Finite Group Databasebeta

← #26219975

Subgroups — #26219975

Order 12032.

1526 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689697C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4C2×#15689120C211yesmaximal, characteristic
5C2×#15689078C211yesmaximal, characteristic, complemented
6#20246012C211yesmaximal, characteristic, complemented
7#20244206C212yesmaximal, complemented
8#20244201C212yesmaximal, complemented
9#20244196C212yesmaximal, complemented
10#20244188C212yesmaximal, complemented
11C22×#103×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14#15689697—24no—
15#15689697C418yescomplemented
16C2×#13094022C2212yes—
17C2×#13094013C2212yes—
18C2×#13094007—22no—
19C2×#13094000—22no—
20C2×#13093983C2211yescharacteristic
21C2×#13093983C2211yescharacteristic
22#15689132—24no—
23#15689122—24no—
24#15689120—24no—
25C2×#13093949C2211yescharacteristic
26C2×#13093949C2211yescharacteristic
27C2×#13093948C2212yes—
28C2×#13093948C2212yes—
29C2×#13093947C2212yes—
30C2×#13093946C2211yescharacteristic
31C2×#13093946C2211yescharacteristic
32C2×#13093944C2212yes—
33#15689078—24no—
34C2×#13093934C2212yes—
35C2×C8×Dic47C2212yes—
36#15689019—22no—
37#15688996—22no—
38#15688992C2212yes—
39#15688988C2212yes—
40C23×Q8×C47C2211yescharacteristic
41C23×C8×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C2×#103×C47—24no—
44C2×#103×C47C2212yes—
45C2×#103×C47C2212yes—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350C2×C414yes—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47—21no—
55C22×C4×D47—21no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329D414yes—
59C22×C47⋊C8C2312yes—
60#13094022—24no—
61#13094022C2×C418yescomplemented
62#13094013—24no—
63#13094013C2×C418yescomplemented
64#13094007—216no—
65#13094000—216no—
66#13093983—22no—
67#13093983—22no—
68#13093983C2×C414yes—
69#13093983D414yes—
70C2×C4⋊Dic47D412yes—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—24no—
74C2×Dic47⋊C4—24no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47D412yes—
77C2×(C2×D47)⋊C4—24no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47C2312yes—
80#13093949—22no—
81#13093949—22no—
82#13093948—24no—
83#13093948—24no—
84#13093947—24no—
85#13093946—22no—
86#13093946—22no—
87#13093944—24no—
88#13093934—24no—
89C8×Dic47—24no—
90C47⋊#95—24no—
91C22×Q8×C47—21no—
92C22×Q8×C47—22no—
93C22×Q8×C47—22no—
94C22×Q8×C47—22no—
95C22×Q8×C47—22no—
96C22×Q8×C47—22no—
97C22×Q8×C47C2×C414yes—
98C22×Q8×C47C2312yes—
99C23×C4×C47—21no—
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102C22×C8×C47—22no—
103C22×C8×C47—22no—
104C22×C8×C47D414yes—
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C2×C4⋊C4×C47—22no—
108C2×C4⋊C4×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47C2312yes—
111#103×C47—24no—
112#103×C47—24no—
113C24×C47C2×D411yescharacteristic
114C23×D47—21no—
115C2×Q8×C47—24no—
116C2×Q8×C47—22no—
117C2×Q8×C47—28no—
118C2×Q8×C47—28no—
119C2×Q8×C47—22no—
120C2×Q8×C47—22no—
121C2×Q8×C47—22no—
122C2×Q8×C47—22no—
123C2×Q8×C47—44no—
124C2×Q8×C47—44no—
125C2×Q8×C47—42no—
126C2×Q8×C47—42no—
127C2×Q8×C47C22×C414yes—
128C22×C4×C47—24no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—82no—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C2×C22⋊D47—24no—
144C2×C22⋊D47—42no—
145C2×C22⋊D47—42no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47C2×D414yes—
150#10625350—28no—
151#10625350—28no—
152#10625350—44no—
153#10625350—44no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188—21no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188C22×C412yes—
160C2×C4×D47—24no—
161C2×C4×D47—42no—
162C2×C4×D47—42no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94C22×C412yes—
169C2×C8×C47—28no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47C2×D412yes—
175C2×C8×C47C2×D412yes—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—42no—
179C22⋊Dic47—42no—
180C22⋊Dic47C22⋊C418yes—
181(C2×D47)⋊C4—216no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47C22⋊C418yes—
187Dic47⋊C4—216no—
188Dic47⋊C4—84no—
189C4×Dic47—42no—
190C4×Dic47C22×C414yes—
191C4×Dic47C2×D414yes—
192#10625329—28no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8C2×D414yes—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47C2×C22⋊C412yes—
199C23×C47#12412yes—
200C23×C47#12712yes—
201C23×C47C22×D411yescharacteristic
202C22×D47—22no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—42no—
206C22×D47—42no—
207Q8×C47—24no—
208Q8×C47—24no—
209Q8×C47—42no—
210Q8×C47—42no—
211Q8×C47—48no—
212C2×C4×C47—24no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—48no—
216C2×C4×C47—42no—
217C2×C4×C47—44no—
218C2×C4×C47—84no—
219C2×C4×C47C2×C22⋊C412yes—
220C2×C4×C47C4×D414yes—
221C2×C4×C47C4×D414yes—
222C2×C4×C47#12114yes—
223C2×C4×C47#12114yes—
224C2×C4×C47#12412yes—
225C2×C4×C47#12712yes—
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C22⋊D47—48no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47C2×C22⋊C418yes—
243D188—22no—
244D188—22no—
245D188—41no—
246D188—41no—
247C4×D47—48no—
248Dic94—22no—
249Dic94—22no—
250Dic94—41no—
251Dic94—41no—
252C8×C47—22no—
253C8×C47—22no—
254C47⋊C8—24no—
255#21205—471nomaximal, Sylow
256C22×C47—24no—
257C22×C47—24no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47#39014yes—
261C22×C47C22×C22⋊C411yescharacteristic
262C22×C47C2×#12411yescharacteristic
263C22×C47C2×#12711yescharacteristic
264C22×C47#57712yes—
265C22×C47#57812yes—
266C2×D47—44no—
267C2×D47—42no—
268C2×D47—82no—
269C2×D47—81no—
270C2×D47—81no—
271C4×C47—44no—
272C4×C47—82no—
273C4×C47#39014yes—
274C4×C47C2×C4×D411yescharacteristic
275C4×C47C2×C4×D411yescharacteristic
276C4×C47C2×#12111yescharacteristic
277C4×C47C2×#12111yescharacteristic
278Dic47—28no—
279Dic47—42no—
280Dic47—81no—
281Dic47—81no—
282C2×#536—471no—
283C2×#417—471no—
284C2×#419—471no—
285C2×#416—471no—
286C22×#103D4711yescharacteristic, complemented
287C2×#405—471no—
288#2317—471no—
289#1960—472no—
290#1959—472no—
291#1958—472no—
292#1956—472no—
293C2×C47—24no—
294C2×C47C2×#39011yescharacteristic
295C2×C47#289812yes—
296C2×C47#313712yes—
297C2×C47#315411yescharacteristic
298C2×C47#315611yescharacteristic
299D47—82no—
300C22×#41—471no—
301C23×Q8C2×D4711yescharacteristic
302C22×M16—471no—
303C23×C8C2×D4711yescharacteristic
304#536—478no—
305#536—944no—
306C2×C4⋊Q8—472no—
307C2×#124—472no—
308C2×#123—942no—
309C2×C22⋊Q8—942no—
310C2×#117—471no—
311C2×#117—471no—
312C22×C4⋊C4C2×D4711yescharacteristic
313#419—944no—
314#417—944no—
315#416—944no—
316C2×#103—24no—
317C2×#103—472no—
318C2×#103—472no—
319C2×#103—471no—
320C2×#103—471no—
321C2×#103—472no—
322C2×#103C2×D4712yes—
323C2×#103C2×D4712yes—
324C2×#102—471no—
325C2×#102—471no—
326C2×#102—472no—
327#405—944no—
328C2×#97—472no—
329C2×C4×C8—472no—
330#389—942no—
331#385—942no—
332#384—472no—
333#382—472no—
334C47#2120511yescharacteristic, complemented, Sylow
335C2×#41—474no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C2×#41—942no—
341C22×Q8—21no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—471no—
348C22×Q8C4×D4714yes—
349C22×Q8C22×D4712yes—
350C22×D4—471no—
351C22×D4—941no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—941no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—474no—
357C2×M16—942no—
358C2×M16—942no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—472no—
362C22×C8C22⋊D4714yes—
363C22×C8C22×D4711yescharacteristic
364C22×C8C22×D4711yescharacteristic
365C4⋊Q8—478no—
366C4⋊Q8—944no—
367#124—478no—
368#124—944no—
369#123—9416no—
370C22⋊Q8—9416no—
371#117—474no—
372#117—474no—
373#117—942no—
374#117—942no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—472no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—944no—
383C2×C4⋊C4C22×D4712yes—
384C2×C22⋊C4—472no—
385C2×C22⋊C4—944no—
386C2×C22⋊C4—941no—
387C2×C42—472no—
388C2×C42—941no—
389#103—24no—
390#103—24no—
391#103—944no—
392#103—944no—
393#103—944no—
394#103—942no—
395#103—942no—
396#102—944no—
397#102—942no—
398#102—942no—
399#97—944no—
400C4×C8—944no—
401#95—944no—
402C24—941no—
403C24D4×D4711yescharacteristic
404#41—948no—
405#41—948no—
406#41—1884no—
407#41—1884no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—28no—
411C2×Q8—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—472no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8C2×C4×D4714yes—
427C2×D4—472no—
428C2×D4—944no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—1882no—
435C2×D4—1882no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—474no—
444C22×C4—941no—
445C22×C4—944no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4C2×C4×D4712yes—
451C22×C4#1062535012yes—
452C22×C4D4×D4711yescharacteristic
453C22×C4D4×D4712yes—
454C22×C4D4×D4712yes—
455C22×C4#1062535212yes—
456C22×C4C2×C22⋊D4712yes—
457C22×C4C23×D4711yescharacteristic
458M16—948no—
459C2×C8—28no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—474no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8C2×C22⋊D4712yes—
468C2×C8C2×C22⋊D4712yes—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—478no—
473C4⋊C4—9416no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—3764no—
479C22⋊C4—478no—
480C22⋊C4—9416no—
481C22⋊C4—1882no—
482C42—474no—
483C42—474no—
484C42—1882no—
485C23—22no—
486C23—22no—
487C23—942no—
488C23—1881no—
489C23—1881no—
490C23—1882no—
491C23—1882no—
492C23C22⋊C4×D4712yes—
493C23#1309401012yes—
494C23#1309402012yes—
495C23C2×D4×D4711yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—48no—
499Q8—42no—
500Q8—42no—
501Q8—942no—
502Q8—942no—
503Q8—1881no—
504Q8—1881no—
505D4—942no—
506D4—942no—
507D4—1888no—
508D4—1881no—
509D4—1881no—
510C2×C4—24no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—84no—
517C2×C4—478no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1888no—
528C2×C4C22⋊C4×D4712yes—
529C2×C4(C22⋊D47)⋊C414yes—
530C2×C4C4×C22⋊D4714yes—
531C2×C4#1309399614yes—
532C2×C4#1309399714yes—
533C2×C4#1309401012yes—
534C2×C4#1309402012yes—
535C2×C4C22×C4×D4711yescharacteristic
536C2×C4C2×D4×D4711yescharacteristic
537C2×C4C2×D4×D4711yescharacteristic
538C2×C4C22×C22⋊D4711yescharacteristic
539C2×C4C2×#1062535211yescharacteristic
540C2×C4C2×#1062535011yescharacteristic
541C8—22no—
542C8—22no—
543C8—944no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—3762no—
549C22—3761no—
550C22—3761no—
551C22—1884no—
552C22—1882no—
553C22#1568902314yes—
554C22C2×C22⋊C4×D4711yescharacteristic
555C22C2×#1309401011yescharacteristic
556C22C2×#1309402011yescharacteristic
557C22#1568992912yes—
558C22#1568993612yes—
559C4—44no—
560C4—82no—
561C4—948no—
562C4—3761no—
563C4—3761no—
564C4—1882no—
565C4#1568902314yes—
566C4C2×(C22⋊D47)⋊C411yescharacteristic
567C4C2×C4×C22⋊D4711yescharacteristic
568C4C2×#1309399611yescharacteristic
569C4C2×#1309399711yescharacteristic
570C2—24no—
571C2—3762no—
572C2C2×#1568902311yescharacteristic
573C2#2025191612yes—
574C2#2025375112yes—
575C2#2025385811yescharacteristic
576C2#2025387711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#103×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC22×#103D47
hypocenterC47#21205
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C2×#15689132C2semidirect
5. C2×#15689078C2semidirect
6. #20246012C2semidirect
7. #20244206C2semidirect
8. #20244201C2semidirect
9. #20244196C2semidirect
10. #20244188C2semidirect
11. C22×#103×C47C2semidirect
15. #15689697C4semidirect
61. #13094022C2×C4semidirect
63. #13094013C2×C4semidirect
286. C22×#103D47semidirect
334. C47#21205semidirect

Maximal subgroups

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

subgroupindex
C2×#156896972
C2×#156891222
C2×#156891322
C2×#156891202
C2×#156890782
#202460122
#202442062
#202442012
#202441962
#202441882
C22×#103×C472
#2120547