101C2×Q8×C43—28no—
102C2×Q8×C43—48no—
103C2×Q8×C43—44no—
104C2×Q8×C43—44no—
105C2×Q8×C43—48no—
106C22×C4×C43—22no—
107C22×C4×C43—21no—
108C22×C4×C43—21no—
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—22no—
117C22×C4×C43—22no—
118C22×C4×C43—22no—
119C22×C4×C43—22no—
120C22×C4×C43—22no—
121C22×C4×C43—22no—
122C22×C4×C43—22no—
123C22×C4×C43—22no—
124C22×C4×C43—22no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—22no—
128C22×C4×C43—21no—
129C22×C4×C43—21no—
130C22×C4×C43—22no—
131C22×C4×C43C22×C414yes—
132C22×C4×C43C22×C412yes—
133C22×C4×C43C2×D412yes—
134C22×C4×C43#4112yes—
135C22×Dic43—21no—
136C22×Dic43—22no—
137C22×Dic43—21no—
138C22×Dic43—21no—
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×Dic43—21no—
153C22×Dic43—21no—
154C22×Dic43—22no—
155C22×Dic43—21no—
156C22×Dic43—22no—
157C22×Dic43—22no—
158C22×Dic43—22no—
159C22×Dic43—22no—
160C22×Dic43C22×C412yes—
161C22×Dic43C22×C414yes—
162C22×Dic43C2×D412yes—
163C22×Dic43C2×D412yes—
164(C2×D43)⋊C4—28no—
165(C2×D43)⋊C4—48no—
166(C2×D43)⋊C4—44no—
167(C2×D43)⋊C4—44no—
168(C2×D43)⋊C4—48no—
169C4⋊Dic43—44no—
170Dic43⋊C4—44no—
171Dic43⋊C4—48no—
172C4×Dic43—44no—
173C23×C43C23×C411yescharacteristic
174C23×C43C22×D411yescharacteristic
175C23×C43C2×#4111yescharacteristic
176C23×C43H(5,𝔽2)12yes—
177C23×C43H(5,𝔽2)12yes—
178C23×C43H(5,𝔽2)12yes—
179C23×C43H(5,𝔽2)11yescharacteristic
180C23×C432−1+411yescharacteristic
181C23×C432−1+411yescharacteristic
182C23×C432−1+411yescharacteristic
183C23×C432−1+412yes—
184C22×D43—22no—
185C22×D43—24no—
186C22×D43—22no—
187C22×D43—22no—
188C22×D43—24no—
189C22×D43—41no—
190C22×D43—41no—
191C22×D43—41no—
192C22×D43—41no—
193C22×D43—41no—
194C22×D43—41no—
195C22×D43—42no—
196C22×D43—41no—
197C22×D43—41no—
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—
207Q8×C43—416no—
208C2×C4×C43—24no—
209C2×C4×C43—22no—
210C2×C4×C43—22no—
211C2×C4×C43—24no—
212C2×C4×C43—24no—
213C2×C4×C43—24no—
214C2×C4×C43—24no—
215C2×C4×C43—42no—
216C2×C4×C43—42no—
217C2×C4×C43—42no—
218C2×C4×C43—42no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—42no—
226C2×C4×C43—42no—
227C2×C4×C43—42no—
228C2×C4×C43—42no—
229C2×C4×C43—42no—
230C2×C4×C43—42no—
231C2×C4×C43—42no—
232C2×C4×C43—42no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43C4×D414yes—
237C2×Dic43—22no—
238C2×Dic43—24no—
239C2×Dic43—24no—
240C2×Dic43—24no—
241C2×Dic43—24no—
242C2×Dic43—22no—
243C2×Dic43—22no—
244C2×Dic43—22no—
245C2×Dic43—22no—
246C2×Dic43—22no—
247C2×Dic43—22no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—42no—
251C2×Dic43—42no—
252C2×Dic43—42no—
253C2×Dic43—42no—
254C2×Dic43—42no—
255C2×Dic43—42no—
256C2×Dic43—42no—
257C2×Dic43—42no—
258C2×Dic43—42no—
259C2×Dic43—42no—
260C2×Dic43—42no—
261C2×Dic43—42no—
262C2×Dic43—42no—
263C2×Dic43—42no—
264C2×Dic43—42no—
265C2×Dic43—42no—
266#14625—431nomaximal, Sylow
267C22×C43C2×C4×D411yescharacteristic
268C22×C43#51812yes—
269C22×C43#51911yescharacteristic
270C22×C43#52011yescharacteristic
271C22×C43#52012yes—
272C22×C43#53411yescharacteristic
273C22×C43#53511yescharacteristic
274C22×C43#53511yescharacteristic
275C22×C43#53611yescharacteristic
276C22×C43#53611yescharacteristic
277C22×C43#53711yescharacteristic
278C22×C43#53812yes—
279C22×C43#54111yescharacteristic
280C22×C43#54211yescharacteristic
281C22×C43#54212yes—
282C22×C43H(3,𝔽4)12yes—
283C22×C43#56212yes—
284C22×C43#56212yes—
285C22×C43#56212yes—
286C22×C43#56212yes—
287C22×C43#56312yes—
288C22×C43#56312yes—
289C22×C43#56312yes—
290C2×D43—42no—
291C2×D43—42no—
292C2×D43—42no—
293C2×D43—44no—
294C2×D43—44no—
295C2×D43—81no—
296C2×D43—81no—
297C2×D43—81no—
298C2×D43—82no—
299C2×D43—82no—
300C2×D43—81no—
301C4×C43—24no—
302C4×C43—44no—
303C4×C43—81no—
304C4×C43—81no—
305C4×C43—82no—
306Dic43—42no—
307Dic43—42no—
308Dic43—44no—
309Dic43—42no—
310Dic43—82no—
311C2×#393—431no—
312C2×#391—431no—
313C2×#388—431no—
314#1446D4311yescharacteristic, complemented
315#1444—431no—
316#1442—432no—
317#1442—432no—
318#1440—432no—
319#1439—432no—
320#1439—431no—
321#1434—431no—
322C2×C43#232111yescharacteristic
323C2×C43#232311yescharacteristic
324C2×C43#232611yescharacteristic
325C2×C43#237812yes—
326C2×C43#237912yes—
327C2×C43#281411yescharacteristic
328C2×C43#281611yescharacteristic
329C2×C43#281911yescharacteristic
330C2×C43#282111yescharacteristic
331C2×C43#283112yes—
332C2×C43#283212yes—
333D43—82no—
334C23×Q8C2×D4311yescharacteristic
335C22×C4⋊C4—431no—
336C22×C4⋊C4—431no—
337C22×C4⋊C4—432no—
338C22×C22⋊C4—431no—
339C22×C22⋊C4—432no—
340C22×C22⋊C4—431no—
341C22×C22⋊C4—431no—
342C22×C22⋊C4—432no—
343C22×C42—431no—
344#393—438no—
345#391—864no—
346#388—864no—
347C2×#95—431no—
348C2×#95—432no—
349C2×#95—432no—
350C2×#95—432no—
351C2×#95—431no—
352C2×#95—432no—
353C2×#95—432no—
354C2×#95—432no—
355C2×#95—432no—
356C2×#95C2×D4311yescharacteristic
357C2×#95C2×D4311yescharacteristic
358C2×#95C2×D4312yes—
359C2×#95C2×D4312yes—
360C43#1462511yescharacteristic, complemented, Sylow
361C25—431no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—42no—
365C22×Q8—42no—
366C22×Q8C4×D4314yes—
367C23×C4—431no—
368C23×C4—432no—
369C23×C4—431no—
370C23×C4—431no—
371C23×C4—432no—
372C23×C4C22×D4311yescharacteristic
373C23×C4C22×D4311yescharacteristic
374C23×C4C22×D4312yes—
375C23×C4C22×D4312yes—
376C23×C4C22×D4311yescharacteristic
377C2×C4⋊C4—862no—
378C2×C4⋊C4—862no—
379C2×C4⋊C4—1722no—
380C2×C4⋊C4—1722no—
381C2×C4⋊C4—1722no—
382C2×C4⋊C4—1722no—
383C2×C4⋊C4—1722no—
384C2×C22⋊C4—434no—
385C2×C22⋊C4—438no—
386C2×C22⋊C4—862no—
387C2×C22⋊C4—862no—
388C2×C22⋊C4—862no—
389C2×C22⋊C4—862no—
390C2×C22⋊C4—1722no—
391C2×C22⋊C4—1722no—
392C2×C22⋊C4—1722no—
393C2×C22⋊C4—1722no—
394C2×C22⋊C4—1722no—
395C2×C22⋊C4—1722no—
396C2×C22⋊C4—1722no—
397C2×C42—434no—
398C2×C42—1722no—
399#95—22no—
400#95—22no—
401#95—42no—
402#95—42no—
403#95—438no—
404#95—434no—
405#95—862no—
406#95—864no—
407#95—864no—
408#95—1722no—
409#95—1722no—
410#95—1722no—
411#95—1722no—
412C24—432no—
413C24—861no—
414C24—861no—
415C24—862no—
416C24—862no—
417C24—861no—
418C24—862no—
419C24—861no—
420C24—861no—
421C24—861no—
422C24—862no—
423C24C23×D4311yescharacteristic
424C2×Q8—28no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—48no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—432no—
455C22×C4—434no—
456C22×C4—432no—
457C22×C4—432no—
458C22×C4—861no—
459C22×C4—861no—
460C22×C4—862no—
461C22×C4—862no—
462C22×C4—862no—
463C22×C4—862no—
464C22×C4—861no—
465C22×C4—862no—
466C22×C4—862no—
467C22×C4—862no—
468C22×C4—862no—
469C22×C4—861no—
470C22×C4—862no—
471C22×C4—862no—
472C22×C4—862no—
473C22×C4—861no—
474C22×C4—862no—
475C22×C4—861no—
476C22×C4—862no—
477C22×C4—862no—
478C22×C4—862no—
479C22×C4—862no—
480C22×C4—862no—
481C22×C4—862no—
482C22×C4—862no—
483C22×C4C2×C4×D4314yes—
484C22×C4C2×C4×D4312yes—
485C22×C4D4×D4312yes—
486C22×C4#1062190012yes—
487C4⋊C4—1724no—
488C4⋊C4—1724no—
489C4⋊C4—1728no—
490C22⋊C4—868no—
491C22⋊C4—1728no—
492C22⋊C4—1724no—
493C22⋊C4—1724no—
494C22⋊C4—1728no—
495C42—1724no—
496C23—862no—
497C23—864no—
498C23—862no—
499C23—862no—
500C23—864no—
501C23—1722no—
502C23—1722no—
503C23—1722no—
504C23—1722no—
505C23—1722no—
506C23—1722no—
507C23—1722no—
508C23—1722no—
509C23—1721no—
510C23—1721no—
511C23—1721no—
512C23—1721no—
513C23—1721no—
514C23—1721no—
515C23—1722no—
516C23—1721no—
517C23—1721no—
518C23—1722no—
519C23C22×C4×D4311yescharacteristic
520C23C2×D4×D4311yescharacteristic
521C23C2×#1062190011yescharacteristic
522C23#1306337012yes—
523C23#1306337012yes—
524C23#1306337111yescharacteristic
525C23#1306337112yes—
526C23#1306337211yescharacteristic
527C23#1306337311yescharacteristic
528C23#1306337311yescharacteristic
529C23#1306337312yes—
530Q8—416no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—862no—
560C2×C4—864no—
561C2×C4—864no—
562C2×C4—864no—
563C2×C4—864no—
564C2×C4—862no—
565C2×C4—862no—
566C2×C4—862no—
567C2×C4—862no—
568C2×C4—862no—
569C2×C4—862no—
570C2×C4—1722no—
571C2×C4—1722no—
572C2×C4—1722no—
573C2×C4—1722no—
574C2×C4—1722no—
575C2×C4—1722no—
576C2×C4—1722no—
577C2×C4—1722no—
578C2×C4—1722no—
579C2×C4—1722no—
580C2×C4—1722no—
581C2×C4—1722no—
582C2×C4—1722no—
583C2×C4—1722no—
584C2×C4—1722no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1722no—
588C2×C4(C22⋊D43)⋊C414yes—
589C22—3441no—
590C22—3442no—
591C22—3442no—
592C22—3441no—
593C22—3441no—
594C22—3441no—
595C22—1722no—
596C22—1722no—
597C22—1724no—
598C22—1722no—
599C22—1724no—
600C22C2×(C22⋊D43)⋊C411yescharacteristic
601C22#1458337212yes—
602C22#1458337511yescharacteristic
603C22#1458338111yescharacteristic
604C22#1458338112yes—
605C22#1458345911yescharacteristic
606C22#1458346411yescharacteristic
607C22#1458346711yescharacteristic
608C22#1458347211yescharacteristic
609C22#1458347411yescharacteristic
610C22#1458347911yescharacteristic
611C22#1458348412yes—
612C22#1458351011yescharacteristic
613C22#1458351811yescharacteristic
614C22#1458352212yes—
615C22#1458364112yes—
616C22#1458364312yes—
617C22#1458364512yes—
618C22#1458364512yes—
619C22#1458364512yes—
620C22#1458364812yes—
621C22#1458364812yes—
622C22#1458365012yes—
623C4—24no—
624C4—44no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—3442no—
629C4—1722no—
630C4—1722no—
631C4—1722no—
632C4—1724no—
633C2—3442no—
634C2#1913725011yescharacteristic
635C2#1913728011yescharacteristic
636C2#1913730911yescharacteristic
637C2#1913775612yes—
638C2#1913776712yes—
639C2#1914260711yescharacteristic
640C2#1914262111yescharacteristic
641C2#1914264311yescharacteristic
642C2#1914265911yescharacteristic
643C2#1914275512yes—
644C2#1914276512yes—
subgroupG/H
center, FrattiniC24C23×D43
commutatorC23×C43C23×C4
FittingC43×#1446C2
socle, cosocleC24×C43C24
prime radicalC24×D43C23
hypercenter#1446D43
hypocenterC43#14625
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
2. C2×#14582806C2semidirect
4. #19132859C2semidirect
5. #19132853C2semidirect
6. #19132850C2semidirect
10. #19132820C2semidirect
11. C43×#1446C2semidirect
21. #14582810C4semidirect
314. #1446D43semidirect
360. C43#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#145828102
C2×#145828062
C2×#145827882
#191328592
#191328532
#191328502
#191328432
#191328412
#191328392
#191328202
C43×#14462
#1462543