Finite Group Databasebeta

← #25159723

Subgroups — #25159723

Order 11008.

932 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583188C211yesmaximal, characteristic, complemented
2C2×#14582981C211yesmaximal, characteristic
3#19138443C211yesmaximal, characteristic, complemented
4#19137503C211yesmaximal, characteristic, complemented
5#19136119C211yesmaximal, characteristic, complemented
6#19135970C211yesmaximal, characteristic, complemented
7#19135921C211yesmaximal, characteristic, complemented
8#19135322C211yesmaximal, characteristic, complemented
9#19135151C211yesmaximal, characteristic, complemented
10#19134944C211yesmaximal, characteristic
11#19134942C211yesmaximal, characteristic
12#19134855C211yesmaximal, characteristic, complemented
13#19134853C211yesmaximal, characteristic, complemented
14#19134843C211yesmaximal, characteristic, complemented
15C43×#2085C211yesmaximal, characteristic, complemented
16C22×D344C2211yescharacteristic
17C2×#13063324C2211yescharacteristic
18C2×#13063305—21no—
19C2×C4×D172C2211yescharacteristic
20C2×#13063287C2211yescharacteristic
21C2×#13063287C2211yescharacteristic
22C2×#13063287C2211yescharacteristic
23#14583188—24no—
24#14582981—24no—
25C2×#13063258C2211yescharacteristic
26C2×#13063256C2211yescharacteristic
27C2×#13063256C2211yescharacteristic
28C2×Dic43⋊C8C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31C2×#13063245C2211yescharacteristic
32C2×#13063245C2211yescharacteristic
33C2×#13063245C2211yescharacteristic
34C2×#13063245C2211yescharacteristic
35C2×C8×Dic43C2211yescharacteristic
36#14582821—21no—
37#14582812—21no—
38#14582801C2211yescharacteristic
39#14582800C2211yescharacteristic
40#14582798C2211yescharacteristic
41#14582795C2211yescharacteristic
42#14582795C2211yescharacteristic
43#14582773—21no—
44#14582773—21no—
45#14582773—21no—
46#14582758—21no—
47C4×C4⋊Dic43C2211yescharacteristic
48#95×D43—21no—
49#14582662C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582654C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×#107×C43C2211yescharacteristic
55C2×#106×C43C2211yescharacteristic
56C43×#389C2211yescharacteristic
57C43×#382C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86—21no—
62C22×D172C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C22×C4×D43—21no—
67C22×C4×D43—21no—
68C2×D344—22no—
69C2×D344—22no—
70C2×D344Q814yescomplemented
71C22×C43⋊C8C2311yescharacteristic
72C22×C43⋊C8C2311yescharacteristic
73#13063324—22no—
74#13063324—41no—
75#13063305—44no—
76C4×D172—22no—
77C4×D172—22no—
78#13063287—22no—
79#13063287—22no—
80#13063287—22no—
81#13063287—41no—
82#13063287—41no—
83#13063287—41no—
84C2×C4⋊Dic43—21no—
85C2×C4⋊Dic43D411yescharacteristic
86C2×C4⋊Dic43D411yescharacteristic
87C2×C4⋊Dic43C2311yescharacteristic
88C2×C4⋊Dic43C2311yescharacteristic
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4—21no—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4—21no—
97C2×(C2×D43)⋊C4—21no—
98C2×(C2×D43)⋊C4—21no—
99C2×(C2×D43)⋊C4—21no—
100C2×C4×Dic43C2311yescharacteristic
101C2×C4×Dic43C2311yescharacteristic
102#13063258—41no—
103#13063256—22no—
104#13063256—22no—
105Dic43⋊C8—22no—
106#13063247—22no—
107#13063247—22no—
108#13063245—22no—
109#13063245—22no—
110#13063245—41no—
111#13063245D414yes—
112C8×Dic43—22no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95D411yescharacteristic
117C43⋊#95D411yescharacteristic
118C22×C8×C43C2311yescharacteristic
119C22×C8×C43C2311yescharacteristic
120C2×C4⋊C4×C43—21no—
121C2×C4⋊C4×C43C2311yescharacteristic
122C2×C4⋊C4×C43C2311yescharacteristic
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C42×C43C2311yescharacteristic
126#107×C43—22no—
127#106×C43—22no—
128#95×C43—21no—
129#95×C43—21no—
130#95×C43—21no—
131C23×D43—21no—
132C23×D43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2411yescharacteristic
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43C2×D411yescharacteristic
143C22×Dic43C2×D411yescharacteristic
144C22×Dic43C2×D411yescharacteristic
145C22×Dic43C2×D411yescharacteristic
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×D172—21no—
151C2×D172—21no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172C2×D412yes—
155C2×D172C2×Q812yes—
156C2×D172C2×Q812yes—
157C2×D172#4112yes—
158C2×C4×D43—24no—
159C2×C4×D43—42no—
160C2×C4×D43—42no—
161C2×C4×D43—42no—
162C2×C4×D43—42no—
163C2×C4×D43—42no—
164C2×C4×D43—42no—
165C2×C4×D43—81no—
166C2×C4×D43—81no—
167C2×C4×D43—81no—
168C2×C4×D43—81no—
169C2×C4×D43—81no—
170C2×Dic86—42no—
171C2×Dic86—42no—
172C2×Dic86—81no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43—21no—
178C2×C8×C43C2×Q812yes—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—42no—
187C4⋊C4×C43C2×D412yes—
188C42×C43—21no—
189C42×C43—21no—
190(C2×D43)⋊C4—24no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—42no—
193(C2×D43)⋊C4—42no—
194(C2×D43)⋊C4—42no—
195(C2×D43)⋊C4—42no—
196(C2×D43)⋊C4—42no—
197(C2×D43)⋊C4—81no—
198C4⋊Dic43—21no—
199C4⋊Dic43—22no—
200C4⋊Dic43—22no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—42no—
204C4⋊Dic43#4112yes—
205Dic43⋊C4—42no—
206Dic43⋊C4—81no—
207Dic43⋊C4—81no—
208C4×Dic43—21no—
209C4×Dic43—21no—
210C4×Dic43—21no—
211C4×Dic43—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8C2×D412yes—
218D344—28no—
219C23×C43C22×D411yescharacteristic
220C22×D43—22no—
221C22×D43—22no—
222C22×D43—22no—
223C22×D43—22no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43—41no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—41no—
233C22×D43—41no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—22no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—41no—
252C2×C4×C43C22×D411yescharacteristic
253C2×C4×C43C22×Q811yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43H(5,𝔽2)11yescharacteristic
257C2×C4×C432−1+411yescharacteristic
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—22no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43#13512yes—
275C2×Dic43AGL(1,ℤ/8)12yes—
276D172—22no—
277D172—22no—
278D172—22no—
279D172—22no—
280D172—22no—
281D172—41no—
282D172—41no—
283D172C22⋊Q814yescomplemented
284C4×D43—84no—
285C4×D43—84no—
286C4×D43—84no—
287C4×D43—84no—
288Dic86—84no—
289C8×C43—22no—
290C8×C43—41no—
291C43⋊C8—22no—
292C43⋊C8—41no—
293#26380—431nomaximal, Sylow
294C22×C43#52511yescharacteristic
295C22×C43D4211yescharacteristic
296C22×C43D4×Q811yescharacteristic
297C22×C43C2×#13511yescharacteristic
298C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C43#57611yescharacteristic
300C22×C43#57711yescharacteristic
301C2×D43—24no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—42no—
305C2×D43—42no—
306C2×D43—42no—
307C2×D43—42no—
308C2×D43—42no—
309C2×D43—42no—
310C2×D43—42no—
311C2×D43—81no—
312C2×D43—81no—
313C2×D43—81no—
314C2×D43—81no—
315C4×C43—41no—
316C4×C43—41no—
317C4×C43—42no—
318C4×C43—81no—
319C4×C43—81no—
320C4×C43—81no—
321C4×C43C2×C22⋊Q811yescharacteristic
322C4×C43#53911yescharacteristic
323C4×C43#54711yescharacteristic
324C4×C43#55411yescharacteristic
325Dic43—22no—
326Dic43—22no—
327Dic43—42no—
328Dic43—41no—
329Dic43—41no—
330Dic43—81no—
331C2×#476—431no—
332C2×C4×D8—431no—
333#2452—431no—
334#2349—431no—
335#2092—431no—
336#2085D4311yescharacteristic, complemented
337#2065—431no—
338#2057—431no—
339#1941—431no—
340#1916—431no—
341#1879—431no—
342#1879—431no—
343#1864—431no—
344#1864—431no—
345#1862—431no—
346C2×C43#244911yescharacteristic
347C2×C43#310611yescharacteristic
348C2×C43#311811yescharacteristic
349C2×C43#326411yescharacteristic
350C2×C43#328211yescharacteristic
351C2×C43#338011yescharacteristic
352C2×C43#338411yescharacteristic
353D43—44no—
354D43—82no—
355C22×D8—431no—
356C2×C4⋊Q8—431no—
357C2×C22⋊Q8—861no—
358C2×C4×D4—431no—
359C2×C4×D4—431no—
360C2×C4×D4—431no—
361C2×C4×D4—431no—
362#476—864no—
363C4×D8—864no—
364C2×#107—431no—
365C2×#107C2×D4311yescharacteristic
366C2×#106—431no—
367C2×#106—431no—
368C2×#106C2×D4311yescharacteristic
369C2×C4⋊C8—431no—
370C2×#102—431no—
371C2×#102—431no—
372C2×#102—431no—
373C2×#102—431no—
374C2×#102—431no—
375C2×#102—431no—
376C2×C4×C8—431no—
377#396—861no—
378#393—861no—
379#390—431no—
380#390—431no—
381#389—431no—
382#389—431no—
383#389—431no—
384#389C2×D4311yescharacteristic
385#385—861no—
386#385—861no—
387#385—861no—
388#382C2×D4311yescharacteristic
389#380—861no—
390C4×C4⋊C4—431no—
391C2×#95—861no—
392#340—431no—
393#340—431no—
394#340C2×D4311yescharacteristic
395#340C2×D4311yescharacteristic
396#336—431no—
397#336—431no—
398#336—431no—
399#336C2×D4311yescharacteristic
400C43#2638011yescharacteristic, complemented, Sylow
401C22×Q8—861no—
402C22×D4—431no—
403C22×D4—431no—
404C23×C4—861no—
405C23×C4—861no—
406C23×C4—861no—
407C23×C4—861no—
408C2×D8—434no—
409C2×D8—862no—
410C2×D8—862no—
411C22×C8—431no—
412C22×C8—431no—
413C22×C8C22×D4311yescharacteristic
414C22×C8C22×D4311yescharacteristic
415C4⋊Q8—862no—
416C4⋊Q8—1721no—
417C22⋊Q8—1724no—
418C4×D4—862no—
419C4×D4—862no—
420C4×D4—862no—
421C4×D4—862no—
422C4×D4—862no—
423C4×D4—1721no—
424C4×D4—1721no—
425C4×D4—1721no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—431no—
428C2×C4⋊C4—431no—
429C2×C4⋊C4—431no—
430C2×C4⋊C4—431no—
431C2×C4⋊C4—861no—
432C2×C4⋊C4—861no—
433C2×C4⋊C4—861no—
434C2×C4⋊C4—861no—
435C2×C4⋊C4C22×D4311yescharacteristic
436C2×C4⋊C4C22×D4311yescharacteristic
437C2×C4⋊C4C22×D4311yescharacteristic
438C2×C4⋊C4C22×D4311yescharacteristic
439C2×C22⋊C4—861no—
440C2×C22⋊C4—861no—
441C2×C22⋊C4—861no—
442C2×C22⋊C4—861no—
443C2×C22⋊C4—861no—
444C2×C22⋊C4—861no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C42—431no—
448C2×C42—431no—
449C2×C42C22×D4311yescharacteristic
450#107—22no—
451#107—1721no—
452#106—22no—
453#106—862no—
454#106—862no—
455C4⋊C8—862no—
456#102—434no—
457#102—862no—
458#102—862no—
459#102—862no—
460#102—862no—
461#102—1721no—
462C4×C8—862no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—431no—
467#95—431no—
468#95—861no—
469#95—861no—
470#95—861no—
471C24—861no—
472C24—861no—
473C2×Q8—3441no—
474C2×Q8—1722no—
475C2×Q8—1722no—
476C2×D4—432no—
477C2×D4—432no—
478C2×D4—432no—
479C2×D4—432no—
480C2×D4—861no—
481C2×D4—861no—
482C2×D4—861no—
483C2×D4—861no—
484C2×D4—861no—
485C2×D4—861no—
486C2×D4—861no—
487C2×D4—861no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—431no—
493C22×C4—431no—
494C22×C4—431no—
495C22×C4—431no—
496C22×C4—864no—
497C22×C4—861no—
498C22×C4—861no—
499C22×C4—3441no—
500C22×C4—3441no—
501C22×C4—3441no—
502C22×C4—3441no—
503C22×C4—3441no—
504C22×C4—1722no—
505C22×C4—1722no—
506C22×C4—1722no—
507C22×C4—1722no—
508C22×C4—1722no—
509C22×C4—1722no—
510C22×C4D4×D4311yescharacteristic
511C22×C4D4×D4311yescharacteristic
512C22×C4C23×D4311yescharacteristic
513D8—868no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—432no—
520C2×C8—861no—
521C2×C8—861no—
522C2×C8—861no—
523C2×C8—861no—
524C2×C8—861no—
525C2×C8Q8×D4312yes—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—42no—
534C4⋊C4—432no—
535C4⋊C4—862no—
536C4⋊C4—862no—
537C4⋊C4—861no—
538C4⋊C4—861no—
539C4⋊C4—861no—
540C4⋊C4—3441no—
541C4⋊C4—3441no—
542C4⋊C4—1722no—
543C4⋊C4—1722no—
544C4⋊C4D4×D4312yes—
545C22⋊C4—864no—
546C22⋊C4—3441no—
547C22⋊C4—1722no—
548C22⋊C4—1722no—
549C22⋊C4—1722no—
550C22⋊C4—1722no—
551C22⋊C4—1722no—
552C22⋊C4—1722no—
553C42—21no—
554C42—21no—
555C42—861no—
556C42—861no—
557C42—861no—
558C42—861no—
559C23—862no—
560C23—862no—
561C23—862no—
562C23—862no—
563C23—1721no—
564C23—1721no—
565C23—1721no—
566C23—1721no—
567C23—1721no—
568C23—1721no—
569C23—1721no—
570C23—1721no—
571C23—1721no—
572C23—1721no—
573C23C2×D4×D4311yescharacteristic
574Q8—3444no—
575D4—434no—
576D4—862no—
577D4—862no—
578D4—862no—
579D4—862no—
580D4—862no—
581D4—1721no—
582D4—1721no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—432no—
602C2×C4—432no—
603C2×C4—861no—
604C2×C4—861no—
605C2×C4—861no—
606C2×C4—861no—
607C2×C4—862no—
608C2×C4—861no—
609C2×C4—861no—
610C2×C4—861no—
611C2×C4—861no—
612C2×C4—861no—
613C2×C4—861no—
614C2×C4—3444no—
615C2×C4—3444no—
616C2×C4—3444no—
617C2×C4—3444no—
618C2×C4—1721no—
619C2×C4—1721no—
620C2×C4—1721no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4C2×D4×D4311yescharacteristic
624C2×C4C2×Q8×D4311yescharacteristic
625C2×C4#41×D4311yescharacteristic
626C2×C4#41×D4311yescharacteristic
627C2×C4#1306337111yescharacteristic
628C2×C4#1306337311yescharacteristic
629C8—22no—
630C8—41no—
631C8—862no—
632C8—1721no—
633C22—864no—
634C22—3441no—
635C22—3441no—
636C22—3441no—
637C22—3441no—
638C22—1722no—
639C22—1722no—
640C22—1722no—
641C22—1722no—
642C22—1722no—
643C22—1722no—
644C22—1722no—
645C22—1722no—
646C22—1722no—
647C22#1458340711yescharacteristic
648C22#1458353311yescharacteristic
649C22#1458356411yescharacteristic
650C22#135×D4311yescharacteristic
651C22AGL(1,ℤ/8)×D4311yescharacteristic
652C22#1458370111yescharacteristic
653C22#1458370811yescharacteristic
654C4—41no—
655C4—42no—
656C4—41no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—862no—
661C4—862no—
662C4—3441no—
663C4—1721no—
664C4—1722no—
665C4—1721no—
666C4C22⋊Q8×D4311yescharacteristic
667C4#1458349711yescharacteristic
668C4#1458355111yescharacteristic
669C4#1458360511yescharacteristic
670C2—3442no—
671C2—1724no—
672C2#1913839911yescharacteristic
673C2#1914465011yescharacteristic
674C2#1914478211yescharacteristic
675C2#1914608111yescharacteristic
676C2#1914629711yescharacteristic
677C2#1914745911yescharacteristic
678C2#1914749111yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2085C2
socleC23×C43C22×D4
prime radicalC22×D344C22
hypercenter#2085D43
hypocenterC43#26380
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2085

Complements

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

NH ≅ G/Ntype
1. C2×#14583188C2semidirect
3. #19138443C2semidirect
4. #19137503C2semidirect
5. #19136119C2semidirect
6. #19135970C2semidirect
7. #19135921C2semidirect
8. #19135322C2semidirect
9. #19135151C2semidirect
12. #19134855C2semidirect
13. #19134853C2semidirect
14. #19134843C2semidirect
15. C43×#2085C2semidirect
70. C2×D344Q8semidirect
283. D172C22⋊Q8semidirect
336. #2085D43semidirect
400. C43#26380semidirect

Maximal subgroups

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

subgroupindex
C2×#145831882
C2×#145829812
#191384432
#191375032
#191361192
#191359702
#191359212
#191353222
#191351512
#191349442
#191349422
#191348552
#191348532
#191348432
C43×#20852
#2638043