Finite Group Databasebeta

← #26225335

Subgroups — #26225335

Order 12032.

1538 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094004C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic
3#380×D47C211yesmaximal, characteristic
4#20243820C214yesmaximal, complemented
5#20243694C214yesmaximal, complemented
6#20243474C211yesmaximal, characteristic, complemented
7#20243324C211yesmaximal, characteristic, complemented
8#20243322C211yesmaximal, characteristic, complemented
9C47×#1823C211yesmaximal, characteristic, complemented
10C23×Dic94C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C22×#10625324C2212yes—
13C2×#13094004—24no—
14C2×#13094004—21no—
15C2×#13094004—21no—
16C2×#13094004C418yescomplemented
17C2×#13094004C2212yes—
18C2×#13094004C2212yes—
19C22×C4⋊Dic47—21no—
20C2×C4⋊C4×D47C2212yes—
21C22×(C2×D47)⋊C4—21no—
22C2×C22⋊C4×D47—21no—
23C2×#13093949C2214yescomplemented
24C2×#13093947C2214yescomplemented
25#15689085—24no—
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×(C2×D47)⋊C8C2212yes—
28#15688983C2211yescharacteristic
29#15688982—21no—
30#15688981—21no—
31#15688980C2212yes—
32#95×D47—21no—
33#15688885C2214yescomplemented
34C22×C4⋊C4×C47C2211yescharacteristic
35C2×C22⋊C8×C47C2211yescharacteristic
36C47×#380C2211yescharacteristic
37C47×#340C2214yescomplemented
38C24×D47D411yescharacteristic
39C22×Dic94—21no—
40C22×Dic94—22no—
41C22×Dic94—44no—
42C22×Dic94C2×C414yescomplemented
43C22×Dic94C2312yes—
44C23×Dic47—21no—
45C23×Dic47D411yescharacteristic
46C22×C4×D47—22no—
47C22×C4×D47—44no—
48C22×C4×D47C2×C414yes—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47D414yes—
52C22×C4×D47C2311yescharacteristic
53C22×C4×D47C2311yescharacteristic
54C2×#10625324—24no—
55C2×#10625324—44no—
56C22×C47⋊C8C2312yes—
57#13094004—28no—
58#13094004—28no—
59#13094004—24no—
60#13094004—24no—
61#13094004C2×C418yes—
62#13094004C2×C418yescomplemented
63C2×C4⋊Dic47—28no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47C2312yes—
67C2×Dic47⋊C4—22no—
68C4⋊C4×D47—44no—
69C2×C22⋊Dic47—21no—
70C2×(C2×D47)⋊C4—28no—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—22no—
74C22⋊C4×D47—44no—
75#13093949—44no—
76#13093947—44no—
77C22⋊(C47⋊C8)—22no—
78(C2×D47)⋊C8—24no—
79C47⋊#95—21no—
80C47⋊#95—22no—
81C23×C4×C47—21no—
82C23×C4×C47C2311yescharacteristic
83C22×C8×C47C2312yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47C2×C414yescomplemented
87C2×C4⋊C4×C47C2312yes—
88C2×C4⋊C4×C47C2312yes—
89C2×C22⋊C4×C47—21no—
90C22⋊C8×C47—22no—
91#95×C47—21no—
92C24×C47C2×D411yescharacteristic
93C23×D47—22no—
94C23×D47—22no—
95C23×D47—24no—
96C23×D47—42no—
97C23×D47C22⋊C414yes—
98C23×D47C2×D411yescharacteristic
99C23×D47C2×D411yescharacteristic
100C22×C4×C47—24no—
101C22×C4×C47—24no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—42no—
105C22×C4×C47—42no—
106C22×C4×C47C22×C412yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D412yes—
109C22×C4×C47C2411yescharacteristic
110C22×Dic47—24no—
111C22×Dic47—22no—
112C22×Dic47—22no—
113C22×Dic47—42no—
114C22×Dic47—44no—
115C22×Dic47C22⋊C414yes—
116C22×Dic47C2×D411yescharacteristic
117C22×Dic47C2×D411yescharacteristic
118C2×C4×D47—24no—
119C2×C4×D47—24no—
120C2×C4×D47—48no—
121C2×C4×D47—48no—
122C2×C4×D47—42no—
123C2×C4×D47—42no—
124C2×C4×D47—44no—
125C2×C4×D47—44no—
126C2×C4×D47—44no—
127C2×C4×D47C22⋊C414yes—
128C2×C4×D47C22⋊C414yes—
129C2×C4×D47C22×C412yes—
130C2×C4×D47C22×C412yes—
131C2×C4×D47C2×D412yes—
132C2×C4×D47C2×D412yes—
133C2×Dic94—24no—
134C2×Dic94—24no—
135C2×Dic94—24no—
136C2×Dic94—22no—
137C2×Dic94—48no—
138C2×Dic94—44no—
139C2×Dic94—48no—
140C2×Dic94C22×C414yes—
141C2×C8×C47—22no—
142C2×C8×C47—24no—
143C4⋊C4×C47—24no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—24no—
146C4⋊C4×C47C22×C414yes—
147C22⋊C4×C47—42no—
148C22⋊Dic47—42no—
149(C2×D47)⋊C4—28no—
150(C2×D47)⋊C4—28no—
151(C2×D47)⋊C4—44no—
152C4⋊Dic47—28no—
153C4⋊Dic47—28no—
154C4⋊Dic47—24no—
155Dic47⋊C4—44no—
156C2×C47⋊C8—22no—
157C2×C47⋊C8—24no—
158#10625324—48no—
159C23×C47—22no—
160C23×C47—22no—
161C23×C47C2×C22⋊C412yes—
162C23×C47AGL(1,ℤ/8)12yes—
163C23×C47#13712yes—
164C23×C47C22×D411yescharacteristic
165C22×D47—22no—
166C22×D47—21no—
167C22×D47—21no—
168C22×D47—22no—
169C22×D47—21no—
170C22×D47—22no—
171C22×D47—21no—
172C22×D47—28no—
173C22×D47—28no—
174C22×D47—44no—
175C22×D47—48no—
176C22×D47—48no—
177C22×D47C2×C22⋊C412yes—
178C22×D47C2×C22⋊C412yes—
179C22×D47#13714yes—
180C2×C4×C47—22no—
181C2×C4×C47—24no—
182C2×C4×C47—24no—
183C2×C4×C47—24no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—48no—
187C2×C4×C47—42no—
188C2×C4×C47—42no—
189C2×C4×C47—42no—
190C2×C4×C47C2×C22⋊C412yes—
191C2×C4×C47C23×C411yescharacteristic
192C2×C4×C47C22×D411yescharacteristic
193C2×C4×C47H(5,𝔽2)14yes—
194C2×Dic47—22no—
195C2×Dic47—22no—
196C2×Dic47—24no—
197C2×Dic47—44no—
198C2×Dic47—48no—
199C2×Dic47—44no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—42no—
203C2×Dic47C2×C22⋊C412yes—
204C2×Dic47C2×C22⋊C412yes—
205C4×D47—24no—
206C4×D47—24no—
207C4×D47—24no—
208C4×D47—24no—
209C4×D47—44no—
210C4×D47—44no—
211C4×D47—88no—
212Dic94—24no—
213Dic94—24no—
214Dic94—48no—
215Dic94—44no—
216C8×C47—42no—
217C47⋊C8—42no—
218#21765—471nomaximal, Sylow
219C22×C47—22no—
220C22×C47—22no—
221C22×C47—24no—
222C22×C47—24no—
223C22×C47#41714yes—
224C22×C47C22×C22⋊C411yescharacteristic
225C22×C47#53412yes—
226C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C47C2×#13712yes—
228C22×C47C2×#13711yescharacteristic
229C2×D47—22no—
230C2×D47—22no—
231C2×D47—24no—
232C2×D47—24no—
233C2×D47—22no—
234C2×D47—22no—
235C2×D47—22no—
236C2×D47—416no—
237C2×D47—42no—
238C2×D47—42no—
239C2×D47#41914yes—
240C4×C47—22no—
241C4×C47—44no—
242C4×C47—82no—
243C4×C47#51812yes—
244C4×C47#53412yes—
245Dic47—22no—
246Dic47—22no—
247Dic47—44no—
248Dic47—82no—
249C22×C22⋊Q8—471no—
250C2×#407—471no—
251C2×#380—471no—
252#1894—474no—
253#1872—474no—
254#1823—471no—
255#1823D4711yescharacteristic, complemented
256#1787—471no—
257#1787—471no—
258C2×C47—24no—
259C2×C47#230811yescharacteristic
260C2×C47C2×#41911yescharacteristic
261C2×C47C2×#41711yescharacteristic
262C2×C47#319312yes—
263C2×C47#319512yes—
264D47—24no—
265D47—42no—
266C23×Q8—471no—
267C24×C4—471no—
268C22×M16—472no—
269C2×C22⋊Q8—478no—
270C2×C22⋊Q8—472no—
271C2×C22⋊Q8—472no—
272C2×C22⋊Q8—941no—
273C2×C22⋊Q8—941no—
274C2×C22⋊Q8—944no—
275C22×C4⋊C4—472no—
276C22×C4⋊C4—941no—
277C22×C4⋊C4C2×D4711yescharacteristic
278C22×C22⋊C4—941no—
279C22×C22⋊C4—941no—
280C2×#103—474no—
281C2×#103—474no—
282#407—944no—
283C2×C22⋊C8—471no—
284C2×C22⋊C8—472no—
285C2×C22⋊C8C2×D4711yescharacteristic
286#380—471no—
287#380—472no—
288#380—941no—
289#380—941no—
290#380C2×D4711yescharacteristic
291C2×#95—941no—
292#340—474no—
293#340C2×D4714yescomplemented
294C47#2176511yescharacteristic, complemented, Sylow
295C25—471no—
296C22×Q8—474no—
297C22×Q8—472no—
298C22×Q8—942no—
299C22×Q8—941no—
300C22×Q8—1884no—
301C23×C4—21no—
302C23×C4—471no—
303C23×C4—471no—
304C23×C4—471no—
305C23×C4—474no—
306C23×C4—471no—
307C23×C4—471no—
308C23×C4—474no—
309C23×C4—942no—
310C23×C4—941no—
311C23×C4—1884no—
312C23×C4C22×D4711yescharacteristic
313C2×M16—944no—
314C2×M16—1884no—
315C22×C8—472no—
316C22×C8C22×D4712yes—
317C22⋊Q8—478no—
318C22⋊Q8—478no—
319C22⋊Q8—948no—
320C22⋊Q8—948no—
321C22⋊Q8—944no—
322C22⋊Q8—944no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—472no—
326C2×C4⋊C4—948no—
327C2×C4⋊C4—942no—
328C2×C4⋊C4—942no—
329C2×C4⋊C4—942no—
330C2×C4⋊C4—1884no—
331C2×C4⋊C4C4×D4714yescomplemented
332C2×C4⋊C4C22×D4712yes—
333C2×C4⋊C4C22×D4712yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—948no—
336C2×C22⋊C4—942no—
337C2×C22⋊C4—942no—
338C2×C22⋊C4—942no—
339C2×C22⋊C4—941no—
340C2×C22⋊C4—1884no—
341#103—1884no—
342#103—1884no—
343C22⋊C8—22no—
344C22⋊C8—944no—
345C22⋊C8—942no—
346#95—21no—
347#95—942no—
348#95—941no—
349C24—471no—
350C24—471no—
351C24—474no—
352C24—944no—
353C24—942no—
354C24—942no—
355C24—1882no—
356C24D4×D4711yescharacteristic
357C2×Q8—474no—
358C2×Q8—944no—
359C2×Q8—942no—
360C2×Q8—944no—
361C2×Q8—944no—
362C2×Q8—1888no—
363C2×Q8—1884no—
364C2×Q8—1888no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—471no—
372C22×C4—471no—
373C22×C4—474no—
374C22×C4—472no—
375C22×C4—472no—
376C22×C4—472no—
377C22×C4—472no—
378C22×C4—474no—
379C22×C4—474no—
380C22×C4—944no—
381C22×C4—944no—
382C22×C4—944no—
383C22×C4—942no—
384C22×C4—942no—
385C22×C4—1888no—
386C22×C4—1888no—
387C22×C4—1882no—
388C22×C4—1882no—
389C22×C4—1884no—
390C22×C4—1884no—
391C22×C4—1884no—
392C22×C4—1882no—
393C22×C4—1884no—
394C22×C4C2×C4×D4712yes—
395C22×C4D4×D4711yescharacteristic
396C22×C4D4×D4712yes—
397C22×C4C23×D4711yescharacteristic
398M16—1888no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—942no—
402C2×C8—944no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—948no—
407C4⋊C4—948no—
408C4⋊C4—944no—
409C4⋊C4—1884no—
410C4⋊C4C2×C4×D4714yes—
411C22⋊C4—42no—
412C22⋊C4—948no—
413C22⋊C4—948no—
414C22⋊C4—1884no—
415C22⋊C4—1882no—
416C23—22no—
417C23—22no—
418C23—472no—
419C23—474no—
420C23—472no—
421C23—948no—
422C23—941no—
423C23—941no—
424C23—942no—
425C23—948no—
426C23—941no—
427C23—942no—
428C23—941no—
429C23—942no—
430C23—1884no—
431C23—1888no—
432C23—1888no—
433C23C22⋊C4×D4712yes—
434C23#1309404812yes—
435C23#1309405312yes—
436C23C2×D4×D4711yescharacteristic
437Q8—944no—
438Q8—944no—
439Q8—1888no—
440Q8—1884no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—472no—
452C2×C4—472no—
453C2×C4—942no—
454C2×C4—942no—
455C2×C4—944no—
456C2×C4—944no—
457C2×C4—944no—
458C2×C4—944no—
459C2×C4—944no—
460C2×C4—3768no—
461C2×C4—1884no—
462C2×C4—1884no—
463C2×C4—1884no—
464C2×C4—1882no—
465C2×C4—1882no—
466C2×C4—1882no—
467C2×C4—1888no—
468C2×C4—1884no—
469C2×C4C22⋊C4×D4712yes—
470C2×C4C22×C4×D4711yescharacteristic
471C2×C4C2×D4×D4711yescharacteristic
472C2×C4#1309407014yes—
473C8—42no—
474C8—1882no—
475C22—24no—
476C22—22no—
477C22—22no—
478C22—24no—
479C22—474no—
480C22—944no—
481C22—942no—
482C22—944no—
483C22—942no—
484C22—942no—
485C22—942no—
486C22—942no—
487C22—18816no—
488C22—1882no—
489C22—1882no—
490C22#1568912114yes—
491C22C2×C22⋊C4×D4711yescharacteristic
492C22#1568968412yes—
493C22C2×#1309404811yescharacteristic
494C22#137×D4712yes—
495C22C2×#1309405311yescharacteristic
496C4—22no—
497C4—44no—
498C4—82no—
499C4—942no—
500C4—942no—
501C4—3762no—
502C4—1884no—
503C4#1568959712yes—
504C4#1568968412yes—
505C2—24no—
506C2—944no—
507C2—1882no—
508C2#2024596511yescharacteristic
509C2#419×D4711yescharacteristic
510C2C2×#1568912111yescharacteristic
511C2#2025419612yes—
512C2#2025421112yes—
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#21765
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. #20243820C2semidirect
5. #20243694C2semidirect
6. #20243474C2semidirect
7. #20243324C2semidirect
8. #20243322C2semidirect
9. C47×#1823C2semidirect
16. C2×#13094004C4semidirect
23. C2×#13093949C22semidirect
24. C2×#13093947C22semidirect
33. #15688885C22semidirect
37. C47×#340C22semidirect
42. C22×Dic94C2×C4semidirect
62. #13094004C2×C4semidirect
86. C2×C4⋊C4×C47C2×C4semidirect
255. #1823D47semidirect
293. #340C2×D47semidirect
294. C47#21765semidirect
331. 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×#130940042
C2×#156890852
#380×D472
#202438202
#202436942
#202434742
#202433242
#202433222
C47×#18232
#2176547