Finite Group Databasebeta

← #26112089

Subgroups — #26112089

Order 12032.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252402C211yesmaximal, characteristic, complemented
2#20243198C211yesmaximal, characteristic, complemented
3#20241980C211yesmaximal, characteristic, complemented
4#20241896C211yesmaximal, characteristic, complemented
5#20241280C211yesmaximal, characteristic, complemented
6#20241259C211yesmaximal, characteristic, complemented
7C47×#3115C211yesmaximal, characteristic
8#15689254—21no—
9#15689195—22no—
10D4×C47⋊C8—21no—
11#15689180—21no—
12#15689180C2211yescharacteristic
13C2×#13093951—21no—
14C4×#10625329C2211yescharacteristic
15C4×#10625329C2211yescharacteristic
16C2×#13093935—21no—
17#15688969C2211yescharacteristic
18#15688882—21no—
19#15688863—21no—
20#15688863—21no—
21#15688857—22no—
22C47×#532C2211yescharacteristic
23C4×#41×C47C2211yescharacteristic
24C47×#482—21no—
25C47×#475C411yescharacteristic, complemented
26C47×#474C411yescharacteristic, complemented
27C47×#460C411yescharacteristic, complemented
28C47×#459C411yescharacteristic, complemented
29C47×#449—21no—
30C47×#442—21no—
31C47×#440—21no—
32C47×#432C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329—21no—
35C22×C47⋊C8—22no—
36C2×C4×Dic47—21no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)—21no—
39#13093951—24no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)D412yes—
42#13093935—22no—
43#13093935—21no—
44#13093935—21no—
45#13093935D411yescharacteristic
46#13093935D411yescharacteristic
47C4×C47⋊C8—21no—
48C4×C47⋊C8—21no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C2×#41×C47—21no—
52C2×#41×C47D411yescharacteristic
53C2×SD16×C47—21no—
54C2×D8×C47—21no—
55C4⋊Q8×C47C2×C411yescharacteristic
56#127×C47C2×C411yescharacteristic
57#124×C47—21no—
58#121×C47—21no—
59#121×C47—21no—
60C4×Q8×C47C2×C411yescharacteristic
61C4×Q8×C47D411yescharacteristic
62C4×D4×C47—21no—
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47C2×C411yescharacteristic
67C4×D4×C47D411yescharacteristic
68#117×C47—21no—
69#117×C47D411yescharacteristic
70C2×C42×C47—21no—
71C2×C42×C47C2311yescharacteristic
72#107×C47—21no—
73#106×C47—21no—
74C4⋊C8×C47C2×C411yescharacteristic
75C4⋊C8×C47C2×C411yescharacteristic
76#103×C47—21no—
77#102×C47—21no—
78#102×C47—21no—
79#102×C47—21no—
80C22⋊C8×C47—21no—
81#97×C47—21no—
82#41×C47—22no—
83#41×C47—41no—
84#41×C47—82no—
85C2×Q8×C47—21no—
86C2×Q8×C47C22⋊C411yescharacteristic
87C2×D4×C47—21no—
88C2×D4×C47—21no—
89C2×D4×C47—21no—
90C2×D4×C47—21no—
91C2×D4×C47—41no—
92C2×D4×C47C22⋊C411yescharacteristic
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—41no—
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47#4111yescharacteristic
100C22×Dic47—41no—
101SD16×C47—42no—
102D8×C47—42no—
103C2×C8×C47—21no—
104C2×C8×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47C22⋊C411yescharacteristic
110C4⋊C4×C47C22⋊C411yescharacteristic
111C22⋊C4×C47—21no—
112C22⋊C4×C47—21no—
113C22⋊C4×C47—21no—
114C22⋊C4×C47—41no—
115C22⋊C4×C47—41no—
116C42×C47—21no—
117C42×C47—21no—
118C42×C47—21no—
119C42×C47C22×C411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C4×Dic47—22no—
124C4×Dic47—41no—
125#10625329—22no—
126#10625329—42no—
127#10625329—41no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—44no—
132C2×C47⋊C8—41no—
133C23×C47—21no—
134C23×C47—41no—
135C23×C47#12111yescharacteristic
136Q8×C47—21no—
137Q8×C47—21no—
138Q8×C47—81no—
139D4×C47—21no—
140D4×C47—42no—
141D4×C47—41no—
142D4×C47—41no—
143D4×C47—81no—
144D4×C47—81no—
145D4×C47—82no—
146D4×C47#10412yes—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—42no—
155C2×C4×C47—42no—
156C2×C4×C47—41no—
157C2×C4×C47—81no—
158C2×C4×C47—82no—
159C2×C4×C47C2×C22⋊C411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47C4×D411yescharacteristic
162C2×C4×C47#12011yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12311yescharacteristic
165C2×Dic47—42no—
166C2×Dic47—81no—
167C2×Dic47—81no—
168C8×C47—41no—
169C8×C47—41no—
170C47⋊C8—22no—
171C47⋊C8—42no—
172C47⋊C8—41no—
173C47⋊C8—42no—
174#9161—471nomaximal, Sylow
175C22×C47—21no—
176C22×C47—22no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—41no—
180C22×C47—41no—
181C22×C47—81no—
182C22×C47—81no—
183C22×C47—81no—
184C22×C47#38611yescharacteristic
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—41no—
191C4×C47—41no—
192C4×C47—41no—
193C4×C47—81no—
194C4×C47C2×#10411yescharacteristic
195C4×C47#42111yescharacteristic
196Dic47—82no—
197#3115D4711yescharacteristic
198#2970—471no—
199#1759—471no—
200#1525—471no—
201#1505—471no—
202#1300—471no—
203#1291—471no—
204C2×C47—41no—
205C2×C47—42no—
206C2×C47—161no—
207C2×C47#187111yescharacteristic
208C2×C47#190311yescharacteristic
209C2×C47#190611yescharacteristic
210#532C2×D4711yescharacteristic
211C4×#41C2×D4711yescharacteristic
212#482—21no—
213#475Dic4711yescharacteristic, complemented
214#474Dic4711yescharacteristic, complemented
215#460Dic4711yescharacteristic, complemented
216#459Dic4711yescharacteristic, complemented
217#449—21no—
218#446—941no—
219#442—21no—
220#440—21no—
221#435—942no—
222C8×D4—941no—
223#432—471no—
224#432—941no—
225#432C2×D4711yescharacteristic
226C2×#104—941no—
227C4×M16—471no—
228C4×M16—471no—
229C2×#97—941no—
230#376—471no—
231#339—941no—
232#329—941no—
233#329—941no—
234#327—942no—
235C47#916111yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41C22⋊D4711yescharacteristic
238C2×SD16—21no—
239C2×D8—21no—
240C2×M16—941no—
241C2×M16—941no—
242C22×C8—942no—
243C4⋊Q8C2×Dic4711yescharacteristic
244#127C2×Dic4711yescharacteristic
245#124—21no—
246#121—21no—
247#121—21no—
248C4×Q8C2×Dic4711yescharacteristic
249C4×Q8C22⋊D4711yescharacteristic
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C2×Dic4711yescharacteristic
255C4×D4C22⋊D4711yescharacteristic
256#117—21no—
257#117C22⋊D4711yescharacteristic
258C2×C42—21no—
259C2×C42—941no—
260C2×C42C22×D4711yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—942no—
264C4⋊C8—941no—
265C4⋊C8C2×Dic4711yescharacteristic
266C4⋊C8C2×Dic4711yescharacteristic
267#104—944no—
268#103—21no—
269#102—21no—
270#102—21no—
271#102—21no—
272C22⋊C8—21no—
273C22⋊C8—472no—
274C22⋊C8—942no—
275#97—21no—
276#97—471no—
277#97—471no—
278#97—942no—
279#97—941no—
280#97—941no—
281C4×C8—941no—
282C4×C8—941no—
283#95—941no—
284#95—941no—
285#41—22no—
286#41—41no—
287#41—82no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic4711yescharacteristic
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4C22⋊Dic4711yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—1881no—
301C22×C4D4×D4711yescharacteristic
302C22×C4#1062535211yescharacteristic
303C22×C4C2×C22⋊D4711yescharacteristic
304SD16—42no—
305D8—42no—
306M16—942no—
307M16—1882no—
308M16—1881no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—941no—
312C2×C8—941no—
313C2×C8—942no—
314C2×C8—1884no—
315C2×C8—1881no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4C22⋊Dic4711yescharacteristic
321C4⋊C4C22⋊Dic4711yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—942no—
331C42—1881no—
332C42D4×D4711yescharacteristic
333C42#1062535211yescharacteristic
334C42C22×Dic4711yescharacteristic
335C42C2×C22⋊D4711yescharacteristic
336C23—21no—
337C23—41no—
338C23#1309399611yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—81no—
342D4—21no—
343D4—42no—
344D4—41no—
345D4—41no—
346D4—81no—
347D4—81no—
348D4—82no—
349D4#1309395112yes—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—3761no—
363C2×C4—3761no—
364C2×C4—1882no—
365C2×C4C2×C22⋊Dic4711yescharacteristic
366C2×C4D4×Dic4711yescharacteristic
367C2×C4D4×Dic4711yescharacteristic
368C2×C4#1309399211yescharacteristic
369C2×C4#1309399611yescharacteristic
370C2×C4#1309400511yescharacteristic
371C8—41no—
372C8—41no—
373C8—942no—
374C8—1882no—
375C8—1881no—
376C8—1882no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1568900311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—3762no—
397C4C2×#1309395111yescharacteristic
398C4#1568914011yescharacteristic
399C2—41no—
400C2—42no—
401C2—161no—
402C2#2024369111yescharacteristic
403C2#2024387011yescharacteristic
404C2#2024389111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3115C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3115D47
hypocenterC47#9161
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #20252402C2semidirect
2. #20243198C2semidirect
3. #20241980C2semidirect
4. #20241896C2semidirect
5. #20241280C2semidirect
6. #20241259C2semidirect
25. C47×#475C4semidirect
26. C47×#474C4semidirect
27. C47×#460C4semidirect
28. C47×#459C4semidirect
213. #475Dic47semidirect
214. #474Dic47semidirect
215. #460Dic47semidirect
216. #459Dic47semidirect
235. C47#9161semidirect

Maximal subgroups

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

subgroupindex
#202524022
#202431982
#202419802
#202418962
#202412802
#202412592
C47×#31152
#916147