Finite Group Databasebeta

← #26107724

Subgroups — #26107724

Order 12032.

1276 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C2×#15688861C214yesmaximal, complemented
3#20241611C211yesmaximal, characteristic
4C47×#1779C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic, complemented
6C2×#13094003—28no—
7C2×#13094003—21no—
8C2×#13094003—21no—
9C2×#13094003C412yescomplemented
10C2×#13094003C2212yes—
11C22×C4⋊Dic47C411yescharacteristic, complemented
12C22×Dic47⋊C4—21no—
13C22×C22⋊Dic47—21no—
14C22×C22⋊Dic47—21no—
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2212yes—
17#15688861—28no—
18C24×C4×C47C2211yescharacteristic
19C2×C22⋊C8×C47—21no—
20C2×C22⋊C8×C47—21no—
21C2×C22⋊C8×C47C2212yes—
22C22×Dic94—21no—
23C22×Dic94—24no—
24C22×Dic94—42no—
25C22×Dic94—42no—
26C22×Dic94C2×C412yes—
27C23×Dic47—21no—
28C23×Dic47—21no—
29C23×Dic47—22no—
30#13094003—28no—
31#13094003—28no—
32#13094003—24no—
33#13094003—24no—
34#13094003—24no—
35#13094003D418yes—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—24no—
38C2×C4⋊Dic47C2×C412yes—
39C2×Dic47⋊C4—28no—
40C2×Dic47⋊C4—22no—
41C2×Dic47⋊C4—22no—
42C2×C22⋊Dic47—28no—
43C2×C22⋊Dic47—22no—
44C2×C22⋊Dic47—22no—
45C2×C22⋊Dic47—24no—
46C2×C22⋊Dic47—42no—
47C2×C22⋊Dic47—42no—
48C47⋊#95—24no—
49C47⋊#95—42no—
50C25×C47D411yescharacteristic
51C23×C4×C47—24no—
52C23×C4×C47—42no—
53C23×C4×C47—42no—
54C23×C4×C47C2×C411yescharacteristic
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47D411yescharacteristic
57C23×C4×C47D411yescharacteristic
58C23×C4×C47D411yescharacteristic
59C23×C4×C47C2311yescharacteristic
60C22×C8×C47—22no—
61C22⋊C8×C47—24no—
62C22⋊C8×C47—42no—
63C22⋊C8×C47—42no—
64C24×C47—22no—
65C24×C47—22no—
66C24×C47—22no—
67C24×C47—22no—
68C24×C47—41no—
69C24×C47—41no—
70C24×C47C22⋊C411yescharacteristic
71C24×C47C22⋊C411yescharacteristic
72C24×C47SD1612yes—
73C24×C47Q1612yes—
74C24×C47C2×D411yescharacteristic
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—416no—
93C22×C4×C47C22⋊C411yescharacteristic
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D414yes—
98C22×Dic47—24no—
99C22×Dic47—24no—
100C22×Dic47—22no—
101C22×Dic47—22no—
102C22×Dic47—24no—
103C22×Dic47—42no—
104C22×Dic47—42no—
105C22×Dic47—42no—
106C22×Dic47—42no—
107C22×Dic47—42no—
108C22×Dic47—42no—
109C22×Dic47—42no—
110C22×Dic47—42no—
111C22×Dic47—42no—
112C22×Dic47—42no—
113C2×Dic94—24no—
114C2×Dic94—22no—
115C2×Dic94—42no—
116C2×Dic94—416no—
117C2×Dic94—42no—
118C2×Dic94—42no—
119C2×Dic94—42no—
120C2×Dic94C22⋊C414yescomplemented
121C2×C8×C47—42no—
122C2×C8×C47—42no—
123C2×C8×C47—42no—
124C22⋊Dic47—28no—
125C22⋊Dic47—28no—
126C22⋊Dic47—48no—
127C4⋊Dic47—24no—
128C4⋊Dic47—22no—
129C4⋊Dic47C22⋊C414yescomplemented
130Dic47⋊C4—28no—
131Dic47⋊C4—28no—
132C23×C47—21no—
133C23×C47—21no—
134C23×C47—21no—
135C23×C47—21no—
136C23×C47—21no—
137C23×C47—21no—
138C23×C47—21no—
139C23×C47—21no—
140C23×C47—24no—
141C23×C47—24no—
142C23×C47—21no—
143C23×C47—21no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—24no—
149C23×C47—24no—
150C23×C47—42no—
151C23×C47—42no—
152C23×C47—44no—
153C23×C47—44no—
154C23×C47—44no—
155C23×C47—44no—
156C23×C47#9912yes—
157C23×C47#10312yes—
158C23×C47#10312yes—
159C23×C47#10412yes—
160C23×C47C2×C22⋊C411yescharacteristic
161C23×C47#12012yes—
162C23×C47C2×SD1611yescharacteristic
163C23×C47C2×Q1611yescharacteristic
164C23×C47#13712yes—
165C2×C4×C47—28no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—21no—
169C2×C4×C47—21no—
170C2×C4×C47—21no—
171C2×C4×C47—21no—
172C2×C4×C47—21no—
173C2×C4×C47—21no—
174C2×C4×C47—21no—
175C2×C4×C47—44no—
176C2×C4×C47—44no—
177C2×C4×C47—44no—
178C2×C4×C47—44no—
179C2×C4×C47—44no—
180C2×C4×C47#9912yes—
181C2×C4×C47C2×C22⋊C412yes—
182C2×C4×C47#12012yes—
183C2×Dic47—24no—
184C2×Dic47—24no—
185C2×Dic47—48no—
186C2×Dic47—48no—
187C2×Dic47—42no—
188C2×Dic47—42no—
189C2×Dic47—42no—
190C2×Dic47—42no—
191C2×Dic47—44no—
192C2×Dic47—44no—
193C2×Dic47—44no—
194C2×Dic47—82no—
195C2×Dic47—82no—
196C2×Dic47—82no—
197C2×Dic47—82no—
198Dic94—48no—
199Dic94—44no—
200Dic94—44no—
201C8×C47—82no—
202#8329—471nomaximal, Sylow
203C22×C47—21no—
204C22×C47—21no—
205C22×C47—22no—
206C22×C47—22no—
207C22×C47—22no—
208C22×C47—22no—
209C22×C47—21no—
210C22×C47—22no—
211C22×C47—21no—
212C22×C47—22no—
213C22×C47—22no—
214C22×C47—22no—
215C22×C47—22no—
216C22×C47—22no—
217C22×C47—22no—
218C22×C47—22no—
219C22×C47—22no—
220C22×C47—48no—
221C22×C47—48no—
222C22×C47—41no—
223C22×C47—41no—
224C22×C47—41no—
225C22×C47—41no—
226C22×C47#32814yescomplemented
227C22×C47#37911yescharacteristic
228C22×C47C2×#9911yescharacteristic
229C22×C47C2×#9911yescharacteristic
230C22×C47C2×#10311yescharacteristic
231C22×C47#41911yescharacteristic
232C22×C47C2×#10411yescharacteristic
233C22×C47#42111yescharacteristic
234C22×C47#44812yes—
235C22×C47#45112yes—
236C4×C47—22no—
237C4×C47—22no—
238C4×C47—22no—
239C4×C47—41no—
240Dic47—44no—
241Dic47—44no—
242Dic47—84no—
243C22×C22⋊Q8—471no—
244#1779D4711yescharacteristic
245C2×#328—474no—
246C22⋊#95—471no—
247C2×C47—22no—
248C2×C47—24no—
249C2×C47—22no—
250C2×C47—42no—
251C2×C47C2×#32811yescharacteristic
252C2×C47#150811yescharacteristic
253C2×C47#151111yescharacteristic
254C2×C47#152011yescharacteristic
255C2×C47#178111yescharacteristic
256C2×C47#178711yescharacteristic
257C2×C47#178911yescharacteristic
258C23×Q8—471no—
259C24×C4C2×D4711yescharacteristic
260C2×C22⋊Q8—472no—
261C2×C22⋊Q8—472no—
262C2×C22⋊Q8—948no—
263C2×C22⋊Q8—941no—
264C2×C22⋊Q8—941no—
265C22×C4⋊C4—471no—
266C22×C4⋊C4—941no—
267C22×C22⋊C4—941no—
268C22×C22⋊C4—941no—
269C2×C22⋊C8—21no—
270C2×C22⋊C8—21no—
271C2×C22⋊C8C2×D4712yes—
272C2×#95—472no—
273C2×#95—941no—
274#328—948no—
275C47#832911yescharacteristic, complemented, Sylow
276C25D18811yescharacteristic
277C22×Q8—472no—
278C22×Q8—944no—
279C22×Q8—941no—
280C22×Q8—1882no—
281C22×Q8—1882no—
282C23×C4—24no—
283C23×C4—42no—
284C23×C4—42no—
285C23×C4—941no—
286C23×C4—941no—
287C23×C4—942no—
288C23×C4C4×D4711yescharacteristic
289C23×C4C4×D4711yescharacteristic
290C23×C4D18811yescharacteristic
291C23×C4C22⋊D4711yescharacteristic
292C23×C4C22⋊D4711yescharacteristic
293C23×C4C22×D4711yescharacteristic
294C22×C8—22no—
295C22⋊Q8—478no—
296C22⋊Q8—944no—
297C22⋊Q8—944no—
298C22⋊Q8—948no—
299C22⋊Q8—948no—
300C22⋊Q8—944no—
301C2×C4⋊C4—472no—
302C2×C4⋊C4—944no—
303C2×C4⋊C4—948no—
304C2×C4⋊C4—941no—
305C2×C4⋊C4—942no—
306C2×C4⋊C4—942no—
307C2×C22⋊C4—948no—
308C2×C22⋊C4—942no—
309C2×C22⋊C4—942no—
310C2×C22⋊C4—944no—
311C2×C22⋊C4—1882no—
312C2×C22⋊C4—1882no—
313C22⋊C8—24no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316#95—944no—
317#95—1882no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—41no—
323C24—41no—
324C24#1062532512yes—
325C24Dic18812yes—
326C24(C2×D47)⋊C411yescharacteristic
327C24(C2×D47)⋊C411yescharacteristic
328C24C2×D18811yescharacteristic
329C2×Q8—474no—
330C2×Q8—944no—
331C2×Q8—942no—
332C2×Q8—18816no—
333C2×Q8—1882no—
334C2×Q8—1882no—
335C2×Q8—1882no—
336C2×Q8—1882no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—416no—
355C22×C4—944no—
356C22×C4—944no—
357C22×C4—942no—
358C22×C4—942no—
359C22×C4—944no—
360C22×C4—1882no—
361C22×C4—1882no—
362C22×C4—1882no—
363C22×C4—1882no—
364C22×C4—1882no—
365C22×C4—1882no—
366C22×C4—1882no—
367C22×C4—1882no—
368C22×C4—1882no—
369C22×C4—1882no—
370C22×C4(C2×D47)⋊C411yescharacteristic
371C22×C4(C2×D47)⋊C411yescharacteristic
372C22×C4C2×C4×D4711yescharacteristic
373C22×C4D4×D4714yes—
374C22×C4C2×C22⋊D4711yescharacteristic
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—474no—
379C4⋊C4—948no—
380C4⋊C4—944no—
381C4⋊C4—942no—
382C4⋊C4—948no—
383C22⋊C4—948no—
384C22⋊C4—948no—
385C22⋊C4—1888no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—24no—
396C23—24no—
397C23—21no—
398C23—21no—
399C23—24no—
400C23—24no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23#1309393812yes—
411C23#1309394912yes—
412C23#1309394912yes—
413C23#1309395012yes—
414C23C2×(C2×D47)⋊C411yescharacteristic
415C23#1309399412yes—
416C23C2×#1062532511yescharacteristic
417C23C2×Dic18811yescharacteristic
418C23#1309405712yes—
419Q8—1884no—
420Q8—1888no—
421Q8—1884no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—944no—
438C2×C4—944no—
439C2×C4—3762no—
440C2×C4—3762no—
441C2×C4—3762no—
442C2×C4—3762no—
443C2×C4—1884no—
444C2×C4—1884no—
445C2×C4—1884no—
446C2×C4—1888no—
447C2×C4—1888no—
448C2×C4—1882no—
449C2×C4—1882no—
450C2×C4—1882no—
451C2×C4—1882no—
452C2×C4#1309393812yes—
453C2×C4C22⋊C4×D4712yes—
454C2×C4#1309399212yes—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—22no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—48no—
474C22—48no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#1568886114yescomplemented
480C22#1568897811yescharacteristic
481C22C2×#1309393811yescharacteristic
482C22C2×#1309393811yescharacteristic
483C22C2×#1309394911yescharacteristic
484C22#1568913311yescharacteristic
485C22C2×#1309395011yescharacteristic
486C22#1568913911yescharacteristic
487C22#1568926412yes—
488C22#1568928112yes—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—3764no—
494C4—1884no—
495C4—1884no—
496C2—22no—
497C2—22no—
498C2—24no—
499C2—42no—
500C2C2×#1568886111yescharacteristic
501C2#2024190911yescharacteristic
502C2#2024192211yescharacteristic
503C2#2024196111yescharacteristic
504C2#2024330211yescharacteristic
505C2#2024332411yescharacteristic
506C2#2024333111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1779C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#1779D47
hypocenterC47#8329
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#15688861C2semidirect
5. C23×Dic94C4semidirect
9. C2×#13094003C4semidirect
11. C22×C4⋊Dic47C4semidirect
120. C2×Dic94C22⋊C4semidirect
129. C4⋊Dic47C22⋊C4semidirect
226. C22×C47#328semidirect
275. C47#8329semidirect
479. C22#15688861semidirect

Maximal subgroups

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

subgroupindex
C22×#130940032
C2×#156888612
#202416112
C47×#17792
#832947