Finite Group Databasebeta

← #26143277

Subgroups — #26143277

Order 12032.

1318 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689043C211yesmaximal, characteristic
2C2×#15689028C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20241658C211yesmaximal, characteristic
5#20241654C212yesmaximal
6#20241647C211yesmaximal, characteristic
7#20241644C212yesmaximal
8#20241623C212yesmaximal
9#20241613C212yesmaximal
10#20241613C211yesmaximal, characteristic
11C47×#1441C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×C4×Dic47C2211yescharacteristic
16C22×C4×Dic47C2211yescharacteristic
17C22×C4×Dic47C2211yescharacteristic
18#15689043C418yescomplemented
19#15689028—24no—
20C4⋊C4×Dic47—24no—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C2×#95×C47C2212yes—
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2212yes—
38C22×Dic94—22no—
39C22×Dic94—22no—
40C22×Dic94—42no—
41C22×Dic94—42no—
42C22×Dic94C2×C414yes—
43C23×Dic47C2312yes—
44C23×Dic47C2312yes—
45C23×Dic47C2311yescharacteristic
46C23×Dic47C2311yescharacteristic
47C23×Dic47C2311yescharacteristic
48C23×Dic47C2311yescharacteristic
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—22no—
51C2×C4⋊Dic47—22no—
52C2×Dic47⋊C4—42no—
53C2×Dic47⋊C4—42no—
54C2×Dic47⋊C4C2×C418yes—
55C2×C4×Dic47—22no—
56C2×C4×Dic47—22no—
57C2×C4×Dic47—22no—
58C2×C4×Dic47—42no—
59C2×C4×Dic47—42no—
60C2×C4×Dic47D414yes—
61C2×C4×Dic47Q814yescomplemented
62C47⋊#95—22no—
63C47⋊#95—22no—
64C47⋊#95—24no—
65C47⋊#95—24no—
66C47⋊#95—24no—
67C47⋊#95—24no—
68C47⋊#95—42no—
69C47⋊#95—42no—
70C47⋊#95—42no—
71C47⋊#95—42no—
72C47⋊#95C2×C414yes—
73C47⋊#95C2×C418yes—
74C23×C4×C47C2312yes—
75C23×C4×C47C2312yes—
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—42no—
83C2×C4⋊C4×C47C2×C414yes—
84#95×C47—22no—
85#95×C47—24no—
86#95×C47—42no—
87C24×C47C2411yescharacteristic
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C2×D412yes—
116C22×C4×C47C2×Q812yes—
117C22×C4×C47#4112yes—
118C22×C4×C47#4112yes—
119C22×Dic47—22no—
120C22×Dic47—22no—
121C22×Dic47—21no—
122C22×Dic47—22no—
123C22×Dic47—21no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47C22×C414yes—
146C22×Dic47C22×C414yes—
147C22×Dic47C2×D412yes—
148C22×Dic47C2×D412yes—
149C22×Dic47C2×Q812yes—
150C22×Dic47C2×Q812yes—
151C22×Dic47#4112yes—
152C22×Dic47#4112yes—
153C2×Dic94—28no—
154C2×Dic94—48no—
155C2×Dic94—44no—
156C2×Dic94—44no—
157C2×Dic94—48no—
158C4⋊C4×C47—28no—
159C4⋊C4×C47—44no—
160C4⋊Dic47—28no—
161Dic47⋊C4—48no—
162C4×Dic47—28no—
163C4×Dic47—44no—
164C4×Dic47—44no—
165C23×C47C23×C411yescharacteristic
166C23×C47C22×D411yescharacteristic
167C23×C47C22×Q811yescharacteristic
168C23×C47C2×#4111yescharacteristic
169C23×C47C2×#4111yescharacteristic
170C23×C47C2×#4111yescharacteristic
171C23×C47C2×#4111yescharacteristic
172C23×C47H(5,𝔽2)12yes—
173C23×C47H(5,𝔽2)12yes—
174C23×C472−1+412yes—
175C23×C472−1+412yes—
176C2×C4×C47—22no—
177C2×C4×C47—22no—
178C2×C4×C47—24no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—24no—
186C2×C4×C47—22no—
187C2×C4×C47—24no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47C4×D414yes—
206C2×C4×C47C4×Q814yes—
207C2×Dic47—22no—
208C2×Dic47—22no—
209C2×Dic47—24no—
210C2×Dic47—24no—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—24no—
216C2×Dic47—24no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—41no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47C22⋊Q814yes—
241Dic94—416no—
242#14320—471nomaximal, Sylow
243C22×C47C2×C4×D411yescharacteristic
244C22×C47C2×C4×Q811yescharacteristic
245C22×C47C4×#4111yescharacteristic
246C22×C47#51812yes—
247C22×C47#51912yes—
248C22×C47C2×C22⋊Q811yescharacteristic
249C22×C47#52511yescharacteristic
250C22×C47#52911yescharacteristic
251C22×C47#53311yescharacteristic
252C22×C47#53411yescharacteristic
253C22×C47#53511yescharacteristic
254C22×C47#53611yescharacteristic
255C22×C47#53611yescharacteristic
256C22×C47#53812yes—
257C22×C47#53912yes—
258C22×C47#53912yes—
259C22×C47#54112yes—
260C22×C47#54212yes—
261C22×C47#54212yes—
262C22×C47#54212yes—
263C22×C47#54212yes—
264C22×C47#54412yes—
265C22×C47#54412yes—
266C4×C47—24no—
267C4×C47—24no—
268C4×C47—42no—
269C4×C47—82no—
270C4×C47—82no—
271Dic47—24no—
272Dic47—44no—
273Dic47—44no—
274Dic47—42no—
275Dic47—42no—
276Dic47—42no—
277C2×#395—471no—
278C2×#391—471no—
279C2×C4×C4⋊C4—471no—
280#1446—471no—
281#1445—472no—
282#1442—472no—
283#1442—471no—
284#1441D4711yescharacteristic
285#1436—472no—
286#1434—471no—
287#1434—472no—
288C2×C47C4×C22⋊Q811yescharacteristic
289C2×C47#232111yescharacteristic
290C2×C47#232711yescharacteristic
291C2×C47#233112yes—
292C2×C47#233212yes—
293C2×C47#262711yescharacteristic
294C2×C47#262811yescharacteristic
295C2×C47#263211yescharacteristic
296C2×C47#263811yescharacteristic
297C2×C47#265012yes—
298C2×C47#266612yes—
299C23×Q8—471no—
300C22×C4⋊C4—472no—
301C22×C4⋊C4—471no—
302C22×C4⋊C4C2×D4711yescharacteristic
303C22×C4⋊C4C2×D4711yescharacteristic
304C22×C42—471no—
305C22×C42—471no—
306C22×C42—471no—
307#395—478no—
308#391—944no—
309C4×C4⋊C4—944no—
310C2×#95—472no—
311C2×#95—471no—
312C2×#95—472no—
313C2×#95—472no—
314C2×#95—472no—
315C2×#95—472no—
316C2×#95—471no—
317C2×#95—471no—
318C2×#95—472no—
319C2×#95—472no—
320C2×#95—472no—
321C2×#95—472no—
322C2×#95C2×D4712yes—
323C2×#95C2×D4711yescharacteristic
324C2×#95C2×D4712yes—
325C47#1432011yescharacteristic, complemented, Sylow
326C22×Q8—474no—
327C22×Q8—942no—
328C22×Q8—942no—
329C22×Q8—1882no—
330C22×Q8—1882no—
331C23×C4—471no—
332C23×C4—472no—
333C23×C4—472no—
334C23×C4—471no—
335C23×C4—471no—
336C23×C4—471no—
337C23×C4C22×D4712yes—
338C23×C4C22×D4712yes—
339C23×C4C22×D4711yescharacteristic
340C23×C4C22×D4711yescharacteristic
341C23×C4C22×D4711yescharacteristic
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—478no—
347C2×C4⋊C4—942no—
348C2×C4⋊C4—942no—
349C2×C4⋊C4—942no—
350C2×C4⋊C4—1882no—
351C2×C4⋊C4—1882no—
352C2×C4⋊C4C4×D4714yes—
353C2×C42—474no—
354C2×C42—474no—
355C2×C42—942no—
356C2×C42—942no—
357C2×C42—942no—
358C2×C42—1882no—
359C2×C42—1882no—
360#95—22no—
361#95—24no—
362#95—42no—
363#95—474no—
364#95—478no—
365#95—944no—
366#95—944no—
367#95—942no—
368#95—944no—
369#95—944no—
370#95—942no—
371#95—1882no—
372#95—1882no—
373#95—1882no—
374#95—1882no—
375C24C23×D4711yescharacteristic
376C2×Q8—948no—
377C2×Q8—1888no—
378C2×Q8—1884no—
379C2×Q8—1884no—
380C2×Q8—1888no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—472no—
406C22×C4—472no—
407C22×C4—472no—
408C22×C4—474no—
409C22×C4—474no—
410C22×C4—472no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—941no—
416C22×C4—942no—
417C22×C4—941no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—941no—
421C22×C4—941no—
422C22×C4—941no—
423C22×C4—941no—
424C22×C4—942no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4C2×C4×D4712yes—
440C22×C4C2×C4×D4712yes—
441C22×C4C2×C4×D4712yes—
442C22×C4D4×D4712yes—
443C22×C4#1062535212yes—
444C22×C4#1062535212yes—
445C22×C4Q8×D4712yes—
446C4⋊C4—28no—
447C4⋊C4—44no—
448C4⋊C4—948no—
449C4⋊C4—1888no—
450C42—948no—
451C42—1884no—
452C42—1884no—
453C23C22×C4×D4711yescharacteristic
454C23C2×D4×D4711yescharacteristic
455C23C2×Q8×D4711yescharacteristic
456C23#41×D4711yescharacteristic
457C23#41×D4711yescharacteristic
458C23C2×#1062535211yescharacteristic
459C23C2×#1062535211yescharacteristic
460C23#1309406912yes—
461C23#1309407012yes—
462C23#1309407112yes—
463C23#1309407212yes—
464Q8—18816no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—474no—
495C2×C4—944no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—944no—
499C2×C4—944no—
500C2×C4—944no—
501C2×C4—944no—
502C2×C4—944no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—944no—
512C2×C4—1882no—
513C2×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1881no—
518C2×C4—1882no—
519C2×C4—1881no—
520C2×C4—1882no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4(C22⋊D47)⋊C414yes—
529C2×C4#1309398914yes—
530C22C2×(C22⋊D47)⋊C411yescharacteristic
531C22C2×#1309398911yescharacteristic
532C22#1568959111yescharacteristic
533C22#1568959512yes—
534C22#1568959912yes—
535C22C22⋊Q8×D4711yescharacteristic
536C22#1568963311yescharacteristic
537C22#1568965411yescharacteristic
538C22#1568967511yescharacteristic
539C22#1568968411yescharacteristic
540C22#1568968811yescharacteristic
541C22#1568969311yescharacteristic
542C22#1568969811yescharacteristic
543C22#1568971012yes—
544C22#1568971412yes—
545C22#1568972112yes—
546C22#1568973312yes—
547C22#1568974012yes—
548C22#1568974112yes—
549C22#1568974312yes—
550C22#1568974612yes—
551C22#1568975012yes—
552C22#1568975312yes—
553C4—24no—
554C4—24no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4—944no—
559C4—1884no—
560C4—1882no—
561C4—1884no—
562C4—1882no—
563C4—1882no—
564C2#2024591611yescharacteristic
565C2#2024605111yescharacteristic
566C2#2024610611yescharacteristic
567C2#2024613512yes—
568C2#2024614212yes—
569C2#2024920211yescharacteristic
570C2#2024921111yescharacteristic
571C2#2024925311yescharacteristic
572C2#2024931311yescharacteristic
573C2#2024940712yes—
574C2#2024956812yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1441C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1441D47
hypocenterC47#14320
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
18. #15689043C4semidirect
61. C2×C4×Dic47Q8semidirect
325. C47#14320semidirect

Maximal subgroups

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

subgroupindex
C2×#156890432
C2×#156890282
C2×C4⋊C4×Dic472
#202416582
#202416542
#202416472
#202416442
#202416232
#202416132
C47×#14412
#1432047