Finite Group Databasebeta

← #26216084

Subgroups — #26216084

Order 12032.

1542 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic
3#380×D47C211yesmaximal, characteristic
4#20243482C214yesmaximal, complemented
5#20243479C211yesmaximal, characteristic, complemented
6#20243476C211yesmaximal, characteristic, complemented
7#20243474C211yesmaximal, characteristic, complemented
8#20243106C214yesmaximal, complemented
9C47×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625324C2212yes—
12C2×#13093982—24no—
13C2×#13093982—21no—
14C2×#13093982—21no—
15C2×#13093982C418yescomplemented
16C2×#13093982C2212yes—
17C2×#13093982C2212yes—
18C22×C4⋊Dic47C2211yescharacteristic
19C2×C4⋊C4×D47C2212yes—
20C22×(C2×D47)⋊C4—21no—
21C2×C22⋊C4×D47—21no—
22C22×C4×Dic47—21no—
23#15689085—24no—
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×(C2×D47)⋊C8C2212yes—
26#15688983C2211yescharacteristic
27#15688982—21no—
28#15688981—21no—
29#15688980C2212yes—
30#95×D47—21no—
31#15688886C2214yescomplemented
32#15688885C2214yescomplemented
33#15688885C2214yescomplemented
34C22×C4⋊C4×C47C2211yescharacteristic
35C2×C22⋊C8×C47C2211yescharacteristic
36C47×#380C2211yescharacteristic
37C47×#340C2214yescomplemented
38C24×D47D411yescharacteristic
39C23×Dic47—21no—
40C23×Dic47D411yescharacteristic
41C22×C4×D47—28no—
42C22×C4×D47—22no—
43C22×C4×D47C2×C414yes—
44C22×C4×D47D411yescharacteristic
45C22×C4×D47D411yescharacteristic
46C22×C4×D47D414yes—
47C22×C4×D47C2311yescharacteristic
48C22×C4×D47C2311yescharacteristic
49C2×#10625324—24no—
50C2×#10625324—28no—
51C22×C47⋊C8C2312yes—
52#13093982—28no—
53#13093982—28no—
54#13093982—24no—
55#13093982—24no—
56#13093982C2×C418yes—
57#13093982C2×C418yescomplemented
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47C2×C414yescomplemented
61C2×C4⋊Dic47C2312yes—
62C2×C4⋊Dic47C2312yes—
63C2×Dic47⋊C4—22no—
64C4⋊C4×D47—28no—
65C2×C22⋊Dic47—21no—
66C2×(C2×D47)⋊C4—28no—
67C2×(C2×D47)⋊C4—22no—
68C2×(C2×D47)⋊C4—22no—
69C2×(C2×D47)⋊C4—22no—
70C22⋊C4×D47—44no—
71C2×C4×Dic47—28no—
72C2×C4×Dic47—22no—
73C2×C4×Dic47—22no—
74C22⋊(C47⋊C8)—22no—
75(C2×D47)⋊C8—24no—
76C47⋊#95—21no—
77C47⋊#95—22no—
78C23×C4×C47—21no—
79C23×C4×C47C2311yescharacteristic
80C22×C8×C47C2312yes—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47C2×C414yescomplemented
84C2×C4⋊C4×C47C2312yes—
85C2×C4⋊C4×C47C2312yes—
86C2×C22⋊C4×C47—21no—
87C22⋊C8×C47—22no—
88#95×C47—21no—
89C24×C47C2×D411yescharacteristic
90C23×D47—22no—
91C23×D47—22no—
92C23×D47—24no—
93C23×D47—42no—
94C23×D47C22⋊C414yes—
95C23×D47C2×D411yescharacteristic
96C23×D47C2×D411yescharacteristic
97C22×C4×C47—24no—
98C22×C4×C47—24no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—42no—
102C22×C4×C47—42no—
103C22×C4×C47C22×C412yes—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D412yes—
106C22×C4×C47C2411yescharacteristic
107C22×Dic47—24no—
108C22×Dic47—24no—
109C22×Dic47—22no—
110C22×Dic47—22no—
111C22×Dic47—44no—
112C22×Dic47C22⋊C414yes—
113C22×Dic47C2×D411yescharacteristic
114C22×Dic47C2×D411yescharacteristic
115C2×C4×D47—24no—
116C2×C4×D47—24no—
117C2×C4×D47—24no—
118C2×C4×D47—216no—
119C2×C4×D47—216no—
120C2×C4×D47—44no—
121C2×C4×D47—44no—
122C2×C4×D47—44no—
123C2×C4×D47C22⋊C414yes—
124C2×C4×D47C22⋊C414yes—
125C2×C4×D47C22×C412yes—
126C2×C4×D47C22×C412yes—
127C2×C4×D47C2×D412yes—
128C2×C4×D47C2×D412yes—
129C2×C4×D47#4118yes—
130C2×C8×C47—22no—
131C2×C8×C47—24no—
132C4⋊C4×C47—24no—
133C4⋊C4×C47—22no—
134C4⋊C4×C47—24no—
135C4⋊C4×C47C22×C414yes—
136C22⋊C4×C47—42no—
137C22⋊Dic47—42no—
138(C2×D47)⋊C4—28no—
139(C2×D47)⋊C4—28no—
140(C2×D47)⋊C4—44no—
141C4⋊Dic47—24no—
142C4⋊Dic47—22no—
143C4⋊Dic47—24no—
144C4⋊Dic47C22×C414yes—
145Dic47⋊C4—44no—
146C4×Dic47—28no—
147C4×Dic47—28no—
148C2×C47⋊C8—22no—
149C2×C47⋊C8—24no—
150#10625324—48no—
151C23×C47—22no—
152C23×C47—22no—
153C23×C47C2×C22⋊C412yes—
154C23×C47AGL(1,ℤ/8)12yes—
155C23×C47#13712yes—
156C23×C47C22×D411yescharacteristic
157C22×D47—21no—
158C22×D47—21no—
159C22×D47—22no—
160C22×D47—22no—
161C22×D47—21no—
162C22×D47—21no—
163C22×D47—22no—
164C22×D47—28no—
165C22×D47—28no—
166C22×D47—44no—
167C22×D47—48no—
168C22×D47—48no—
169C22×D47C2×C22⋊C412yes—
170C22×D47C2×C22⋊C412yes—
171C22×D47#12314yes—
172C2×C4×C47—22no—
173C2×C4×C47—24no—
174C2×C4×C47—24no—
175C2×C4×C47—24no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—48no—
179C2×C4×C47—42no—
180C2×C4×C47—42no—
181C2×C4×C47—42no—
182C2×C4×C47C2×C22⋊C412yes—
183C2×C4×C47C23×C411yescharacteristic
184C2×C4×C47C22×D411yescharacteristic
185C2×C4×C47C2×#4114yes—
186C2×Dic47—22no—
187C2×Dic47—28no—
188C2×Dic47—24no—
189C2×Dic47—48no—
190C2×Dic47—44no—
191C2×Dic47—42no—
192C2×Dic47—42no—
193C2×Dic47—42no—
194C2×Dic47C2×C22⋊C412yes—
195C2×Dic47C2×C22⋊C412yes—
196C2×Dic47#12314yes—
197C4×D47—28no—
198C4×D47—24no—
199C4×D47—24no—
200C4×D47—28no—
201C4×D47—88no—
202C4×D47#11718yes—
203C4×D47#12318yes—
204C8×C47—42no—
205C47⋊C8—42no—
206#20702—471nomaximal, Sylow
207C22×C47—24no—
208C22×C47—24no—
209C22×C47—22no—
210C22×C47—22no—
211C22×C47#41714yes—
212C22×C47C22×C22⋊C411yescharacteristic
213C22×C47C2×#12312yes—
214C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
215C22×C47C2×#13711yescharacteristic
216C22×C47#57512yes—
217C2×D47—24no—
218C2×D47—24no—
219C2×D47—22no—
220C2×D47—22no—
221C2×D47—22no—
222C2×D47—22no—
223C2×D47—22no—
224C2×D47—416no—
225C2×D47—42no—
226C2×D47—42no—
227C2×D47#38114yes—
228C4×C47—22no—
229C4×C47—44no—
230C4×C47—82no—
231C4×C47C2×#11712yes—
232C4×C47C2×#12312yes—
233Dic47—22no—
234Dic47—44no—
235Dic47—82no—
236Dic47#38114yes—
237C22×#117—471no—
238C2×#407—471no—
239C2×#380—471no—
240#1825—474no—
241#1824—471no—
242#1824—471no—
243#1823—471no—
244#1823D4711yescharacteristic, complemented
245#1736—474no—
246C2×C47—24no—
247C2×C47C2×#38111yescharacteristic
248C2×C47C2×#41711yescharacteristic
249C2×C47#289611yescharacteristic
250C2×C47#309412yes—
251C2×C47#309512yes—
252D47—24no—
253D47—42no—
254C24×C4—471no—
255C22×M16—472no—
256C2×#117—478no—
257C2×#117—472no—
258C2×#117—472no—
259C2×#117—941no—
260C2×#117—941no—
261C2×#117—944no—
262C22×C4⋊C4—471no—
263C22×C4⋊C4—472no—
264C22×C4⋊C4C2×D4711yescharacteristic
265C22×C22⋊C4—941no—
266C22×C22⋊C4—941no—
267C22×C42—941no—
268#407—944no—
269C2×C22⋊C8—471no—
270C2×C22⋊C8—472no—
271C2×C22⋊C8C2×D4711yescharacteristic
272#380—471no—
273#380—472no—
274#380—941no—
275#380—941no—
276#380C2×D4711yescharacteristic
277C2×#95—941no—
278#340—474no—
279#340—474no—
280#340—474no—
281#340C2×D4714yescomplemented
282C47#2070211yescharacteristic, complemented, Sylow
283C25—471no—
284C23×C4—21no—
285C23×C4—471no—
286C23×C4—471no—
287C23×C4—474no—
288C23×C4—471no—
289C23×C4—471no—
290C23×C4—474no—
291C23×C4—471no—
292C23×C4—948no—
293C23×C4—942no—
294C23×C4—941no—
295C23×C4C22×D4711yescharacteristic
296C2×M16—948no—
297C2×M16—944no—
298C22×C8—472no—
299C22×C8C22×D4712yes—
300#117—478no—
301#117—478no—
302#117—948no—
303#117—948no—
304#117—944no—
305#117—944no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—474no—
309C2×C4⋊C4—472no—
310C2×C4⋊C4—472no—
311C2×C4⋊C4—948no—
312C2×C4⋊C4—941no—
313C2×C4⋊C4—942no—
314C2×C4⋊C4—942no—
315C2×C4⋊C4C4×D4714yescomplemented
316C2×C4⋊C4C22×D4712yes—
317C2×C4⋊C4C22×D4712yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—948no—
320C2×C22⋊C4—942no—
321C2×C22⋊C4—942no—
322C2×C22⋊C4—942no—
323C2×C22⋊C4—941no—
324C2×C22⋊C4—1884no—
325C2×C42—948no—
326C2×C42—942no—
327C2×C42—942no—
328C22⋊C8—22no—
329C22⋊C8—944no—
330C22⋊C8—942no—
331#95—21no—
332#95—942no—
333#95—941no—
334C24—471no—
335C24—471no—
336C24—474no—
337C24—944no—
338C24—942no—
339C24—942no—
340C24—1882no—
341C24D4×D4711yescharacteristic
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—471no—
349C22×C4—472no—
350C22×C4—472no—
351C22×C4—478no—
352C22×C4—474no—
353C22×C4—471no—
354C22×C4—474no—
355C22×C4—474no—
356C22×C4—472no—
357C22×C4—472no—
358C22×C4—944no—
359C22×C4—9416no—
360C22×C4—9416no—
361C22×C4—944no—
362C22×C4—944no—
363C22×C4—944no—
364C22×C4—944no—
365C22×C4—942no—
366C22×C4—942no—
367C22×C4—1884no—
368C22×C4—1884no—
369C22×C4—1884no—
370C22×C4—1884no—
371C22×C4C2×C4×D4712yes—
372C22×C4D4×D4711yescharacteristic
373C22×C4D4×D4712yes—
374C22×C4C23×D4711yescharacteristic
375M16—1888no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—942no—
379C2×C8—944no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—474no—
384C4⋊C4—944no—
385C4⋊C4—942no—
386C4⋊C4—944no—
387C4⋊C4—1884no—
388C4⋊C4C2×C4×D4714yes—
389C22⋊C4—42no—
390C22⋊C4—948no—
391C22⋊C4—948no—
392C22⋊C4—1884no—
393C22⋊C4—1882no—
394C42—948no—
395C42—948no—
396C23—22no—
397C23—22no—
398C23—472no—
399C23—474no—
400C23—472no—
401C23—948no—
402C23—941no—
403C23—941no—
404C23—942no—
405C23—942no—
406C23—948no—
407C23—941no—
408C23—941no—
409C23—942no—
410C23—1884no—
411C23—1888no—
412C23—1888no—
413C23C22⋊C4×D4712yes—
414C23#1309404812yes—
415C23#1309405312yes—
416C23C2×D4×D4711yescharacteristic
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—472no—
428C2×C4—474no—
429C2×C4—478no—
430C2×C4—478no—
431C2×C4—472no—
432C2×C4—948no—
433C2×C4—948no—
434C2×C4—948no—
435C2×C4—942no—
436C2×C4—944no—
437C2×C4—944no—
438C2×C4—944no—
439C2×C4—3768no—
440C2×C4—1882no—
441C2×C4—1882no—
442C2×C4—1882no—
443C2×C4—1888no—
444C2×C4—1884no—
445C2×C4C22⋊C4×D4712yes—
446C2×C4C22×C4×D4711yescharacteristic
447C2×C4C2×D4×D4711yescharacteristic
448C2×C4#41×D4714yes—
449C8—42no—
450C8—1882no—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—474no—
456C22—942no—
457C22—944no—
458C22—944no—
459C22—942no—
460C22—942no—
461C22—942no—
462C22—942no—
463C22—18816no—
464C22—1882no—
465C22—1882no—
466C22#1568912114yes—
467C22C2×C22⋊C4×D4711yescharacteristic
468C22#123×D4712yes—
469C22C2×#1309404811yescharacteristic
470C22C2×#1309405311yescharacteristic
471C22#1568992212yes—
472C4—22no—
473C4—44no—
474C4—82no—
475C4—474no—
476C4—942no—
477C4—3762no—
478C4—1884no—
479C4#117×D4712yes—
480C4#123×D4712yes—
481C2—24no—
482C2—944no—
483C2—1882no—
484C2#381×D4711yescharacteristic
485C2C2×#1568912111yescharacteristic
486C2#2025190611yescharacteristic
487C2#2025332312yes—
488C2#2025333112yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1823C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1823D47
hypocenterC47#20702
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #20243482C2semidirect
5. #20243479C2semidirect
6. #20243476C2semidirect
7. #20243474C2semidirect
8. #20243106C2semidirect
9. C47×#1823C2semidirect
15. C2×#13093982C4semidirect
31. #15688886C22semidirect
32. #15688885C22semidirect
33. #15688885C22semidirect
37. C47×#340C22semidirect
57. #13093982C2×C4semidirect
60. C2×C4⋊Dic47C2×C4semidirect
83. C2×C4⋊C4×C47C2×C4semidirect
244. #1823D47semidirect
281. #340C2×D47semidirect
282. C47#20702semidirect
315. C2×C4⋊C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C2×#156890852
#380×D472
#202434822
#202434792
#202434762
#202434742
#202431062
C47×#18232
#2070247