Finite Group Databasebeta

← #25098239

Subgroups — #25098239

Order 11008.

1742 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583473C211yesmaximal, characteristic
2#19147796C211yesmaximal, characteristic
3C2×#14582872C211yesmaximal, characteristic
4#19137181C211yesmaximal, characteristic, complemented
5#19135005C218yesmaximal, complemented
6#19134328C211yesmaximal, characteristic, complemented
7#19134327C211yesmaximal, characteristic, complemented
8C43×#2310C211yesmaximal, characteristic, complemented
9C22×#10621898C2211yescharacteristic
10#14583473—24no—
11#14583473C418yescomplemented
12C2×#13063323—21no—
13C2×#13063314—21no—
14C2×#13063308C2214yes—
15C2×#13063301—22no—
16#14583380—22no—
17#14583379—22no—
18#14583376—21no—
19#14583373—21no—
20#14583372C2214yes—
21C4×#10621898—22no—
22#14583367—22no—
23C2×#13063284C2211yescharacteristic
24C2×#13063284C2211yescharacteristic
25C2×#13063284C2211yescharacteristic
26#117×D43—22no—
27#14582872—24no—
28C2×#13063239C2214yes—
29#14582806—22no—
30#14582806—22no—
31#14582762C2214yes—
32#14582666C2214yes—
33#14582665C2214yes—
34C23×Q8×C43C2211yescharacteristic
35C2×#117×C43C2211yescharacteristic
36C2×#117×C43C2211yescharacteristic
37C43×#391—22no—
38C43×#391—22no—
39C43×#381C2214yes—
40C2×#10621898—21no—
41C2×#10621898—21no—
42C2×#10621898—21no—
43C2×#10621898—21no—
44C2×#10621898—22no—
45C2×#10621898—22no—
46C2×#10621898C2×C414yes—
47C2×#10621898D414yes—
48C22×Dic86D411yescharacteristic
49C22×D172D411yescharacteristic
50C22×C22⋊D43C2312yes—
51C22×C4×D43D412yes—
52#13063323—28no—
53#13063314—28no—
54#13063308—28no—
55#13063308C2×C4116yescomplemented
56#13063301—216no—
57C4×Dic86—24no—
58#13063290—24no—
59C4×D172—24no—
60#13063287—24no—
61C4×C22⋊D43—28no—
62(C22⋊D43)⋊C4—28no—
63#13063284—21no—
64#13063284—21no—
65#13063284—21no—
66#13063284—21no—
67#13063284—22no—
68#13063284—22no—
69#13063284—22no—
70#13063284C2×C414yescomplemented
71#13063283—24no—
72#13063282—24no—
73#13063281—24no—
74C2×C4⋊Dic43—21no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43D412yes—
77C2×Dic43⋊C4—24no—
78C4⋊C4×D43—24no—
79C2×C22⋊Dic43C2312yes—
80C2×C22⋊Dic43C2312yes—
81C2×C22⋊Dic43C2312yes—
82C2×(C2×D43)⋊C4—24no—
83C22⋊C4×D43—24no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43—21no—
86C2×C4×Dic43D412yes—
87C42×D43—24no—
88#13063239—28no—
89C43⋊#95—28no—
90C22×Q8×C43—21no—
91C22×Q8×C43—21no—
92C22×Q8×C43—21no—
93C22×Q8×C43—21no—
94C22×Q8×C43—21no—
95C22×Q8×C43—21no—
96C22×Q8×C43—22no—
97C22×Q8×C43—22no—
98C22×Q8×C43—22no—
99C22×Q8×C43C2×C414yescomplemented
100C23×C4×C43C2312yes—
101C23×C4×C43C2311yescharacteristic
102#117×C43—21no—
103#117×C43—21no—
104#117×C43—21no—
105#117×C43—21no—
106#117×C43—22no—
107#117×C43—22no—
108C2×C4⋊C4×C43—21no—
109C2×C4⋊C4×C43D412yes—
110C2×C22⋊C4×C43C2312yes—
111C2×C22⋊C4×C43C2312yes—
112C2×C42×C43—21no—
113C2×C42×C43D412yes—
114#95×C43—28no—
115C24×C43C2411yescharacteristic
116C23×D43C2×D412yes—
117C2×Q8×C43—24no—
118C2×Q8×C43—24no—
119C2×Q8×C43—24no—
120C2×Q8×C43—24no—
121C2×Q8×C43—24no—
122C2×Q8×C43—24no—
123C2×Q8×C43—41no—
124C2×Q8×C43—41no—
125C2×Q8×C43—41no—
126C2×Q8×C43—41no—
127C2×Q8×C43—42no—
128C2×Q8×C43—42no—
129C2×Q8×C43—42no—
130C2×Q8×C43—42no—
131C2×Q8×C43—42no—
132C2×Q8×C43—42no—
133C22×C4×C43—22no—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—22no—
137C22×C4×C43—22no—
138C22×C4×C43—22no—
139C22×C4×C43—22no—
140C22×C4×C43C22×C414yes—
141C22×C4×C43C22×C412yes—
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2×D412yes—
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2×D414yes—
148C22×C4×C43C2×D411yescharacteristic
149C22×C4×C43C2×D411yescharacteristic
150C22×C4×C43#4112yes—
151C22×C4×C43#4112yes—
152C2×C22⋊D43—22no—
153C2×C22⋊D43—22no—
154C2×C22⋊D43—22no—
155C2×C22⋊D43—22no—
156C2×C22⋊D43C22×C414yes—
157C2×C22⋊D43C2×D414yes—
158C22×Dic43—22no—
159C22×Dic43—22no—
160C22×Dic43C2×D412yes—
161C22×Dic43C2×D414yes—
162#10621898—24no—
163#10621898—24no—
164#10621898—24no—
165#10621898—24no—
166#10621898—24no—
167#10621898—24no—
168#10621898—24no—
169#10621898—24no—
170C2×D172—21no—
171C2×D172—21no—
172C2×D172—22no—
173C2×D172C22⋊C414yes—
174C2×C4×D43—22no—
175C2×C4×D43—22no—
176C2×C4×D43—24no—
177C2×C4×D43—28no—
178C2×C4×D43C22⋊C418yes—
179C2×Dic86—21no—
180C2×Dic86—21no—
181C2×Dic86—22no—
182C2×Dic86C22⋊C414yes—
183C4⋊C4×C43—22no—
184C4⋊C4×C43—22no—
185C4⋊C4×C43—42no—
186C22⋊C4×C43—22no—
187C22⋊C4×C43—22no—
188C22⋊C4×C43—22no—
189C22⋊C4×C43—22no—
190C42×C43—22no—
191C42×C43—22no—
192C42×C43—42no—
193C22⋊Dic43—22no—
194C22⋊Dic43—22no—
195C22⋊Dic43—22no—
196C22⋊Dic43—22no—
197C22⋊Dic43—22no—
198C22⋊Dic43C22×C414yes—
199(C2×D43)⋊C4—28no—
200(C2×D43)⋊C4—216no—
201C4⋊Dic43—24no—
202C4⋊Dic43—22no—
203C4⋊Dic43—22no—
204C4⋊Dic43—42no—
205Dic43⋊C4—28no—
206Dic43⋊C4—216no—
207C4×Dic43—28no—
208C4×Dic43—24no—
209C4×Dic43—22no—
210C4×Dic43—22no—
211C4×Dic43—42no—
212C23×C43—42no—
213C23×C43C23×C411yescharacteristic
214C23×C43C22×D411yescharacteristic
215C23×C43C22×D411yescharacteristic
216C23×C43C22×D411yescharacteristic
217C23×C43C22×D411yescharacteristic
218C23×C43C2×#4111yescharacteristic
219C23×C43C2×#4111yescharacteristic
220C22×D43—22no—
221C22×D43—22no—
222C22×D43—44no—
223C22×D43C2×C22⋊C414yes—
224Q8×C43—44no—
225Q8×C43—44no—
226Q8×C43—44no—
227Q8×C43—44no—
228C2×C4×C43—24no—
229C2×C4×C43—24no—
230C2×C4×C43—24no—
231C2×C4×C43—24no—
232C2×C4×C43—42no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43—44no—
237C2×C4×C43—44no—
238C2×C4×C43—42no—
239C2×C4×C43C2×C22⋊C412yes—
240C2×C4×C43C2×C22⋊C412yes—
241C2×C4×C43C4×D412yes—
242C2×C4×C43C4×D412yes—
243C2×C4×C43C4×D412yes—
244C2×C4×C43C4×D412yes—
245C2×C4×C43#12011yescharacteristic
246C2×C4×C43#12011yescharacteristic
247C2×C4×C43#12011yescharacteristic
248C2×C4×C43#12011yescharacteristic
249C2×C4×C43#12112yes—
250C2×C4×C43#12112yes—
251C2×C4×C43#12112yes—
252C2×C4×C43#12112yes—
253C2×C4×C43#12312yes—
254C2×C4×C43#12312yes—
255C22⋊D43—24no—
256C22⋊D43—24no—
257C22⋊D43—24no—
258C22⋊D43—24no—
259C2×Dic43—22no—
260C2×Dic43—22no—
261C2×Dic43—24no—
262C2×Dic43—24no—
263C2×Dic43—44no—
264C2×Dic43—44no—
265C2×Dic43—44no—
266C2×Dic43—42no—
267C2×Dic43C2×C22⋊C414yes—
268D172—24no—
269D172—24no—
270C4×D43—28no—
271C4×D43—28no—
272C4×D43—88no—
273Dic86—24no—
274Dic86—24no—
275#20962—431nomaximal, Sylow
276C22×C43—22no—
277C22×C43—22no—
278C22×C43—22no—
279C22×C43—41no—
280C22×C43—41no—
281C22×C43—41no—
282C22×C43—41no—
283C22×C43C22×C22⋊C411yescharacteristic
284C22×C43C2×C4×D411yescharacteristic
285C22×C43C2×C4×D411yescharacteristic
286C22×C43C2×#12011yescharacteristic
287C22×C43C2×#12111yescharacteristic
288C22×C43C2×#12111yescharacteristic
289C22×C43C2×#12311yescharacteristic
290C2×D43—24no—
291C2×D43—44no—
292C2×D43—44no—
293C2×D43—82no—
294C4×C43—44no—
295C4×C43—44no—
296C4×C43—82no—
297C4×C43#38612yes—
298C4×C43#38612yes—
299C4×C43#38612yes—
300C4×C43#38612yes—
301Dic43—24no—
302Dic43—44no—
303Dic43—44no—
304Dic43—82no—
305C2×#536—431no—
306#3428—431no—
307C2×#410—431no—
308#2310—431no—
309#2310D4311yescharacteristic, complemented
310#1890—438no—
311#1740—431no—
312#1740—431no—
313C2×C43—21no—
314C2×C43—21no—
315C2×C43—21no—
316C2×C43—21no—
317C2×C43—21no—
318C2×C43—21no—
319C2×C43C2×#38611yescharacteristic
320C2×C43#302812yes—
321D43—44no—
322C22×#41—431no—
323C23×Q8C2×D4311yescharacteristic
324#536—438no—
325#536—864no—
326C2×C4⋊Q8—861no—
327C2×#124—861no—
328C2×#123—434no—
329C2×C22⋊Q8—862no—
330#520—862no—
331#520—862no—
332#519—861no—
333#518—434no—
334#518—861no—
335C4×#41—862no—
336C4×#41—862no—
337C2×#117—431no—
338C2×#117—431no—
339C2×#117—431no—
340C2×#117—862no—
341C2×#117C2×D4311yescharacteristic
342C2×#117C2×D4311yescharacteristic
343#410—864no—
344C2×#99—434no—
345#391—22no—
346#391—22no—
347#391—862no—
348#391—862no—
349#381—434no—
350#381C2×D4314yes—
351#342—434no—
352#342—434no—
353C43#2096211yescharacteristic, complemented, Sylow
354C2×#41—434no—
355C2×#41—434no—
356C2×#41—862no—
357C2×#41—861no—
358C2×#41—861no—
359C2×#41—862no—
360C2×#41—861no—
361C2×#41—861no—
362C22×Q8—22no—
363C22×Q8—21no—
364C22×Q8—21no—
365C22×Q8—22no—
366C22×Q8—21no—
367C22×Q8—21no—
368C22×Q8—22no—
369C22×Q8—21no—
370C22×Q8—21no—
371C22×Q8—431no—
372C22×Q8C4×D4314yescomplemented
373C22×D4—432no—
374C22×D4—431no—
375C23×C4—432no—
376C23×C4C22×D4312yes—
377C23×C4C22×D4311yescharacteristic
378C4⋊Q8—868no—
379#124—868no—
380#123—4316no—
381#123—868no—
382C22⋊Q8—8616no—
383C4×Q8—864no—
384C4×Q8—864no—
385C4×D4—864no—
386C4×D4—868no—
387C4×D4—864no—
388C4×D4—868no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—22no—
394#117—22no—
395#117—434no—
396#117—864no—
397#117—864no—
398#117—864no—
399#117—861no—
400#117—861no—
401#117—862no—
402#117—861no—
403#117—861no—
404#117—862no—
405#117—862no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—432no—
408C2×C4⋊C4—864no—
409C2×C4⋊C4—864no—
410C2×C4⋊C4—861no—
411C2×C4⋊C4—861no—
412C2×C4⋊C4C22⋊D4312yes—
413C2×C22⋊C4—432no—
414C2×C22⋊C4—432no—
415C2×C22⋊C4—432no—
416C2×C22⋊C4—864no—
417C2×C22⋊C4—864no—
418C2×C22⋊C4C22×D4312yes—
419C2×C22⋊C4C22×D4312yes—
420C2×C42—21no—
421C2×C42—432no—
422C2×C42—864no—
423C2×C42—861no—
424C2×C42—861no—
425C2×C42C22⋊D4312yes—
426#99—868no—
427#95—28no—
428#95—868no—
429C24—432no—
430C24C23×D4311yescharacteristic
431#41—864no—
432#41—864no—
433#41—864no—
434#41—864no—
435#41—864no—
436#41—864no—
437#41—864no—
438#41—864no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—434no—
456C2×Q8—861no—
457C2×Q8—861no—
458C2×Q8—862no—
459C2×D4—434no—
460C2×D4—434no—
461C2×D4—434no—
462C2×D4—862no—
463C2×D4—862no—
464C2×D4—862no—
465C2×D4—862no—
466C2×D4—861no—
467C2×D4—861no—
468C2×D4—862no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—432no—
477C22×C4—438no—
478C22×C4—434no—
479C22×C4—868no—
480C22×C4—862no—
481C22×C4—862no—
482C22×C4—864no—
483C22×C4—862no—
484C22×C4—862no—
485C22×C4C2×C4×D4314yes—
486C22×C4C2×C4×D4312yes—
487C22×C4#1062189812yes—
488C22×C4D4×D4311yescharacteristic
489C22×C4D4×D4311yescharacteristic
490C22×C4D4×D4312yes—
491C22×C4D4×D4311yescharacteristic
492C22×C4D4×D4311yescharacteristic
493C22×C4#1062190012yes—
494C22×C4C2×C22⋊D4314yes—
495C22×C4C2×C22⋊D4311yescharacteristic
496C22×C4C2×C22⋊D4311yescharacteristic
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—8616no—
501C4⋊C4—868no—
502C4⋊C4—864no—
503C4⋊C4—862no—
504C4⋊C4—862no—
505C4⋊C4—1722no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—434no—
511C22⋊C4—8616no—
512C22⋊C4—868no—
513C22⋊C4—862no—
514C22⋊C4—862no—
515C22⋊C4—862no—
516C22⋊C4—862no—
517C22⋊C4—862no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—868no—
522C42—864no—
523C42—862no—
524C42—862no—
525C42—1722no—
526C23—42no—
527C23—434no—
528C23—862no—
529C23—862no—
530C23—1724no—
531C23C22×C4×D4311yescharacteristic
532C23C2×D4×D4311yescharacteristic
533C23C2×D4×D4311yescharacteristic
534C23C2×D4×D4311yescharacteristic
535C23C22×C22⋊D4311yescharacteristic
536C23C2×#1062190011yescharacteristic
537C23C2×#1062189811yescharacteristic
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—864no—
543Q8—864no—
544D4—864no—
545D4—864no—
546D4—864no—
547D4—864no—
548D4—864no—
549D4—864no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—434no—
562C2×C4—862no—
563C2×C4—862no—
564C2×C4—868no—
565C2×C4—868no—
566C2×C4—864no—
567C2×C4—864no—
568C2×C4—3448no—
569C2×C4—1724no—
570C2×C4—1722no—
571C2×C4—1724no—
572C2×C4—1724no—
573C2×C4C22⋊C4×D4312yes—
574C2×C4C22⋊C4×D4312yes—
575C2×C4(C22⋊D43)⋊C412yes—
576C2×C4(C22⋊D43)⋊C412yes—
577C2×C4C4×C22⋊D4312yes—
578C2×C4C4×C22⋊D4312yes—
579C2×C4#1306329311yescharacteristic
580C2×C4#1306329311yescharacteristic
581C2×C4#1306329311yescharacteristic
582C2×C4#1306329311yescharacteristic
583C2×C4#1306329612yes—
584C2×C4#1306329612yes—
585C2×C4#1306329812yes—
586C2×C4#1306329812yes—
587C2×C4#1306330712yes—
588C2×C4#1306330712yes—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—864no—
597C22—3442no—
598C22—1724no—
599C22—1724no—
600C22C2×C22⋊C4×D4311yescharacteristic
601C22C2×(C22⋊D43)⋊C411yescharacteristic
602C22C2×C4×C22⋊D4311yescharacteristic
603C22C2×#1306329311yescharacteristic
604C22C2×#1306329611yescharacteristic
605C22C2×#1306329811yescharacteristic
606C22C2×#1306330711yescharacteristic
607C4—44no—
608C4—44no—
609C4—82no—
610C4—864no—
611C4—3442no—
612C4—1724no—
613C4—1724no—
614C4#1458277512yes—
615C4#1458277512yes—
616C4#1458277512yes—
617C4#1458277512yes—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—1724no—
625C2C2×#1458277511yescharacteristic
626C2#1914400012yes—
subgroupG/H
centerC22C2×#13063293
commutatorC23×C43C23×C4
FrattiniC24C23×D43
FittingC43×#2310C2
socleC22×C43C2×#120
prime radicalC22×#10621898C22
cosocleC24×C43C24
hypercenter#2310D43
hypocenterC43#20962
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. #19137181C2semidirect
5. #19135005C2semidirect
6. #19134328C2semidirect
7. #19134327C2semidirect
8. C43×#2310C2semidirect
11. #14583473C4semidirect
55. #13063308C2×C4semidirect
70. #13063284C2×C4semidirect
99. C22×Q8×C43C2×C4semidirect
309. #2310D43semidirect
353. C43#20962semidirect
372. C22×Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834732
#191477962
C2×#145828722
#191371812
#191350052
#191343282
#191343272
C43×#23102
#2096243