Finite Group Databasebeta

← #26226005

Subgroups — #26226005

Order 12032.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic
3#382×D47C211yesmaximal, characteristic
4#20244359C212yesmaximal, complemented
5#20244206C212yesmaximal, complemented
6#20243694C212yesmaximal
7#20243692C212yesmaximal
8#20243404C211yesmaximal, characteristic, complemented
9#20243403C211yesmaximal, characteristic, complemented
10#20243400C211yesmaximal, characteristic, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12#15689677—24no—
13#15689677C418yescomplemented
14C2×#13094023C2211yescharacteristic
15C2×#13094022C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094004C2211yescharacteristic
18C2×#13094002—21no—
19C2×#13094000C2211yescharacteristic
20C2×C4×Dic94—21no—
21C2×C4×Dic94C2211yescharacteristic
22C2×#13093981—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47C2211yescharacteristic
25#15689179—24no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093949C2212yes—
28C2×#13093948C2212yes—
29C2×#13093947C2212yes—
30C2×(C2×D47)⋊C8C2212yes—
31C2×#13093934C2212yes—
32#15688989C2211yescharacteristic
33#15688988C2212yes—
34#15688987—21no—
35#15688987—21no—
36#95×D47—21no—
37#95×D47—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#382C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C22×C4×D47—22no—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53#13094023—22no—
54#13094023C2×C414yescomplemented
55#13094022—22no—
56#13094022C2×C414yescomplemented
57#13094014—28no—
58#13094004—22no—
59#13094004C2×C414yes—
60#13094002—28no—
61#13094000—22no—
62#13094000C2×C414yes—
63C4×Dic94—22no—
64C4×Dic94—28no—
65C4×Dic94C2×C414yescomplemented
66#13093981—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2312yes—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—22no—
76C4⋊C4×D47—42no—
77C4⋊C4×D47—42no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84C42×D47C2×C414yes—
85C42×D47D414yes—
86C4⋊(C47⋊C8)—22no—
87#13093949—42no—
88#13093948—42no—
89#13093947—42no—
90(C2×D47)⋊C8—24no—
91#13093934—24no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98C4×Q8×C47—22no—
99C4×Q8×C47C2×C414yescomplemented
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C42×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106#103×C47—42no—
107#95×C47—21no—
108#95×C47—21no—
109C23×D47C2×D411yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47C22×C412yes—
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—21no—
124C22×Dic47—22no—
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127C22×Dic47C2×D411yescharacteristic
128C2×C4×D47—24no—
129C2×C4×D47—21no—
130C2×C4×D47—21no—
131C2×C4×D47—21no—
132C2×C4×D47—21no—
133C2×C4×D47—42no—
134C2×C4×D47—44no—
135C2×C4×D47—44no—
136C2×C4×D47—44no—
137C2×C4×D47C22⋊C414yes—
138C2×C4×D47C22⋊C414yes—
139C2×C4×D47C22×C412yes—
140C2×C4×D47C2×D412yes—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—24no—
147C2×Dic94—44no—
148C2×Dic94C22×C412yes—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47C22×C412yes—
157C42×C47—24no—
158C42×C47C22×C412yes—
159C42×C47C2×D412yes—
160(C2×D47)⋊C4—24no—
161(C2×D47)⋊C4—24no—
162C4⋊Dic47—21no—
163C4⋊Dic47—24no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47C22×C412yes—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172Dic47⋊C4—44no—
173C4×Dic47—24no—
174C4×Dic47C22⋊C414yes—
175C4×Dic47C22⋊C414yes—
176C4×Dic47C22×C412yes—
177C4×Dic47C2×D412yes—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C23×C47C22×D411yescharacteristic
182C22×D47—21no—
183C22×D47—21no—
184C22×D47—22no—
185C22×D47C2×C22⋊C412yes—
186C22×D47#13714yes—
187Q8×C47—22no—
188Q8×C47—22no—
189Q8×C47—41no—
190Q8×C47—41no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—41no—
196C2×C4×C47—41no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47C2×C22⋊C412yes—
202C2×C4×C47C2×C22⋊C412yes—
203C2×C4×C47C23×C411yescharacteristic
204C2×C4×C47C22×D411yescharacteristic
205C2×C4×C47H(5,𝔽2)11yescharacteristic
206C2×C4×C47H(5,𝔽2)11yescharacteristic
207C2×C4×C472−1+411yescharacteristic
208C2×C4×C472−1+411yescharacteristic
209C2×Dic47—21no—
210C2×Dic47—21no—
211C2×Dic47—22no—
212C2×Dic47—22no—
213C2×Dic47—22no—
214C2×Dic47—22no—
215C2×Dic47—41no—
216C2×Dic47—41no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47C2×C22⋊C412yes—
222C2×Dic47C2×C22⋊C412yes—
223C2×Dic47C2×C22⋊C412yes—
224C4×D47—28no—
225C4×D47—22no—
226C4×D47—22no—
227C4×D47—22no—
228C4×D47—22no—
229C4×D47—44no—
230C4×D47—88no—
231Dic94—22no—
232Dic94—22no—
233Dic94—48no—
234Dic94—41no—
235Dic94—41no—
236C8×C47—42no—
237C47⋊C8—42no—
238#21866—471nomaximal, Sylow
239C22×C47C22×C22⋊C411yescharacteristic
240C22×C47#53511yescharacteristic
241C22×C47#53611yescharacteristic
242C22×C47C2×#13712yes—
243C22×C47#57512yes—
244C2×D47—22no—
245C2×D47—22no—
246C2×D47—24no—
247C2×D47—24no—
248C2×D47#41914yes—
249C4×C47—22no—
250C4×C47—22no—
251C4×C47—42no—
252C4×C47—42no—
253C4×C47—82no—
254C4×C47#52011yescharacteristic
255C4×C47#52011yescharacteristic
256C4×C47#53711yescharacteristic
257C4×C47#53711yescharacteristic
258Dic47—22no—
259Dic47—22no—
260Dic47—22no—
261Dic47—42no—
262Dic47—42no—
263Dic47—82no—
264C2×#533—471no—
265C2×#432—471no—
266C2×#382—471no—
267#1993—472no—
268#1960—472no—
269#1872—472no—
270#1872—472no—
271#1805—471no—
272#1805—471no—
273#1805—471no—
274#1805D4711yescharacteristic, complemented
275C2×C47#231111yescharacteristic
276C2×C47C2×#41911yescharacteristic
277C2×C47#289611yescharacteristic
278C2×C47#321512yes—
279C2×C47#323612yes—
280D47—24no—
281#533—478no—
282#533—944no—
283C2×C4⋊Q8—471no—
284C2×C4⋊Q8—471no—
285C2×#125—941no—
286C2×C22⋊Q8—471no—
287C2×C22⋊Q8—471no—
288C2×C22⋊Q8—941no—
289C2×C4×Q8—471no—
290C2×C4×Q8—941no—
291C2×C4×Q8C2×D4711yescharacteristic
292C2×#117—941no—
293C22×C4⋊C4—472no—
294C22×C42—471no—
295#432—944no—
296C2×C4⋊C8—471no—
297C2×C4⋊C8C2×D4711yescharacteristic
298C2×#103—472no—
299C2×#103—472no—
300C2×#103—472no—
301C2×#103C2×D4712yes—
302C2×C22⋊C8—472no—
303C2×#97—472no—
304#382—472no—
305#382—471no—
306#382—941no—
307#382—941no—
308#382C2×D4711yescharacteristic
309C2×#95—941no—
310C2×#95—941no—
311#340—472no—
312#340—472no—
313#340—472no—
314#340C2×D4712yes—
315C47#2186611yescharacteristic, complemented, Sylow
316C22×Q8—471no—
317C22×Q8—941no—
318C22×Q8C22×D4711yescharacteristic
319C23×C4—471no—
320C23×C4—471no—
321C23×C4—471no—
322C23×C4—942no—
323C22×C8—472no—
324C22×C8C22×D4712yes—
325C4⋊Q8—474no—
326C4⋊Q8—474no—
327C4⋊Q8—942no—
328C4⋊Q8—942no—
329#125—948no—
330C22⋊Q8—474no—
331C22⋊Q8—474no—
332C22⋊Q8—942no—
333C22⋊Q8—948no—
334C22⋊Q8—942no—
335C4×Q8—22no—
336C4×Q8—474no—
337C4×Q8—948no—
338C4×Q8—942no—
339C4×Q8C4×D4714yescomplemented
340#117—948no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—471no—
343C2×C4⋊C4—472no—
344C2×C4⋊C4—941no—
345C2×C4⋊C4—941no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4—941no—
348C2×C4⋊C4—941no—
349C2×C4⋊C4—941no—
350C2×C4⋊C4—942no—
351C2×C4⋊C4—1882no—
352C2×C4⋊C4—1882no—
353C2×C4⋊C4C22×D4712yes—
354C2×C4⋊C4C22×D4711yescharacteristic
355C2×C22⋊C4—941no—
356C2×C22⋊C4—941no—
357C2×C42—21no—
358C2×C42—471no—
359C2×C42—471no—
360C2×C42—471no—
361C2×C42—474no—
362C2×C42—474no—
363C2×C42—941no—
364C2×C42C22×D4711yescharacteristic
365C4⋊C8—22no—
366C4⋊C8—942no—
367#103—42no—
368#103—1882no—
369#103—1882no—
370#103—1882no—
371C22⋊C8—944no—
372#97—944no—
373#95—21no—
374#95—21no—
375#95—942no—
376#95—942no—
377#95—941no—
378#95—941no—
379C24—471no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—472no—
386C2×Q8—944no—
387C2×Q8—941no—
388C2×Q8—941no—
389C2×Q8—941no—
390C2×Q8—941no—
391C2×Q8—941no—
392C2×Q8—1884no—
393C2×Q8C2×C4×D4712yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—471no—
398C22×C4—471no—
399C22×C4—471no—
400C22×C4—474no—
401C22×C4—472no—
402C22×C4—472no—
403C22×C4—474no—
404C22×C4—944no—
405C22×C4—941no—
406C22×C4—941no—
407C22×C4—941no—
408C22×C4—941no—
409C22×C4—941no—
410C22×C4—941no—
411C22×C4—942no—
412C22×C4—1882no—
413C22×C4—1884no—
414C22×C4—1884no—
415C22×C4—1884no—
416C22×C4D4×D4711yescharacteristic
417C22×C4D4×D4711yescharacteristic
418C22×C4C23×D4711yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—942no—
423C2×C8—942no—
424C2×C8—942no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—472no—
430C4⋊C4—944no—
431C4⋊C4—944no—
432C4⋊C4—944no—
433C4⋊C4—944no—
434C4⋊C4—944no—
435C4⋊C4—944no—
436C4⋊C4—941no—
437C4⋊C4—942no—
438C4⋊C4—942no—
439C4⋊C4—1884no—
440C4⋊C4C2×C4×D4712yes—
441C22⋊C4—944no—
442C22⋊C4—944no—
443C42—24no—
444C42—474no—
445C42—474no—
446C42—472no—
447C42—472no—
448C42—944no—
449C42C2×C4×D4712yes—
450C42D4×D4712yes—
451C23—472no—
452C23—474no—
453C23—942no—
454C23—941no—
455C23—941no—
456C23C2×D4×D4711yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—942no—
462Q8—942no—
463Q8—1888no—
464Q8—1881no—
465Q8—1881no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—472no—
477C2×C4—472no—
478C2×C4—472no—
479C2×C4—948no—
480C2×C4—942no—
481C2×C4—942no—
482C2×C4—941no—
483C2×C4—941no—
484C2×C4—942no—
485C2×C4—942no—
486C2×C4—942no—
487C2×C4—942no—
488C2×C4—942no—
489C2×C4—942no—
490C2×C4—3768no—
491C2×C4—1884no—
492C2×C4—1882no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1881no—
497C2×C4—1881no—
498C2×C4C22⋊C4×D4712yes—
499C2×C4C22⋊C4×D4712yes—
500C2×C4C22×C4×D4711yescharacteristic
501C2×C4C2×D4×D4711yescharacteristic
502C2×C4#1309406911yescharacteristic
503C2×C4#1309407011yescharacteristic
504C2×C4#1309407111yescharacteristic
505C2×C4#1309407211yescharacteristic
506C8—42no—
507C8—1882no—
508C22—474no—
509C22—944no—
510C22—944no—
511C22—942no—
512C22—942no—
513C22C2×C22⋊C4×D4711yescharacteristic
514C22#1568968911yescharacteristic
515C22#1568969611yescharacteristic
516C22#137×D4712yes—
517C22#1568992012yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—942no—
524C4—942no—
525C4—942no—
526C4—3762no—
527C4—1882no—
528C4—1882no—
529C4#1568960311yescharacteristic
530C4#1568960411yescharacteristic
531C4#1568970211yescharacteristic
532C4#1568970311yescharacteristic
533C2—944no—
534C2#2024597711yescharacteristic
535C2#419×D4711yescharacteristic
536C2#2025190311yescharacteristic
537C2#2025438912yes—
538C2#2025459212yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47#21866
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #20244359C2semidirect
5. #20244206C2semidirect
8. #20243404C2semidirect
9. #20243403C2semidirect
10. #20243400C2semidirect
11. C47×#1805C2semidirect
13. #15689677C4semidirect
54. #13094023C2×C4semidirect
56. #13094022C2×C4semidirect
65. C4×Dic94C2×C4semidirect
99. C4×Q8×C47C2×C4semidirect
274. #1805D47semidirect
315. C47#21866semidirect
339. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×#156891792
#382×D472
#202443592
#202442062
#202436942
#202436922
#202434042
#202434032
#202434002
C47×#18052
#2186647