Finite Group Databasebeta

← #26221323

Subgroups — #26221323

Order 12032.

1258 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689675C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20244334C212yesmaximal, complemented
5#20244309C212yesmaximal, complemented
6#20243696C211yesmaximal, characteristic
7#20243695C211yesmaximal, characteristic
8#20243693C211yesmaximal, characteristic
9#103×Dic47C212yesmaximal, complemented
10#20243207C212yesmaximal, complemented
11C47×#1872C211yesmaximal, characteristic
12#15689675—24no—
13#15689675C418yescomplemented
14C2×#13094023C2211yescharacteristic
15C2×#13094022C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094003C2211yescharacteristic
18C2×#13094001—21no—
19C2×Q8×Dic47C2211yescharacteristic
20C2×#13093989—21no—
21C2×#13093989C2211yescharacteristic
22C2×#13093980—21no—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4×Dic47—21no—
25C22×C4×Dic47C2211yescharacteristic
26#15689173—24no—
27C2×Dic47⋊C8C2212yes—
28C2×#13093949C2212yes—
29C2×#13093948C2212yes—
30C2×#13093947C2212yes—
31C2×C22⋊(C47⋊C8)C2211yescharacteristic
32C2×C8×Dic47C2212yes—
33C4⋊C4×Dic47—21no—
34C4⋊C4×Dic47—21no—
35C4⋊C4×Dic47C2212yes—
36C4⋊C4×Dic47C2212yes—
37C2×C47⋊#95—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C22⋊Q8×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×C22⋊C8×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C47⋊C8C2312yes—
52#13094023—22no—
53#13094023C2×C414yescomplemented
54#13094022—22no—
55#13094022C2×C414yescomplemented
56#13094014—28no—
57#13094003—22no—
58#13094003C2×C414yes—
59#13094001—28no—
60Q8×Dic47—22no—
61Q8×Dic47C2×C414yescomplemented
62#13093989—28no—
63#13093989—22no—
64#13093989C2×C414yescomplemented
65#13093980—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—22no—
80C2×C4×Dic47—22no—
81C2×C4×Dic47—44no—
82C2×C4×Dic47C2×C414yes—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D414yes—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88Dic47⋊C8—24no—
89#13093949—24no—
90#13093948—24no—
91#13093947—24no—
92C22⋊(C47⋊C8)—22no—
93C8×Dic47—24no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47C2312yes—
100C22⋊Q8×C47—22no—
101C22⋊Q8×C47C2×C414yes—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—24no—
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2312yes—
107C2×C22⋊C4×C47—21no—
108#103×C47—24no—
109C22⋊C8×C47—22no—
110C24×C47C2×D411yescharacteristic
111C2×Q8×C47—21no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22×C412yes—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—24no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—22no—
121C22×C4×C47—42no—
122C22×C4×C47—42no—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2411yescharacteristic
127C22×Dic47—24no—
128C22×Dic47—22no—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—41no—
132C22×Dic47—41no—
133C22×Dic47—42no—
134C22×Dic47—42no—
135C22×Dic47C22⋊C414yes—
136C22×Dic47C22⋊C414yes—
137C22×Dic47SD1614yes—
138C22×Dic47Q1614yes—
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C2×Dic94—22no—
144C2×Dic94—22no—
145C2×Dic94—21no—
146C2×Dic94—24no—
147C2×Dic94—42no—
148C2×Dic94—42no—
149C2×Dic94C22×C412yes—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47C22×C412yes—
158C4⋊C4×C47#4114yes—
159C22⋊C4×C47—24no—
160C22⋊Dic47—24no—
161C4⋊Dic47—24no—
162C4⋊Dic47—21no—
163C4⋊Dic47—22no—
164C4⋊Dic47—24no—
165C4⋊Dic47C22×C412yes—
166C4⋊Dic47#4114yes—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172C4×Dic47—24no—
173C4×Dic47—24no—
174C4×Dic47—44no—
175C4×Dic47—44no—
176C4×Dic47C22⋊C414yes—
177C4×Dic47C22⋊C414yes—
178C4×Dic47C22×C412yes—
179C4×Dic47C22×C412yes—
180C4×Dic47C2×D412yes—
181C4×Dic47C2×D412yes—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47—21no—
189C23×C47C2×C22⋊C412yes—
190C23×C47C2×SD1612yes—
191C23×C47C2×Q1612yes—
192C23×C47C22×D411yescharacteristic
193Q8×C47—24no—
194Q8×C47—42no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—48no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47C2×C22⋊C412yes—
210C2×C4×C47C23×C411yescharacteristic
211C2×C4×C47C22×D411yescharacteristic
212C2×C4×C47C2×#4112yes—
213C2×C4×C47C2×#4112yes—
214C2×Dic47—21no—
215C2×Dic47—21no—
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—24no—
221C2×Dic47—44no—
222C2×Dic47—48no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—42no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—42no—
229C2×Dic47#103116yescomplemented
230C2×Dic47C2×C22⋊C412yes—
231C2×Dic47C2×C22⋊C412yes—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C2×C22⋊C412yes—
234C2×Dic47C2×SD1612yes—
235C2×Dic47C2×SD1612yes—
236C2×Dic47C2×Q1612yes—
237C2×Dic47C2×Q1612yes—
238Dic94—24no—
239Dic94—48no—
240Dic94—42no—
241C8×C47—42no—
242C47⋊C8—42no—
243#21385—471nomaximal, Sylow
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—22no—
250C22×C47C2×#10314yes—
251C22×C47C22×C22⋊C411yescharacteristic
252C22×C47#52511yescharacteristic
253C22×C47#52511yescharacteristic
254C22×C47C22×SD1611yescharacteristic
255C22×C47C22×Q1611yescharacteristic
256C22×C47#57511yescharacteristic
257C22×C47#57511yescharacteristic
258C4×C47—22no—
259C4×C47—44no—
260C4×C47—42no—
261C4×C47—42no—
262C4×C47C4×#4112yes—
263C4×C47#53212yes—
264Dic47—22no—
265Dic47—22no—
266Dic47—44no—
267Dic47—42no—
268Dic47—42no—
269Dic47C2×#10314yes—
270Dic47C2×#10314yes—
271C2×#533—471no—
272C2×#431—471no—
273C2×C4×C4⋊C4—471no—
274#1987—472no—
275#1981—472no—
276#1872—471no—
277#1872—471no—
278#1872—471no—
279#1872D4711yescharacteristic
280C4×#103—472no—
281C4×#103—472no—
282C2×C47—24no—
283C2×C47#229711yescharacteristic
284C2×C47C22×#10311yescharacteristic
285C2×C47#289611yescharacteristic
286C2×C47#310111yescharacteristic
287C2×C47#310111yescharacteristic
288C2×C47#310411yescharacteristic
289C2×C47#310411yescharacteristic
290#533—478no—
291#533—944no—
292C2×C4⋊Q8—471no—
293C2×C4⋊Q8—471no—
294C2×#125—941no—
295C2×C22⋊Q8—471no—
296C2×C22⋊Q8—941no—
297C2×C22⋊Q8C2×D4711yescharacteristic
298C2×C4×Q8—471no—
299C2×C4×Q8—471no—
300C2×C4×Q8—941no—
301C2×#117—941no—
302C22×C4⋊C4—471no—
303C22×C4⋊C4C2×D4711yescharacteristic
304C22×C42—471no—
305C22×C42—941no—
306#431—944no—
307C2×C4⋊C8—472no—
308C2×#103—472no—
309C2×#103—472no—
310C2×#103—472no—
311C2×#103C2×D4712yes—
312C2×C22⋊C8—471no—
313C2×C22⋊C8C2×D4711yescharacteristic
314C2×C4×C8—472no—
315C4×C4⋊C4—472no—
316C4×C4⋊C4—472no—
317C4×C4⋊C4—941no—
318C4×C4⋊C4—941no—
319C2×#95—941no—
320#340—472no—
321#340—472no—
322#340—472no—
323#340C2×D4712yes—
324C47#2138511yescharacteristic, complemented, Sylow
325C22×Q8—471no—
326C22×Q8—941no—
327C22×Q8C22×D4711yescharacteristic
328C23×C4—21no—
329C23×C4—471no—
330C23×C4—471no—
331C23×C4—941no—
332C23×C4C22×D4711yescharacteristic
333C22×C8—472no—
334C22×C8C22×D4712yes—
335C4⋊Q8—474no—
336C4⋊Q8—474no—
337C4⋊Q8—942no—
338C4⋊Q8—942no—
339#125—948no—
340C22⋊Q8—22no—
341C22⋊Q8—474no—
342C22⋊Q8—948no—
343C22⋊Q8—942no—
344C22⋊Q8C4×D4714yes—
345C4×Q8—474no—
346C4×Q8—474no—
347C4×Q8—942no—
348C4×Q8—948no—
349C4×Q8—942no—
350#117—948no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—471no—
356C2×C4⋊C4—944no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—941no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C4⋊C4C22×D4712yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—941no—
368C2×C42—474no—
369C2×C42—471no—
370C2×C42—471no—
371C2×C42—474no—
372C2×C42—471no—
373C2×C42—471no—
374C2×C42—941no—
375C2×C42—941no—
376C2×C42—942no—
377C2×C42—942no—
378C2×C42—1884no—
379C4⋊C8—944no—
380#103—24no—
381#103—944no—
382#103—944no—
383#103—944no—
384C22⋊C8—22no—
385C22⋊C8—942no—
386C4×C8—944no—
387#95—942no—
388#95—942no—
389C24D4×D4711yescharacteristic
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—472no—
394C2×Q8—944no—
395C2×Q8—941no—
396C2×Q8—942no—
397C2×Q8—942no—
398C2×Q8—1882no—
399C2×Q8—1882no—
400C2×Q8C2×C4×D4712yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—474no—
410C22×C4—474no—
411C22×C4—474no—
412C22×C4—471no—
413C22×C4—471no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—474no—
417C22×C4—944no—
418C22×C4—942no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—1881no—
422C22×C4—1881no—
423C22×C4—1882no—
424C22×C4—1882no—
425C22×C4C2×C4×D4712yes—
426C22×C4D4×D4711yescharacteristic
427C22×C4D4×D4712yes—
428C22×C4C23×D4711yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—942no—
433C2×C8—942no—
434C2×C8—942no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—472no—
440C4⋊C4—474no—
441C4⋊C4—944no—
442C4⋊C4—944no—
443C4⋊C4—944no—
444C4⋊C4—944no—
445C4⋊C4—944no—
446C4⋊C4—944no—
447C4⋊C4—944no—
448C4⋊C4—941no—
449C4⋊C4—942no—
450C4⋊C4C2×C4×D4712yes—
451C4⋊C4#1062535214yes—
452C22⋊C4—24no—
453C22⋊C4—944no—
454C42—474no—
455C42—474no—
456C42—472no—
457C42—472no—
458C42—472no—
459C42—472no—
460C42—944no—
461C42—944no—
462C42—1884no—
463C42—1884no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C22⋊C4×D4712yes—
469C23SD16×D4712yes—
470C23Q16×D4712yes—
471C23C2×D4×D4711yescharacteristic
472Q8—24no—
473Q8—42no—
474Q8—944no—
475Q8—1888no—
476Q8—1882no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—48no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—472no—
492C2×C4—472no—
493C2×C4—4716no—
494C2×C4—472no—
495C2×C4—472no—
496C2×C4—472no—
497C2×C4—472no—
498C2×C4—472no—
499C2×C4—472no—
500C2×C4—941no—
501C2×C4—941no—
502C2×C4—941no—
503C2×C4—941no—
504C2×C4—942no—
505C2×C4—942no—
506C2×C4—944no—
507C2×C4—1884no—
508C2×C4—1888no—
509C2×C4—1881no—
510C2×C4—1881no—
511C2×C4—1882no—
512C2×C4—1881no—
513C2×C4—1882no—
514C2×C4—1881no—
515C2×C4C22⋊C4×D4712yes—
516C2×C4C22×C4×D4711yescharacteristic
517C2×C4C2×D4×D4711yescharacteristic
518C2×C4#41×D4712yes—
519C2×C4C2×#1062535212yes—
520C8—42no—
521C8—1882no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22#103×D4714yes—
529C22C2×C22⋊C4×D4711yescharacteristic
530C22#1568962911yescharacteristic
531C22#1568963111yescharacteristic
532C22C2×SD16×D4711yescharacteristic
533C22C2×Q16×D4711yescharacteristic
534C22#1568991911yescharacteristic
535C22#1568992211yescharacteristic
536C4—22no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—474no—
541C4—474no—
542C4—942no—
543C4—942no—
544C4—1882no—
545C4—1882no—
546C4—1884no—
547C4C4×#1062535212yes—
548C4#1568966812yes—
549C2—24no—
550C2#2024587311yescharacteristic
551C2C2×#103×D4711yescharacteristic
552C2#2025190411yescharacteristic
553C2#2025339111yescharacteristic
554C2#2025339311yescharacteristic
555C2#2025342211yescharacteristic
556C2#2025342411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1872C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1872D47
hypocenterC47#21385
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #20244334C2semidirect
5. #20244309C2semidirect
9. #103×Dic47C2semidirect
10. #20243207C2semidirect
13. #15689675C4semidirect
53. #13094023C2×C4semidirect
55. #13094022C2×C4semidirect
61. Q8×Dic47C2×C4semidirect
64. #13093989C2×C4semidirect
229. C2×Dic47#103semidirect
324. C47#21385semidirect

Maximal subgroups

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

subgroupindex
C2×#156896752
C2×#156891732
C2×C4⋊C4×Dic472
#202443342
#202443092
#202436962
#202436952
#202436932
#103×Dic472
#202432072
C47×#18722
#2138547