Finite Group Databasebeta

← #26215768

Subgroups — #26215768

Order 12032.

1598 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625352C211yesmaximal, characteristic
2C2×#15689145C211yesmaximal, characteristic, complemented
3C2×#15689121C212yesmaximal
4#384×D47C211yesmaximal, characteristic, complemented
5#20243394C211yesmaximal, characteristic, complemented
6#20243392C212yesmaximal, complemented
7#20243390C211yesmaximal, characteristic, complemented
8#20243388C211yesmaximal, characteristic, complemented
9#20243386C212yesmaximal, complemented
10#20243106C212yesmaximal, complemented
11C47×#1802C211yesmaximal, characteristic, complemented
12C22×#10625352C2211yescharacteristic
13C22×#10625324C2212yes—
14C4×#10625352—24no—
15C4×#10625352C418yescomplemented
16C2×C4×Dic94C2211yescharacteristic
17C2×#13093989C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C2×C4×C22⋊D47—21no—
20C2×(C22⋊D47)⋊C4—21no—
21C2×#13093982C2211yescharacteristic
22C2×#13093980—21no—
23C2×C4⋊C4×D47—21no—
24C2×C4⋊C4×D47—21no—
25C2×C4⋊C4×D47C2212yes—
26C22×C4×Dic47—21no—
27C2×C42×D47C2211yescharacteristic
28#15689145—24no—
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×Dic47⋊C8C2212yes—
31#15689121—28no—
32C2×#13093949C2212yes—
33C2×#13093947C2212yes—
34C2×#13093945C2212yes—
35#15688993C2211yescharacteristic
36#15688992—21no—
37#15688992—21no—
38#15688992C2212yes—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×D4×C47C2211yescharacteristic
43C2×C4⋊C8×C47C2211yescharacteristic
44C2×#102×C47C2212yes—
45C47×#384C2211yescharacteristic
46C47×#340C2212yes—
47C2×#10625352—22no—
48C2×#10625352—22no—
49C2×#10625352—22no—
50C2×#10625352C2×C414yes—
51C2×#10625352D414yes—
52C2×#10625352Q814yes—
53C22×Dic94C2311yescharacteristic
54C22×C22⋊D47—21no—
55C23×Dic47—21no—
56C23×Dic47—21no—
57C22×C4×D47—22no—
58C22×C4×D47D411yescharacteristic
59C22×C4×D47D411yescharacteristic
60C22×C4×D47C2311yescharacteristic
61C2×#10625324—24no—
62C2×#10625324—24no—
63C2×#10625324—24no—
64C22×C47⋊C8C2312yes—
65C4×Dic94—22no—
66C4×Dic94C2×C414yescomplemented
67#13093989—22no—
68#13093989C2×C414yescomplemented
69D4×Dic47—22no—
70D4×Dic47C2×C414yescomplemented
71C4×C22⋊D47—28no—
72(C22⋊D47)⋊C4—28no—
73#13093982—22no—
74#13093982C2×C414yes—
75#13093980—28no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47C2312yes—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—22no—
83C2×Dic47⋊C4—22no—
84C2×Dic47⋊C4—22no—
85C4⋊C4×D47—24no—
86C4⋊C4×D47—24no—
87C4⋊C4×D47—44no—
88C4⋊C4×D47—44no—
89C2×C22⋊Dic47—21no—
90C2×(C2×D47)⋊C4—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—28no—
95C2×C4×Dic47D411yescharacteristic
96C2×C4×Dic47D411yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98C42×D47C2×C414yes—
99C42×D47D414yes—
100C4⋊(C47⋊C8)—22no—
101Dic47⋊C8—24no—
102#13093949—24no—
103#13093947—24no—
104#13093945—24no—
105C22×D4×C47C2311yescharacteristic
106C23×C4×C47—21no—
107C22×C8×C47C2312yes—
108C4×D4×C47—22no—
109C4×D4×C47C2×C414yescomplemented
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47C2312yes—
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C22⋊C4×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C4⋊C8×C47—22no—
117#102×C47—24no—
118C24×C47—21no—
119C23×D47C2×D411yescharacteristic
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47C22×C412yes—
124C2×D4×C47C2×D412yes—
125C2×D4×C47C2×Q812yes—
126C22×C4×C47—21no—
127C22×C4×C47—24no—
128C22×C4×C47—22no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—24no—
135C2×C22⋊D47—24no—
136C2×C22⋊D47—24no—
137C22×Dic47—21no—
138C22×Dic47—24no—
139C22×Dic47—24no—
140C22×Dic47—24no—
141C22×Dic47—21no—
142C22×Dic47—24no—
143C22×Dic47—22no—
144C22×Dic47—42no—
145C22×Dic47—42no—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149#10625352—28no—
150#10625352—28no—
151#10625352—28no—
152#10625352C4⋊C4116yescomplemented
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—44no—
158C2×C4×D47—44no—
159C2×C4×D47—44no—
160C2×C4×D47C22⋊C414yes—
161C2×C4×D47C22⋊C414yes—
162C2×C4×D47C22×C412yes—
163C2×C4×D47C2×D412yes—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×Q812yes—
166C2×C4×D47#4112yes—
167C2×C4×D47#4112yes—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94C22×C412yes—
172C2×Dic94C2×D412yes—
173C2×Dic94C2×Q812yes—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—22no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—42no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47C22×C412yes—
183C22⋊C4×C47—24no—
184C42×C47C22×C412yes—
185C42×C47C2×D412yes—
186C22⋊Dic47—24no—
187(C2×D47)⋊C4—24no—
188C4⋊Dic47—21no—
189C4⋊Dic47—22no—
190C4⋊Dic47—22no—
191C4⋊Dic47—42no—
192C4⋊Dic47—42no—
193C4⋊Dic47C22×C412yes—
194Dic47⋊C4—24no—
195Dic47⋊C4—24no—
196Dic47⋊C4—44no—
197Dic47⋊C4—44no—
198Dic47⋊C4—44no—
199C4×Dic47—24no—
200C4×Dic47—24no—
201C4×Dic47—24no—
202C4×Dic47C22⋊C414yes—
203C4×Dic47C22⋊C414yes—
204C4×Dic47C22×C412yes—
205C4×Dic47C2×D412yes—
206C2×C47⋊C8—22no—
207C2×C47⋊C8—22no—
208C2×C47⋊C8—22no—
209#10625324—48no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47—22no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47C22×D411yescharacteristic
218C22×D47—21no—
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—21no—
222C22×D47C2×C22⋊C412yes—
223C22×D47#12112yes—
224C22×D47C22⋊Q812yes—
225D4×C47—22no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47C2×C4⋊C414yescomplemented
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—22no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—48no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47AGL(1,ℤ/8)12yes—
241C2×C4×C47#13712yes—
242C2×C4×C47C23×C411yescharacteristic
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×Q811yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C22⋊D47—216no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—216no—
255C2×Dic47—22no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—48no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47C2×C22⋊C412yes—
263C2×Dic47C2×C22⋊C412yes—
264C2×Dic47C2×C22⋊C412yes—
265C2×Dic47#12112yes—
266C2×Dic47C22⋊Q812yes—
267C2×Dic47AGL(1,ℤ/8)12yes—
268C2×Dic47#13712yes—
269C4×D47—28no—
270C4×D47—28no—
271C4×D47—88no—
272C4×D47C2×C4⋊C414yes—
273C4×D47#11714yes—
274C4×D47#12114yes—
275C4×D47C22⋊Q814yes—
276Dic94—22no—
277Dic94—22no—
278Dic94—22no—
279Dic94C2×C4⋊C414yescomplemented
280C8×C47—42no—
281C47⋊C8—42no—
282#20648—471nomaximal, Sylow
283C22×C47—24no—
284C22×C47—42no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47C22×C22⋊C411yescharacteristic
291C22×C47C2×#12111yescharacteristic
292C22×C47C2×C22⋊Q811yescharacteristic
293C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C47C2×#13711yescharacteristic
296C22×C47C2×#13711yescharacteristic
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—22no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—22no—
303C2×D47#38014yes—
304C4×C47—22no—
305C4×C47—42no—
306C4×C47—82no—
307C4×C47#41714yescomplemented
308C4×C47C22×C4⋊C411yescharacteristic
309C4×C47C2×#11711yescharacteristic
310C4×C47C2×#12111yescharacteristic
311C4×C47C2×C22⋊Q811yescharacteristic
312Dic47—22no—
313Dic47—24no—
314Dic47—42no—
315Dic47—82no—
316Dic47#38014yes—
317Dic47#41714yescomplemented
318C2×C4×#41—471no—
319C2×#423—471no—
320C2×#417—472no—
321C2×#384—471no—
322#1803—471no—
323#1803—471no—
324#1803—472no—
325#1802—472no—
326#1802—471no—
327#1802D4711yescharacteristic, complemented
328#1736—472no—
329C2×C47—44no—
330C2×C47C2×#38011yescharacteristic
331C2×C47C2×#41711yescharacteristic
332C2×C47C2×#41711yescharacteristic
333C2×C47#303811yescharacteristic
334C2×C47#303911yescharacteristic
335C2×C47#307911yescharacteristic
336C2×C47#307911yescharacteristic
337D47—24no—
338C22×#41—471no—
339C22×M16—472no—
340C4×#41—478no—
341C4×#41—944no—
342C2×C4×Q8—471no—
343C2×C4×Q8—471no—
344C2×C4×D4—471no—
345C2×C4×D4—941no—
346C2×C4×D4—941no—
347C2×C4×D4C2×D4711yescharacteristic
348C2×#117—471no—
349C2×#117—941no—
350C22×C4⋊C4—472no—
351C22×C4⋊C4—941no—
352C22×C4⋊C4—941no—
353C22×C42—471no—
354C22×C42—941no—
355#423—944no—
356C2×C4⋊C8—472no—
357C2×C4⋊C8—471no—
358C2×C4⋊C8C2×D4711yescharacteristic
359#417—948no—
360C2×#103—472no—
361C2×#103—472no—
362C2×#102—472no—
363C2×#102C2×D4712yes—
364#384—472no—
365#384—471no—
366#384—941no—
367#384—941no—
368#384C2×D4711yescharacteristic
369#340—472no—
370#340—472no—
371#340—472no—
372#340C2×D4712yes—
373C47#2064811yescharacteristic, complemented, Sylow
374C2×#41—474no—
375C2×#41—474no—
376C2×#41—474no—
377C2×#41—942no—
378C2×#41—942no—
379C2×#41—942no—
380C22×Q8—471no—
381C22×D4—941no—
382C22×D4C22×D4711yescharacteristic
383C23×C4—21no—
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—471no—
387C23×C4—942no—
388C23×C4—941no—
389C23×C4—941no—
390C2×M16—944no—
391C2×M16—944no—
392C2×M16—944no—
393C22×C8—472no—
394C22×C8C22×D4712yes—
395C4×Q8—474no—
396C4×Q8—474no—
397C4×Q8—942no—
398C4×Q8—942no—
399C4×D4—22no—
400C4×D4—474no—
401C4×D4—942no—
402C4×D4—948no—
403C4×D4—948no—
404C4×D4C4×D4714yescomplemented
405#117—474no—
406#117—948no—
407#117—942no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—472no—
412C2×C4⋊C4—944no—
413C2×C4⋊C4—944no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—942no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—1884no—
422C2×C4⋊C4—1884no—
423C2×C4⋊C4C22×D4712yes—
424C2×C4⋊C4C22×D4711yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—941no—
427C2×C22⋊C4—941no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—474no—
432C2×C42—474no—
433C2×C42—948no—
434C2×C42—941no—
435C2×C42—941no—
436C2×C42—941no—
437C2×C42C22×D4711yescharacteristic
438C4⋊C8—22no—
439C4⋊C8—944no—
440C4⋊C8—942no—
441#103—944no—
442#103—944no—
443#102—24no—
444#102—944no—
445C24—21no—
446C24—471no—
447#41—4716no—
448#41—948no—
449#41—948no—
450#41—948no—
451C2×Q8—472no—
452C2×Q8—472no—
453C2×Q8—472no—
454C2×Q8—941no—
455C2×Q8—941no—
456C2×Q8—941no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—944no—
461C2×D4—944no—
462C2×D4—944no—
463C2×D4C2×C4×D4712yes—
464C2×D4D4×D4712yes—
465C2×D4Q8×D4712yes—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—471no—
472C22×C4—471no—
473C22×C4—472no—
474C22×C4—472no—
475C22×C4—472no—
476C22×C4—472no—
477C22×C4—472no—
478C22×C4—472no—
479C22×C4—471no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—944no—
485C22×C4—944no—
486C22×C4—944no—
487C22×C4—944no—
488C22×C4—941no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—1884no—
494C22×C4—1884no—
495C22×C4—1884no—
496C22×C4—1882no—
497C22×C4—1882no—
498C22×C4D4×D4711yescharacteristic
499C22×C4D4×D4711yescharacteristic
500C22×C4C23×D4711yescharacteristic
501M16—1888no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—942no—
506C2×C8—942no—
507C2×C8—942no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—472no—
514C4⋊C4—944no—
515C4⋊C4—944no—
516C4⋊C4—941no—
517C4⋊C4—942no—
518C4⋊C4—942no—
519C4⋊C4—1882no—
520C4⋊C4—1884no—
521C4⋊C4—1884no—
522C4⋊C4—1884no—
523C4⋊C4—1882no—
524C4⋊C4C2×C4×D4712yes—
525C22⋊C4—24no—
526C22⋊C4—944no—
527C22⋊C4—944no—
528C42—474no—
529C42—474no—
530C42—472no—
531C42—472no—
532C42—944no—
533C42—944no—
534C42—944no—
535C42C2×C4×D4712yes—
536C42D4×D4712yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—472no—
545C23—472no—
546C23—472no—
547C23—941no—
548C23—941no—
549C23—941no—
550C23—941no—
551C23C2×D4×D4711yescharacteristic
552Q8—474no—
553Q8—942no—
554Q8—942no—
555Q8—942no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—9416no—
560D4C4⋊C4×D4714yescomplemented
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—472no—
571C2×C4—472no—
572C2×C4—472no—
573C2×C4—472no—
574C2×C4—472no—
575C2×C4—472no—
576C2×C4—474no—
577C2×C4—474no—
578C2×C4—474no—
579C2×C4—474no—
580C2×C4—472no—
581C2×C4—948no—
582C2×C4—9416no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—948no—
589C2×C4—942no—
590C2×C4—3768no—
591C2×C4—1888no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4C22⋊C4×D4712yes—
598C2×C4C22⋊C4×D4712yes—
599C2×C4#1309404812yes—
600C2×C4#1309405312yes—
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4C2×D4×D4711yescharacteristic
603C2×C4C2×D4×D4711yescharacteristic
604C2×C4C2×Q8×D4711yescharacteristic
605C2×C4#41×D4711yescharacteristic
606C2×C4#41×D4711yescharacteristic
607C8—42no—
608C8—1882no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—474no—
617C22—942no—
618C22—942no—
619C22—942no—
620C22—942no—
621C22—942no—
622C22—942no—
623C22C2×C22⋊C4×D4711yescharacteristic
624C22#121×D4711yescharacteristic
625C22C22⋊Q8×D4711yescharacteristic
626C22AGL(1,ℤ/8)×D4711yescharacteristic
627C22C2×#1309404811yescharacteristic
628C22#137×D4711yescharacteristic
629C22C2×#1309405311yescharacteristic
630C4—22no—
631C4—42no—
632C4—82no—
633C4—474no—
634C4—474no—
635C4—942no—
636C4—944no—
637C4—3762no—
638C4—1882no—
639C4#1568912114yescomplemented
640C4C2×C4⋊C4×D4711yescharacteristic
641C4#117×D4711yescharacteristic
642C4#121×D4711yescharacteristic
643C4C22⋊Q8×D4711yescharacteristic
644C2—44no—
645C2—944no—
646C2#380×D4711yescharacteristic
647C2#417×D4711yescharacteristic
648C2C2×#1568912111yescharacteristic
649C2#2025286311yescharacteristic
650C2#2025287011yescharacteristic
651C2#2025319611yescharacteristic
652C2#2025320011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC22×#10625352C22
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#20648
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#15689145C2semidirect
4. #384×D47C2semidirect
5. #20243394C2semidirect
6. #20243392C2semidirect
7. #20243390C2semidirect
8. #20243388C2semidirect
9. #20243386C2semidirect
10. #20243106C2semidirect
11. C47×#1802C2semidirect
15. C4×#10625352C4semidirect
66. C4×Dic94C2×C4semidirect
68. #13093989C2×C4semidirect
70. D4×Dic47C2×C4semidirect
109. C4×D4×C47C2×C4semidirect
152. #10625352C4⋊C4semidirect
228. D4×C47C2×C4⋊C4semidirect
279. Dic94C2×C4⋊C4semidirect
307. C4×C47#417semidirect
317. Dic47#417semidirect
327. #1802D47semidirect
373. C47#20648semidirect
404. C4×D4C4×D47semidirect
560. D4C4⋊C4×D47semidirect
639. C4#15689121semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253522
C2×#156891452
C2×#156891212
#384×D472
#202433942
#202433922
#202433902
#202433882
#202433862
#202431062
C47×#18022
#2064847