Finite Group Databasebeta

← #25146455

Subgroups — #25146455

Order 11008.

894 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583248C211yesmaximal, characteristic, complemented
2C2×#14583102C211yesmaximal, characteristic
3#19139546C211yesmaximal, characteristic
4#19137652C211yesmaximal, characteristic, complemented
5#19136007C211yesmaximal, characteristic, complemented
6#19136003C211yesmaximal, characteristic, complemented
7#19135911C211yesmaximal, characteristic
8#19135545C211yesmaximal, characteristic, complemented
9#19135310C211yesmaximal, characteristic, complemented
10#19134953C211yesmaximal, characteristic
11#19134952C211yesmaximal, characteristic
12#19134601C211yesmaximal, characteristic, complemented
13#19134598C211yesmaximal, characteristic, complemented
14#102×Dic43C211yesmaximal, characteristic
15C43×#2055C211yesmaximal, characteristic
16C22×#10621883C2211yescharacteristic
17C2×#13063324C2211yescharacteristic
18C2×#13063323C2211yescharacteristic
19C2×#13063302—21no—
20C2×#13063302—21no—
21C2×Q8×Dic43C2211yescharacteristic
22C2×C4×Dic86C2211yescharacteristic
23C2×D4×Dic43C2211yescharacteristic
24#14583248—24no—
25#14583102—24no—
26C2×Dic43⋊C8C2211yescharacteristic
27C2×Dic43⋊C8C2211yescharacteristic
28C2×#13063250C2211yescharacteristic
29C2×#13063250C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063249C2211yescharacteristic
32C2×#13063248C2211yescharacteristic
33C2×#13063246C2211yescharacteristic
34C2×C8×Dic43C2211yescharacteristic
35#14582835—21no—
36#14582835—21no—
37#14582804C2211yescharacteristic
38#14582802C2211yescharacteristic
39#14582796—21no—
40#14582786—21no—
41#14582781—21no—
42#14582769C2211yescharacteristic
43#14582768C2211yescharacteristic
44C4⋊C4×Dic43C2211yescharacteristic
45C4×Dic43⋊C4C2211yescharacteristic
46#14582750—21no—
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582661C2211yescharacteristic
50#14582661C2211yescharacteristic
51#14582661C2211yescharacteristic
52#14582654C2211yescharacteristic
53#14582653C2211yescharacteristic
54C22×SD16×C43C2211yescharacteristic
55C2×C4⋊C8×C43C2211yescharacteristic
56C2×#103×C43C2211yescharacteristic
57C2×#102×C43C2211yescharacteristic
58C43×#391C2211yescharacteristic
59C43×#390C2211yescharacteristic
60C43×#340C2211yescharacteristic
61C22×Dic86—21no—
62C22×Dic86C2311yescharacteristic
63C23×Dic43—21no—
64C2×#10621883—22no—
65C2×#10621883—22no—
66C2×#10621883—41no—
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69#13063324—22no—
70#13063324—22no—
71#13063323—22no—
72#13063323D414yes—
73#13063302—44no—
74#13063302—44no—
75Q8×Dic43—22no—
76Q8×Dic43—41no—
77C4×Dic86—22no—
78C4×Dic86—41no—
79D4×Dic43—22no—
80D4×Dic43—22no—
81C2×C4⋊Dic43—21no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43—21no—
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×C4⋊Dic43C2311yescharacteristic
87C2×Dic43⋊C4—21no—
88C2×Dic43⋊C4—21no—
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4D411yescharacteristic
94C2×Dic43⋊C4D411yescharacteristic
95C2×C22⋊Dic43—21no—
96C2×C22⋊Dic43—21no—
97C2×C22⋊Dic43—21no—
98C2×C4×Dic43—21no—
99C2×C4×Dic43—21no—
100C2×C4×Dic43—21no—
101C2×C4×Dic43C2311yescharacteristic
102C2×C4×Dic43C2311yescharacteristic
103Dic43⋊C8—22no—
104Dic43⋊C8—41no—
105#13063250—22no—
106#13063250—22no—
107#13063249—22no—
108#13063249—22no—
109#13063248—22no—
110#13063246—22no—
111C8×Dic43—22no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95D411yescharacteristic
116C43⋊#95D411yescharacteristic
117C22×Q8×C43C2311yescharacteristic
118C22×D4×C43C2311yescharacteristic
119C2×SD16×C43—22no—
120C2×SD16×C43—22no—
121C2×SD16×C43—41no—
122C22×C8×C43C2311yescharacteristic
123C22×C8×C43C2311yescharacteristic
124C2×C4⋊C4×C43C2311yescharacteristic
125C2×C4⋊C4×C43C2311yescharacteristic
126C2×C22⋊C4×C43—21no—
127C2×C22⋊C4×C43—21no—
128C2×C42×C43C2311yescharacteristic
129C4⋊C8×C43—41no—
130#103×C43—22no—
131#102×C43—22no—
132#95×C43—21no—
133#95×C43—21no—
134C24×C43—21no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—21no—
138C2×Q8×C43—21no—
139C2×Q8×C43—21no—
140C2×Q8×C43C2×D412yes—
141C2×D4×C43—21no—
142C2×D4×C43—21no—
143C2×D4×C43—21no—
144C2×D4×C43—21no—
145C2×D4×C43—21no—
146C2×D4×C43#4112yes—
147C22×C4×C43—21no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43C2×D411yescharacteristic
151C22×C4×C43C2×D411yescharacteristic
152C22×C4×C43C2411yescharacteristic
153C22×Dic43—21no—
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C22×Dic43—42no—
158C22×Dic43—42no—
159C22×Dic43—42no—
160C22×Dic43C2×D411yescharacteristic
161C22×Dic43C2×D411yescharacteristic
162C22×Dic43C2×D411yescharacteristic
163C22×Dic43C2×D411yescharacteristic
164C2×Dic86—21no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—21no—
168C2×Dic86—21no—
169C2×Dic86—42no—
170C2×Dic86—42no—
171C2×Dic86—42no—
172C2×Dic86C2×D412yes—
173SD16×C43—44no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C22⋊C4×C43—42no—
185C22⋊C4×C43—42no—
186C42×C43—21no—
187C42×C43—21no—
188C22⋊Dic43—42no—
189C22⋊Dic43—42no—
190C22⋊Dic43—42no—
191#10621883—44no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—42no—
198C4⋊Dic43—42no—
199C4⋊Dic43—42no—
200C4⋊Dic43#4112yes—
201Dic43⋊C4—24no—
202Dic43⋊C4—24no—
203Dic43⋊C4—22no—
204Dic43⋊C4—42no—
205Dic43⋊C4—42no—
206Dic43⋊C4—42no—
207Dic43⋊C4—42no—
208Dic43⋊C4SD1614yescomplemented
209C4×Dic43—21no—
210C4×Dic43—21no—
211C4×Dic43—42no—
212C4×Dic43—42no—
213C4×Dic43—42no—
214C4×Dic43C2×D412yes—
215C4×Dic43#4112yes—
216C2×C43⋊C8—21no—
217C2×C43⋊C8—21no—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C23×C43—22no—
223C23×C43—41no—
224C23×C43—41no—
225C23×C43—41no—
226C23×C43—41no—
227C23×C43—41no—
228C23×C43—41no—
229C23×C43C22×D411yescharacteristic
230Q8×C43—22no—
231Q8×C43—22no—
232Q8×C43—41no—
233Q8×C43—41no—
234D4×C43—22no—
235D4×C43—22no—
236D4×C43—41no—
237D4×C43—41no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—22no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
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×C43C2×SD1612yes—
253C2×C4×C43C22×D411yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43H(5,𝔽2)11yescharacteristic
258C2×C4×C432−1+411yescharacteristic
259C2×Dic43—21no—
260C2×Dic43—21no—
261C2×Dic43—21no—
262C2×Dic43—21no—
263C2×Dic43—22no—
264C2×Dic43—21no—
265C2×Dic43—22no—
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#12112yes—
280C2×Dic43#12112yes—
281C2×Dic43C2×SD1612yes—
282C2×Dic43C2×SD1612yes—
283C2×Dic43AGL(1,ℤ/8)12yes—
284Dic86—22no—
285Dic86—22no—
286Dic86—41no—
287Dic86—41no—
288Dic86—84no—
289C8×C43—41no—
290C8×C43—41no—
291C43⋊C8—41no—
292C43⋊C8—41no—
293#25320—431nomaximal, Sylow
294C22×C43—42no—
295C22×C43—42no—
296C22×C43—42no—
297C22×C43—81no—
298C22×C43—81no—
299C22×C43—81no—
300C22×C43—81no—
301C22×C43C2×#12111yescharacteristic
302C22×C43#54611yescharacteristic
303C22×C43#54811yescharacteristic
304C22×C43C22×SD1611yescharacteristic
305C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C43#57811yescharacteristic
307C22×C43#57811yescharacteristic
308C4×C43—22no—
309C4×C43—41no—
310C4×C43—42no—
311C4×C43—81no—
312C4×C43—81no—
313C4×C43#52911yescharacteristic
314C4×C43#54211yescharacteristic
315C4×C43#54611yescharacteristic
316C4×C43#54711yescharacteristic
317Dic43—22no—
318Dic43—22no—
319Dic43—41no—
320Dic43—42no—
321Dic43—42no—
322Dic43—81no—
323Dic43—81no—
324Dic43#46014yes—
325C2×#487—431no—
326C2×#461—431no—
327#2542—431no—
328#2365—431no—
329#2072—431no—
330#2072—431no—
331#2055—431no—
332#2055D4311yescharacteristic
333#1988—431no—
334#1939—431no—
335#1880—431no—
336#1880—431no—
337#1803—431no—
338#1803—431no—
339C4×#102—431no—
340C2×C43—82no—
341C2×C43#242711yescharacteristic
342C2×C43C2×#46011yescharacteristic
343C2×C43#304411yescharacteristic
344C2×C43#329811yescharacteristic
345C2×C43#333011yescharacteristic
346C2×C43#335911yescharacteristic
347C2×C43#336711yescharacteristic
348C22×SD16—431no—
349C22×SD16C2×D4311yescharacteristic
350C2×C4⋊Q8—431no—
351C2×C4⋊Q8—431no—
352C2×C22⋊Q8—861no—
353C2×C22⋊Q8—861no—
354C2×C4×Q8—431no—
355C2×C4×Q8—431no—
356C2×C4×D4—431no—
357#487—864no—
358#461—864no—
359C2×C4⋊C8—431no—
360C2×C4⋊C8—431no—
361C2×C4⋊C8C2×D4311yescharacteristic
362C2×#103—431no—
363C2×#103—431no—
364C2×#103—431no—
365C2×#103—431no—
366C2×#103—431no—
367C2×#103C2×D4311yescharacteristic
368C2×#102—431no—
369C2×#102C2×D4311yescharacteristic
370C2×C4×C8—431no—
371#399—861no—
372#399—861no—
373#391—431no—
374#391C2×D4311yescharacteristic
375#390—431no—
376#390C2×D4311yescharacteristic
377#389—861no—
378#387—861no—
379#387—861no—
380#384—431no—
381#384—431no—
382C4×C4⋊C4—431no—
383C4×C4⋊C4—431no—
384C4×C4⋊C4—861no—
385#340—431no—
386#340—431no—
387#340—431no—
388#340—431no—
389#340—431no—
390#340C2×D4311yescharacteristic
391#336—431no—
392#336—431no—
393C43#2532011yescharacteristic, complemented, Sylow
394C22×Q8—431no—
395C22×Q8—861no—
396C22×Q8C22×D4311yescharacteristic
397C22×D4C22×D4311yescharacteristic
398C23×C4—861no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16—41no—
402C2×SD16—862no—
403C2×SD16—862no—
404C2×SD16—1721no—
405C22×C8—431no—
406C22×C8—431no—
407C22×C8C22×D4311yescharacteristic
408C22×C8C22×D4311yescharacteristic
409C4⋊Q8—434no—
410C4⋊Q8—862no—
411C4⋊Q8—862no—
412C4⋊Q8—862no—
413C22⋊Q8—1724no—
414C22⋊Q8—1724no—
415C4×Q8—862no—
416C4×Q8—862no—
417C4×Q8—1721no—
418C4×Q8—1721no—
419C4×D4—862no—
420C4×D4—862no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—431no—
424C2×C4⋊C4—431no—
425C2×C4⋊C4—431no—
426C2×C4⋊C4—861no—
427C2×C4⋊C4—861no—
428C2×C4⋊C4—861no—
429C2×C4⋊C4—861no—
430C2×C4⋊C4—861no—
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×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—861no—
440C2×C22⋊C4—861no—
441C2×C22⋊C4—861no—
442C2×C42—431no—
443C2×C42—431no—
444C2×C42—861no—
445C2×C42—861no—
446C2×C42—861no—
447C2×C42C22×D4311yescharacteristic
448C4⋊C8—41no—
449C4⋊C8—862no—
450C4⋊C8—1721no—
451#103—22no—
452#103—862no—
453#103—862no—
454#103—862no—
455#103—862no—
456#103—862no—
457#102—22no—
458#102—862no—
459C4×C8—862no—
460#95—21no—
461#95—21no—
462#95—431no—
463#95—431no—
464#95—861no—
465#95—861no—
466#95—861no—
467C24—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—432no—
474C2×Q8—861no—
475C2×Q8—861no—
476C2×Q8—861no—
477C2×Q8—861no—
478C2×Q8—861no—
479C2×Q8—1722no—
480C2×Q8—1722no—
481C2×Q8—1722no—
482C2×Q8D4×D4312yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4#1062190012yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—431no—
493C22×C4—431no—
494C22×C4—431no—
495C22×C4—431no—
496C22×C4—861no—
497C22×C4—861no—
498C22×C4—861no—
499C22×C4—861no—
500C22×C4—1722no—
501C22×C4—1722no—
502C22×C4—1722no—
503C22×C4D4×D4311yescharacteristic
504C22×C4D4×D4311yescharacteristic
505C22×C4C23×D4311yescharacteristic
506SD16—44no—
507SD16—1724no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—861no—
515C2×C8—861no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—434no—
525C4⋊C4—432no—
526C4⋊C4—862no—
527C4⋊C4—864no—
528C4⋊C4—864no—
529C4⋊C4—861no—
530C4⋊C4—861no—
531C4⋊C4—861no—
532C4⋊C4—861no—
533C4⋊C4—861no—
534C4⋊C4—1722no—
535C4⋊C4—1722no—
536C4⋊C4—1722no—
537C4⋊C4—1722no—
538C4⋊C4—1722no—
539C4⋊C4—1722no—
540C4⋊C4—1722no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—1722no—
544C22⋊C4—1722no—
545C22⋊C4—1722no—
546C42—21no—
547C42—21no—
548C42—432no—
549C42—432no—
550C42—861no—
551C42—861no—
552C42—1722no—
553C42—1722no—
554C42—1722no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D4×D4311yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—862no—
568Q8—862no—
569Q8—3444no—
570Q8—1721no—
571Q8—1721no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—432no—
591C2×C4—432no—
592C2×C4—432no—
593C2×C4—432no—
594C2×C4—432no—
595C2×C4—861no—
596C2×C4—861no—
597C2×C4—861no—
598C2×C4—861no—
599C2×C4—862no—
600C2×C4—861no—
601C2×C4—861no—
602C2×C4—861no—
603C2×C4—862no—
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×C4SD16×D4312yes—
616C2×C4C2×D4×D4311yescharacteristic
617C2×C4#41×D4311yescharacteristic
618C2×C4#41×D4311yescharacteristic
619C2×C4C2×#1062190011yescharacteristic
620C2×C4#1306337011yescharacteristic
621C2×C4#1306337211yescharacteristic
622C8—41no—
623C8—41no—
624C8—1721no—
625C8—1721no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22#121×D4311yescharacteristic
634C22#1458353811yescharacteristic
635C22#1458355511yescharacteristic
636C22C2×SD16×D4311yescharacteristic
637C22AGL(1,ℤ/8)×D4311yescharacteristic
638C22#1458371311yescharacteristic
639C22#1458371511yescharacteristic
640C4—22no—
641C4—42no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—434no—
646C4—862no—
647C4—862no—
648C4—3441no—
649C4—3441no—
650C4—1722no—
651C4—1721no—
652C4—1722no—
653C4#1458342611yescharacteristic
654C4#1458351311yescharacteristic
655C4#1458353611yescharacteristic
656C4#1458354711yescharacteristic
657C2—82no—
658C2#1913816511yescharacteristic
659C2#460×D4311yescharacteristic
660C2#1914412311yescharacteristic
661C2#1914645511yescharacteristic
662C2#1914689511yescharacteristic
663C2#1914723211yescharacteristic
664C2#1914730811yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2055C2
socleC23×C43C22×D4
prime radicalC22×D4×C43C23
hypercenter#2055D43
hypocenterC43#25320
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
1. C2×#14583248C2semidirect
4. #19137652C2semidirect
5. #19136007C2semidirect
6. #19136003C2semidirect
8. #19135545C2semidirect
9. #19135310C2semidirect
12. #19134601C2semidirect
13. #19134598C2semidirect
208. Dic43⋊C4SD16semidirect
393. C43#25320semidirect

Maximal subgroups

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

subgroupindex
C2×#145832482
C2×#145831022
#191395462
#191376522
#191360072
#191360032
#191359112
#191355452
#191353102
#191349532
#191349522
#191346012
#191345982
#102×Dic432
C43×#20552
#2532043