Finite Group Databasebeta

← #26113851

Subgroups — #26113851

Order 12032.

414 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252384C211yesmaximal, characteristic
2#20242003C211yesmaximal, characteristic, complemented
3#20241952C211yesmaximal, characteristic, complemented
4#20241926C211yesmaximal, characteristic
5#20241287C211yesmaximal, characteristic, complemented
6#20241270C211yesmaximal, characteristic, complemented
7C47×#3181C211yesmaximal, characteristic
8#15689254—21no—
9#15689195—21no—
10#15689195—21no—
11D4×C47⋊C8—21no—
12#15689184—21no—
13#15689180C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689079—21no—
16C2×#13093935C2211yescharacteristic
17#15688989C2211yescharacteristic
18#15688865—21no—
19#15688865—21no—
20#15688860—21no—
21#15688860—21no—
22#15688860—21no—
23#15688857—21no—
24C47×#533C2211yescharacteristic
25C47×#529C2211yescharacteristic
26C47×#501C411yescharacteristic, complemented
27C47×#497C411yescharacteristic, complemented
28C47×#491C411yescharacteristic, complemented
29C47×#489C411yescharacteristic, complemented
30C47×#467—21no—
31C47×#465—21no—
32C47×#441—21no—
33C47×#440—21no—
34C2×#97×C47C2211yescharacteristic
35C2×#10625329—21no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8—21no—
38C2×C4⋊Dic47—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)D411yescharacteristic
47C22⋊(C47⋊C8)D411yescharacteristic
48#13093935—21no—
49#13093935—21no—
50#13093935—21no—
51#13093935D411yescharacteristic
52#13093935D411yescharacteristic
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C47⋊#95—21no—
56C47⋊#95—21no—
57C2×Q16×C47—21no—
58C2×SD16×C47—21no—
59C22×C8×C47—21no—
60C4⋊Q8×C47C2×C411yescharacteristic
61C4⋊Q8×C47C2×C411yescharacteristic
62#126×C47—21no—
63#125×C47—21no—
64#125×C47C2×C411yescharacteristic
65#124×C47C2×C411yescharacteristic
66#123×C47—21no—
67C22⋊Q8×C47—21no—
68C22⋊Q8×C47—21no—
69C22⋊Q8×C47D411yescharacteristic
70#121×C47D411yescharacteristic
71C4×Q8×C47—21no—
72C4×Q8×C47—21no—
73C4×Q8×C47D411yescharacteristic
74C4×D4×C47—21no—
75C4×D4×C47D411yescharacteristic
76#117×C47—21no—
77#117×C47—21no—
78C2×C42×C47C2311yescharacteristic
79#107×C47—21no—
80#106×C47—21no—
81#103×C47—21no—
82#103×C47—21no—
83#103×C47—21no—
84#102×C47—21no—
85#97×C47—21no—
86#97×C47C2×C411yescharacteristic
87#97×C47C2×C411yescharacteristic
88C2×Q8×C47—21no—
89C2×Q8×C47—21no—
90C2×Q8×C47C22⋊C411yescharacteristic
91C2×D4×C47—21no—
92C2×D4×C47C22⋊C411yescharacteristic
93C22×C4×C47—21no—
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47#4111yescharacteristic
97C22×Dic47—41no—
98Q16×C47—42no—
99SD16×C47—42no—
100C2×C8×C47—21no—
101C2×C8×C47—21no—
102C2×C8×C47—42no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47C22⋊C411yescharacteristic
114C4⋊C4×C47C22⋊C411yescharacteristic
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C22⋊C4×C47—21no—
118C22⋊C4×C47—21no—
119C22⋊C4×C47—21no—
120C22⋊C4×C47—41no—
121C42×C47—21no—
122C42×C47—41no—
123C42×C47C22×C411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47C2×D411yescharacteristic
126C42×C47#4111yescharacteristic
127C4⋊Dic47—81no—
128#10625329—42no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—21no—
132C2×C47⋊C8—21no—
133C2×C47⋊C8—42no—
134C2×C47⋊C8—42no—
135C2×C47⋊C8—41no—
136C23×C47—21no—
137C23×C47#12111yescharacteristic
138Q8×C47—41no—
139Q8×C47—41no—
140Q8×C47—41no—
141Q8×C47—81no—
142D4×C47—41no—
143D4×C47—41no—
144D4×C47—41no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47C2×C22⋊C411yescharacteristic
157C2×C4×C47C4×D411yescharacteristic
158C2×C4×C47C4×D411yescharacteristic
159C2×C4×C47#12011yescharacteristic
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12311yescharacteristic
162C2×Dic47—81no—
163C2×Dic47—81no—
164C2×Dic47—81no—
165C8×C47—22no—
166C8×C47—41no—
167C47⋊C8—22no—
168C47⋊C8—41no—
169C47⋊C8—41no—
170C47⋊C8—41no—
171C47⋊C8—81no—
172#9431—471nomaximal, Sylow
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—41no—
176C22×C47—41no—
177C22×C47—41no—
178C22×C47#38611yescharacteristic
179C22×C47#47111yescharacteristic
180C22×C47#47311yescharacteristic
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—41no—
188C4×C47—41no—
189C4×C47—41no—
190C4×C47—81no—
191C4×C47—81no—
192Dic47—161no—
193#3181D4711yescharacteristic
194#2968—471no—
195#1531—471no—
196#1518—471no—
197#1512—471no—
198#1302—471no—
199#1295—471no—
200C2×C47—21no—
201C2×C47—21no—
202C2×C47—81no—
203C2×C47#188411yescharacteristic
204C2×C47#190611yescharacteristic
205C2×C47#190811yescharacteristic
206#533C2×D4711yescharacteristic
207#529C2×D4711yescharacteristic
208#501Dic4711yescharacteristic, complemented
209#497Dic4711yescharacteristic, complemented
210#491Dic4711yescharacteristic, complemented
211#489Dic4711yescharacteristic, complemented
212#467—21no—
213#465—21no—
214#446—941no—
215#441—21no—
216#440—21no—
217#435—941no—
218#435—941no—
219C8×D4—941no—
220#433—941no—
221#432—471no—
222#431—471no—
223#405—941no—
224C2×#97—471no—
225C2×#97C2×D4711yescharacteristic
226#382—471no—
227#330—941no—
228#330—941no—
229#328—941no—
230#328—941no—
231#328—941no—
232#327—941no—
233C47#943111yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×SD16—21no—
236C2×M16—941no—
237C22×C8—21no—
238C22×C8—941no—
239C22×C8—941no—
240C4⋊Q8C2×Dic4711yescharacteristic
241C4⋊Q8C2×Dic4711yescharacteristic
242#126—21no—
243#125—21no—
244#125C2×Dic4711yescharacteristic
245#124C2×Dic4711yescharacteristic
246#123—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C22⋊D4711yescharacteristic
250#121C22⋊D4711yescharacteristic
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D4711yescharacteristic
254C4×D4—21no—
255C4×D4C22⋊D4711yescharacteristic
256#117—21no—
257#117—21no—
258C2×C4⋊C4—941no—
259C2×C42C22×D4711yescharacteristic
260#107—21no—
261#106—21no—
262C4⋊C8—941no—
263C4⋊C8—941no—
264C4⋊C8—941no—
265C4⋊C8—941no—
266#103—21no—
267#103—21no—
268#103—21no—
269#102—21no—
270C22⋊C8—471no—
271C22⋊C8—471no—
272C22⋊C8—941no—
273C22⋊C8—941no—
274C22⋊C8—941no—
275#97—21no—
276#97—471no—
277#97—471no—
278#97—941no—
279#97—941no—
280#97—941no—
281#97C2×Dic4711yescharacteristic
282#97C2×Dic4711yescharacteristic
283C4×C8—941no—
284C4×C8—941no—
285#95—941no—
286#95—941no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic4711yescharacteristic
290C2×D4—21no—
291C2×D4C22⋊Dic4711yescharacteristic
292C22×C4—21no—
293C22×C4—1881no—
294C22×C4D4×D4711yescharacteristic
295C22×C4#1062535211yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297Q16—42no—
298SD16—42no—
299M16—1882no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—42no—
303C2×C8—941no—
304C2×C8—941no—
305C2×C8—941no—
306C2×C8—941no—
307C2×C8—1882no—
308C2×C8—1882no—
309C2×C8—1881no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—3761no—
321C4⋊C4C22⋊Dic4711yescharacteristic
322C4⋊C4C22⋊Dic4711yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C42—21no—
330C42—41no—
331C42D4×D4711yescharacteristic
332C42#1062535211yescharacteristic
333C42C22×Dic4711yescharacteristic
334C42C2×C22⋊D4711yescharacteristic
335C23—21no—
336C23#1309399611yescharacteristic
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—81no—
341D4—41no—
342D4—41no—
343D4—41no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—3761no—
356C2×C4—3761no—
357C2×C4—3761no—
358C2×C4C2×C22⋊Dic4711yescharacteristic
359C2×C4D4×Dic4711yescharacteristic
360C2×C4D4×Dic4711yescharacteristic
361C2×C4#1309399211yescharacteristic
362C2×C4#1309399611yescharacteristic
363C2×C4#1309400511yescharacteristic
364C8—22no—
365C8—41no—
366C8—942no—
367C8—3761no—
368C8—1881no—
369C8—1881no—
370C8—1881no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22#1568900311yescharacteristic
377C22#1568938811yescharacteristic
378C22#1568939611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—7521no—
391C2—21no—
392C2—21no—
393C2—81no—
394C2#2024377211yescharacteristic
395C2#2024389111yescharacteristic
396C2#2024390511yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3181C2
socleC22×C47#386
prime radical#121×C47D4
cosocleC2×C42×C47C23
hypercenter#3181D47
hypocenterC47#9431
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3181

Complements

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

NH ≅ G/Ntype
2. #20242003C2semidirect
3. #20241952C2semidirect
5. #20241287C2semidirect
6. #20241270C2semidirect
26. C47×#501C4semidirect
27. C47×#497C4semidirect
28. C47×#491C4semidirect
29. C47×#489C4semidirect
208. #501Dic47semidirect
209. #497Dic47semidirect
210. #491Dic47semidirect
211. #489Dic47semidirect
233. C47#9431semidirect

Maximal subgroups

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

subgroupindex
#202523842
#202420032
#202419522
#202419262
#202412872
#202412702
C47×#31812
#943147