Finite Group Databasebeta

← #25164636

Subgroups — #25164636

Order 11008.

832 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19138932C211yesmaximal, characteristic, complemented
2#19137615C211yesmaximal, characteristic
3#19135911C211yesmaximal, characteristic, complemented
4#19135739C211yesmaximal, characteristic, complemented
5#19135598C211yesmaximal, characteristic
6#19135551C211yesmaximal, characteristic, complemented
7#19135272C211yesmaximal, characteristic, complemented
8#19135235C211yesmaximal, characteristic, complemented
9#19135204C211yesmaximal, characteristic, complemented
10#19135168C211yesmaximal, characteristic, complemented
11#19134927C211yesmaximal, characteristic, complemented
12#19134900C211yesmaximal, characteristic, complemented
13#19134683C211yesmaximal, characteristic, complemented
14#102×Dic43C211yesmaximal, characteristic, complemented
15C43×#2033C211yesmaximal, characteristic
16C22×#10621883C2211yescharacteristic
17C2×#13063316C2211yescharacteristic
18C2×#13063304C2211yescharacteristic
19C2×D4×Dic43—21no—
20C2×D4×Dic43C2211yescharacteristic
21C22×C4⋊Dic43C2211yescharacteristic
22C2×Dic43⋊C8C2211yescharacteristic
23C2×#13063250C2211yescharacteristic
24C2×#13063250C2211yescharacteristic
25C2×#13063248C2211yescharacteristic
26C2×#13063246C2211yescharacteristic
27C2×C22⋊(C43⋊C8)C2211yescharacteristic
28C2×C8×Dic43C2211yescharacteristic
29#14582841—21no—
30#14582827—21no—
31#14582804C2211yescharacteristic
32#14582804C2211yescharacteristic
33#14582802C2211yescharacteristic
34#14582798C2211yescharacteristic
35#14582792—21no—
36#14582792—21no—
37#14582779—21no—
38#14582774—21no—
39#14582763—21no—
40C4⋊C4×Dic43C2211yescharacteristic
41C4⋊C4×Dic43C2211yescharacteristic
42C4×C22⋊Dic43C2211yescharacteristic
43#14582662C2211yescharacteristic
44#14582662C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582661C2211yescharacteristic
48#14582661C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582653C2211yescharacteristic
51#14582653C2211yescharacteristic
52C2×#121×C43C2211yescharacteristic
53C2×#107×C43C2211yescharacteristic
54C2×#102×C43C2211yescharacteristic
55C2×#102×C43C2211yescharacteristic
56C2×C22⋊C8×C43C2211yescharacteristic
57C43×#380C2211yescharacteristic
58C43×#340C2211yescharacteristic
59C22×Dic86C2311yescharacteristic
60C23×Dic43—21no—
61C23×Dic43—21no—
62C2×#10621883—22no—
63C2×#10621883—41no—
64C2×#10621883—41no—
65C22×C43⋊C8C2311yescharacteristic
66C22×C43⋊C8C2311yescharacteristic
67#13063316—22no—
68#13063316—41no—
69#13063304—22no—
70#13063304—41no—
71D4×Dic43—22no—
72D4×Dic43—41no—
73D4×Dic43—44no—
74C2×C4⋊Dic43—22no—
75C2×C4⋊Dic43—21no—
76C2×C4⋊Dic43—21no—
77C2×C4⋊Dic43—21no—
78C2×C4⋊Dic43—41no—
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×C22⋊Dic43—21no—
85C2×C22⋊Dic43—21no—
86C2×C22⋊Dic43—21no—
87C2×C22⋊Dic43—21no—
88C2×C22⋊Dic43D411yescharacteristic
89C2×C22⋊Dic43D411yescharacteristic
90C2×C4×Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43C2311yescharacteristic
93C2×C4×Dic43C2311yescharacteristic
94Dic43⋊C8—41no—
95#13063250—22no—
96#13063250—41no—
97#13063248—22no—
98#13063246—22no—
99C22⋊(C43⋊C8)—41no—
100C8×Dic43—41no—
101C43⋊#95—21no—
102C43⋊#95—21no—
103C43⋊#95—21no—
104C43⋊#95—21no—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95D411yescharacteristic
109C43⋊#95D411yescharacteristic
110C22×D4×C43—21no—
111C22×D4×C43C2311yescharacteristic
112C23×C4×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115#121×C43—22no—
116#121×C43—22no—
117C2×C4⋊C4×C43C2311yescharacteristic
118C2×C4⋊C4×C43C2311yescharacteristic
119C2×C4⋊C4×C43C2311yescharacteristic
120C2×C22⋊C4×C43—21no—
121C2×C22⋊C4×C43—21no—
122#107×C43—22no—
123#102×C43—22no—
124#102×C43—41no—
125C22⋊C8×C43—41no—
126#95×C43—21no—
127C24×C43—21no—
128C24×C43C2×D411yescharacteristic
129C2×D4×C43—24no—
130C2×D4×C43—21no—
131C2×D4×C43—21no—
132C2×D4×C43—21no—
133C2×D4×C43—21no—
134C2×D4×C43—21no—
135C2×D4×C43—42no—
136C2×D4×C43—42no—
137C2×D4×C43#4112yes—
138C22×C4×C43—21no—
139C22×C4×C43—21no—
140C22×C4×C43—21no—
141C22×C4×C43—21no—
142C22×C4×C43—21no—
143C22×C4×C43—21no—
144C22×C4×C43—21no—
145C22×C4×C43—21no—
146C22×C4×C43—21no—
147C22×C4×C43C2×D411yescharacteristic
148C22×C4×C43C2411yescharacteristic
149C22×Dic43—21no—
150C22×Dic43—21no—
151C22×Dic43—21no—
152C22×Dic43—21no—
153C22×Dic43—42no—
154C22×Dic43—42no—
155C22×Dic43—42no—
156C22×Dic43—42no—
157C22×Dic43—81no—
158C22×Dic43—81no—
159C22×Dic43C2×D411yescharacteristic
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
162C22×Dic43C2×D411yescharacteristic
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43#4112yes—
181C22⋊C4×C43—24no—
182C22⋊C4×C43—42no—
183C22⋊Dic43—22no—
184C22⋊Dic43—22no—
185C22⋊Dic43—42no—
186C22⋊Dic43—42no—
187C22⋊Dic43—42no—
188C22⋊Dic43—81no—
189#10621883—44no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—42no—
195C4⋊Dic43—42no—
196C4⋊Dic43—42no—
197C4⋊Dic43#4112yes—
198C4⋊Dic43#4112yes—
199Dic43⋊C4—42no—
200Dic43⋊C4—42no—
201C4×Dic43—21no—
202C4×Dic43—21no—
203C4×Dic43—21no—
204C4×Dic43—21no—
205C4×Dic43—42no—
206C4×Dic43—42no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C23×C43—21no—
214C23×C43—21no—
215C23×C43—21no—
216C23×C43—21no—
217C23×C43—21no—
218C23×C43—21no—
219C23×C43—22no—
220C23×C43—41no—
221C23×C43—41no—
222C23×C43—41no—
223C23×C43—41no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43#13512yes—
227C23×C43C22×D411yescharacteristic
228D4×C43—22no—
229D4×C43—22no—
230D4×C43—48no—
231D4×C43—41no—
232D4×C43—41no—
233C2×C4×C43—21no—
234C2×C4×C43—22no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—41no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
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×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43C2×#4111yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43H(5,𝔽2)11yescharacteristic
254C2×C4×C43H(5,𝔽2)11yescharacteristic
255C2×Dic43—21no—
256C2×Dic43—21no—
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—22no—
261C2×Dic43—22no—
262C2×Dic43—21no—
263C2×Dic43—21no—
264C2×Dic43—21no—
265C2×Dic43—21no—
266C2×Dic43—21no—
267C2×Dic43—21no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272C2×Dic43—41no—
273C2×Dic43—41no—
274C2×Dic43—41no—
275C2×Dic43—41no—
276C2×Dic43—41no—
277C2×Dic43—41no—
278C2×Dic43—84no—
279C2×Dic43—84no—
280C2×Dic43AGL(1,ℤ/8)12yes—
281Dic86—41no—
282Dic86—41no—
283Dic86—41no—
284Dic86—41no—
285C8×C43—41no—
286C8×C43—41no—
287C43⋊C8—41no—
288C43⋊C8—41no—
289#26721—431nomaximal, Sylow
290C22×C43—22no—
291C22×C43—22no—
292C22×C43—22no—
293C22×C43—41no—
294C22×C43—41no—
295C22×C43—41no—
296C22×C43—41no—
297C22×C43—42no—
298C22×C43—42no—
299C22×C43—42no—
300C22×C43—81no—
301C22×C43—81no—
302C22×C43—81no—
303C22×C43—81no—
304C22×C43#52511yescharacteristic
305C22×C43#54611yescharacteristic
306C22×C43#54611yescharacteristic
307C22×C43C2×#13511yescharacteristic
308C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C43#57711yescharacteristic
310C22×C43#57811yescharacteristic
311C4×C43—42no—
312C4×C43—41no—
313C4×C43—81no—
314C4×C43—81no—
315C4×C43#52911yescharacteristic
316C4×C43#53811yescharacteristic
317C4×C43#55511yescharacteristic
318C4×C43#55911yescharacteristic
319Dic43—22no—
320Dic43—41no—
321Dic43—42no—
322Dic43—41no—
323Dic43—41no—
324Dic43—42no—
325Dic43—81no—
326Dic43—81no—
327#2489—431no—
328#2361—431no—
329#2055—431no—
330#2033D4311yescharacteristic
331#2025—431no—
332#2000—431no—
333#1989—431no—
334#1933—431no—
335#1928—431no—
336#1923—431no—
337#1918—431no—
338#1877—431no—
339#1872—431no—
340#1824—431no—
341C4×#102—431no—
342C2×C43—42no—
343C2×C43—82no—
344C2×C43#245911yescharacteristic
345C2×C43#310511yescharacteristic
346C2×C43#312011yescharacteristic
347C2×C43#330011yescharacteristic
348C2×C43#330111yescharacteristic
349C2×C43#331011yescharacteristic
350C2×C43#333011yescharacteristic
351C22×SD16—431no—
352C2×#125—431no—
353C2×C22⋊Q8—431no—
354C2×#121C2×D4311yescharacteristic
355C2×C4×D4—431no—
356C2×C4×D4—861no—
357C22×C4⋊C4—431no—
358C2×#107C2×D4311yescharacteristic
359C2×C4⋊C8—431no—
360C2×#103—431no—
361C2×#103—431no—
362C2×#103—431no—
363C2×#102—431no—
364C2×#102C2×D4311yescharacteristic
365C2×#102C2×D4311yescharacteristic
366C2×C22⋊C8—431no—
367C2×C22⋊C8C2×D4311yescharacteristic
368C2×C4×C8—431no—
369#400—861no—
370#397—861no—
371#391—431no—
372#391—431no—
373#390—431no—
374#389—431no—
375#388—861no—
376#388—861no—
377#386—861no—
378#385—861no—
379#382—861no—
380#380C2×D4311yescharacteristic
381C4×C4⋊C4—431no—
382C4×C4⋊C4—431no—
383C4×C22⋊C4—431no—
384#340—431no—
385#340—431no—
386#340—431no—
387#340—431no—
388#340—431no—
389#340—431no—
390#340—431no—
391#340C2×D4311yescharacteristic
392#336—431no—
393#336—431no—
394C43#2672111yescharacteristic, complemented, Sylow
395C22×Q8—431no—
396C22×D4—21no—
397C22×D4C22×D4311yescharacteristic
398C23×C4—861no—
399C23×C4—861no—
400C23×C4C22×D4311yescharacteristic
401C2×SD16—862no—
402C2×SD16—1721no—
403C2×SD16—1721no—
404C22×C8—431no—
405C22×C8—431no—
406C22×C8C22×D4311yescharacteristic
407C22×C8C22×D4311yescharacteristic
408#125—862no—
409#125—1721no—
410C22⋊Q8—862no—
411C22⋊Q8—1721no—
412#121—22no—
413#121—22no—
414C4×D4—862no—
415C4×D4—1721no—
416C4×D4—1724no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—431no—
419C2×C4⋊C4—431no—
420C2×C4⋊C4—862no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C4⋊C4—861no—
426C2×C4⋊C4—1721no—
427C2×C4⋊C4C22×D4311yescharacteristic
428C2×C4⋊C4C22×D4311yescharacteristic
429C2×C4⋊C4C22×D4311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—431no—
433C2×C22⋊C4—431no—
434C2×C22⋊C4—861no—
435C2×C22⋊C4—861no—
436C2×C22⋊C4—861no—
437C2×C22⋊C4—861no—
438C2×C42—431no—
439C2×C42—431no—
440C2×C42—861no—
441C2×C42—861no—
442#107—22no—
443C4⋊C8—1721no—
444#103—862no—
445#103—862no—
446#103—1721no—
447#102—22no—
448#102—41no—
449#102—862no—
450C22⋊C8—41no—
451C22⋊C8—1721no—
452C4×C8—1721no—
453#95—21no—
454#95—431no—
455#95—431no—
456#95—861no—
457#95—861no—
458#95—861no—
459#95—861no—
460#95—861no—
461#95—861no—
462#95—861no—
463C24—21no—
464C24D4×D4311yescharacteristic
465C2×Q8—861no—
466C2×Q8—861no—
467C2×Q8—861no—
468C2×Q8—861no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4#1062190012yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—431no—
490C22×C4—431no—
491C22×C4—431no—
492C22×C4—431no—
493C22×C4—861no—
494C22×C4—861no—
495C22×C4—861no—
496C22×C4—861no—
497C22×C4—3441no—
498C22×C4—3441no—
499C22×C4—1722no—
500C22×C4—1722no—
501C22×C4—1722no—
502C22×C4—1722no—
503C22×C4D4×D4311yescharacteristic
504C22×C4C23×D4311yescharacteristic
505SD16—1724no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—861no—
513C2×C8—861no—
514C2×C8—861no—
515C2×C8—861no—
516C2×C8—861no—
517C2×C8—861no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—432no—
524C4⋊C4—432no—
525C4⋊C4—861no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—1722no—
530C4⋊C4—1722no—
531C4⋊C4—1722no—
532C4⋊C4—1722no—
533C4⋊C4—1722no—
534C4⋊C4#1062190012yes—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—862no—
538C22⋊C4—862no—
539C22⋊C4—3441no—
540C22⋊C4—1722no—
541C22⋊C4—1722no—
542C22⋊C4—1722no—
543C42—861no—
544C42—861no—
545C42—861no—
546C42—861no—
547C42—1722no—
548C42—1722no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1306334412yes—
563C23C2×D4×D4311yescharacteristic
564Q8—1721no—
565Q8—1721no—
566Q8—1721no—
567Q8—1721no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—41no—
572D4—41no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—432no—
590C2×C4—861no—
591C2×C4—861no—
592C2×C4—861no—
593C2×C4—861no—
594C2×C4—861no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—861no—
598C2×C4—861no—
599C2×C4—861no—
600C2×C4—862no—
601C2×C4—861no—
602C2×C4—862no—
603C2×C4—3444no—
604C2×C4—3444no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4—1721no—
610C2×C4—1721no—
611C2×C4—1721no—
612C2×C4—1721no—
613C2×C4—1721no—
614C2×C4—1721no—
615C2×C4#41×D4311yescharacteristic
616C2×C4#41×D4311yescharacteristic
617C2×C4C2×#1062190011yescharacteristic
618C2×C4C2×#1062190011yescharacteristic
619C2×C4#1306337011yescharacteristic
620C2×C4#1306337111yescharacteristic
621C8—41no—
622C8—41no—
623C8—1721no—
624C8—1721no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22#1458340511yescharacteristic
640C22#1458353811yescharacteristic
641C22#1458354111yescharacteristic
642C22C2×#1306334411yescharacteristic
643C22AGL(1,ℤ/8)×D4311yescharacteristic
644C22#1458370611yescharacteristic
645C22#1458371511yescharacteristic
646C4—42no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—862no—
651C4—3441no—
652C4—3441no—
653C4—1721no—
654C4—1722no—
655C4—1721no—
656C4—1722no—
657C4—1721no—
658C4#1458342611yescharacteristic
659C4#1458348611yescharacteristic
660C4#1458361311yescharacteristic
661C4#1458363111yescharacteristic
662C2—42no—
663C2—82no—
664C2#1913852511yescharacteristic
665C2#1914463711yescharacteristic
666C2#1914480211yescharacteristic
667C2#1914648011yescharacteristic
668C2#1914650311yescharacteristic
669C2#1914661911yescharacteristic
670C2#1914689511yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2033C2
socleC23×C43C22×D4
prime radicalC2×#121×C43C22
hypercenter#2033D43
hypocenterC43#26721
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. #19138932C2semidirect
3. #19135911C2semidirect
4. #19135739C2semidirect
6. #19135551C2semidirect
7. #19135272C2semidirect
8. #19135235C2semidirect
9. #19135204C2semidirect
10. #19135168C2semidirect
11. #19134927C2semidirect
12. #19134900C2semidirect
13. #19134683C2semidirect
14. #102×Dic43C2semidirect
394. C43#26721semidirect

Maximal subgroups

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

subgroupindex
#191389322
#191376152
#191359112
#191357392
#191355982
#191355512
#191352722
#191352352
#191352042
#191351682
#191349272
#191349002
#191346832
#102×Dic432
C43×#20332
#2672143