Finite Group Databasebeta

← #26109178

Subgroups — #26109178

Order 12032.

524 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689195C211yesmaximal, characteristic
2#20243415C211yesmaximal, characteristic, complemented
3#20243411C211yesmaximal, characteristic, complemented
4#20243376C211yesmaximal, characteristic
5#20241255C211yesmaximal, characteristic, complemented
6#20241255C211yesmaximal, characteristic, complemented
7C47×#1799C211yesmaximal, characteristic
8C22×#10625329C411yescharacteristic, complemented
9C23×C47⋊C8C2211yescharacteristic
10#15689195—21no—
11#15689195—21no—
12#15689195—21no—
13#15689195—21no—
14C2×#13093957—21no—
15C2×#13093955—21no—
16C2×C4⋊(C47⋊C8)C411yescharacteristic, complemented
17C2×#13093939—21no—
18C2×C22⋊(C47⋊C8)C411yescharacteristic, complemented
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093935C411yescharacteristic, complemented
21#15688983C2211yescharacteristic
22#15688983C2211yescharacteristic
23#15688890—21no—
24#15688885—21no—
25#15688885—21no—
26C2×C4×D4×C47C2211yescharacteristic
27C2×#99×C47—21no—
28C47×#380C2211yescharacteristic
29C47×#380C2211yescharacteristic
30C47×#342—21no—
31C2×#10625329—21no—
32C2×#10625329—21no—
33C2×#10625329—42no—
34C22×C47⋊C8—42no—
35C22×C47⋊C8C2×C411yescharacteristic
36C22×C47⋊C8C2×C411yescharacteristic
37C22×C47⋊C8C2×C411yescharacteristic
38C22×C47⋊C8C2×C411yescharacteristic
39C22×C47⋊C8D411yescharacteristic
40C22×C47⋊C8D411yescharacteristic
41C22×C47⋊C8D411yescharacteristic
42C22×C47⋊C8D411yescharacteristic
43C2×C4⋊Dic47—21no—
44C2×C4⋊Dic47—21no—
45C2×C22⋊Dic47—21no—
46C2×C22⋊Dic47—21no—
47#13093957—42no—
48#13093955—42no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)—21no—
51#13093939—42no—
52C22⋊(C47⋊C8)—21no—
53C22⋊(C47⋊C8)—21no—
54C22⋊(C47⋊C8)—21no—
55C22⋊(C47⋊C8)—21no—
56#13093935—21no—
57#13093935—21no—
58C47⋊#95—21no—
59C47⋊#95—21no—
60C22×D4×C47D411yescharacteristic
61C23×C4×C47C2×C411yescharacteristic
62C23×C4×C47C2311yescharacteristic
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47Q811yescharacteristic
70C2×C22⋊C4×C47—21no—
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47D411yescharacteristic
73C2×C22⋊C4×C47Q811yescharacteristic
74C2×C42×C47C2×C411yescharacteristic
75#99×C47—42no—
76#95×C47—21no—
77#95×C47—21no—
78C24×C47C4⋊C411yescharacteristic
79C24×C47C2×D411yescharacteristic
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—41no—
94C22×C4×C47C4⋊C411yescharacteristic
95C22×C4×C47C4⋊C411yescharacteristic
96C22×C4×C47C4⋊C411yescharacteristic
97C22×C4×C47C22×C411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×Q811yescharacteristic
101C22×C4×C47#4111yescharacteristic
102C22×C4×C47#4111yescharacteristic
103C22×Dic47—41no—
104C22×Dic47—41no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C22⋊C4×C47—21no—
112C22⋊C4×C47—21no—
113C22⋊C4×C47—21no—
114C22⋊C4×C47—21no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47—41no—
117C22⋊C4×C47—41no—
118C22⋊C4×C47—41no—
119C42×C47—21no—
120C42×C47—21no—
121C22⋊Dic47—41no—
122C22⋊Dic47—41no—
123C22⋊Dic47—41no—
124C22⋊Dic47—41no—
125C4⋊Dic47—41no—
126C4⋊Dic47—41no—
127C4⋊Dic47—41no—
128C4⋊Dic47—41no—
129#10625329—42no—
130#10625329—42no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—22no—
134C2×C47⋊C8—21no—
135C2×C47⋊C8—21no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—41no—
138C2×C47⋊C8—41no—
139C2×C47⋊C8—41no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8C22⋊C411yescharacteristic
142C2×C47⋊C8C22⋊C411yescharacteristic
143C2×C47⋊C8C22⋊C411yescharacteristic
144C2×C47⋊C8C22⋊C411yescharacteristic
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—41no—
149C23×C47—41no—
150C23×C47—41no—
151C23×C47—41no—
152C23×C47C2×C4⋊C411yescharacteristic
153C23×C47#12111yescharacteristic
154C23×C47#12111yescharacteristic
155C23×C47AGL(1,ℤ/8)11yescharacteristic
156C23×C47AGL(1,ℤ/8)11yescharacteristic
157C23×C47#13711yescharacteristic
158C23×C47#13711yescharacteristic
159D4×C47—41no—
160D4×C47—41no—
161D4×C47—41no—
162D4×C47—41no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
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—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—42no—
178C2×C4×C47—41no—
179C2×C4×C47—41no—
180C2×C4×C47—41no—
181C2×C4×C47—41no—
182C2×C4×C47—81no—
183C2×C4×C47—81no—
184C2×C4×C47—81no—
185C2×C4×C47—81no—
186C2×C4×C47—81no—
187C2×C4×C47—81no—
188C2×C4×C47C2×C22⋊C411yescharacteristic
189C2×C4×C47#11711yescharacteristic
190C2×C4×C47C22⋊Q811yescharacteristic
191C2×C4×C47C22⋊Q811yescharacteristic
192C2×Dic47—81no—
193C2×Dic47—81no—
194C2×Dic47—81no—
195C2×Dic47—81no—
196C2×Dic47—81no—
197C2×Dic47—81no—
198C47⋊C8—21no—
199C47⋊C8—21no—
200C47⋊C8—21no—
201C47⋊C8—41no—
202C47⋊C8—41no—
203C47⋊C8#9912yescomplemented
204#8610—471nomaximal, Sylow
205C22×C47—21no—
206C22×C47—21no—
207C22×C47—21no—
208C22×C47—21no—
209C22×C47—21no—
210C22×C47—21no—
211C22×C47—41no—
212C22×C47—41no—
213C22×C47—41no—
214C22×C47—41no—
215C22×C47—41no—
216C22×C47—41no—
217C22×C47—41no—
218C22×C47—81no—
219C22×C47—81no—
220C22×C47#38011yescharacteristic
221C22×C47#42811yescharacteristic
222C22×C47#42811yescharacteristic
223C22×C47#46811yescharacteristic
224C22×C47#46811yescharacteristic
225C22×C47#46911yescharacteristic
226C22×C47#47011yescharacteristic
227C4×C47—21no—
228C4×C47—21no—
229C4×C47—21no—
230C4×C47—41no—
231C4×C47—41no—
232C4×C47—41no—
233C4×C47—161no—
234C4×C47—161no—
235C4×C47C2×#9911yescharacteristic
236C4×C47#41011yescharacteristic
237Dic47—161no—
238Dic47—161no—
239C2×#435—471no—
240#1808—471no—
241#1807—471no—
242#1799—471no—
243#1799D4711yescharacteristic
244#1289—471no—
245#1289—471no—
246C2×C47—21no—
247C2×C47—21no—
248C2×C47—21no—
249C2×C47—21no—
250C2×C47—21no—
251C2×C47—21no—
252C2×C47—81no—
253C2×C47#179911yescharacteristic
254C2×C47#180911yescharacteristic
255C2×C47#181211yescharacteristic
256C22×M16—471no—
257C23×C8—471no—
258C2×C4×D4C2×D4711yescharacteristic
259#435—941no—
260#435—941no—
261#435—941no—
262#435—941no—
263C2×#107—941no—
264C2×#106—941no—
265C2×C4⋊C8—471no—
266C2×#99—21no—
267C2×#99—941no—
268C2×C22⋊C8—471no—
269C2×C22⋊C8—471no—
270C2×#97—471no—
271#380—471no—
272#380—471no—
273#380C2×D4711yescharacteristic
274#380C2×D4711yescharacteristic
275#342—21no—
276#342—941no—
277#340—941no—
278#340—941no—
279C47#861011yescharacteristic, complemented, Sylow
280C22×D4C22⋊D4711yescharacteristic
281C23×C4C4×D4711yescharacteristic
282C23×C4C22×D4711yescharacteristic
283C2×M16—941no—
284C2×M16—941no—
285C2×M16—1882no—
286C22×C8—471no—
287C22×C8—471no—
288C22×C8—471no—
289C22×C8—471no—
290C22×C8—471no—
291C22×C8—471no—
292C22×C8—471no—
293C22×C8—471no—
294C22×C8—1882no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—941no—
302C2×C4⋊C4—941no—
303C2×C4⋊C4Dic9411yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—941no—
307C2×C22⋊C4—941no—
308C2×C22⋊C4Dic9411yescharacteristic
309C2×C22⋊C4C22⋊D4711yescharacteristic
310C2×C42C4×D4711yescharacteristic
311#107—1882no—
312#106—1882no—
313C4⋊C8—941no—
314C4⋊C8—941no—
315#99—42no—
316#99—1882no—
317C22⋊C8—941no—
318C22⋊C8—941no—
319C22⋊C8—941no—
320C22⋊C8—941no—
321#97—941no—
322#97—941no—
323#95—21no—
324#95—21no—
325#95—941no—
326#95—941no—
327C24Dic47⋊C411yescharacteristic
328C24C2×C22⋊D4711yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—1881no—
344C22×C4—1881no—
345C22×C4Dic47⋊C411yescharacteristic
346C22×C4Dic47⋊C411yescharacteristic
347C22×C4Dic47⋊C411yescharacteristic
348C22×C4C2×Dic9411yescharacteristic
349C22×C4C2×C4×D4711yescharacteristic
350C22×C4D4×D4711yescharacteristic
351C22×C4D4×D4711yescharacteristic
352C22×C4#1062535211yescharacteristic
353C22×C4#1062535211yescharacteristic
354M16—1882no—
355M16—1882no—
356C2×C8—471no—
357C2×C8—471no—
358C2×C8—471no—
359C2×C8—471no—
360C2×C8—942no—
361C2×C8—942no—
362C2×C8—942no—
363C2×C8—941no—
364C2×C8—941no—
365C2×C8—1881no—
366C2×C8—1881no—
367C2×C8—1881no—
368C2×C8—1881no—
369C2×C8—1881no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—1881no—
377C4⋊C4—1881no—
378C4⋊C4—1881no—
379C4⋊C4—1881no—
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—1881no—
389C22⋊C4—1881no—
390C22⋊C4—1881no—
391C22⋊C4—1881no—
392C42—21no—
393C42—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C2×Dic47⋊C411yescharacteristic
402C23#1309399611yescharacteristic
403C23#1309399611yescharacteristic
404C23#1309404911yescharacteristic
405C23#1309405011yescharacteristic
406C23#1309405411yescharacteristic
407C23#1309405511yescharacteristic
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—3761no—
438C2×C4—3761no—
439C2×C4—3761no—
440C2×C4—3761no—
441C2×C4—3761no—
442C2×C4—3761no—
443C2×C4C22⋊C4×D4711yescharacteristic
444C2×C4#1309398011yescharacteristic
445C2×C4#1309400111yescharacteristic
446C2×C4#1309400111yescharacteristic
447C8—472no—
448C8—941no—
449C8—941no—
450C8—941no—
451C8—1881no—
452C8—1881no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—81no—
468C22#1568898111yescharacteristic
469C22#1568916311yescharacteristic
470C22#1568916611yescharacteristic
471C22#1568936711yescharacteristic
472C22#1568937111yescharacteristic
473C22#1568937411yescharacteristic
474C22#1568937911yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—161no—
482C4—161no—
483C4—7521no—
484C4—7521no—
485C4#99×D4711yescharacteristic
486C4#1568909511yescharacteristic
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—81no—
494C2#2024337411yescharacteristic
495C2#2024341911yescharacteristic
496C2#2024343211yescharacteristic
subgroupG/H
centerC22#15688981
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1799C2
socleC22×C47#380
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#1799D47
hypocenterC47#8610
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #20243415C2semidirect
3. #20243411C2semidirect
5. #20241255C2semidirect
6. #20241255C2semidirect
8. C22×#10625329C4semidirect
16. C2×C4⋊(C47⋊C8)C4semidirect
18. C2×C22⋊(C47⋊C8)C4semidirect
20. C2×#13093935C4semidirect
203. C47⋊C8#99semidirect
279. C47#8610semidirect

Maximal subgroups

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

subgroupindex
C2×#156891952
#202434152
#202434112
#202433762
#202412552
C47×#17992
#861047