Finite Group Databasebeta

← #25245427

Subgroups — #25245427

Order 11008.

1436 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D43C211yesmaximal, characteristic
2C2×#14583110C211yesmaximal, characteristic, complemented
3C2×#14583109C211yesmaximal, characteristic, complemented
4C2×#14583056C212yesmaximal
5#19142948C211yesmaximal, characteristic
6#19134937C212yesmaximal, complemented
7#19134934C212yesmaximal, complemented
8#19134901C212yesmaximal
9#19134597C211yesmaximal, characteristic, complemented
10#19134572C211yesmaximal, characteristic
11C43×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D43C2211yescharacteristic
13C22×Dic172C2212yes—
14C22×#10621885C2212yes—
15C2×#13063325C2211yescharacteristic
16C2×#13063305C2212yes—
17C2×Q8×Dic43C2211yescharacteristic
18C2×C4×Dic86—21no—
19C2×C4×Dic86C2211yescharacteristic
20C2×#13063290—21no—
21C2×#13063290C2211yescharacteristic
22C4×Q8×D43—24no—
23C4×Q8×D43—24no—
24C2×C4⋊C4×D43—21no—
25C2×C4⋊C4×D43C2211yescharacteristic
26C2×C42×D43—21no—
27C2×C42×D43C2211yescharacteristic
28#14583110—24no—
29#14583109—24no—
30#14583056—28no—
31C2×C4⋊(C43⋊C8)C2211yescharacteristic
32C2×#13063250C2212yes—
33C2×#13063248C2212yes—
34C2×#13063248C2212yes—
35C2×(C2×D43)⋊C8C2212yes—
36#14582804C2212yes—
37#14582798—21no—
38#14582798—21no—
39#14582756—21no—
40#14582756—21no—
41#14582661C2212yes—
42#14582653C2212yes—
43C2×C4×Q8×C43C2211yescharacteristic
44C2×C4⋊C8×C43C2211yescharacteristic
45C2×#103×C43C2212yes—
46C43×#384C2211yescharacteristic
47C43×#340C2212yes—
48C22×Dic86—21no—
49C22×Dic86C2312yes—
50C22×Dic86C2311yescharacteristic
51C2×Q8×D43—22no—
52C2×Q8×D43—22no—
53C2×Q8×D43—22no—
54C2×Q8×D43—22no—
55C2×Q8×D43—22no—
56C2×Q8×D43D414yes—
57C22×C4×D43—21no—
58C22×C4×D43—21no—
59C22×C4×D43D411yescharacteristic
60C22×C4×D43D411yescharacteristic
61C22×C4×D43C2311yescharacteristic
62C2×Dic172—24no—
63C2×Dic172—24no—
64C2×Dic172—24no—
65C2×#10621885—24no—
66C2×#10621885—24no—
67C2×#10621885—24no—
68C22×C43⋊C8C2312yes—
69#13063325—22no—
70#13063325—22no—
71#13063305—24no—
72#13063305—24no—
73Q8×Dic43—22no—
74Q8×Dic43—22no—
75C4×Dic86—28no—
76C4×Dic86—22no—
77C4×Dic86D414yes—
78#13063290—28no—
79#13063290—22no—
80#13063290—22no—
81C2×C4⋊Dic43—21no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43—21no—
84C2×C4⋊Dic43—22no—
85C2×C4⋊Dic43C2311yescharacteristic
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C4⋊C4×D43—28no—
90C4⋊C4×D43—22no—
91C4⋊C4×D43—22no—
92C2×(C2×D43)⋊C4—22no—
93C2×C4×Dic43—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43D411yescharacteristic
97C2×C4×Dic43D411yescharacteristic
98C2×C4×Dic43C2311yescharacteristic
99C42×D43—28no—
100C42×D43—22no—
101C42×D43—22no—
102C4⋊(C43⋊C8)—22no—
103#13063250—24no—
104#13063248—24no—
105#13063248—24no—
106(C2×D43)⋊C8—24no—
107C43⋊#95—22no—
108C43⋊#95—22no—
109C22×Q8×C43C2311yescharacteristic
110C22×C8×C43C2312yes—
111C4×Q8×C43—22no—
112C4×Q8×C43D414yes—
113C2×C4⋊C4×C43—21no—
114C2×C4⋊C4×C43—21no—
115C2×C4⋊C4×C43—21no—
116C2×C4⋊C4×C43C2312yes—
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C42×C43—21no—
119C2×C42×C43C2311yescharacteristic
120C4⋊C8×C43—22no—
121#103×C43—24no—
122C23×D43C2×D411yescharacteristic
123C2×Q8×C43—21no—
124C2×Q8×C43—21no—
125C2×Q8×C43—21no—
126C2×Q8×C43C2×D412yes—
127C2×Q8×C43C2×D412yes—
128C2×Q8×C43#4112yes—
129C22×C4×C43—21no—
130C22×C4×C43—21no—
131C22×C4×C43—22no—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2411yescharacteristic
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—22no—
138C22×Dic43C2×D411yescharacteristic
139C22×Dic43C2×D411yescharacteristic
140C22×Dic43C2×D411yescharacteristic
141Q8×D43—28no—
142Q8×D43—28no—
143Q8×D43—28no—
144Q8×D43—28no—
145C2×C4×D43—24no—
146C2×C4×D43—24no—
147C2×C4×D43—22no—
148C2×C4×D43—22no—
149C2×C4×D43—21no—
150C2×C4×D43—21no—
151C2×C4×D43—21no—
152C2×C4×D43—21no—
153C2×C4×D43—24no—
154C2×C4×D43—22no—
155C2×C4×D43—24no—
156C2×C4×D43—22no—
157C2×C4×D43—41no—
158C2×C4×D43—42no—
159C2×C4×D43—42no—
160C2×C4×D43Q1614yescomplemented
161C2×C4×D43C2×D412yes—
162C2×C4×D43#4112yes—
163C2×Dic86—24no—
164C2×Dic86—24no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—24no—
169C2×Dic86—22no—
170C2×Dic86—22no—
171C2×Dic86—22no—
172C2×Dic86—22no—
173C2×Dic86—22no—
174C2×Dic86—22no—
175C2×Dic86C2×D412yes—
176C2×Dic86C2×D412yes—
177C2×Dic86#4112yes—
178C2×C8×C43—22no—
179C2×C8×C43—22no—
180C2×C8×C43—22no—
181C4⋊C4×C43—24no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—22no—
184C4⋊C4×C43—22no—
185C4⋊C4×C43—42no—
186C4⋊C4×C43—42no—
187C4⋊C4×C43C2×D412yes—
188C42×C43—24no—
189C42×C43C2×D412yes—
190C42×C43C2×D412yes—
191#10621885—48no—
192(C2×D43)⋊C4—44no—
193C4⋊Dic43—24no—
194C4⋊Dic43—21no—
195C4⋊Dic43—42no—
196C4⋊Dic43—42no—
197C4⋊Dic43—44no—
198C4⋊Dic43C2×D412yes—
199Dic43⋊C4—24no—
200Dic43⋊C4—24no—
201Dic43⋊C4—24no—
202C4×Dic43—24no—
203C4×Dic43—24no—
204C4×Dic43—24no—
205C4×Dic43—22no—
206C4×Dic43—22no—
207C4×Dic43—21no—
208C4×Dic43—21no—
209C2×C43⋊C8—22no—
210C2×C43⋊C8—22no—
211C2×C43⋊C8—22no—
212Dic172—48no—
213C23×C43C22×D411yescharacteristic
214C22×D43—21no—
215C22×D43—21no—
216C22×D43—21no—
217C22×D43—21no—
218C22×D43#12112yes—
219C22×D43C2×Q1612yes—
220C22×D43#13712yes—
221Q8×C43—22no—
222Q8×C43—22no—
223Q8×C43—22no—
224Q8×C43#12114yes—
225C2×C4×C43—22no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—22no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43—42no—
234C2×C4×C43—42no—
235C2×C4×C43—42no—
236C2×C4×C43#12012yes—
237C2×C4×C43#12012yes—
238C2×C4×C43C2×Q1612yes—
239C2×C4×C43#13712yes—
240C2×C4×C43C22×D411yescharacteristic
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43C22×D411yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×C4×C43C2×#4111yescharacteristic
245C2×C4×C43C2×#4111yescharacteristic
246C2×Dic43—21no—
247C2×Dic43—21no—
248C2×Dic43—22no—
249C2×Dic43—22no—
250C2×Dic43—21no—
251C2×Dic43—22no—
252C2×Dic43—21no—
253C2×Dic43—21no—
254C2×Dic43—22no—
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—42no—
260C2×Dic43—42no—
261C2×Dic43—42no—
262C2×Dic43#12112yes—
263C2×Dic43C2×Q1612yes—
264C4×D43—216no—
265C4×D43—22no—
266C4×D43—22no—
267C4×D43—22no—
268C4×D43—22no—
269C4×D43—28no—
270C4×D43—44no—
271C4×D43—48no—
272Dic86—216no—
273Dic86—22no—
274Dic86—22no—
275Dic86—22no—
276Dic86—42no—
277Dic86—42no—
278Dic86—42no—
279Dic86—42no—
280Dic86#12114yes—
281C8×C43—42no—
282C43⋊C8—42no—
283#32906—431nomaximal, Sylow
284C22×C43C2×#12011yescharacteristic
285C22×C43C2×#12111yescharacteristic
286C22×C43#52511yescharacteristic
287C22×C43C22×Q1611yescharacteristic
288C22×C43C2×#13711yescharacteristic
289C22×C43C2×#13711yescharacteristic
290C22×C43#57511yescharacteristic
291C2×D43—22no—
292C2×D43—22no—
293C2×D43—22no—
294C2×D43—22no—
295C2×D43—22no—
296C2×D43—22no—
297C2×D43#46214yescomplemented
298C4×C43—24no—
299C4×C43—22no—
300C4×C43—42no—
301C4×C43—82no—
302C4×C43#45114yescomplemented
303C4×C43C2×#12111yescharacteristic
304C4×C43C2×#12111yescharacteristic
305C4×C43#52511yescharacteristic
306C4×C43#52511yescharacteristic
307Dic43—22no—
308Dic43—24no—
309Dic43—22no—
310Dic43—41no—
311Dic43—42no—
312Dic43—82no—
313C22×C4×Q8—431no—
314C2×#462—431no—
315C2×#462—431no—
316C2×#451—432no—
317#2857—431no—
318#1878—432no—
319#1878—432no—
320#1872—432no—
321#1803—431no—
322#1803D4311yescharacteristic, complemented
323#1797—431no—
324C2×C43#284711yescharacteristic
325C2×C43C2×#45111yescharacteristic
326C2×C43#300711yescharacteristic
327C2×C43C2×#46211yescharacteristic
328C2×C43#303711yescharacteristic
329C2×C43#310411yescharacteristic
330C2×C43#311311yescharacteristic
331D43—24no—
332C23×Q8—431no—
333C22×Q16—432no—
334C22×Q16—432no—
335C2×C4⋊Q8—431no—
336C2×C22⋊Q8—432no—
337C2×C4×Q8—431no—
338C2×C4×Q8—431no—
339C2×C4×Q8—431no—
340C2×C4×Q8—861no—
341C2×C4×Q8—861no—
342C2×C4×Q8—864no—
343C2×C4×Q8—864no—
344C2×C4×Q8C2×D4311yescharacteristic
345C22×C4⋊C4—431no—
346C22×C4⋊C4—861no—
347C22×C42—431no—
348C22×C42—861no—
349#462—864no—
350#462—864no—
351#451—868no—
352C2×C4⋊C8—431no—
353C2×C4⋊C8C2×D4311yescharacteristic
354C2×#103—432no—
355C2×#103—432no—
356C2×#103—432no—
357C2×#103C2×D4312yes—
358C2×C22⋊C8—432no—
359#391—432no—
360#389—861no—
361#389—861no—
362#384C2×D4311yescharacteristic
363#380—861no—
364#380—861no—
365#340—432no—
366#340C2×D4312yes—
367#336—432no—
368C43#3290611yescharacteristic, complemented, Sylow
369C22×Q8—434no—
370C22×Q8—431no—
371C22×Q8—432no—
372C22×Q8—861no—
373C22×Q8—862no—
374C22×Q8—862no—
375C22×Q8—862no—
376C22×Q8—862no—
377C22×Q8—862no—
378C22×Q8C22×D4311yescharacteristic
379C23×C4—431no—
380C23×C4—431no—
381C23×C4—431no—
382C23×C4—861no—
383C23×C4—861no—
384C2×Q16—864no—
385C2×Q16—864no—
386C2×Q16—864no—
387C2×Q16—864no—
388C2×Q16—864no—
389C2×Q16—864no—
390C22×C8—432no—
391C22×C8C22×D4312yes—
392C4⋊Q8—862no—
393C4⋊Q8—862no—
394C22⋊Q8—864no—
395C22⋊Q8—864no—
396C4×Q8—22no—
397C4×Q8—434no—
398C4×Q8—868no—
399C4×Q8—862no—
400C4×Q8—868no—
401C4×Q8—862no—
402C4×Q8—862no—
403C4×Q8—862no—
404C4×Q8—862no—
405C4×Q8D17214yes—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—862no—
411C2×C4⋊C4—868no—
412C2×C4⋊C4—861no—
413C2×C4⋊C4—861no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—862no—
417C2×C4⋊C4—862no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4C22×D4312yes—
421C2×C4⋊C4C22×D4311yescharacteristic
422C2×C22⋊C4—862no—
423C2×C42—21no—
424C2×C42—431no—
425C2×C42—431no—
426C2×C42—431no—
427C2×C42—868no—
428C2×C42—861no—
429C2×C42—861no—
430C2×C42—861no—
431C2×C42—862no—
432C2×C42—862no—
433C2×C42C22×D4311yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—862no—
436#103—24no—
437#103—864no—
438#103—864no—
439#103—864no—
440C22⋊C8—864no—
441#95—862no—
442#95—862no—
443C24—431no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—432no—
448C2×Q8—432no—
449C2×Q8—432no—
450C2×Q8—864no—
451C2×Q8—864no—
452C2×Q8—864no—
453C2×Q8—868no—
454C2×Q8—868no—
455C2×Q8—868no—
456C2×Q8—868no—
457C2×Q8—861no—
458C2×Q8—861no—
459C2×Q8—861no—
460C2×Q8—862no—
461C2×Q8—862no—
462C2×Q8—862no—
463C2×Q8—862no—
464C2×Q8—862no—
465C2×Q8—862no—
466C2×Q8C2×D17212yes—
467C2×Q8D4×D4312yes—
468C2×Q8#1062190212yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—431no—
473C22×C4—431no—
474C22×C4—434no—
475C22×C4—432no—
476C22×C4—432no—
477C22×C4—431no—
478C22×C4—864no—
479C22×C4—864no—
480C22×C4—864no—
481C22×C4—862no—
482C22×C4—864no—
483C22×C4—861no—
484C22×C4—862no—
485C22×C4—862no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4—861no—
490C22×C4—861no—
491C22×C4—862no—
492C22×C4—862no—
493C22×C4—1721no—
494C22×C4—1722no—
495C22×C4—1722no—
496C22×C4D4×D4311yescharacteristic
497C22×C4D4×D4311yescharacteristic
498C22×C4C23×D4311yescharacteristic
499Q16—1728no—
500Q16—1728no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—862no—
505C2×C8—862no—
506C2×C8—862no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—432no—
514C4⋊C4—864no—
515C4⋊C4—864no—
516C4⋊C4—864no—
517C4⋊C4—864no—
518C4⋊C4—861no—
519C4⋊C4—1724no—
520C4⋊C4—1722no—
521C4⋊C4—1722no—
522C4⋊C4C2×D17212yes—
523C22⋊C4—1724no—
524C42—24no—
525C42—864no—
526C42—864no—
527C42—862no—
528C42—862no—
529C42—864no—
530C42—861no—
531C42—861no—
532C42C2×D17212yes—
533C42D4×D4312yes—
534C23—432no—
535C23—432no—
536C23—432no—
537C23—861no—
538C23—861no—
539C23—861no—
540C23—861no—
541C23C2×D4×D4311yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—434no—
546Q8—8616no—
547Q8—862no—
548Q8—862no—
549Q8—862no—
550Q8—1722no—
551Q8—1722no—
552Q8—1722no—
553Q8—1722no—
554Q8#1306330014yes—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—432no—
567C2×C4—432no—
568C2×C4—8616no—
569C2×C4—861no—
570C2×C4—861no—
571C2×C4—862no—
572C2×C4—862no—
573C2×C4—861no—
574C2×C4—862no—
575C2×C4—861no—
576C2×C4—868no—
577C2×C4—862no—
578C2×C4—862no—
579C2×C4—862no—
580C2×C4—862no—
581C2×C4—862no—
582C2×C4—861no—
583C2×C4—861no—
584C2×C4—861no—
585C2×C4—1724no—
586C2×C4—1728no—
587C2×C4—1722no—
588C2×C4—1722no—
589C2×C4—1722no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4#1306329512yes—
593C2×C4#1306329512yes—
594C2×C4Q16×D4312yes—
595C2×C4#1306335412yes—
596C2×C4C2×D4×D4311yescharacteristic
597C2×C4C2×D4×D4311yescharacteristic
598C2×C4C22×D17211yescharacteristic
599C2×C4#41×D4311yescharacteristic
600C2×C4#41×D4311yescharacteristic
601C2×C4C2×#1062190211yescharacteristic
602C8—42no—
603C8—1722no—
604C22—434no—
605C22—862no—
606C22—862no—
607C22—862no—
608C22—862no—
609C22—862no—
610C22—862no—
611C22C2×#1306329511yescharacteristic
612C22#121×D4311yescharacteristic
613C22#1458341211yescharacteristic
614C22C2×Q16×D4311yescharacteristic
615C22#137×D4311yescharacteristic
616C22C2×#1306335411yescharacteristic
617C22#1458369511yescharacteristic
618C4—22no—
619C4—24no—
620C4—42no—
621C4—82no—
622C4—862no—
623C4—862no—
624C4—864no—
625C4—3442no—
626C4—1721no—
627C4—1722no—
628C4#1458305614yescomplemented
629C4#121×D4311yescharacteristic
630C4C2×#1306330011yescharacteristic
631C4#1458341111yescharacteristic
632C4#1458341211yescharacteristic
633C2—864no—
634C2#1914287811yescharacteristic
635C2C2×#1458305611yescharacteristic
636C2#1914385611yescharacteristic
637C2#462×D4311yescharacteristic
638C2#1914406211yescharacteristic
639C2#1914463611yescharacteristic
640C2#1914472711yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1803C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenter#1803D43
hypocenterC43#32906
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#14583110C2semidirect
3. C2×#14583109C2semidirect
6. #19134937C2semidirect
7. #19134934C2semidirect
9. #19134597C2semidirect
11. C43×#1803C2semidirect
160. C2×C4×D43Q16semidirect
297. C2×D43#462semidirect
302. C4×C43#451semidirect
322. #1803D43semidirect
368. C43#32906semidirect
628. C4#14583056semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D432
C2×#145831102
C2×#145831092
C2×#145830562
#191429482
#191349372
#191349342
#191349012
#191345972
#191345722
C43×#18032
#3290643