Finite Group Databasebeta

← #26228435

Subgroups — #26228435

Order 12032.

784 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689423C211yesmaximal, characteristic
2C4×#13094016C211yesmaximal, characteristic, complemented
3#20244345C211yesmaximal, characteristic, complemented
4#20244156C211yesmaximal, characteristic, complemented
5#20244128C211yesmaximal, characteristic, complemented
6#20244123C211yesmaximal, characteristic, complemented
7#20244088C211yesmaximal, characteristic
8#20244088C211yesmaximal, characteristic
9#20244012C211yesmaximal, characteristic
10#20244005C211yesmaximal, characteristic
11#20243963C211yesmaximal, characteristic, complemented
12#20243956C211yesmaximal, characteristic, complemented
13#20243398C211yesmaximal, characteristic
14#20243241C211yesmaximal, characteristic, complemented
15C47×#2293C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17#15689423C418yescomplemented
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30#15689021C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689007—21no—
33#15689007—21no—
34#15688989C2211yescharacteristic
35C4×Dic47⋊C4—21no—
36C4×Dic47⋊C4—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×C4×D4×C47C2211yescharacteristic
50C2×#117×C47—21no—
51C47×#390C2211yescharacteristic
52C47×#390C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C47×#388—21no—
56C47×#388—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C43×C47C2211yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094016—22no—
65#13094016C2×C414yes—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47—21no—
76#13093957—41no—
77#13093957C2×C414yes—
78#13093955—41no—
79#13093955C2×C414yes—
80C4⋊(C47⋊C8)—22no—
81C4⋊(C47⋊C8)C2×C414yes—
82#13093945—41no—
83#13093945—41no—
84#13093945C2×C414yes—
85#13093945C2×C414yes—
86#13093935—41no—
87#13093935—41no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C22×D4×C47C2311yescharacteristic
95C23×C4×C47—21no—
96C4×D4×C47—22no—
97C4×D4×C47C2×C414yes—
98#117×C47—44no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C42×C47—21no—
110C2×C42×C47D411yescharacteristic
111C2×C42×C47D411yescharacteristic
112C2×C42×C47D411yescharacteristic
113C2×C42×C47D411yescharacteristic
114C2×C42×C47C2311yescharacteristic
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117#95×C47—21no—
118#95×C47—21no—
119#95×C47—21no—
120C24×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47—21no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47C22×C412yes—
127C22×C4×C47—21no—
128C22×C4×C47—24no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—42no—
150C4⋊C4×C47—42no—
151C4⋊C4×C47C22×C412yes—
152C22⋊C4×C47—24no—
153C22⋊C4×C47—42no—
154C22⋊C4×C47—42no—
155C22⋊C4×C47—42no—
156C22⋊C4×C47—42no—
157C42×C47—22no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—41no—
163C42×C47—41no—
164C42×C47—41no—
165C42×C47—42no—
166C42×C47C22×C412yes—
167C42×C47#4112yes—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47C22×C412yes—
175C4⋊Dic47C22×C412yes—
176Dic47⋊C4—24no—
177Dic47⋊C4—24no—
178Dic47⋊C4—42no—
179Dic47⋊C4—42no—
180C4×Dic47—42no—
181C4×Dic47—42no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8C22×C412yes—
193C2×C47⋊C8C22×C412yes—
194C23×C47—22no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47C22×D411yescharacteristic
202D4×C47—22no—
203D4×C47—22no—
204D4×C47—41no—
205D4×C47—41no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—22no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—48no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47C4×D412yes—
232C2×C4×C47C4×D412yes—
233C2×C4×C47AGL(1,ℤ/8)12yes—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47H(5,𝔽2)11yescharacteristic
237C2×C4×C472−1+411yescharacteristic
238C2×C4×C472−1+411yescharacteristic
239C2×C4×C472−1+411yescharacteristic
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C47⋊C8—22no—
253C47⋊C8—22no—
254C47⋊C8—41no—
255C47⋊C8—41no—
256#22120—471nomaximal, Sylow
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—81no—
261C22×C47—81no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47C2×C4×D411yescharacteristic
265C22×C47#53611yescharacteristic
266C22×C47#53711yescharacteristic
267C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C47#57511yescharacteristic
269C22×C47#57511yescharacteristic
270C22×C47#57511yescharacteristic
271C4×C47—22no—
272C4×C47—22no—
273C4×C47—22no—
274C4×C47—42no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—41no—
278C4×C47—81no—
279C4×C47—81no—
280C4×C47#51911yescharacteristic
281C4×C47#52011yescharacteristic
282C4×C47#54111yescharacteristic
283C4×C47#54211yescharacteristic
284Dic47—42no—
285Dic47—42no—
286Dic47—81no—
287Dic47—81no—
288C2×#478—471no—
289C4×#125—471no—
290#2293D4711yescharacteristic
291#1989—471no—
292#1949—471no—
293#1944—471no—
294#1943—471no—
295#1936—471no—
296#1936—471no—
297#1926—471no—
298#1925—471no—
299#1919—471no—
300#1918—471no—
301#1804—471no—
302#1767—471no—
303C2×C47—82no—
304C2×C47#232311yescharacteristic
305C2×C47#294311yescharacteristic
306C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C47#323411yescharacteristic
308C2×C47#324011yescharacteristic
309C2×C47#325111yescharacteristic
310C2×C47#325611yescharacteristic
311C2×#125—471no—
312C2×C4×D4C2×D4711yescharacteristic
313C2×#117—21no—
314#478—478no—
315C2×#107—471no—
316C2×#107—471no—
317C2×#106—471no—
318C2×#106—471no—
319C2×C4⋊C8—471no—
320C2×C4⋊C8—471no—
321C2×#102—471no—
322C2×#102—471no—
323C2×#102—471no—
324C2×#102—471no—
325C2×#97—471no—
326C2×#97—471no—
327#390C2×D4711yescharacteristic
328#390C2×D4711yescharacteristic
329#389—471no—
330#389—471no—
331#389C2×D4711yescharacteristic
332#389C2×D4711yescharacteristic
333#388—21no—
334#388—21no—
335#387—941no—
336#387—941no—
337#382—471no—
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4—941no—
341C4×C4⋊C4—941no—
342C4×C4⋊C4C2×D4711yescharacteristic
343C4×C22⋊C4—21no—
344C43C2×D4711yescharacteristic
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#336—471no—
354#336—471no—
355C47#2212011yescharacteristic, complemented, Sylow
356C22×D4C22×D4711yescharacteristic
357C23×C4—21no—
358C22×C8—471no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—471no—
362#125—474no—
363#125—942no—
364C4×D4—22no—
365C4×D4C4×D4714yes—
366#117—44no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42C22⋊D4711yescharacteristic
389C2×C42C22⋊D4711yescharacteristic
390C2×C42C22⋊D4711yescharacteristic
391C2×C42C22⋊D4711yescharacteristic
392C2×C42C22×D4711yescharacteristic
393C2×C42C22×D4711yescharacteristic
394C2×C42C22×D4711yescharacteristic
395#107—474no—
396#107—1881no—
397#106—474no—
398#106—1881no—
399C4⋊C8—474no—
400C4⋊C8—942no—
401#102—474no—
402#102—474no—
403#102—1881no—
404#102—1881no—
405#97—1881no—
406#97—1881no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D4712yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4C2×C22⋊D4711yescharacteristic
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C2×C22⋊D4711yescharacteristic
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C2×C22⋊D4711yescharacteristic
439C22×C4C23×D4711yescharacteristic
440C2×C8—472no—
441C2×C8—472no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—472no—
460C4⋊C4—472no—
461C4⋊C4—944no—
462C4⋊C4—944no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—1882no—
470C4⋊C4—1882no—
471C4⋊C4C2×C4×D4712yes—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—1882no—
487C42—1882no—
488C42C2×C4×D4712yes—
489C42#1062535012yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C22⋊D4711yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4C4×C22⋊D4712yes—
540C2×C4C4×C22⋊D4712yes—
541C2×C4#1309405012yes—
542C2×C4C22×C4×D4711yescharacteristic
543C2×C4C2×#1062535011yescharacteristic
544C2×C4#1309407011yescharacteristic
545C2×C4#1309407111yescharacteristic
546C2×C4#1309407111yescharacteristic
547C2×C4#1309407111yescharacteristic
548C8—942no—
549C8—942no—
550C8—1881no—
551C8—1881no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×C4×C22⋊D4711yescharacteristic
560C22#1568969411yescharacteristic
561C22#1568970111yescharacteristic
562C22C2×#1309405011yescharacteristic
563C22#1568992111yescharacteristic
564C22#1568992111yescharacteristic
565C22#1568992111yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—3761no—
576C4—3761no—
577C4—1882no—
578C4—1882no—
579C4#1568960011yescharacteristic
580C4#1568960311yescharacteristic
581C4#1568973511yescharacteristic
582C4#1568974411yescharacteristic
583C2—82no—
584C2#2024607011yescharacteristic
585C2#2025218611yescharacteristic
586C2C4×#1309405011yescharacteristic
587C2#2025457411yescharacteristic
588C2#2025462411yescharacteristic
589C2#2025474411yescharacteristic
590C2#2025478411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2293C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#2293D47
hypocenterC47#22120
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C4×#13094016C2semidirect
3. #20244345C2semidirect
4. #20244156C2semidirect
5. #20244128C2semidirect
6. #20244123C2semidirect
11. #20243963C2semidirect
12. #20243956C2semidirect
14. #20243241C2semidirect
17. #15689423C4semidirect
355. C47#22120semidirect

Maximal subgroups

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

subgroupindex
C2×#156894232
C4×#130940162
#202443452
#202441562
#202441282
#202441232
#202440882
#202440122
#202440052
#202439632
#202439562
#202433982
#202432412
C47×#22932
#2212047