Finite Group Databasebeta

← #26216313

Subgroups — #26216313

Order 12032.

1402 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2C2×#15689174C211yesmaximal, characteristic, complemented
3#20245792C211yesmaximal, characteristic, complemented
4#20243481C212yesmaximal, complemented
5#20243391C212yesmaximal, complemented
6#20243387C212yesmaximal, complemented
7#20243327C212yesmaximal
8C4×#13093948C211yesmaximal, characteristic, complemented
9C4×#13093944C211yesmaximal, characteristic, complemented
10C4×#13093946C211yesmaximal, characteristic, complemented
11C4×#103×C47C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C4×#10625354—24no—
14C4×#10625354C418yescomplemented
15C2×Q8×Dic47C2211yescharacteristic
16C2×C4×D188—21no—
17C2×C4×D188C2211yescharacteristic
18C2×#13093986—21no—
19C2×#13093986C2211yescharacteristic
20C2×#13093982—21no—
21C2×#13093982C2211yescharacteristic
22C2×#13093981—21no—
23C2×C42×D47—21no—
24C2×C42×D47C2211yescharacteristic
25#15689174—24no—
26C2×Dic47⋊C8C2212yes—
27C2×#13093948C2212yes—
28C2×#13093946C2212yes—
29C2×#13093944C2212yes—
30C2×(C2×D47)⋊C8C2212yes—
31C2×C4×C47⋊C8C2211yescharacteristic
32#15688992C2212yes—
33#15688987—21no—
34#15688982—21no—
35#15688980C2212yes—
36C4×Dic47⋊C4—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×C4×C8×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C2×#10625354—24no—
47C2×#10625354—24no—
48C2×#10625354—22no—
49C2×#10625354C2×C414yes—
50C22×D188—21no—
51C22×D188C2311yescharacteristic
52C22×C4×D47—21no—
53C22×C4×D47—21no—
54C22×C4×D47—21no—
55C22×C4×D47D411yescharacteristic
56C22×C4×D47D411yescharacteristic
57C22×C4×D47C2311yescharacteristic
58C22×C47⋊C8C2312yes—
59Q8×Dic47—22no—
60Q8×Dic47C2×C414yes—
61C4×D188—28no—
62C4×D188—22no—
63C4×D188C2×C414yes—
64#13093986—28no—
65#13093986—22no—
66#13093986C2×C414yes—
67#13093982—22no—
68#13093982—28no—
69#13093982C2×C414yes—
70#13093981—44no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2312yes—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—22no—
75C2×Dic47⋊C4—22no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86C42×D47—28no—
87C42×D47C2×C414yes—
88C42×D47D414yes—
89Dic47⋊C8—24no—
90#13093948—24no—
91#13093946—24no—
92#13093944—24no—
93(C2×D47)⋊C8—24no—
94C4×C47⋊C8—22no—
95C47⋊#95—21no—
96C47⋊#95—22no—
97C22×Q8×C47C2311yescharacteristic
98C22×C8×C47C2312yes—
99C4×Q8×C47—22no—
100C4×Q8×C47C2×C414yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2312yes—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C42×C47—21no—
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107#103×C47—24no—
108C4×C8×C47—22no—
109#95×C47—21no—
110C23×D47—21no—
111C23×D47C2×D411yescharacteristic
112C2×Q8×C47—21no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22×C412yes—
115C2×Q8×C47#4114yes—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2411yescharacteristic
122C22×Dic47—21no—
123C22×Dic47—22no—
124C22×Dic47C2×D411yescharacteristic
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127#10625354—216no—
128#10625354—28no—
129#10625354—28no—
130C2×D188—28no—
131C2×D188—24no—
132C2×D188—21no—
133C2×D188—22no—
134C2×D188C22×C412yes—
135C2×D188#4114yes—
136C2×C4×D47—28no—
137C2×C4×D47—22no—
138C2×C4×D47—22no—
139C2×C4×D47—22no—
140C2×C4×D47—22no—
141C2×C4×D47—24no—
142C2×C4×D47—24no—
143C2×C4×D47—24no—
144C2×C4×D47—24no—
145C2×C4×D47—42no—
146C2×C4×D47—42no—
147C2×C4×D47—44no—
148C2×C4×D47C22⋊C414yes—
149C2×C4×D47C22⋊C414yes—
150C2×C4×D47C22×C412yes—
151C2×C4×D47C2×D412yes—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C4⋊C4×C47—24no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47C22×C412yes—
160C42×C47—24no—
161C42×C47—42no—
162C42×C47C22×C412yes—
163C42×C47C2×D412yes—
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—24no—
166(C2×D47)⋊C4—24no—
167(C2×D47)⋊C4—44no—
168C4⋊Dic47—21no—
169C4⋊Dic47—24no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47C22×C412yes—
173Dic47⋊C4—44no—
174Dic47⋊C4—44no—
175C4×Dic47—24no—
176C4×Dic47—24no—
177C4×Dic47—42no—
178C4×Dic47C22⋊C414yes—
179C4×Dic47C22⋊C414yes—
180C4×Dic47C22×C412yes—
181C4×Dic47C2×D412yes—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C23×C47C22×D411yescharacteristic
186C22×D47—22no—
187C22×D47—22no—
188C22×D47—24no—
189C22×D47—22no—
190C22×D47—21no—
191C22×D47—21no—
192C22×D47—42no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47C2×C22⋊C412yes—
196Q8×C47—22no—
197Q8×C47—24no—
198Q8×C47#11714yescomplemented
199C2×C4×C47—22no—
200C2×C4×C47—24no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C2×C22⋊C412yes—
209C2×C4×C47AGL(1,ℤ/8)12yes—
210C2×C4×C47#13712yes—
211C2×C4×C47C23×C411yescharacteristic
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C2×#4112yes—
214C2×C4×C47C2×#4112yes—
215C2×Dic47—22no—
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—22no—
219C2×Dic47—41no—
220C2×Dic47—41no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47C2×C22⋊C412yes—
225C2×Dic47C2×C22⋊C412yes—
226C2×Dic47C2×C22⋊C412yes—
227C2×Dic47#13514yes—
228D188—216no—
229D188—22no—
230D188—24no—
231D188#11714yescomplemented
232C4×D47—216no—
233C4×D47—22no—
234C4×D47—22no—
235C4×D47—24no—
236C4×D47—28no—
237C4×D47—28no—
238C4×D47—48no—
239C4×D47—48no—
240C8×C47—42no—
241C47⋊C8—42no—
242#20737—471nomaximal, Sylow
243C22×C47C22×C22⋊C411yescharacteristic
244C22×C47#52512yes—
245C22×C47C2×#13512yes—
246C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C47C2×#13711yescharacteristic
248C2×D47—22no—
249C2×D47—22no—
250C2×D47—22no—
251C2×D47—24no—
252C2×D47—42no—
253C2×D47—44no—
254C2×D47—42no—
255C2×D47—44no—
256C2×D47—42no—
257C2×D47—42no—
258C4×C47—22no—
259C4×C47—24no—
260C4×C47—42no—
261C4×C47—82no—
262C4×C47#41714yes—
263C4×C47C2×#11711yescharacteristic
264C4×C47C2×#11711yescharacteristic
265C4×C47#52512yes—
266Dic47—22no—
267Dic47—22no—
268Dic47—42no—
269Dic47—82no—
270Dic47#41614yes—
271C2×C4×#41—471no—
272C2×#431—471no—
273#2285—471no—
274#1825—472no—
275#1803—472no—
276#1802—472no—
277#1788—472no—
278C4×#103—471no—
279C4×#103D4711yescharacteristic, complemented
280C4×#102—471no—
281C4×#102—471no—
282C2×C47#228011yescharacteristic
283C2×C47C2×#41611yescharacteristic
284C2×C47C2×#41711yescharacteristic
285C2×C47#310612yes—
286C2×C47#310712yes—
287D47—42no—
288D47—44no—
289C22×#41—471no—
290C4×#41—478no—
291C4×#41—944no—
292C2×C4×Q8—471no—
293C2×C4×Q8C2×D4711yescharacteristic
294C2×C4×D4—471no—
295C2×C4×D4—471no—
296C2×C4×D4—941no—
297C2×C4×D4—941no—
298C2×#117—471no—
299C2×#117—941no—
300C2×#117—941no—
301C22×C42—471no—
302C22×C42—941no—
303#431—944no—
304C2×C4⋊C8—472no—
305C2×#103—472no—
306C2×#103C2×D4712yes—
307C2×#102—472no—
308C2×#102—472no—
309C2×C22⋊C8—472no—
310C2×C4×C8—471no—
311C2×C4×C8C2×D4711yescharacteristic
312#384—472no—
313#382—941no—
314#380—472no—
315#380—941no—
316C4×C4⋊C4—471no—
317C4×C4⋊C4—941no—
318C4×C4⋊C4C2×D4711yescharacteristic
319#340—472no—
320#340—472no—
321#340—472no—
322#340C2×D4712yes—
323C47#2073711yescharacteristic, complemented, Sylow
324C2×#41—474no—
325C2×#41—942no—
326C2×#41—944no—
327C2×#41—944no—
328C22×Q8C22×D4711yescharacteristic
329C22×D4—471no—
330C22×D4—941no—
331C23×C4—471no—
332C23×C4—471no—
333C23×C4—471no—
334C23×C4—941no—
335C23×C4—941no—
336C23×C4—941no—
337C22×C8—472no—
338C22×C8C22×D4712yes—
339C4×Q8—22no—
340C4×Q8—474no—
341C4×Q8—942no—
342C4×Q8C4×D4714yes—
343C4×D4—474no—
344C4×D4—474no—
345C4×D4—948no—
346C4×D4—942no—
347C4×D4—948no—
348C4×D4—942no—
349#117—474no—
350#117—948no—
351#117—942no—
352#117—1884no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4C22×D4712yes—
360C2×C4⋊C4C22×D4711yescharacteristic
361C2×C22⋊C4—941no—
362C2×C22⋊C4—941no—
363C2×C22⋊C4—941no—
364C2×C22⋊C4—942no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—471no—
368C2×C42—471no—
369C2×C42—471no—
370C2×C42—474no—
371C2×C42—474no—
372C2×C42—948no—
373C2×C42—941no—
374C2×C42—941no—
375C2×C42—941no—
376C2×C42C22×D4711yescharacteristic
377C4⋊C8—944no—
378#103—24no—
379#103—944no—
380#102—944no—
381#102—944no—
382C22⋊C8—944no—
383C4×C8—22no—
384C4×C8—942no—
385#95—21no—
386#95—942no—
387#95—941no—
388C24—471no—
389C24—941no—
390#41—948no—
391#41—9416no—
392#41—948no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8C2×C4×D4712yes—
396C2×Q8#1062535014yes—
397C2×D4—472no—
398C2×D4—474no—
399C2×D4—944no—
400C2×D4—948no—
401C2×D4—941no—
402C2×D4—942no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—471no—
407C22×C4—471no—
408C22×C4—471no—
409C22×C4—474no—
410C22×C4—474no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—944no—
414C22×C4—944no—
415C22×C4—944no—
416C22×C4—948no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—941no—
422C22×C4—944no—
423C22×C4—942no—
424C22×C4—1884no—
425C22×C4—1882no—
426C22×C4—1882no—
427C22×C4D4×D4711yescharacteristic
428C22×C4D4×D4711yescharacteristic
429C22×C4C23×D4711yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—942no—
434C2×C8—942no—
435C2×C8—942no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—472no—
441C4⋊C4—944no—
442C4⋊C4—941no—
443C4⋊C4—942no—
444C4⋊C4—942no—
445C4⋊C4—1884no—
446C4⋊C4—1884no—
447C4⋊C4C2×C4×D4712yes—
448C22⋊C4—944no—
449C22⋊C4—944no—
450C22⋊C4—944no—
451C22⋊C4—1884no—
452C42—24no—
453C42—42no—
454C42—474no—
455C42—474no—
456C42—472no—
457C42—472no—
458C42—944no—
459C42—944no—
460C42—1882no—
461C42C2×C4×D4712yes—
462C42D4×D4712yes—
463C23—472no—
464C23—942no—
465C23—944no—
466C23—942no—
467C23—942no—
468C23—941no—
469C23—941no—
470C23—1882no—
471C23—1881no—
472C23—1881no—
473C23C2×D4×D4711yescharacteristic
474Q8—22no—
475Q8—24no—
476Q8#1309398114yescomplemented
477D4—474no—
478D4—9416no—
479D4—942no—
480D4—944no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—472no—
490C2×C4—474no—
491C2×C4—472no—
492C2×C4—472no—
493C2×C4—948no—
494C2×C4—9416no—
495C2×C4—942no—
496C2×C4—941no—
497C2×C4—941no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—944no—
501C2×C4—948no—
502C2×C4—942no—
503C2×C4—1888no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4—1888no—
510C2×C4C22⋊C4×D4712yes—
511C2×C4C22⋊C4×D4712yes—
512C2×C4#1309405112yes—
513C2×C4#1309405612yes—
514C2×C4C22×C4×D4711yescharacteristic
515C2×C4C2×D4×D4711yescharacteristic
516C2×C4#41×D4712yes—
517C2×C4C2×#1062535012yes—
518C8—42no—
519C8—1882no—
520C22—944no—
521C22—942no—
522C22—942no—
523C22—942no—
524C22—1882no—
525C22—1884no—
526C22—1882no—
527C22—1884no—
528C22—1882no—
529C22—1882no—
530C22C2×C22⋊C4×D4711yescharacteristic
531C22#1568963212yes—
532C22#135×D4712yes—
533C22C2×#1309405111yescharacteristic
534C22C2×#1309405611yescharacteristic
535C4—24no—
536C4—22no—
537C4—42no—
538C4—82no—
539C4—474no—
540C4—942no—
541C4—942no—
542C4—3762no—
543C4—1882no—
544C4#1568912314yes—
545C4#117×D4711yescharacteristic
546C4C2×#1309398111yescharacteristic
547C4#1568963212yes—
548C2—1882no—
549C2—1884no—
550C2#2024576911yescharacteristic
551C2#416×D4711yescharacteristic
552C2C2×#1568912311yescharacteristic
553C2#2025344612yes—
554C2#2025345712yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#103×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
cosocleC22×C4×C47C24
hypercenterC4×#103D47
hypocenterC47#20737
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#15689174C2semidirect
3. #20245792C2semidirect
4. #20243481C2semidirect
5. #20243391C2semidirect
6. #20243387C2semidirect
8. C4×#13093948C2semidirect
9. C4×#13093944C2semidirect
10. C4×#13093946C2semidirect
11. C4×#103×C47C2semidirect
14. C4×#10625354C4semidirect
198. Q8×C47#117semidirect
231. D188#117semidirect
279. C4×#103D47semidirect
323. C47#20737semidirect
476. Q8#13093981semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×#156891742
#202457922
#202434812
#202433912
#202433872
#202433272
C4×#130939482
C4×#130939442
C4×#130939462
C4×#103×C472
#2073747