Finite Group Databasebeta

← #26217644

Subgroups — #26217644

Order 12032.

1802 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×#13094055C211yesmaximal, characteristic
2C22×Q8×Dic47C211yesmaximal, characteristic, complemented
3C2×#15689125C211yesmaximal, characteristic
4#20245876C211yesmaximal, characteristic
5#20243741C214yesmaximal, complemented
6#20243727C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243081C211yesmaximal, characteristic, complemented
9C2×C47×#419C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C2×#13094055—24no—
12C2×#13094055—24no—
13C2×#13094055C418yescomplemented
14C22×#10625337C2212yes—
15C22×#10625336C2212yes—
16C22×#10625329C2211yescharacteristic
17C2×Q8×Dic47—24no—
18C2×Q8×Dic47—22no—
19C2×Q8×Dic47—24no—
20C2×Q8×Dic47C2214yes—
21C22×C4⋊Dic47—21no—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C4×Dic47—21no—
24C22×C4×Dic47C2211yescharacteristic
25#15689125—24no—
26C2×#13093949C2212yes—
27C2×#13093946C2212yes—
28#15689028C2212yes—
29#15689024C2212yes—
30#15689000—21no—
31#15689000—21no—
32C4×(C2×D47)⋊C4—22no—
33#15688885C2214yes—
34#15688877C2214yes—
35C23×Q8×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C47×#419—24no—
39C2×#103×C47C2214yes—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350C2×C414yes—
46C22×Dic94C2311yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—21no—
49C23×Dic47—21no—
50C23×Dic47D411yescharacteristic
51C23×Dic47D411yescharacteristic
52C22×C4×D47—21no—
53#13094055—232no—
54C2×#10625337—24no—
55C2×#10625337—24no—
56C2×#10625337C2×C418yescomplemented
57C2×#10625336—24no—
58C2×#10625336—24no—
59C2×#10625336C2×C418yescomplemented
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329C2×C414yes—
63C22×C47⋊C8C2312yes—
64Q8×Dic47—216no—
65Q8×Dic47—28no—
66Q8×Dic47—28no—
67C2×C4⋊Dic47—28no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—22no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47D414yes—
72C2×C4⋊Dic47C2312yes—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4—22no—
75C2×C4×Dic47—28no—
76C2×C4×Dic47—22no—
77C2×C4×Dic47—22no—
78C2×C4×Dic47—24no—
79C2×C4×Dic47D412yes—
80C2×C4×Dic47C2312yes—
81#13093949—24no—
82#13093946—24no—
83C47⋊#95—22no—
84C47⋊#95—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47—21no—
87C22×Q8×C47—22no—
88C22×Q8×C47—22no—
89C22×Q8×C47C2×C414yes—
90C22×Q8×C47D414yes—
91C22×Q8×C47D414yes—
92C22×Q8×C47C2312yes—
93C23×C4×C47—21no—
94C23×C4×C47C2311yescharacteristic
95C2×M16×C47—22no—
96C2×M16×C47—22no—
97C2×M16×C47—22no—
98C22×C8×C47C2312yes—
99#117×C47—22no—
100#117×C47—22no—
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—21no—
103C2×C42×C47—21no—
104#103×C47—28no—
105C24×C47C2×D411yescharacteristic
106C23×D47—21no—
107C2×Q8×C47—24no—
108C2×Q8×C47—28no—
109C2×Q8×C47—24no—
110C2×Q8×C47—24no—
111C2×Q8×C47—28no—
112C2×Q8×C47—28no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47C22⋊C4116yescomplemented
117C2×Q8×C47C22×C414yes—
118C2×Q8×C47C2×D414yes—
119C2×Q8×C47C2×D414yes—
120C22×C4×C47—24no—
121C22×C4×C47—24no—
122C22×C4×C47—24no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D412yes—
130C22×C4×C47#4112yes—
131C22×C4×C47#4112yes—
132C22×C4×C47C2411yescharacteristic
133C2×C22⋊D47—24no—
134C2×C22⋊D47—42no—
135C2×C22⋊D47—42no—
136C22×Dic47—22no—
137C22×Dic47—24no—
138C22×Dic47—22no—
139C22×Dic47—24no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—42no—
143C22×Dic47—44no—
144C22×Dic47C2×D412yes—
145C22×Dic47C2×D412yes—
146#10625350—28no—
147#10625350—28no—
148#10625350—44no—
149#10625350—44no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188C22×C412yes—
156C2×C4×D47—24no—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94C22×C412yes—
165M16×C47—44no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C22⋊C4×C47—42no—
172C42×C47—42no—
173#10625337—216no—
174#10625336—216no—
175(C2×D47)⋊C4—44no—
176(C2×D47)⋊C4—44no—
177C4⋊Dic47—216no—
178C4⋊Dic47—24no—
179C4⋊Dic47—22no—
180C4⋊Dic47C2×D414yes—
181C4×Dic47—216no—
182C4×Dic47—24no—
183C4×Dic47—22no—
184C4×Dic47—22no—
185#10625329—28no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8C22×C414yes—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47C4×D412yes—
192C23×C47#12012yes—
193C23×C47#12112yes—
194C23×C47C22×D411yescharacteristic
195C22×D47—22no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—42no—
199C22×D47—42no—
200Q8×C47—24no—
201Q8×C47—24no—
202Q8×C47—24no—
203Q8×C47—216no—
204Q8×C47C2×C22⋊C418yes—
205C2×C4×C47—24no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—216no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47C2×C22⋊C414yes—
213C2×C4×C47C4×D414yes—
214C2×C4×C47C4×D412yes—
215C2×C4×C47#12012yes—
216C2×C4×C47#12114yes—
217C2×C4×C47#12112yes—
218C2×C4×C47#12314yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47C2×#4111yescharacteristic
225C22⋊D47—48no—
226C2×Dic47—28no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—44no—
235C2×Dic47—48no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—44no—
239C2×Dic47#13714yescomplemented
240D188—22no—
241D188—22no—
242D188—41no—
243D188—41no—
244C4×D47—48no—
245Dic94—22no—
246Dic94—22no—
247Dic94—41no—
248Dic94—41no—
249C8×C47—42no—
250C47⋊C8—24no—
251#20926—471nomaximal, Sylow
252C22×C47—24no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—24no—
256C22×C47#38614yes—
257C22×C47C2×C4×D411yescharacteristic
258C22×C47C2×#12011yescharacteristic
259C22×C47C2×#12111yescharacteristic
260C22×C47C2×#13712yes—
261C22×C47#57512yes—
262C2×D47—44no—
263C2×D47—42no—
264C2×D47—81no—
265C2×D47—81no—
266C2×D47—82no—
267C4×C47—28no—
268C4×C47—82no—
269C4×C47#38614yes—
270C4×C47C22×C22⋊C411yescharacteristic
271C4×C47C2×C4×D411yescharacteristic
272C4×C47C2×#12111yescharacteristic
273C4×C47C2×#12311yescharacteristic
274Dic47—24no—
275Dic47—42no—
276Dic47—44no—
277Dic47—42no—
278C22×#137—471no—
279C22×C4×Q8—471no—
280C2×#417—471no—
281C2×#419D4711yescharacteristic, complemented
282#2297—471no—
283#1880—474no—
284#1878—474no—
285#1736—471no—
286#1728—471no—
287C2×C47—24no—
288C2×C47C2×#38611yescharacteristic
289C2×C47C4×#13712yes—
290C2×C47#300411yescharacteristic
291C2×C47#300511yescharacteristic
292C2×C47#303712yes—
293D47—82no—
294C22×#41—471no—
295C23×Q8C2×D4711yescharacteristic
296C2×#137—478no—
297C2×#137—944no—
298C2×#137—944no—
299C22×Q16—472no—
300C22×SD16—472no—
301C22×M16—471no—
302C22×M16C2×D4711yescharacteristic
303C2×C4×Q8—474no—
304C2×C4×Q8—942no—
305C2×C4×Q8—944no—
306C2×C4×Q8—944no—
307C2×#117C2×D4711yescharacteristic
308C22×C4⋊C4—471no—
309C22×C4⋊C4—941no—
310C22×C42—471no—
311C22×C42—941no—
312#419—24no—
313#417—944no—
314C2×#103—472no—
315C2×#103C2×D4714yes—
316C2×#102—472no—
317#391—472no—
318#390—472no—
319#386—941no—
320#386—941no—
321C4×C22⋊C4—942no—
322#340—474no—
323#336—474no—
324C47#2092611yescharacteristic, complemented, Sylow
325C2×#41—474no—
326C2×#41—942no—
327C2×#41—942no—
328C2×#41—942no—
329C2×#41—942no—
330C2×#41—942no—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—471no—
336C22×Q8C4×D4714yes—
337C22×Q8D18814yes—
338C22×Q8C22⋊D4714yes—
339C22×Q8C22×D4712yes—
340C22×D4—471no—
341C22×D4—941no—
342C23×C4—21no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—941no—
346C23×C4—941no—
347C23×C4C22×D4711yescharacteristic
348#137—9432no—
349C2×Q16—478no—
350C2×Q16—944no—
351C2×Q16—944no—
352C2×SD16—478no—
353C2×SD16—944no—
354C2×SD16—944no—
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—22no—
358C2×M16—474no—
359C2×M16—942no—
360C2×M16—942no—
361C22×C8—472no—
362C22×C8C22×D4712yes—
363C4×Q8—948no—
364C4×Q8—9416no—
365C4×Q8—948no—
366#117—22no—
367#117—22no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—472no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—948no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4C22×D4712yes—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4—942no—
378C2×C42—21no—
379C2×C42—472no—
380C2×C42—472no—
381C2×C42—944no—
382C2×C42—948no—
383C2×C42—942no—
384C2×C42—942no—
385#103—28no—
386#103—944no—
387#102—944no—
388#95—942no—
389#95—942no—
390C24—941no—
391C24D4×D4711yescharacteristic
392#41—948no—
393#41—948no—
394#41—1884no—
395#41—1884no—
396C2×Q8—28no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—472no—
406C2×Q8—941no—
407C2×Q8—941no—
408C2×Q8—941no—
409C2×Q8—941no—
410C2×Q8—941no—
411C2×Q8(C2×D47)⋊C4116yescomplemented
412C2×Q8C2×C4×D4714yes—
413C2×Q8C2×D18814yes—
414C2×Q8C2×C22⋊D4714yes—
415C2×D4—472no—
416C2×D4—944no—
417C2×D4—941no—
418C2×D4—941no—
419C2×D4—941no—
420C2×D4—941no—
421C2×D4—941no—
422C2×D4—1882no—
423C2×D4—1882no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—472no—
430C22×C4—472no—
431C22×C4—944no—
432C22×C4—942no—
433C22×C4—944no—
434C22×C4—942no—
435C22×C4—941no—
436C22×C4—942no—
437C22×C4—944no—
438C22×C4—1882no—
439C22×C4—1884no—
440C22×C4—1882no—
441C22×C4—1882no—
442C22×C4C2×C4×D4712yes—
443C22×C4C2×D18812yes—
444C22×C4D4×D4711yescharacteristic
445C22×C4D4×D4712yes—
446C22×C4#1062535212yes—
447C22×C4#1062535212yes—
448C22×C4C2×C22⋊D4712yes—
449C22×C4C23×D4711yescharacteristic
450Q16—9416no—
451SD16—9416no—
452M16—44no—
453M16—948no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—474no—
458C2×C8—942no—
459C2×C8—942no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—474no—
463C4⋊C4—9416no—
464C4⋊C4—944no—
465C4⋊C4—942no—
466C22⋊C4—42no—
467C22⋊C4—1884no—
468C22⋊C4—1884no—
469C42—42no—
470C42—9416no—
471C42—944no—
472C42—942no—
473C42—942no—
474C23—22no—
475C23—22no—
476C23—942no—
477C23—1881no—
478C23—1881no—
479C23—1882no—
480C23—1882no—
481C23(C22⋊D47)⋊C412yes—
482C23#1309399412yes—
483C23#1309399612yes—
484C23C2×D4×D4711yescharacteristic
485Q8—216no—
486Q8—24no—
487Q8—24no—
488Q8—24no—
489Q8—942no—
490Q8—942no—
491Q8—1881no—
492Q8—1881no—
493Q8C2×(C2×D47)⋊C418yes—
494D4—942no—
495D4—942no—
496D4—1888no—
497D4—1881no—
498D4—1881no—
499C2×C4—216no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—474no—
507C2×C4—948no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—944no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—1884no—
516C2×C4—1888no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4—1884no—
520C2×C4—1888no—
521C2×C4C2×(C2×D47)⋊C414yes—
522C2×C4(C22⋊D47)⋊C412yes—
523C2×C4(C22⋊D47)⋊C414yes—
524C2×C4#1309399412yes—
525C2×C4#1309399612yes—
526C2×C4#1309399614yes—
527C2×C4#1309400814yes—
528C2×C4C22×C4×D4711yescharacteristic
529C2×C4C2×D4×D4711yescharacteristic
530C2×C4C22×C22⋊D4711yescharacteristic
531C2×C4C22×D18811yescharacteristic
532C2×C4C2×#1062535211yescharacteristic
533C2×C4C2×#1062535211yescharacteristic
534C8—42no—
535C8—944no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—3762no—
541C22—3761no—
542C22—3761no—
543C22—1884no—
544C22—1882no—
545C22#1568900014yes—
546C22C2×(C22⋊D47)⋊C411yescharacteristic
547C22C2×#1309399411yescharacteristic
548C22C2×#1309399611yescharacteristic
549C22#137×D4712yes—
550C22#1568992212yes—
551C4—28no—
552C4—82no—
553C4—944no—
554C4—1882no—
555C4—1882no—
556C4—1884no—
557C4#1568900014yes—
558C4C22×(C2×D47)⋊C411yescharacteristic
559C4C2×(C22⋊D47)⋊C411yescharacteristic
560C4C2×#1309399611yescharacteristic
561C4C2×#1309400811yescharacteristic
562C2—24no—
563C2—3762no—
564C2C2×#1568900011yescharacteristic
565C2#2025220312yes—
566C2#2025262411yescharacteristic
567C2#2025263411yescharacteristic
568C2#2025285712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#419C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#419D47
hypocenterC47#20926
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic47C2semidirect
5. #20243741C2semidirect
6. #20243727C2semidirect
7. #20243109C2semidirect
8. #20243081C2semidirect
9. C2×C47×#419C2semidirect
13. C2×#13094055C4semidirect
56. C2×#10625337C2×C4semidirect
59. C2×#10625336C2×C4semidirect
116. C2×Q8×C47C22⋊C4semidirect
239. C2×Dic47#137semidirect
281. C2×#419D47semidirect
324. C47#20926semidirect
411. C2×Q8(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940552
C22×Q8×Dic472
C2×#156891252
#202458762
#202437412
#202437272
#202431092
#202430812
C2×C47×#4192
#2092647