Finite Group Databasebeta

← #26226551

Subgroups — #26226551

Order 12032.

1194 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D47C211yesmaximal, characteristic
2#20245770C211yesmaximal, characteristic
3C2×#15688980C211yesmaximal, characteristic
4#20243590C212yesmaximal, complemented
5#20243584C212yesmaximal, complemented
6#20243581C211yesmaximal, characteristic, complemented
7#20243580C211yesmaximal, characteristic, complemented
8#20243578C211yesmaximal, characteristic, complemented
9#20243414C212yesmaximal, complemented
10#20243412C212yesmaximal, complemented
11C47×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×C8×D47C2212yes—
14C2×#13093982C2211yescharacteristic
15C2×#13093982C2211yescharacteristic
16C22×C4⋊Dic47C2211yescharacteristic
17C22×(C2×D47)⋊C4—21no—
18C2×#13093958C2212yescomplemented
19C2×#13093955C2212yescomplemented
20C2×#13093955C2212yescomplemented
21C2×C22⋊(C47⋊C8)C2211yescharacteristic
22C2×(C2×D47)⋊C8C2212yes—
23C22⋊C8×D47C418yescomplemented
24#15688985—21no—
25#15688983C2211yescharacteristic
26#15688980—21no—
27#15688980—21no—
28#15688980C2211yescharacteristic
29#15688980C2211yescharacteristic
30#15688980C2212yes—
31#15688980C2212yes—
32#15688978—21no—
33C22⋊C4×Dic47—21no—
34#15688970—21no—
35C2×C47⋊#95—21no—
36#15688886C2212yescomplemented
37#15688885C2212yescomplemented
38#15688885C2212yescomplemented
39C22×C4⋊C4×C47C2211yescharacteristic
40C2×#107×C47C2212yescomplemented
41C2×C22⋊C8×C47C2211yescharacteristic
42C47×#380C2211yescharacteristic
43C47×#340C2212yescomplemented
44C24×D47D411yescharacteristic
45C23×Dic47—21no—
46C23×Dic47D411yescharacteristic
47C22×C4×D47—42no—
48C22×C4×D47—42no—
49C22×C4×D47C2×C414yes—
50C22×C4×D47D411yescharacteristic
51C22×C4×D47D411yescharacteristic
52C22×C4×D47Q814yes—
53C22×C4×D47C2311yescharacteristic
54C22×C4×D47C2311yescharacteristic
55C22×C47⋊C8C2312yes—
56C2×C8×D47—42no—
57C2×C8×D47—42no—
58C2×C8×D47C2×C418yescomplemented
59#13093982—42no—
60#13093982—42no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—42no—
63C2×C4⋊Dic47C2312yes—
64C2×C4⋊Dic47C2311yescharacteristic
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C22⋊Dic47—21no—
67C2×(C2×D47)⋊C4—21no—
68C2×(C2×D47)⋊C4—21no—
69C2×(C2×D47)⋊C4—21no—
70C2×(C2×D47)⋊C4—21no—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—44no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47—21no—
76#13093958—42no—
77#13093955—42no—
78#13093955—42no—
79C22⋊(C47⋊C8)C2×C414yes—
80(C2×D47)⋊C8C2×C418yescomplemented
81C47⋊#95—21no—
82C47⋊#95—22no—
83C47⋊#95—22no—
84C47⋊#95—21no—
85C47⋊#95—21no—
86C23×C4×C47—21no—
87C23×C4×C47C2311yescharacteristic
88C22×C8×C47C2312yes—
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47—42no—
91C2×C4⋊C4×C47C2312yes—
92C2×C4⋊C4×C47C2311yescharacteristic
93C2×C4⋊C4×C47C2311yescharacteristic
94C2×C22⋊C4×C47—21no—
95#107×C47—42no—
96C22⋊C8×C47C2×C414yes—
97#95×C47—21no—
98C24×C47C2×D411yescharacteristic
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—22no—
103C23×D47—41no—
104C23×D47—41no—
105C23×D47C4⋊C414yes—
106C23×D47C2×D411yescharacteristic
107C23×D47C2×D411yescharacteristic
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—42no—
114C22×C4×C47—42no—
115C22×C4×C47—42no—
116C22×C4×C47C22×C412yes—
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×Q812yes—
119C22×C4×C47C2411yescharacteristic
120C22×Dic47—21no—
121C22×Dic47—21no—
122C22×Dic47—22no—
123C22×Dic47—41no—
124C22×Dic47—41no—
125C22×Dic47—42no—
126C22×Dic47—42no—
127C22×Dic47—42no—
128C22×Dic47C4⋊C414yes—
129C22×Dic47C2×D411yescharacteristic
130C22×Dic47C2×D411yescharacteristic
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—48no—
136C2×C4×D47—48no—
137C2×C4×D47—41no—
138C2×C4×D47—41no—
139C2×C4×D47—41no—
140C2×C4×D47—41no—
141C2×C4×D47C4⋊C414yes—
142C2×C4×D47C4⋊C414yes—
143C2×C4×D47C22×C412yes—
144C2×C4×D47C22×C412yes—
145C2×C4×D47C2×Q812yes—
146C2×C4×D47C2×Q812yes—
147C2×C8×C47—22no—
148C2×C8×C47—22no—
149C2×C8×C47C22×C414yes—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—42no—
155C22⋊C4×C47—42no—
156C22⋊Dic47—42no—
157(C2×D47)⋊C4—44no—
158(C2×D47)⋊C4—42no—
159(C2×D47)⋊C4—42no—
160(C2×D47)⋊C4—42no—
161(C2×D47)⋊C4—44no—
162(C2×D47)⋊C4—42no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—42no—
168C4×Dic47—42no—
169C4×Dic47—42no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8C22×C414yes—
173C8×D47—48no—
174C23×C47—21no—
175C23×C47—21no—
176C23×C47—21no—
177C23×C47—21no—
178C23×C47C2×C4⋊C412yes—
179C23×C47#13512yes—
180C23×C47#13512yes—
181C23×C47C22×D411yescharacteristic
182C22×D47—21no—
183C22×D47—21no—
184C22×D47—21no—
185C22×D47—21no—
186C22×D47—21no—
187C22×D47—21no—
188C22×D47—21no—
189C22×D47—21no—
190C22×D47—21no—
191C22×D47—21no—
192C22×D47—28no—
193C22×D47—28no—
194C22×D47—44no—
195C22×D47—44no—
196C22×D47—44no—
197C22×D47—44no—
198C22×D47—44no—
199C22×D47C2×C4⋊C412yes—
200C22×D47C2×C4⋊C412yes—
201C22×D47AGL(1,ℤ/8)12yes—
202C22×D47#13712yes—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—84no—
217C2×C4×C47C2×C4⋊C412yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C22×Q811yescharacteristic
220C2×C4×C47H(5,𝔽2)12yes—
221C2×C4×C47H(5,𝔽2)12yes—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—44no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—42no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—84no—
237C2×Dic47C2×C4⋊C412yes—
238C2×Dic47C2×C4⋊C412yes—
239C4×D47—24no—
240C4×D47—24no—
241C4×D47—24no—
242C4×D47—24no—
243C4×D47—44no—
244C4×D47—44no—
245C8×C47—24no—
246C47⋊C8—24no—
247#21941—471nomaximal, Sylow
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47#42714yes—
255C22×C47C22×C4⋊C411yescharacteristic
256C22×C47#53411yescharacteristic
257C22×C47#53411yescharacteristic
258C22×C47C2×#13511yescharacteristic
259C22×C47C2×#13511yescharacteristic
260C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C47C2×#13711yescharacteristic
262C2×D47—22no—
263C2×D47—22no—
264C2×D47—22no—
265C2×D47—22no—
266C2×D47—22no—
267C2×D47—22no—
268C2×D47—22no—
269C2×D47—22no—
270C2×D47—22no—
271C2×D47—416no—
272C2×D47—41no—
273C2×D47—41no—
274C2×D47—41no—
275C2×D47—41no—
276C2×D47#42814yes—
277C4×C47—22no—
278C4×C47—81no—
279C4×C47—82no—
280C4×C47—81no—
281C4×C47#51812yes—
282C4×C47#54312yes—
283Dic47—22no—
284Dic47—22no—
285Dic47—81no—
286Dic47—82no—
287Dic47—81no—
288C22×C22⋊C8—471no—
289#2280—471no—
290C2×#380—471no—
291#1854—472no—
292#1854—472no—
293#1853—471no—
294#1853—471no—
295#1852—471no—
296#1852D4711yescharacteristic, complemented
297#1808—472no—
298#1807—472no—
299C2×C47—24no—
300C2×C47#231511yescharacteristic
301C2×C47C2×#42711yescharacteristic
302C2×C47C2×#42811yescharacteristic
303C2×C47#318811yescharacteristic
304C2×C47#318811yescharacteristic
305C2×C47#319011yescharacteristic
306C2×C47#319011yescharacteristic
307D47—24no—
308D47—42no—
309C24×C4—471no—
310C23×C8—472no—
311C2×#117—471no—
312C2×#117—471no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C22⋊C4—941no—
316C2×#107—472no—
317C2×#107C2×D4712yescomplemented
318C2×#106—472no—
319C2×#106—472no—
320C2×C22⋊C8—471no—
321C2×C22⋊C8—472no—
322C2×C22⋊C8—478no—
323C2×C22⋊C8C2×D4711yescharacteristic
324#381—941no—
325#380—471no—
326#380—471no—
327#380—472no—
328#380—471no—
329#380—472no—
330#380—941no—
331#380—941no—
332#380C2×D4711yescharacteristic
333#379—941no—
334C4×C22⋊C4—941no—
335C4×C22⋊C4—941no—
336C2×#95—941no—
337#340—472no—
338#340—472no—
339#340—472no—
340#340C2×D4712yescomplemented
341C47#2194111yescharacteristic, complemented, Sylow
342C25—471no—
343C23×C4—21no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—474no—
348C23×C4—471no—
349C23×C4—471no—
350C23×C4—474no—
351C23×C4—941no—
352C23×C4—1882no—
353C23×C4—1882no—
354C23×C4C22×D4711yescharacteristic
355C22×C8—472no—
356C22×C8—478no—
357C22×C8—1882no—
358C22×C8—1882no—
359C22×C8C22×D4712yes—
360#117—1882no—
361#117—1882no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4—1882no—
369C2×C4⋊C4C22×D4712yes—
370C2×C4⋊C4C22×D4711yescharacteristic
371C2×C4⋊C4C22×D4711yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—942no—
374C2×C22⋊C4—942no—
375C2×C22⋊C4—941no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—1884no—
381C2×C42—941no—
382C2×C42—941no—
383#107—42no—
384#107—1882no—
385#106—1882no—
386#106—1882no—
387C22⋊C8—474no—
388C22⋊C8—478no—
389C22⋊C8C4×D4714yes—
390#95—21no—
391#95—942no—
392#95—942no—
393#95—941no—
394#95—941no—
395#95—941no—
396C24—471no—
397C24—471no—
398C24—474no—
399C24—942no—
400C24—942no—
401C24—942no—
402C24—942no—
403C24—1881no—
404C24—1881no—
405C24D4×D4711yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—474no—
417C22×C4—472no—
418C22×C4—472no—
419C22×C4—472no—
420C22×C4—472no—
421C22×C4—474no—
422C22×C4—474no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—942no—
430C22×C4—1888no—
431C22×C4—1888no—
432C22×C4—1881no—
433C22×C4—1881no—
434C22×C4—1881no—
435C22×C4—1881no—
436C22×C4—1881no—
437C22×C4—1881no—
438C22×C4—1882no—
439C22×C4—1882no—
440C22×C4—1882no—
441C22×C4C2×C4×D4712yes—
442C22×C4D4×D4711yescharacteristic
443C22×C4Q8×D4712yes—
444C22×C4C23×D4711yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—474no—
448C2×C8—942no—
449C2×C8—942no—
450C2×C8—1888no—
451C2×C8C2×C4×D4714yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—942no—
458C4⋊C4—942no—
459C4⋊C4—942no—
460C4⋊C4—942no—
461C4⋊C4—1882no—
462C22⋊C4—42no—
463C22⋊C4—1882no—
464C22⋊C4—1882no—
465C22⋊C4—1884no—
466C22⋊C4—1884no—
467C22⋊C4—1882no—
468C22⋊C4—1882no—
469C22⋊C4—1882no—
470C42—1882no—
471C42—1882no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—472no—
477C23—472no—
478C23—472no—
479C23—472no—
480C23—948no—
481C23—941no—
482C23—941no—
483C23—941no—
484C23—941no—
485C23—941no—
486C23—941no—
487C23—941no—
488C23—941no—
489C23—948no—
490C23—941no—
491C23—941no—
492C23—1884no—
493C23—1884no—
494C23—1884no—
495C23—1884no—
496C23—1884no—
497C23C4⋊C4×D4712yes—
498C23#1309404312yes—
499C23#1309404712yes—
500C23C2×D4×D4711yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—3764no—
526C2×C4—1884no—
527C2×C4—1884no—
528C2×C4—1884no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1882no—
538C2×C4C4⋊C4×D4712yes—
539C2×C4C22×C4×D4711yescharacteristic
540C2×C4C2×Q8×D4711yescharacteristic
541C2×C4#1309406912yes—
542C2×C4#1309407012yes—
543C8—24no—
544C8—944no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—474no—
552C22—942no—
553C22—942no—
554C22—942no—
555C22—942no—
556C22—942no—
557C22—942no—
558C22—942no—
559C22—942no—
560C22—942no—
561C22—18816no—
562C22—1881no—
563C22—1881no—
564C22—1881no—
565C22—1881no—
566C22#1568916014yes—
567C22C2×C4⋊C4×D4711yescharacteristic
568C22#1568968311yescharacteristic
569C22#1568968411yescharacteristic
570C22C2×#1309404311yescharacteristic
571C22C2×#1309404711yescharacteristic
572C22AGL(1,ℤ/8)×D4711yescharacteristic
573C22#137×D4711yescharacteristic
574C4—22no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—942no—
579C4—942no—
580C4—3762no—
581C4—3761no—
582C4—3761no—
583C4#1568959512yes—
584C4#1568974812yes—
585C2—24no—
586C2—944no—
587C2—1882no—
588C2#2024599911yescharacteristic
589C2C2×#1568916011yescharacteristic
590C2#428×D4711yescharacteristic
591C2#2025415011yescharacteristic
592C2#2025415811yescharacteristic
593C2#2025416811yescharacteristic
594C2#2025417211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1852C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1852D47
hypocenterC47#21941
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #20243590C2semidirect
5. #20243584C2semidirect
6. #20243581C2semidirect
7. #20243580C2semidirect
8. #20243578C2semidirect
9. #20243414C2semidirect
10. #20243412C2semidirect
11. C47×#1852C2semidirect
18. C2×#13093958C22semidirect
19. C2×#13093955C22semidirect
20. C2×#13093955C22semidirect
23. C22⋊C8×D47C4semidirect
36. #15688886C22semidirect
37. #15688885C22semidirect
38. #15688885C22semidirect
40. C2×#107×C47C22semidirect
43. C47×#340C22semidirect
58. C2×C8×D47C2×C4semidirect
80. (C2×D47)⋊C8C2×C4semidirect
296. #1852D47semidirect
317. C2×#107C2×D47semidirect
340. #340C2×D47semidirect
341. C47#21941semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D472
#202457702
C2×#156889802
#202435902
#202435842
#202435812
#202435802
#202435782
#202434142
#202434122
C47×#18522
#2194147