Finite Group Databasebeta

← #26252338

Subgroups — #26252338

Order 12032.

1378 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic, complemented
2C2×#15689401C212yesmaximal, complemented
3#121×Dic47C211yesmaximal, characteristic
4#20244715C212yesmaximal, complemented
5#20244522C212yesmaximal, complemented
6#20244338C212yesmaximal, complemented
7#20243706C211yesmaximal, characteristic
8#20243578C211yesmaximal, characteristic, complemented
9#20243576C211yesmaximal, characteristic, complemented
10#20243366C211yesmaximal, characteristic, complemented
11C47×#1874C211yesmaximal, characteristic
12C23×Dic94—21no—
13C2×#13094023—28no—
14C2×#13094023—21no—
15C2×#13094023—21no—
16C2×#13094023C2212yes—
17C2×#13094023C2212yes—
18C2×D4×Dic47—21no—
19C2×D4×Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4—21no—
22C22×C4×Dic47C2211yescharacteristic
23#15689401—28no—
24C2×#13093958C2212yes—
25C2×#13093957C2212yes—
26C2×Dic47⋊C8C2212yes—
27C2×#13093945C2212yes—
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29#15689026C2211yescharacteristic
30#15689022C2211yescharacteristic
31#15689016—21no—
32#15689003—21no—
33#15688983C2211yescharacteristic
34C4⋊C4×Dic47C2211yescharacteristic
35C22⋊C4×Dic47—21no—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39#15688877C2212yes—
40C2×#121×C47C2211yescharacteristic
41C22×C4⋊C4×C47C2211yescharacteristic
42C2×#102×C47C2212yes—
43C2×C22⋊C8×C47C2211yescharacteristic
44C47×#340C2212yes—
45C22×Dic94—28no—
46C22×Dic94—22no—
47C22×Dic94—22no—
48C22×Dic94—44no—
49C22×Dic94—44no—
50C23×Dic47—21no—
51C23×Dic47—21no—
52C23×Dic47D411yescharacteristic
53C23×Dic47D411yescharacteristic
54C22×C47⋊C8C2312yes—
55#13094023—28no—
56#13094023—28no—
57#13094023—24no—
58#13094023—24no—
59#13094023—24no—
60#13094023D418yescomplemented
61D4×Dic47—24no—
62D4×Dic47—44no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—24no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2312yes—
69C2×Dic47⋊C4—28no—
70C2×Dic47⋊C4—22no—
71C2×Dic47⋊C4—22no—
72C2×C22⋊Dic47—21no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C4×Dic47—24no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47D411yescharacteristic
79C2×C4×Dic47D411yescharacteristic
80C2×C4×Dic47C2311yescharacteristic
81C2×C4×Dic47C2311yescharacteristic
82#13093958—24no—
83#13093957—24no—
84Dic47⋊C8—24no—
85#13093945—24no—
86C22⋊(C47⋊C8)—22no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C22×D4×C47—21no—
91C22×D4×C47C2311yescharacteristic
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94C22×C8×C47C2312yes—
95#121×C47—22no—
96#121×C47Q814yescomplemented
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—24no—
99C2×C4⋊C4×C47C2312yes—
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C22⋊C4×C47—21no—
102#102×C47—24no—
103C22⋊C8×C47—22no—
104C24×C47—21no—
105C24×C47C2×D411yescharacteristic
106C2×D4×C47—24no—
107C2×D4×C47—22no—
108C2×D4×C47—21no—
109C2×D4×C47—22no—
110C2×D4×C47—42no—
111C2×D4×C47—42no—
112C2×D4×C47C2×Q812yescomplemented
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—24no—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47—42no—
119C22×C4×C47—42no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×Q812yes—
122C22×C4×C47#4112yes—
123C22×C4×C47C2411yescharacteristic
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—24no—
130C22×Dic47—22no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—42no—
134C22×Dic47—42no—
135C22×Dic47—42no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47D814yes—
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C2×Dic94—28no—
144C2×Dic94—28no—
145C2×Dic94—44no—
146C2×Dic94—432no—
147C2×Dic94—44no—
148C2×Dic94—44no—
149C2×Dic94—44no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47C2×D414yes—
157C4⋊C4×C47C2×Q812yescomplemented
158C22⋊C4×C47—24no—
159C22⋊Dic47—42no—
160C22⋊Dic47—42no—
161C22⋊Dic47—42no—
162C22⋊Dic47—42no—
163C4⋊Dic47—24no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—42no—
168C4⋊Dic47C2×D414yes—
169Dic47⋊C4—28no—
170Dic47⋊C4—28no—
171C4×Dic47—22no—
172C4×Dic47—22no—
173C4×Dic47—22no—
174C4×Dic47—42no—
175C4×Dic47C2×D414yes—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—21no—
183C23×C47—22no—
184C23×C47—41no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47C22⋊Q812yes—
191C23×C47C2×D812yes—
192C23×C47#13712yes—
193C23×C47C22×D411yescharacteristic
194D4×C47—24no—
195D4×C47—48no—
196D4×C47—42no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—24no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—22no—
203C2×C4×C47—48no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—41no—
208C2×C4×C47C22⋊Q812yes—
209C2×C4×C47C22×D412yes—
210C2×C4×C47C22×Q811yescharacteristic
211C2×C4×C47C2×#4111yescharacteristic
212C2×C4×C472−1+412yes—
213C2×Dic47—28no—
214C2×Dic47—21no—
215C2×Dic47—21no—
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—21no—
219C2×Dic47—21no—
220C2×Dic47—28no—
221C2×Dic47—24no—
222C2×Dic47—48no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—84no—
232C2×Dic47#12012yes—
233C2×Dic47#12012yes—
234C2×Dic47C2×D812yes—
235C2×Dic47C2×D812yes—
236C2×Dic47AGL(1,ℤ/8)12yes—
237C2×Dic47#13712yes—
238Dic94—416no—
239Dic94—416no—
240C8×C47—42no—
241C47⋊C8—42no—
242#24298—471nomaximal, Sylow
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—81no—
253C22×C47—81no—
254C22×C47—81no—
255C22×C47—81no—
256C22×C47#47414yescomplemented
257C22×C47C2×#12011yescharacteristic
258C22×C47C2×C22⋊Q811yescharacteristic
259C22×C47#53611yescharacteristic
260C22×C47C22×D811yescharacteristic
261C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C47C2×#13711yescharacteristic
263C22×C47C2×#13711yescharacteristic
264C4×C47—22no—
265C4×C47—44no—
266C4×C47—42no—
267C4×C47#54812yes—
268C4×C47D4×Q812yes—
269Dic47—22no—
270Dic47—22no—
271Dic47—22no—
272Dic47—44no—
273Dic47—81no—
274Dic47—81no—
275Dic47#44714yes—
276C22×C4⋊Q8—471no—
277C2×#474—472no—
278C4×#121—471no—
279#2057—472no—
280#2023—472no—
281#1988—472no—
282#1874—471no—
283#1874D4711yescharacteristic
284#1852—471no—
285#1852—471no—
286#1797—471no—
287C2×C47—24no—
288C2×C47—82no—
289C2×C47#240911yescharacteristic
290C2×C47C2×#44711yescharacteristic
291C2×C47#300411yescharacteristic
292C2×C47C2×#47411yescharacteristic
293C2×C47#307911yescharacteristic
294C2×C47#322611yescharacteristic
295C2×C47#323911yescharacteristic
296C23×Q8—941no—
297C2×C4⋊Q8—472no—
298C2×C4⋊Q8—472no—
299C2×C4⋊Q8—941no—
300C2×C4⋊Q8—941no—
301C2×C4⋊Q8—948no—
302C2×#121C2×D4711yescharacteristic
303C2×C4×D4—471no—
304C2×C4×D4—941no—
305C22×C4⋊C4—471no—
306C22×C4⋊C4—941no—
307C22×C4⋊C4C2×D4711yescharacteristic
308C22×C42—471no—
309#474—948no—
310C2×#107—472no—
311C2×#107—472no—
312C2×C4⋊C8—472no—
313C2×#102—472no—
314C2×#102C2×D4712yes—
315C2×C22⋊C8—471no—
316C2×C22⋊C8C2×D4711yescharacteristic
317#390—471no—
318#389—471no—
319#388—941no—
320#386—941no—
321#380—471no—
322C4×C4⋊C4—471no—
323C4×C22⋊C4—941no—
324#340—472no—
325#340—472no—
326#340—472no—
327#340C2×D4712yes—
328#336—472no—
329C47#2429811yescharacteristic, complemented, Sylow
330C22×Q8—942no—
331C22×Q8—942no—
332C22×Q8—948no—
333C22×Q8—1884no—
334C22×Q8—1884no—
335C22×D4—21no—
336C22×D4C22×D4711yescharacteristic
337C23×C4—21no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—941no—
341C23×C4—941no—
342C23×C4C22×D4711yescharacteristic
343C22×C8—472no—
344C22×C8C22×D4712yes—
345C4⋊Q8—478no—
346C4⋊Q8—944no—
347C4⋊Q8—944no—
348C4⋊Q8—948no—
349C4⋊Q8—948no—
350C4⋊Q8—944no—
351#121—22no—
352#121Dic9414yescomplemented
353C4×D4—944no—
354C4×D4—1884no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—942no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—948no—
364C2×C4⋊C4—941no—
365C2×C4⋊C4—941no—
366C2×C4⋊C4C22×D4711yescharacteristic
367C2×C4⋊C4C22×D4712yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—941no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—941no—
372C2×C22⋊C4—941no—
373C2×C42—471no—
374C2×C42—471no—
375C2×C42—471no—
376C2×C42—471no—
377C2×C42—944no—
378C2×C42—941no—
379#107—944no—
380#107—944no—
381C4⋊C8—944no—
382#102—24no—
383#102—944no—
384C22⋊C8—22no—
385C22⋊C8—942no—
386#95—941no—
387#95—941no—
388#95—941no—
389C24—21no—
390C24D4×D4711yescharacteristic
391C2×Q8—948no—
392C2×Q8—948no—
393C2×Q8—1884no—
394C2×Q8—1884no—
395C2×Q8—1884no—
396C2×Q8—18832no—
397C2×Q8—1884no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C2×Dic9412yescomplemented
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—471no—
413C22×C4—474no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—471no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—944no—
423C22×C4—942no—
424C22×C4—941no—
425C22×C4—941no—
426C22×C4—1882no—
427C22×C4—1882no—
428C22×C4—1882no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4C2×Dic9412yes—
432C22×C4D4×D4711yescharacteristic
433C22×C4#1062535212yes—
434C22×C4C23×D4711yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—942no—
439C2×C8—942no—
440C2×C8—942no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—474no—
445C4⋊C4—948no—
446C4⋊C4—948no—
447C4⋊C4—944no—
448C4⋊C4—942no—
449C4⋊C4—942no—
450C4⋊C4—942no—
451C4⋊C4—1882no—
452C4⋊C4C2×Dic9412yescomplemented
453C4⋊C4D4×D4714yes—
454C22⋊C4—24no—
455C22⋊C4—1882no—
456C22⋊C4—1882no—
457C22⋊C4—1882no—
458C22⋊C4—1882no—
459C42—474no—
460C42—942no—
461C42—942no—
462C42—942no—
463C42—1882no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#1309400112yes—
476C23D8×D4712yes—
477C23#1309405312yes—
478C23C2×D4×D4711yescharacteristic
479Q8—18816no—
480Q8—18816no—
481D4—24no—
482D4—48no—
483D4—42no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—472no—
496C2×C4—472no—
497C2×C4—472no—
498C2×C4—472no—
499C2×C4—472no—
500C2×C4—472no—
501C2×C4—948no—
502C2×C4—948no—
503C2×C4—941no—
504C2×C4—941no—
505C2×C4—941no—
506C2×C4—941no—
507C2×C4—941no—
508C2×C4—941no—
509C2×C4—944no—
510C2×C4—3764no—
511C2×C4—1881no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1888no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4#1309400112yes—
521C2×C4C2×D4×D4712yes—
522C2×C4C22×Dic9411yescharacteristic
523C2×C4C2×#1062535211yescharacteristic
524C2×C4#1309407112yes—
525C8—42no—
526C8—1882no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22#1568940114yescomplemented
541C22#120×D4711yescharacteristic
542C22C2×#1309400111yescharacteristic
543C22#1568969511yescharacteristic
544C22C2×D8×D4711yescharacteristic
545C22AGL(1,ℤ/8)×D4711yescharacteristic
546C22#137×D4711yescharacteristic
547C22C2×#1309405311yescharacteristic
548C4—22no—
549C4—44no—
550C4—42no—
551C4—474no—
552C4—942no—
553C4—942no—
554C4—942no—
555C4—3761no—
556C4—3761no—
557C4—1884no—
558C4#1568977912yes—
559C4D4×Dic9412yes—
560C2—24no—
561C2—82no—
562C2#2024675011yescharacteristic
563C2#447×D4711yescharacteristic
564C2#2025261911yescharacteristic
565C2C2×#1568940111yescharacteristic
566C2#2025319911yescharacteristic
567C2#2025450711yescharacteristic
568C2#2025461611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1874C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenter#1874D47
hypocenterC47#24298
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
2. C2×#15689401C2semidirect
4. #20244715C2semidirect
5. #20244522C2semidirect
6. #20244338C2semidirect
8. #20243578C2semidirect
9. #20243576C2semidirect
10. #20243366C2semidirect
60. #13094023D4semidirect
96. #121×C47Q8semidirect
112. C2×D4×C47C2×Q8semidirect
157. C4⋊C4×C47C2×Q8semidirect
256. C22×C47#474semidirect
329. C47#24298semidirect
352. #121Dic94semidirect
404. C2×D4C2×Dic94semidirect
452. C4⋊C4C2×Dic94semidirect
540. C22#15689401semidirect

Maximal subgroups

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

subgroupindex
C22×#130940232
C2×#156894012
#121×Dic472
#202447152
#202445222
#202443382
#202437062
#202435782
#202435762
#202433662
C47×#18742
#2429847