Finite Group Databasebeta

← #26252368

Subgroups — #26252368

Order 12032.

1294 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic
2C2×#15689409C211yesmaximal, characteristic, complemented
3C2×#15689405C211yesmaximal, characteristic, complemented
4C4×#13094004C211yesmaximal, characteristic
5#20244720C212yesmaximal, complemented
6#20244528C212yesmaximal, complemented
7#20244340C211yesmaximal, characteristic, complemented
8#20243694C212yesmaximal
9#20243582C212yesmaximal
10#20243365C211yesmaximal, characteristic
11C47×#1988C211yesmaximal, characteristic, complemented
12C22×Q8×D47—21no—
13C2×#13094024C2211yescharacteristic
14C2×#13094023—21no—
15C2×#13094023C2212yes—
16C2×#13094022—21no—
17C4⋊Q8×D47—24no—
18C4⋊Q8×D47—24no—
19C2×#13094004C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21C2×C4⋊C4×D47—21no—
22C2×C4⋊C4×D47C2212yes—
23C2×C42×D47C2211yescharacteristic
24#15689409—24no—
25#15689405—24no—
26C2×#13093956C2212yes—
27C2×#13093955C2212yes—
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093949C2212yes—
30C2×#13093947C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32#15689028C2211yescharacteristic
33#15689022C2211yescharacteristic
34#15689008—21no—
35#15688997—21no—
36#15688980C2211yescharacteristic
37C4×C4⋊Dic47—21no—
38C4×(C2×D47)⋊C4—21no—
39#15688885C2212yes—
40#15688885C2212yes—
41#15688877C2212yes—
42C2×C4⋊Q8×C47C2211yescharacteristic
43C2×C4⋊C8×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C47×#340C2212yes—
47C22×Dic94—21no—
48C22×Dic94—22no—
49C22×Dic94C2311yescharacteristic
50C2×Q8×D47—28no—
51C2×Q8×D47—44no—
52C2×Q8×D47—44no—
53C22×C4×D47—22no—
54C22×C4×D47D411yescharacteristic
55C22×C4×D47D411yescharacteristic
56C22×C4×D47C2311yescharacteristic
57C22×C47⋊C8C2312yes—
58#13094024—22no—
59#13094024D414yes—
60#13094023—28no—
61#13094023—24no—
62#13094023—24no—
63#13094022—28no—
64#13094004—22no—
65#13094004Q814yescomplemented
66C4×Dic94—22no—
67C4×Dic94—22no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×Dic47⋊C4—22no—
74C2×Dic47⋊C4—22no—
75C2×Dic47⋊C4—21no—
76C4⋊C4×D47—28no—
77C4⋊C4×D47—24no—
78C4⋊C4×D47—24no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47—22no—
86C42×D47—22no—
87#13093956—24no—
88#13093955—24no—
89C4⋊(C47⋊C8)—22no—
90#13093949—24no—
91#13093947—24no—
92(C2×D47)⋊C8—24no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C22×Q8×C47—21no—
98C22×C8×C47C2312yes—
99C4⋊Q8×C47—22no—
100C4⋊Q8×C47D414yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2312yes—
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107C4⋊C8×C47—22no—
108#95×C47—21no—
109C23×D47C2×D411yescharacteristic
110C2×Q8×C47—24no—
111C2×Q8×C47—42no—
112C2×Q8×C47—42no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—21no—
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2411yescharacteristic
119C22×Dic47—21no—
120C22×Dic47—22no—
121C22×Dic47—21no—
122C22×Dic47C2×D411yescharacteristic
123C22×Dic47C2×D411yescharacteristic
124C22×Dic47C2×D411yescharacteristic
125Q8×D47—432no—
126C2×C4×D47—22no—
127C2×C4×D47—22no—
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—21no—
131C2×C4×D47—21no—
132C2×C4×D47—21no—
133C2×C4×D47—21no—
134C2×C4×D47—22no—
135C2×C4×D47—28no—
136C2×C4×D47—44no—
137C2×C4×D47—44no—
138C2×C4×D47SD1614yes—
139C2×C4×D47C2×Q812yes—
140C2×C4×D47#4112yes—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—28no—
146C2×Dic94—24no—
147C2×Dic94—21no—
148C2×Dic94—44no—
149C2×Dic94—42no—
150C2×Dic94—42no—
151C2×Dic94—44no—
152C2×Dic94C2×Q812yescomplemented
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C2×C8×C47—22no—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47C2×D414yes—
162C4⋊C4×C47C2×Q812yescomplemented
163C42×C47—42no—
164C42×C47C2×D412yes—
165C42×C47C2×D412yes—
166(C2×D47)⋊C4—24no—
167(C2×D47)⋊C4—42no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47—24no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—42no—
174C4⋊Dic47C2×D414yes—
175Dic47⋊C4—28no—
176Dic47⋊C4—28no—
177Dic47⋊C4—42no—
178C4×Dic47—22no—
179C4×Dic47—22no—
180C4×Dic47—21no—
181C4×Dic47—21no—
182C4×Dic47—42no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C23×C47C22×D411yescharacteristic
187C22×D47—21no—
188C22×D47—21no—
189C22×D47—21no—
190C22×D47—21no—
191C22×D47C22⋊Q812yes—
192C22×D47C2×SD1612yes—
193C22×D47#13712yes—
194Q8×C47—48no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—24no—
198C2×C4×C47—22no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—42no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47#12012yes—
207C2×C4×C47#12012yes—
208C2×C4×C47C2×SD1612yes—
209C2×C4×C47AGL(1,ℤ/8)12yes—
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C22×D411yescharacteristic
212C2×C4×C47C22×Q811yescharacteristic
213C2×C4×C47C2×#4111yescharacteristic
214C2×C4×C472−1+411yescharacteristic
215C2×C4×C472−1+411yescharacteristic
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—21no—
219C2×Dic47—21no—
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—24no—
223C2×Dic47—22no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47C22⋊Q812yes—
232C2×Dic47C2×SD1612yes—
233C2×Dic47#13712yes—
234C4×D47—28no—
235C4×D47—22no—
236C4×D47—22no—
237C4×D47—22no—
238C4×D47—22no—
239C4×D47—28no—
240C4×D47—416no—
241Dic94—22no—
242Dic94—22no—
243Dic94—416no—
244Dic94—41no—
245Dic94—41no—
246Dic94—48no—
247C8×C47—42no—
248C47⋊C8—42no—
249#24301—471nomaximal, Sylow
250C22×C47C2×#12011yescharacteristic
251C22×C47C2×C22⋊Q811yescharacteristic
252C22×C47#53611yescharacteristic
253C22×C47C22×SD1611yescharacteristic
254C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C47C2×#13711yescharacteristic
256C22×C47C2×#13711yescharacteristic
257C2×D47—22no—
258C2×D47—22no—
259C2×D47—22no—
260C2×D47—22no—
261C2×D47—22no—
262C2×D47—22no—
263C2×D47#47514yescomplemented
264C4×C47—22no—
265C4×C47—42no—
266C4×C47—44no—
267C4×C47—81no—
268C4×C47#45014yes—
269C4×C47#54811yescharacteristic
270C4×C47#54811yescharacteristic
271C4×C47D4×Q811yescharacteristic
272C4×C47D4×Q811yescharacteristic
273Dic47—22no—
274Dic47—22no—
275Dic47—22no—
276Dic47—42no—
277Dic47—44no—
278Dic47—81no—
279C22×C4⋊Q8—471no—
280C2×#475—471no—
281C2×#475—471no—
282C4×C22⋊Q8—471no—
283#2058—472no—
284#2024—472no—
285#1988—471no—
286#1988D4711yescharacteristic, complemented
287#1872—472no—
288#1853—472no—
289#1797—471no—
290C2×C47#240911yescharacteristic
291C2×C47C2×#45011yescharacteristic
292C2×C47#300411yescharacteristic
293C2×C47C2×#47511yescharacteristic
294C2×C47#307911yescharacteristic
295C2×C47#322911yescharacteristic
296C2×C47#323911yescharacteristic
297D47—24no—
298C23×Q8—941no—
299C2×C4⋊Q8—472no—
300C2×C4⋊Q8—471no—
301C2×C4⋊Q8—941no—
302C2×C4⋊Q8—941no—
303C2×C4⋊Q8—944no—
304C2×C4⋊Q8—944no—
305C2×C4⋊Q8C2×D4711yescharacteristic
306C2×C22⋊Q8—471no—
307C2×C4×Q8—471no—
308C22×C4⋊C4—472no—
309C22×C4⋊C4—941no—
310C22×C42—471no—
311#475—944no—
312#475—944no—
313C2×#106—472no—
314C2×#106—472no—
315C2×C4⋊C8—471no—
316C2×C4⋊C8C2×D4711yescharacteristic
317C2×#103—472no—
318C2×#103—472no—
319C2×C22⋊C8—472no—
320#391—471no—
321#389—471no—
322#387—941no—
323#385—941no—
324#380—471no—
325C4×C4⋊C4—941no—
326C4×C4⋊C4C2×D4711yescharacteristic
327C4×C22⋊C4—941no—
328#340—472no—
329#340—472no—
330#340C2×D4712yes—
331#340C2×D4712yes—
332#336—472no—
333C47#2430111yescharacteristic, complemented, Sylow
334C22×Q8—21no—
335C22×Q8—471no—
336C22×Q8—942no—
337C22×Q8—948no—
338C22×Q8—941no—
339C22×Q8—1884no—
340C22×Q8—1884no—
341C23×C4—471no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—942no—
345C22×C8—472no—
346C22×C8C22×D4712yes—
347C4⋊Q8—22no—
348C4⋊Q8—474no—
349C4⋊Q8—944no—
350C4⋊Q8—948no—
351C4⋊Q8—948no—
352C4⋊Q8—944no—
353C4⋊Q8—942no—
354C4⋊Q8D18814yes—
355C22⋊Q8—474no—
356C22⋊Q8—942no—
357C4×Q8—942no—
358C4×Q8—942no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—472no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—944no—
363C2×C4⋊C4—948no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—942no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4—944no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4C22×D4711yescharacteristic
372C2×C4⋊C4C22×D4712yes—
373C2×C4⋊C4C22×D4711yescharacteristic
374C2×C22⋊C4—941no—
375C2×C22⋊C4—941no—
376C2×C42—21no—
377C2×C42—471no—
378C2×C42—471no—
379C2×C42—471no—
380C2×C42—942no—
381C2×C42—942no—
382C2×C42—941no—
383C2×C42C22×D4711yescharacteristic
384#106—944no—
385#106—944no—
386C4⋊C8—22no—
387C4⋊C8—942no—
388#103—944no—
389#103—944no—
390C22⋊C8—944no—
391#95—21no—
392#95—941no—
393#95—941no—
394#95—941no—
395#95—941no—
396C24—471no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—472no—
401C2×Q8—948no—
402C2×Q8—944no—
403C2×Q8—941no—
404C2×Q8—941no—
405C2×Q8—941no—
406C2×Q8—941no—
407C2×Q8—941no—
408C2×Q8—1884no—
409C2×Q8—1884no—
410C2×Q8—18832no—
411C2×Q8—1882no—
412C2×Q8—1882no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—471no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—474no—
420C22×C4—472no—
421C22×C4—472no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—948no—
425C22×C4—942no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—942no—
434C22×C4—941no—
435C22×C4—1884no—
436C22×C4—1884no—
437C22×C4D4×D4711yescharacteristic
438C22×C4D4×D4711yescharacteristic
439C22×C4C23×D4711yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—942no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—474no—
452C4⋊C4—944no—
453C4⋊C4—948no—
454C4⋊C4—948no—
455C4⋊C4—944no—
456C4⋊C4—942no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—1882no—
460C4⋊C4—1882no—
461C4⋊C4C2×D18814yes—
462C4⋊C4Q8×D4712yescomplemented
463C22⋊C4—944no—
464C22⋊C4—1882no—
465C42—42no—
466C42—942no—
467C42—942no—
468C42—941no—
469C42—941no—
470C42—1882no—
471C42C2×D18812yes—
472C42D4×D4712yes—
473C23—472no—
474C23—472no—
475C23—472no—
476C23—941no—
477C23—941no—
478C23—941no—
479C23—941no—
480C23C2×D4×D4711yescharacteristic
481Q8—48no—
482Q8—942no—
483Q8—942no—
484Q8—18816no—
485Q8—1881no—
486Q8—1881no—
487Q8—1888no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—472no—
500C2×C4—472no—
501C2×C4—472no—
502C2×C4—948no—
503C2×C4—941no—
504C2×C4—941no—
505C2×C4—941no—
506C2×C4—941no—
507C2×C4—941no—
508C2×C4—941no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—948no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—944no—
515C2×C4—942no—
516C2×C4—18816no—
517C2×C4—1881no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4—1881no—
521C2×C4—1882no—
522C2×C4—1882no—
523C2×C4—1881no—
524C2×C4#1309399412yes—
525C2×C4#1309399412yes—
526C2×C4SD16×D4712yes—
527C2×C4#1309404812yes—
528C2×C4C2×D4×D4711yescharacteristic
529C2×C4C22×D18811yescharacteristic
530C2×C4C2×Q8×D4711yescharacteristic
531C2×C4#41×D4711yescharacteristic
532C2×C4#1309407111yescharacteristic
533C2×C4#1309407211yescharacteristic
534C8—42no—
535C8—1882no—
536C22—474no—
537C22—942no—
538C22—942no—
539C22—942no—
540C22—942no—
541C22—942no—
542C22—942no—
543C22C2×#1309399411yescharacteristic
544C22C22⋊Q8×D4711yescharacteristic
545C22#1568969611yescharacteristic
546C22C2×SD16×D4711yescharacteristic
547C22C2×#1309404811yescharacteristic
548C22#137×D4711yescharacteristic
549C22#137×D4711yescharacteristic
550C4—22no—
551C4—42no—
552C4—44no—
553C4—81no—
554C4—942no—
555C4—942no—
556C4—942no—
557C4—3761no—
558C4—1884no—
559C4—1882no—
560C4#1568927514yes—
561C4#1568978111yescharacteristic
562C4#1568978511yescharacteristic
563C4#1568978811yescharacteristic
564C4Q8×D18811yescharacteristic
565C2—944no—
566C2#2024675111yescharacteristic
567C2C2×#1568927511yescharacteristic
568C2#2025262411yescharacteristic
569C2#475×D4711yescharacteristic
570C2#2025319611yescharacteristic
571C2#2025453111yescharacteristic
572C2#2025461711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1988C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenter#1988D47
hypocenterC47#24301
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#15689409C2semidirect
3. C2×#15689405C2semidirect
5. #20244720C2semidirect
6. #20244528C2semidirect
7. #20244340C2semidirect
11. C47×#1988C2semidirect
65. #13094004Q8semidirect
152. C2×Dic94C2×Q8semidirect
162. C4⋊C4×C47C2×Q8semidirect
263. C2×D47#475semidirect
286. #1988D47semidirect
333. C47#24301semidirect
462. C4⋊C4Q8×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D472
C2×#156894092
C2×#156894052
C4×#130940042
#202447202
#202445282
#202443402
#202436942
#202435822
#202433652
C47×#19882
#2430147