Finite Group Databasebeta

← #25156603

Subgroups — #25156603

Order 11008.

908 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583113C211yesmaximal, characteristic
2C2×C4×#10621882C211yesmaximal, characteristic
3#19138376C211yesmaximal, characteristic, complemented
4#19137504C211yesmaximal, characteristic, complemented
5#19136144C211yesmaximal, characteristic, complemented
6#19136087C211yesmaximal, characteristic, complemented
7#19136011C211yesmaximal, characteristic, complemented
8#19135835C211yesmaximal, characteristic, complemented
9#19135270C211yesmaximal, characteristic, complemented
10#19135150C211yesmaximal, characteristic, complemented
11#19134940C211yesmaximal, characteristic
12#19134930C211yesmaximal, characteristic, complemented
13#19134928C211yesmaximal, characteristic, complemented
14#19134853C211yesmaximal, characteristic, complemented
15C43×#1864C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C22×#10621882C2211yescharacteristic
18C2×#13063312C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20C2×#13063287C2211yescharacteristic
21C2×#13063287C2211yescharacteristic
22#14583113—24no—
23C4×#10621882—24no—
24C2×#13063258C2211yescharacteristic
25C2×Dic43⋊C8C2211yescharacteristic
26C2×#13063250C2211yescharacteristic
27C2×#13063248C2211yescharacteristic
28C2×#13063248C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063246C2211yescharacteristic
31C2×#13063245C2211yescharacteristic
32C2×C4×C43⋊C8C2211yescharacteristic
33#14582827—21no—
34#14582827—21no—
35#14582805C2211yescharacteristic
36#14582804C2211yescharacteristic
37#14582802C2211yescharacteristic
38#14582798C2211yescharacteristic
39#14582795C2211yescharacteristic
40#14582773—21no—
41#14582772—21no—
42#14582770—21no—
43#14582764C2211yescharacteristic
44#14582762—21no—
45#14582756—21no—
46C4×Dic43⋊C4C2211yescharacteristic
47#95×D43—21no—
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582653C2211yescharacteristic
52#14582653C2211yescharacteristic
53#14582653C2211yescharacteristic
54C2×C4×D4×C43C2211yescharacteristic
55C2×#106×C43C2211yescharacteristic
56C2×#102×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C43×#389C2211yescharacteristic
59C43×#340C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86C2311yescharacteristic
62C22×D172C2311yescharacteristic
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C2×#10621873—22no—
67C2×#10621873—22no—
68C2×#10621873—22no—
69C2×#10621882—22no—
70C2×#10621882—22no—
71C2×#10621882D414yes—
72C22×C43⋊C8C2311yescharacteristic
73C22×C43⋊C8C2311yescharacteristic
74#13063312—22no—
75#13063312—41no—
76C4×D172—22no—
77C4×D172—41no—
78#13063287—22no—
79#13063287—22no—
80#13063287—22no—
81#13063287—41no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×Dic43⋊C4—21no—
86C2×Dic43⋊C4D411yescharacteristic
87C2×Dic43⋊C4D411yescharacteristic
88C2×C22⋊Dic43—21no—
89C2×C22⋊Dic43—21no—
90C2×C22⋊Dic43—21no—
91C2×C22⋊Dic43—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×(C2×D43)⋊C4—21no—
95C2×C4×Dic43C2311yescharacteristic
96C2×C4×Dic43C2311yescharacteristic
97#13063258—22no—
98Dic43⋊C8—22no—
99#13063250—41no—
100#13063248—22no—
101#13063248—22no—
102#13063247—22no—
103#13063246—22no—
104#13063245—22no—
105C4×C43⋊C8—22no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95D411yescharacteristic
115C43⋊#95D411yescharacteristic
116C22×D4×C43C2311yescharacteristic
117C23×C4×C43—21no—
118C22×C8×C43C2311yescharacteristic
119C22×C8×C43C2311yescharacteristic
120C4×D4×C43—22no—
121C4×D4×C43—41no—
122C2×C4⋊C4×C43—21no—
123C2×C4⋊C4×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C22⋊C4×C43—21no—
127C2×C42×C43C2311yescharacteristic
128#106×C43—41no—
129#102×C43—22no—
130#102×C43—22no—
131#95×C43—21no—
132C24×C43—21no—
133C23×D43—21no—
134C2×D4×C43—21no—
135C2×D4×C43—21no—
136C2×D4×C43—21no—
137C2×D4×C43—21no—
138C2×D4×C43—21no—
139C2×D4×C43C2×D412yes—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—42no—
144C22×C4×C43—42no—
145C22×C4×C43—81no—
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2×D411yescharacteristic
148C22×C4×C43C2411yescharacteristic
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—21no—
152C22×Dic43C2×D411yescharacteristic
153C22×Dic43C2×D411yescharacteristic
154C22×Dic43C2×D411yescharacteristic
155C22×Dic43C2×D411yescharacteristic
156C2×D172—21no—
157C2×D172—21no—
158C2×D172—21no—
159C2×D172C2×D412yes—
160C2×D172#4112yes—
161C2×D172#4112yes—
162C2×C4×D43—24no—
163C2×C4×D43—42no—
164C2×C4×D43—42no—
165C2×C4×D43—42no—
166C2×C4×D43—42no—
167C2×C4×D43—81no—
168C2×C4×D43—81no—
169C2×C4×D43—81no—
170C2×C4×D43—81no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86—21no—
174C2×Dic86—21no—
175C2×Dic86—21no—
176C2×Dic86—21no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43—21no—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43—81no—
189C4⋊C4×C43#4112yes—
190C22⋊C4×C43—42no—
191C42×C43—21no—
192C42×C43—21no—
193C22⋊Dic43—42no—
194C22⋊Dic43—42no—
195C22⋊Dic43—42no—
196C22⋊Dic43—81no—
197#10621882—28no—
198(C2×D43)⋊C4—24no—
199(C2×D43)⋊C4—42no—
200(C2×D43)⋊C4—42no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—81no—
206Dic43⋊C4—22no—
207Dic43⋊C4—41no—
208Dic43⋊C4—81no—
209C4×Dic43—21no—
210C4×Dic43—21no—
211C4×Dic43—21no—
212C4×Dic43—21no—
213C2×C43⋊C8—21no—
214C2×C43⋊C8—21no—
215C2×C43⋊C8—21no—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8C2×D412yes—
219#10621873—44no—
220C23×C43—22no—
221C23×C43—41no—
222C23×C43—41no—
223C23×C43—41no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43—41no—
227C23×C43C22×D411yescharacteristic
228C22×D43—22no—
229C22×D43—22no—
230C22×D43—22no—
231C22×D43—41no—
232C22×D43—41no—
233C22×D43—41no—
234C22×D43—41no—
235D4×C43—22no—
236D4×C43—22no—
237D4×C43—41no—
238D4×C43—41no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—22no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43—21no—
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×C43—41no—
253C2×C4×C43—84no—
254C2×C4×C43#13512yes—
255C2×C4×C43C22×D411yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×C4×C43H(5,𝔽2)11yescharacteristic
260C2×C4×C432−1+411yescharacteristic
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—21no—
270C2×Dic43—21no—
271C2×Dic43—21no—
272C2×Dic43—21no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—41no—
279C2×Dic43—41no—
280C2×Dic43—41no—
281C2×Dic43—41no—
282D172—22no—
283D172—22no—
284D172—22no—
285D172#12314yes—
286C4×D43—84no—
287C4×D43—84no—
288C4×D43—84no—
289Dic86—41no—
290Dic86—41no—
291Dic86—41no—
292Dic86—41no—
293C8×C43—41no—
294C8×C43—41no—
295C43⋊C8—22no—
296C43⋊C8—41no—
297#26160—431nomaximal, Sylow
298C22×C43—42no—
299C22×C43—42no—
300C22×C43—42no—
301C22×C43—81no—
302C22×C43—81no—
303C22×C43—81no—
304C22×C43—81no—
305C22×C43#52511yescharacteristic
306C22×C43D4211yescharacteristic
307C22×C43#54811yescharacteristic
308C22×C43C2×#13511yescharacteristic
309C22×C43#57511yescharacteristic
310C22×C43#57611yescharacteristic
311C22×C43#57711yescharacteristic
312C2×D43—24no—
313C2×D43—42no—
314C2×D43—42no—
315C2×D43—42no—
316C2×D43—42no—
317C2×D43—42no—
318C2×D43—42no—
319C4×C43—22no—
320C4×C43—42no—
321C4×C43—41no—
322C4×C43—81no—
323C4×C43—81no—
324C4×C43C2×#12311yescharacteristic
325C4×C43#53911yescharacteristic
326C4×C43#54711yescharacteristic
327C4×C43#55311yescharacteristic
328Dic43—41no—
329Dic43—41no—
330Dic43—41no—
331Dic43—41no—
332Dic43—81no—
333Dic43—81no—
334Dic43—81no—
335C2×#463—431no—
336C2×C4×D8—431no—
337#2447—431no—
338#2349—431no—
339#2097—431no—
340#2086—431no—
341#2073—431no—
342#2042—431no—
343#1933—431no—
344#1916—431no—
345#1879—431no—
346#1877—431no—
347#1877—431no—
348#1864—431no—
349#1864D4311yescharacteristic, complemented
350C2×C43—82no—
351C2×C43#244311yescharacteristic
352C2×C43#310211yescharacteristic
353C2×C43#311811yescharacteristic
354C2×C43#326711yescharacteristic
355C2×C43#328011yescharacteristic
356C2×C43#336011yescharacteristic
357C2×C43#337011yescharacteristic
358D43—44no—
359C22×SD16—431no—
360C22×D8—431no—
361C2×#124—431no—
362C2×C4×D4—431no—
363C2×C4×D4—431no—
364C2×C4×D4—431no—
365C2×C4×D4C2×D4311yescharacteristic
366#463—864no—
367C4×D8—864no—
368C2×#107—431no—
369C2×#106C2×D4311yescharacteristic
370C2×C4⋊C8—431no—
371C2×#103—431no—
372C2×#103—431no—
373C2×#103—431no—
374C2×#102—431no—
375C2×#102—431no—
376C2×#102—431no—
377C2×#102C2×D4311yescharacteristic
378C2×#102C2×D4311yescharacteristic
379C2×C4×C8—431no—
380#397—861no—
381#397—861no—
382#391—431no—
383#391—431no—
384#390—431no—
385#389—431no—
386#389—431no—
387#389C2×D4311yescharacteristic
388#385—861no—
389#385—861no—
390#385—861no—
391#382—431no—
392#381—861no—
393#380—861no—
394C4×C4⋊C4—431no—
395C2×#95—861no—
396#340—431no—
397#340—431no—
398#340—431no—
399#340C2×D4311yescharacteristic
400#336—431no—
401#336—431no—
402#336—431no—
403#336C2×D4311yescharacteristic
404C43#2616011yescharacteristic, complemented, Sylow
405C22×Q8—431no—
406C22×D4—431no—
407C22×D4C22×D4311yescharacteristic
408C23×C4—21no—
409C23×C4—861no—
410C23×C4—861no—
411C23×C4—861no—
412C2×SD16—862no—
413C2×SD16—862no—
414C2×SD16—862no—
415C2×D8—434no—
416C2×D8—862no—
417C2×D8—862no—
418C22×C8—431no—
419C22×C8—431no—
420C22×C8C22×D4311yescharacteristic
421C22×C8C22×D4311yescharacteristic
422#124—862no—
423#124—1721no—
424C4×D4—22no—
425C4×D4—41no—
426C4×D4—862no—
427C4×D4—862no—
428C4×D4—862no—
429C4×D4—862no—
430C4×D4—1721no—
431C4×D4—1721no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—431no—
434C2×C4⋊C4—431no—
435C2×C4⋊C4—431no—
436C2×C4⋊C4—431no—
437C2×C4⋊C4—861no—
438C2×C4⋊C4—861no—
439C2×C4⋊C4C22×D4311yescharacteristic
440C2×C4⋊C4C22×D4311yescharacteristic
441C2×C4⋊C4C22×D4311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—861no—
444C2×C22⋊C4—861no—
445C2×C22⋊C4—861no—
446C2×C22⋊C4—861no—
447C2×C22⋊C4—861no—
448C2×C22⋊C4—861no—
449C2×C22⋊C4—861no—
450C2×C42—431no—
451C2×C42—431no—
452C2×C42C22×D4311yescharacteristic
453#107—862no—
454#106—41no—
455C4⋊C8—862no—
456#103—862no—
457#103—862no—
458#103—1721no—
459#102—22no—
460#102—22no—
461#102—862no—
462#102—862no—
463#102—862no—
464C4×C8—862no—
465#95—21no—
466#95—431no—
467#95—431no—
468#95—861no—
469#95—861no—
470#95—861no—
471#95—861no—
472#95—861no—
473#95—861no—
474#95—861no—
475#95—861no—
476C24—21no—
477C24—861no—
478C2×Q8—861no—
479C2×Q8—861no—
480C2×Q8—861no—
481C2×Q8—861no—
482C2×Q8—861no—
483C2×Q8—861no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—432no—
490C2×D4—432no—
491C2×D4—432no—
492C2×D4—861no—
493C2×D4—861no—
494C2×D4—861no—
495C2×D4D4×D4312yes—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—81no—
502C22×C4—431no—
503C22×C4—431no—
504C22×C4—431no—
505C22×C4—431no—
506C22×C4—864no—
507C22×C4—861no—
508C22×C4—861no—
509C22×C4—861no—
510C22×C4—3441no—
511C22×C4—3441no—
512C22×C4—3441no—
513C22×C4—3441no—
514C22×C4—1722no—
515C22×C4—1722no—
516C22×C4—1722no—
517C22×C4—1722no—
518C22×C4D4×D4311yescharacteristic
519C22×C4D4×D4311yescharacteristic
520C22×C4C23×D4311yescharacteristic
521SD16—1724no—
522D8—868no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—432no—
530C2×C8—861no—
531C2×C8—861no—
532C2×C8—861no—
533C2×C8—861no—
534C2×C8—861no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—81no—
541C4⋊C4—862no—
542C4⋊C4—861no—
543C4⋊C4—861no—
544C4⋊C4—861no—
545C4⋊C4—861no—
546C4⋊C4—3441no—
547C4⋊C4—3441no—
548C4⋊C4—1721no—
549C4⋊C4#1062189812yes—
550C22⋊C4—42no—
551C22⋊C4—864no—
552C22⋊C4—3441no—
553C22⋊C4—1722no—
554C22⋊C4—1722no—
555C22⋊C4—1722no—
556C22⋊C4—1722no—
557C22⋊C4—1722no—
558C42—21no—
559C42—21no—
560C42—861no—
561C42—861no—
562C42—861no—
563C42—861no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—862no—
572C23—862no—
573C23—862no—
574C23—1721no—
575C23—1721no—
576C23—1721no—
577C23—1721no—
578C23C2×D4×D4311yescharacteristic
579Q8—1721no—
580Q8—1721no—
581Q8—1721no—
582Q8—1721no—
583D4—22no—
584D4—22no—
585D4—41no—
586D4—41no—
587D4—434no—
588D4—862no—
589D4—862no—
590D4—862no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—861no—
607C2×C4—861no—
608C2×C4—861no—
609C2×C4—861no—
610C2×C4—861no—
611C2×C4—861no—
612C2×C4—861no—
613C2×C4—861no—
614C2×C4—861no—
615C2×C4—861no—
616C2×C4—861no—
617C2×C4—861no—
618C2×C4—3444no—
619C2×C4—3444no—
620C2×C4—3444no—
621C2×C4—1721no—
622C2×C4—1721no—
623C2×C4—1721no—
624C2×C4—1721no—
625C2×C4—1721no—
626C2×C4—1721no—
627C2×C4—1721no—
628C2×C4—1721no—
629C2×C4—1721no—
630C2×C4#1306334612yes—
631C2×C4C2×D4×D4311yescharacteristic
632C2×C4#41×D4311yescharacteristic
633C2×C4#41×D4311yescharacteristic
634C2×C4C2×#1062189811yescharacteristic
635C2×C4#1306337011yescharacteristic
636C2×C4#1306337311yescharacteristic
637C8—41no—
638C8—41no—
639C8—862no—
640C8—1721no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—864no—
649C22—1722no—
650C22—1722no—
651C22—1722no—
652C22—1722no—
653C22—1722no—
654C22—1722no—
655C22#1458340811yescharacteristic
656C22#1458353111yescharacteristic
657C22#1458355911yescharacteristic
658C22C2×#1306334611yescharacteristic
659C22#1458369611yescharacteristic
660C22#1458370311yescharacteristic
661C22#1458370911yescharacteristic
662C4—22no—
663C4—42no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—3441no—
668C4—3441no—
669C4—3441no—
670C4—1721no—
671C4—1721no—
672C4—1721no—
673C4—1721no—
674C4#123×D4311yescharacteristic
675C4#1458349511yescharacteristic
676C4#1458355011yescharacteristic
677C4#1458359911yescharacteristic
678C2—82no—
679C2—1724no—
680C2#1913832911yescharacteristic
681C2#1914461111yescharacteristic
682C2#1914477711yescharacteristic
683C2#1914611311yescharacteristic
684C2#1914627311yescharacteristic
685C2#1914724311yescharacteristic
686C2#1914734611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1864C2
socleC23×C43C22×D4
prime radicalC22×#10621882C22
hypercenter#1864D43
hypocenterC43#26160
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
3. #19138376C2semidirect
4. #19137504C2semidirect
5. #19136144C2semidirect
6. #19136087C2semidirect
7. #19136011C2semidirect
8. #19135835C2semidirect
9. #19135270C2semidirect
10. #19135150C2semidirect
12. #19134930C2semidirect
13. #19134928C2semidirect
14. #19134853C2semidirect
15. C43×#1864C2semidirect
349. #1864D43semidirect
404. C43#26160semidirect

Maximal subgroups

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

subgroupindex
C2×#145831132
C2×C4×#106218822
#191383762
#191375042
#191361442
#191360872
#191360112
#191358352
#191352702
#191351502
#191349402
#191349302
#191349282
#191348532
C43×#18642
#2616043