Finite Group Databasebeta

← #26115660

Subgroups — #26115660

Order 12032.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253710C211yesmaximal, characteristic
2#20243518C211yesmaximal, characteristic, complemented
3#20241953C211yesmaximal, characteristic, complemented
4#20241951C211yesmaximal, characteristic, complemented
5#20241948C211yesmaximal, characteristic
6#20241672C211yesmaximal, characteristic, complemented
7C47×#1517C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689477—21no—
10#15689477—21no—
11#15689473C411yescharacteristic, complemented
12#15689473C411yescharacteristic, complemented
13#15689469C411yescharacteristic, complemented
14#15689469C411yescharacteristic, complemented
15#15689176C2211yescharacteristic
16#15689119—21no—
17#15689119—21no—
18C2×C4×C47⋊C8C2211yescharacteristic
19#15688989C2211yescharacteristic
20#15688885—21no—
21#15688885—21no—
22#15688861—22no—
23#15688859—22no—
24#15688857—22no—
25#15688845—21no—
26C8×C47⋊C8—21no—
27C47×#532C2211yescharacteristic
28C47×#431C2211yescharacteristic
29C47×#382C2211yescharacteristic
30C47×#327—22no—
31C22×C47⋊C8D411yescharacteristic
32C22×C47⋊C8D411yescharacteristic
33#13094024C2×C411yescharacteristic
34#13094024C2×C411yescharacteristic
35#13094016—21no—
36#13094003—21no—
37#13094003—21no—
38C4×Dic94—21no—
39C4×Dic94—21no—
40#13093983—21no—
41C2×C4⋊Dic47—21no—
42C4⋊(C47⋊C8)—21no—
43#13093948—21no—
44#13093948—21no—
45#13093948—21no—
46#13093948—21no—
47#13093945—21no—
48#13093945—21no—
49#13093945—21no—
50#13093945—21no—
51C22⋊(C47⋊C8)D412yes—
52C4×C47⋊C8—21no—
53C4×C47⋊C8C2×C411yescharacteristic
54C4×C47⋊C8C2×C411yescharacteristic
55C4×C47⋊C8D411yescharacteristic
56C4×C47⋊C8D411yescharacteristic
57C47⋊#95—21no—
58C47⋊#95—21no—
59C2×#41×C47—21no—
60C4⋊Q8×C47C2×C411yescharacteristic
61#127×C47C2×C411yescharacteristic
62#124×C47—21no—
63#121×C47—21no—
64#121×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47—21no—
70C22⋊C8×C47D412yes—
71C4×C8×C47—21no—
72#95×C47—21no—
73#95×C47—21no—
74#41×C47—41no—
75#41×C47—41no—
76#41×C47—41no—
77#41×C47—41no—
78C2×Q8×C47—21no—
79C2×D4×C47—21no—
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×Dic47—41no—
89C2×Dic94—21no—
90C2×Dic94—21no—
91C2×C8×C47—22no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C42×C47C22×C411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47#4111yescharacteristic
101C42×C47#4111yescharacteristic
102C22⋊Dic47—41no—
103C4⋊Dic47—21no—
104C4⋊Dic47—21no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107C4⋊Dic47—41no—
108C4⋊Dic47—41no—
109Dic47⋊C4—41no—
110Dic47⋊C4—41no—
111C4×Dic47—41no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—21no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8—22no—
117C2×C47⋊C8C22⋊C411yescharacteristic
118C2×C47⋊C8C22⋊C411yescharacteristic
119C2×C47⋊C8C22⋊C411yescharacteristic
120C2×C47⋊C8C22⋊C411yescharacteristic
121C23×C47—41no—
122C23×C47#12711yescharacteristic
123Q8×C47—41no—
124Q8×C47—41no—
125D4×C47—41no—
126D4×C47—41no—
127D4×C47—81no—
128D4×C47—81no—
129D4×C47—81no—
130D4×C47—81no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—81no—
135C2×C4×C47—81no—
136C2×C4×C47—81no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47C2×C22⋊C411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12411yescharacteristic
145C2×C4×C47#13511yescharacteristic
146C2×C4×C47#13511yescharacteristic
147C2×C4×C47#13511yescharacteristic
148C2×C4×C47#13511yescharacteristic
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—81no—
152C2×Dic47—81no—
153C2×Dic47—81no—
154Dic94—81no—
155Dic94—81no—
156C8×C47—42no—
157C47⋊C8—21no—
158C47⋊C8—21no—
159C47⋊C8—21no—
160C47⋊C8—21no—
161C47⋊C8—42no—
162#9757—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47#39011yescharacteristic
169C22×C47#49511yescharacteristic
170C22×C47#49511yescharacteristic
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—81no—
176C4×C47—161no—
177C4×C47#41611yescharacteristic
178C4×C47#41611yescharacteristic
179C4×C47#48811yescharacteristic
180C4×C47#48811yescharacteristic
181Dic47—81no—
182Dic47—81no—
183Dic47—161no—
184#3131—471no—
185#1835—471no—
186#1518—471no—
187#1518—471no—
188#1517—471no—
189#1517D4711yescharacteristic
190#1450—471no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—161no—
194C2×C47#195811yescharacteristic
195C2×C47#197611yescharacteristic
196C2×C47#197611yescharacteristic
197#533—471no—
198#532C2×D4711yescharacteristic
199#488—941no—
200#488—941no—
201#487—471no—
202#487—471no—
203#486—471no—
204#486—471no—
205#431—471no—
206#431C2×D4711yescharacteristic
207#416—941no—
208#416—941no—
209C2×C4×C8—471no—
210#382—471no—
211#382C2×D4711yescharacteristic
212#340—941no—
213#340—941no—
214#328—942no—
215#328—942no—
216#327—22no—
217#327—942no—
218#322—941no—
219C82—941no—
220C47#975711yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C22×C8—471no—
223C22×C8—471no—
224C4⋊Q8—471no—
225C4⋊Q8—471no—
226C4⋊Q8C4×D4711yescharacteristic
227#127C4×D4711yescharacteristic
228#125—941no—
229#124—21no—
230C22⋊Q8—941no—
231C22⋊Q8—941no—
232#121—21no—
233#121—21no—
234C4×Q8—941no—
235C4×Q8—941no—
236C4×D4—21no—
237C4×D4—21no—
238#117—941no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—941no—
241C2×C42C22×D4711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—941no—
244#103—941no—
245#103—941no—
246#103—941no—
247#103—941no—
248#102—941no—
249#102—941no—
250#102—941no—
251#102—941no—
252C22⋊C8—472no—
253C22⋊C8C22⋊D4712yes—
254C4×C8—21no—
255C4×C8—471no—
256C4×C8—471no—
257C4×C8—471no—
258C4×C8—471no—
259C4×C8—941no—
260#95—21no—
261#95—21no—
262#95—941no—
263#95—941no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—941no—
270C2×Q8—941no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—1881no—
278C22×C4D4×D4711yescharacteristic
279C22×C4D4×D4711yescharacteristic
280C22×C4C2×C22⋊D4711yescharacteristic
281C2×C8—22no—
282C2×C8—471no—
283C2×C8—471no—
284C2×C8—471no—
285C2×C8—471no—
286C2×C8—941no—
287C2×C8—941no—
288C2×C8—941no—
289C2×C8—941no—
290C2×C8—942no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—941no—
296C4⋊C4—941no—
297C4⋊C4—941no—
298C4⋊C4—941no—
299C4⋊C4—1881no—
300C4⋊C4—1881no—
301C4⋊C4—1881no—
302C4⋊C4—1881no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—1881no—
306C42—1881no—
307C42C2×C4×D4711yescharacteristic
308C42#1062535011yescharacteristic
309C42D4×D4711yescharacteristic
310C42#1062535211yescharacteristic
311C23—41no—
312C23#1309402011yescharacteristic
313Q8—41no—
314Q8—41no—
315Q8—3761no—
316Q8—3761no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—3761no—
332C2×C4—3761no—
333C2×C4—3761no—
334C2×C4—1881no—
335C2×C4—1881no—
336C2×C4C22⋊C4×D4711yescharacteristic
337C2×C4(C22⋊D47)⋊C411yescharacteristic
338C2×C4C4×C22⋊D4711yescharacteristic
339C2×C4#1309399611yescharacteristic
340C2×C4#1309399711yescharacteristic
341C2×C4#1309401011yescharacteristic
342C2×C4#1309404311yescharacteristic
343C2×C4#1309404311yescharacteristic
344C2×C4#1309404511yescharacteristic
345C2×C4#1309404511yescharacteristic
346C8—42no—
347C8—941no—
348C8—941no—
349C8—941no—
350C8—941no—
351C8—1882no—
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#1568902311yescharacteristic
358C22#1568950411yescharacteristic
359C22#1568950611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—161no—
366C4—3761no—
367C4—3761no—
368C4—7521no—
369C4#1568911611yescharacteristic
370C4#1568911611yescharacteristic
371C4#1568947611yescharacteristic
372C4#1568947611yescharacteristic
373C2—21no—
374C2—21no—
375C2—161no—
376C2#2024419811yescharacteristic
377C2#2024428311yescharacteristic
378C2#2024428511yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1517C2
socleC22×C47#390
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#1517D47
hypocenterC47#9757
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #20243518C2semidirect
3. #20241953C2semidirect
4. #20241951C2semidirect
6. #20241672C2semidirect
11. #15689473C4semidirect
12. #15689473C4semidirect
13. #15689469C4semidirect
14. #15689469C4semidirect
220. C47#9757semidirect

Maximal subgroups

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

subgroupindex
#202537102
#202435182
#202419532
#202419512
#202419482
#202416722
C47×#15172
#975747