Finite Group Databasebeta

← #26182212

Subgroups — #26182212

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689603C211yesmaximal, characteristic
2#20252472C212yesmaximal, complemented
3#20241728C212yesmaximal, complemented
4#20241727C212yesmaximal, complemented
5#20241722C211yesmaximal, characteristic, complemented
6#20241721C212yesmaximal, complemented
7#20241707C212yesmaximal, complemented
8#20241701C212yesmaximal, complemented
9C47×#1464C211yesmaximal, characteristic, complemented
10C22×#10625350C411yescharacteristic
11#15689603—24no—
12#15689603—24no—
13C2×C4×Dic94C411yescharacteristic
14C2×C4×Dic94C2211yescharacteristic
15C2×C4×D188C411yescharacteristic
16C2×C4×D188C2211yescharacteristic
17C2×(C22⋊D47)⋊C4C411yescharacteristic
18C2×(C22⋊D47)⋊C4C411yescharacteristic
19C2×(C22⋊D47)⋊C4C2211yescharacteristic
20C2×(C22⋊D47)⋊C4C2211yescharacteristic
21C2×#13093982C2211yescharacteristic
22C2×#13093982C2211yescharacteristic
23C22×C4⋊Dic47C411yescharacteristic
24C2×C4⋊C4×D47C411yescharacteristic
25C2×C4⋊C4×D47C411yescharacteristic
26#15689252—24no—
27#15689251—24no—
28#15689180C2212yes—
29#15689177C2214yes—
30#15689173C2214yes—
31#15688849C2212yes—
32#15688847C2212yes—
33#15688846C2214yes—
34#15688846C2214yes—
35C2×#117×C47C2211yescharacteristic
36C47×#432C2212yes—
37C47×#324C2212yes—
38C47×#323C2212yes—
39C2×#10625350—24no—
40C2×#10625350—24no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47C2×C411yescharacteristic
46C22×C22⋊D47C2×C411yescharacteristic
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2×C411yescharacteristic
54C22×C4×D47C2×C411yescharacteristic
55C4×Dic94—22no—
56C4×Dic94—22no—
57C4×Dic94D414yes—
58C4×Dic94D414yes—
59C4×D188—22no—
60C4×D188—22no—
61C4×D188—22no—
62C4×D188—22no—
63(C22⋊D47)⋊C4—24no—
64(C22⋊D47)⋊C4—24no—
65(C22⋊D47)⋊C4—24no—
66(C22⋊D47)⋊C4—24no—
67#13093982—24no—
68#13093982—24no—
69C2×C4⋊Dic47—22no—
70C2×C4⋊Dic47—22no—
71C2×C4⋊Dic47C2×C411yescharacteristic
72C2×C4⋊Dic47C2×C411yescharacteristic
73C2×C4⋊Dic47C2×C411yescharacteristic
74C2×C4⋊Dic47C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×Dic47⋊C4C2×C411yescharacteristic
79C4⋊C4×D47—24no—
80C4⋊C4×D47—24no—
81C2×(C2×D47)⋊C4C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2311yescharacteristic
84C2×(C2×D47)⋊C4C2311yescharacteristic
85C2×C4×Dic47C2×C411yescharacteristic
86C2×C4×Dic47C2×C411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89C4⋊(C47⋊C8)—22no—
90Dic47⋊C8—24no—
91Dic47⋊C8—24no—
92C22⋊(C47⋊C8)C2314yes—
93#13093935—22no—
94#13093934—24no—
95C8×Dic47—24no—
96C23×C4×C47C2×C411yescharacteristic
97#117×C47—22no—
98#117×C47—22no—
99C2×C4⋊C4×C47C2×C411yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C22⋊C4×C47C2×C411yescharacteristic
102C2×C22⋊C4×C47C2311yescharacteristic
103C2×C42×C47C2×C411yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106C22⋊C8×C47C2314yes—
107#97×C47—22no—
108C24×C47C22×C411yescharacteristic
109C23×D47C22×C411yescharacteristic
110C23×D47C22×C411yescharacteristic
111C22×C4×C47—22no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47C22×C411yescharacteristic
116C22×C4×C47C22×C411yescharacteristic
117C22×C4×C47C22×C411yescharacteristic
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C22×C411yescharacteristic
120C22×C4×C47C2411yescharacteristic
121C2×C22⋊D47—22no—
122C2×C22⋊D47—22no—
123C2×C22⋊D47—22no—
124C2×C22⋊D47—22no—
125C2×C22⋊D47—22no—
126C2×C22⋊D47M1614yes—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—81no—
130C22×Dic47—81no—
131C22×Dic47C22×C411yescharacteristic
132C22×Dic47C22×C411yescharacteristic
133C22×Dic47C22×C411yescharacteristic
134C22×Dic47C22×C411yescharacteristic
135C22×Dic47C22×C411yescharacteristic
136C22×Dic47C22×C411yescharacteristic
137#10625350—216no—
138#10625350—216no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—21no—
142C2×D188—21no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—81no—
153C2×C4×D47—81no—
154C2×C4×D47—81no—
155C2×C4×D47—81no—
156C2×C4×D47M1614yes—
157C2×Dic94—22no—
158C2×Dic94—22no—
159C2×Dic94C22⋊C412yes—
160C2×Dic94C22⋊C412yes—
161C2×C8×C47—24no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—22no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47C2×D412yes—
169C42×C47C2×D412yes—
170(C2×D47)⋊C4—22no—
171(C2×D47)⋊C4—22no—
172(C2×D47)⋊C4—22no—
173(C2×D47)⋊C4—22no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—22no—
177C4⋊Dic47—22no—
178C4⋊Dic47C22⋊C412yes—
179C4⋊Dic47C22⋊C412yes—
180Dic47⋊C4—22no—
181Dic47⋊C4—22no—
182Dic47⋊C4C22⋊C414yes—
183Dic47⋊C4C22⋊C414yes—
184C4×Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47C2×D414yes—
187C4×Dic47C2×D414yes—
188C2×C47⋊C8—24no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47C2×M1612yes—
194C23×C47C23×C411yescharacteristic
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47C2×M1612yes—
204C22×D47C2×M1612yes—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47C2×C22⋊C412yes—
218C2×C4×C47C2×C22⋊C411yescharacteristic
219C2×C4×C47C2×C22⋊C411yescharacteristic
220C2×C4×C47C2×M1612yes—
221C2×C4×C47C22×D412yes—
222C22⋊D47—28no—
223C22⋊D47—28no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—82no—
233C2×Dic47—82no—
234C2×Dic47C2×C22⋊C412yes—
235C2×Dic47C2×C22⋊C412yes—
236C2×Dic47C2×C22⋊C412yes—
237C2×Dic47C2×C22⋊C412yes—
238C2×Dic47C2×M1612yes—
239C2×Dic47C2×M1612yes—
240D188—24no—
241D188—24no—
242C4×D47—28no—
243C4×D47—28no—
244C4×D47—82no—
245C4×D47—82no—
246C4×D47—82no—
247C4×D47—82no—
248Dic94—24no—
249Dic94—24no—
250C8×C47—44no—
251C47⋊C8—44no—
252#17372—471nomaximal, Sylow
253C22×C47—21no—
254C22×C47—21no—
255C22×C47—22no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47C22×C22⋊C411yescharacteristic
261C22×C47C22×M1611yescharacteristic
262C22×C47C22×M1611yescharacteristic
263C2×D47—22no—
264C2×D47—22no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—41no—
268C2×D47—41no—
269C2×D47—41no—
270C2×D47—41no—
271C2×D47—42no—
272C2×D47—42no—
273C4×C47—21no—
274C4×C47—21no—
275C4×C47—22no—
276C4×C47—22no—
277C4×C47—22no—
278C4×C47—41no—
279C4×C47—41no—
280Dic47—22no—
281Dic47—22no—
282Dic47—22no—
283Dic47—41no—
284Dic47—41no—
285Dic47#40714yes—
286C2×#520—471no—
287#2983—472no—
288#1466—472no—
289#1466—472no—
290#1464—472no—
291#1464—471no—
292#1464D4711yescharacteristic, complemented
293#1460—472no—
294#1458—472no—
295C2×C47—21no—
296C2×C47—21no—
297C2×C47—42no—
298C2×C47C2×#40711yescharacteristic
299C2×C47#288411yescharacteristic
300C2×C47#288811yescharacteristic
301D47—42no—
302D47—42no—
303C22×#41—471no—
304#520—944no—
305#520—944no—
306C2×C4×Q8—471no—
307C2×C4×Q8—471no—
308C2×C4×D4—471no—
309C2×C4×D4—471no—
310C2×C4×D4—471no—
311C2×C4×D4—471no—
312C2×C4×D4—471no—
313C2×C4×D4—471no—
314C2×#117—471no—
315C2×#117—471no—
316C2×#117C2×D4711yescharacteristic
317C22×C4⋊C4—471no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4—471no—
320#446—944no—
321#446—944no—
322#432—472no—
323#432—474no—
324#432C2×D4712yes—
325#431—474no—
326#324—472no—
327#324C2×D4712yes—
328#323—472no—
329#323—474no—
330#323—474no—
331#323C2×D4712yes—
332C47#1737211yescharacteristic, complemented, Sylow
333C2×#41—944no—
334C2×#41—944no—
335C2×#41—942no—
336C2×#41—942no—
337C22×Q8—471no—
338C22×D4—471no—
339C22×D4—471no—
340C22×D4—471no—
341C23×C4—471no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—471no—
348C23×C4—471no—
349C23×C4C4×D4711yescharacteristic
350C4×Q8—474no—
351C4×Q8—474no—
352C4×Q8—942no—
353C4×Q8—942no—
354C4×D4—942no—
355C4×D4—942no—
356C4×D4—944no—
357C4×D4—944no—
358C4×D4—944no—
359C4×D4—944no—
360C4×D4—942no—
361C4×D4—942no—
362#117—22no—
363#117—22no—
364#117—944no—
365#117—944no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—944no—
375C2×C4⋊C4—944no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4C4×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C22⋊C4—471no—
381C2×C22⋊C4—471no—
382C2×C22⋊C4—471no—
383C2×C22⋊C4—471no—
384C2×C22⋊C4C4×D4711yescharacteristic
385C2×C22⋊C4C22×D4711yescharacteristic
386C2×C42—471no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42C4×D4711yescharacteristic
391C2×C42C22×D4711yescharacteristic
392C4⋊C8—22no—
393C4⋊C8—944no—
394C4⋊C8—944no—
395C4⋊C8—942no—
396C22⋊C8—474no—
397C22⋊C8C22×D4714yes—
398#97—22no—
399#97—944no—
400#97—942no—
401C4×C8—944no—
402C24—471no—
403C24—471no—
404C24C2×C4×D4711yescharacteristic
405#41—9416no—
406#41—9416no—
407C2×Q8—472no—
408C2×Q8—472no—
409C2×Q8—942no—
410C2×Q8—942no—
411C2×D4—474no—
412C2×D4—942no—
413C2×D4—942no—
414C2×D4—942no—
415C2×D4—942no—
416C2×D4—942no—
417C2×D4—942no—
418C2×D4—942no—
419C2×D4—941no—
420C2×D4—941no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—471no—
426C22×C4—471no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—474no—
430C22×C4—471no—
431C22×C4—471no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—3761no—
444C22×C4—3761no—
445C22×C4—3761no—
446C22×C4—3761no—
447C22×C4—3761no—
448C22×C4—3761no—
449C22×C4C2×C4×D4711yescharacteristic
450C22×C4C2×C4×D4711yescharacteristic
451C22×C4C2×C4×D4711yescharacteristic
452C22×C4C2×C4×D4711yescharacteristic
453C22×C4C2×C4×D4711yescharacteristic
454C22×C4C23×D4711yescharacteristic
455C2×C8—24no—
456C2×C8—944no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—474no—
460C4⋊C4—474no—
461C4⋊C4—472no—
462C4⋊C4—472no—
463C4⋊C4—942no—
464C4⋊C4—942no—
465C4⋊C4—942no—
466C4⋊C4—942no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—942no—
472C22⋊C4—942no—
473C22⋊C4—942no—
474C22⋊C4—942no—
475C42—21no—
476C42—21no—
477C42—474no—
478C42—474no—
479C42—942no—
480C42—942no—
481C42D4×D4712yes—
482C42D4×D4712yes—
483C23—21no—
484C23—21no—
485C23—41no—
486C23—41no—
487C23—472no—
488C23—472no—
489C23—941no—
490C23—941no—
491C23—941no—
492C23—941no—
493C23—1881no—
494C23—1881no—
495C23—1881no—
496C23—1881no—
497C23M16×D4712yes—
498C23C22×C4×D4711yescharacteristic
499Q8—944no—
500Q8—944no—
501D4—944no—
502D4—944no—
503D4—948no—
504D4—948no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—472no—
520C2×C4—472no—
521C2×C4—472no—
522C2×C4—472no—
523C2×C4—948no—
524C2×C4—948no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—3762no—
530C2×C4—3762no—
531C2×C4—3762no—
532C2×C4—3762no—
533C2×C4—3762no—
534C2×C4—3762no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4C22⋊C4×D4711yescharacteristic
540C2×C4C22⋊C4×D4711yescharacteristic
541C2×C4C22⋊C4×D4712yes—
542C2×C4M16×D4712yes—
543C2×C4C2×D4×D4712yes—
544C8—44no—
545C8—1884no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—942no—
554C22—942no—
555C22—1882no—
556C22—1882no—
557C22—1882no—
558C22—1882no—
559C22—1881no—
560C22—1881no—
561C22—1881no—
562C22—1881no—
563C22C2×C22⋊C4×D4711yescharacteristic
564C22C2×M16×D4711yescharacteristic
565C22C2×M16×D4711yescharacteristic
566C4—22no—
567C4—21no—
568C4—21no—
569C4—22no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—474no—
574C4—942no—
575C4—942no—
576C4—942no—
577C4—1881no—
578C4—1881no—
579C2—21no—
580C2—21no—
581C2—42no—
582C2—1882no—
583C2—1882no—
584C2#407×D4711yescharacteristic
585C2#2025185011yescharacteristic
586C2#2025186511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1464C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625350C4
cosocleC22×C4×C47C24
hypercenter#1464D47
hypocenterC47#17372
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20252472C2semidirect
3. #20241728C2semidirect
4. #20241727C2semidirect
5. #20241722C2semidirect
6. #20241721C2semidirect
7. #20241707C2semidirect
8. #20241701C2semidirect
9. C47×#1464C2semidirect
292. #1464D47semidirect
332. C47#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#156896032
#202524722
#202417282
#202417272
#202417222
#202417212
#202417072
#202417012
C47×#14642
#1737247