Finite Group Databasebeta

← #26115531

Subgroups — #26115531

Order 12032.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244337C211yesmaximal, characteristic, complemented
2#20244316C211yesmaximal, characteristic
3C8×C4⋊Dic47C211yesmaximal, characteristic, complemented
4C2×#15688874C211yesmaximal, characteristic, complemented
5C2×#15688872C211yesmaximal, characteristic, complemented
6C2×#15688867C211yesmaximal, characteristic
7C2×C47×#486C211yesmaximal, characteristic
8C2×C4⋊(C47⋊C8)C2211yescharacteristic
9C2×C4⋊(C47⋊C8)C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×#13093945—22no—
12C2×C8×Dic47—21no—
13C4×C4⋊Dic47C2211yescharacteristic
14#15688885—22no—
15#15688877—21no—
16#15688874—22no—
17#15688872—22no—
18#15688867—22no—
19C2×C4⋊Q8×C47C2211yescharacteristic
20C2×#127×C47C2211yescharacteristic
21C47×#486—22no—
22C47×#486C414yescomplemented
23C2×#102×C47—21no—
24C2×#102×C47—21no—
25C2×C4×C8×C47C2211yescharacteristic
26C22×C47⋊C8—21no—
27C22×C47⋊C8—21no—
28C22×C47⋊C8—21no—
29C2×C4⋊Dic47D412yes—
30C2×C4×Dic47—21no—
31C4⋊(C47⋊C8)—21no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)D412yes—
37#13093945—44no—
38C8×Dic47—42no—
39C47⋊#95—21no—
40C22×Q8×C47—21no—
41C22×D4×C47—21no—
42C22×D4×C47—21no—
43C22×D4×C47—21no—
44C22×C8×C47D411yescharacteristic
45C22×C8×C47D411yescharacteristic
46C4⋊Q8×C47—21no—
47C4⋊Q8×C47—21no—
48C4⋊Q8×C47—21no—
49C4⋊Q8×C47C2×C412yes—
50#127×C47—21no—
51#127×C47—21no—
52#127×C47—21no—
53#127×C47C2×C412yes—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C42×C47C2311yescharacteristic
57#102×C47—24no—
58#102×C47—24no—
59C4×C8×C47C2×C412yes—
60C4×C8×C47D412yes—
61C24×C47—41no—
62C2×Q8×C47—22no—
63C2×Q8×C47—41no—
64C2×Q8×C47—82no—
65C2×D4×C47—22no—
66C2×D4×C47—22no—
67C2×D4×C47—22no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—42no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C2×D4×C47—42no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—82no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C22×Dic47—22no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—21no—
86C2×C8×C47—21no—
87C2×C8×C47C22⋊C412yes—
88C2×C8×C47C22⋊C412yes—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C42×C47C22×C411yescharacteristic
94C42×C47C2×D411yescharacteristic
95C42×C47C2×D411yescharacteristic
96C42×C47#4111yescharacteristic
97C4⋊Dic47—22no—
98C4⋊Dic47SD1614yes—
99C4×Dic47—41no—
100C4×Dic47—41no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C2×C47⋊C8—42no—
107C2×C47⋊C8—42no—
108C23×C47—42no—
109C23×C47—81no—
110C23×C47—81no—
111C23×C47—82no—
112C23×C47—82no—
113C23×C47#12111yescharacteristic
114Q8×C47—82no—
115Q8×C47—82no—
116D4×C47—42no—
117D4×C47—42no—
118D4×C47—44no—
119D4×C47—82no—
120D4×C47—82no—
121D4×C47—81no—
122D4×C47—81no—
123D4×C47—81no—
124D4×C47—81no—
125C2×C4×C47—22no—
126C2×C4×C47—42no—
127C2×C4×C47—82no—
128C2×C4×C47C2×C22⋊C411yescharacteristic
129C2×C4×C47C4×D411yescharacteristic
130C2×C4×C47C4×D411yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12411yescharacteristic
133C2×C4×C47#12711yescharacteristic
134C2×C4×C47C2×SD1612yes—
135C2×C4×C47#13511yescharacteristic
136C2×C4×C47#13511yescharacteristic
137C2×C4×C47#13512yes—
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C47#13711yescharacteristic
140C2×Dic47—24no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C8×C47—22no—
144C8×C47—22no—
145C47⋊C8—42no—
146C47⋊C8—81no—
147C47⋊C8—81no—
148#9735—471nomaximal, Sylow
149C22×C47—82no—
150C22×C47—82no—
151C22×C47—82no—
152C22×C47—162no—
153C22×C47—162no—
154C22×C47#39011yescharacteristic
155C22×C47#46311yescharacteristic
156C22×C47#46411yescharacteristic
157C22×C47#46512yes—
158C22×C47#47212yes—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—82no—
162C4×C47#41611yescharacteristic
163C4×C47#41711yescharacteristic
164C4×C47C4×SD1612yes—
165C4×C47#46311yescharacteristic
166C4×C47#46411yescharacteristic
167C4×C47#49211yescharacteristic
168C4×C47#49511yescharacteristic
169Dic47—44no—
170C2×#486D4711yescharacteristic
171#1987—471no—
172#1982—471no—
173C8×C4⋊C4—471no—
174C2×#334—471no—
175C2×#333—471no—
176C2×#331—471no—
177C2×C47—162no—
178C2×C47#167712yes—
179C2×C47#167812yes—
180C2×C47#196811yescharacteristic
181C2×C47#197811yescharacteristic
182C2×C47#198311yescharacteristic
183C2×C4⋊Q8C2×D4711yescharacteristic
184C2×#127C2×D4711yescharacteristic
185#486—22no—
186#486Dic4714yescomplemented
187C2×C4⋊C8—471no—
188C2×C4⋊C8—471no—
189C2×C4⋊C8—471no—
190C2×#102—21no—
191C2×#102—21no—
192C2×#102—942no—
193C2×C4×C8—941no—
194C2×C4×C8C2×D4711yescharacteristic
195C4×C4⋊C4—471no—
196#340—942no—
197#336—941no—
198#334—942no—
199#333—942no—
200#331—942no—
201C47#973511yescharacteristic, complemented, Sylow
202C22×Q8—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C22×C8—941no—
207C22×C8—941no—
208C22×C8—941no—
209C22×C8C22⋊D4711yescharacteristic
210C22×C8C22⋊D4711yescharacteristic
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C2×Dic4712yes—
215#127—21no—
216#127—21no—
217#127—21no—
218#127C2×Dic4712yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—472no—
222C2×C42—941no—
223C2×C42C22×D4711yescharacteristic
224C4⋊C8—472no—
225C4⋊C8—941no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228C4⋊C8—941no—
229C4⋊C8—941no—
230#102—24no—
231#102—24no—
232#102—1884no—
233C4×C8—1882no—
234C4×C8C2×Dic4712yes—
235C4×C8C22⋊D4712yes—
236#95—941no—
237C24—41no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C22×C4—41no—
255C22×C4—942no—
256C22×C4D4×D4711yescharacteristic
257C22×C4#1062535211yescharacteristic
258C22×C4C2×C22⋊D4711yescharacteristic
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—942no—
264C2×C8—1881no—
265C2×C8—1881no—
266C2×C8—1882no—
267C2×C8—1882no—
268C2×C8—1881no—
269C2×C8—1881no—
270C2×C8C22⋊Dic4712yes—
271C2×C8C22⋊Dic4712yes—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—474no—
277C4⋊C4—942no—
278C42—1881no—
279C42—1881no—
280C42D4×D4711yescharacteristic
281C42#1062535211yescharacteristic
282C42C22×Dic4711yescharacteristic
283C42C2×C22⋊D4711yescharacteristic
284C23—42no—
285C23—81no—
286C23—81no—
287C23—82no—
288C23—82no—
289C23#1309399511yescharacteristic
290Q8—82no—
291Q8—82no—
292D4—42no—
293D4—42no—
294D4—44no—
295D4—82no—
296D4—82no—
297D4—81no—
298D4—81no—
299D4—81no—
300D4—81no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—944no—
305C2×C4—1882no—
306C2×C4—1882no—
307C2×C4C2×C22⋊Dic4711yescharacteristic
308C2×C4D4×Dic4711yescharacteristic
309C2×C4D4×Dic4711yescharacteristic
310C2×C4#1309399511yescharacteristic
311C2×C4#1309401111yescharacteristic
312C2×C4#1309402011yescharacteristic
313C2×C4SD16×D4712yes—
314C2×C4#1309404411yescharacteristic
315C2×C4#1309404411yescharacteristic
316C2×C4#1309404512yes—
317C2×C4#1309404911yescharacteristic
318C2×C4#1309405411yescharacteristic
319C8—22no—
320C8—22no—
321C8—3761no—
322C8—3761no—
323C8—1882no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—162no—
328C22—162no—
329C22#1568902611yescharacteristic
330C22#1568933511yescharacteristic
331C22#1568934211yescharacteristic
332C22#1568934912yes—
333C22#1568939012yes—
334C4—21no—
335C4—21no—
336C4—82no—
337C4—1884no—
338C4#1568911911yescharacteristic
339C4#1568912411yescharacteristic
340C4SD16×Dic4712yes—
341C4#1568933511yescharacteristic
342C4#1568934211yescharacteristic
343C4#1568949411yescharacteristic
344C4#1568950411yescharacteristic
345C2—162no—
346C2#2024279412yes—
347C2#2024280112yes—
348C2#2024424411yescharacteristic
349C2#2024429911yescharacteristic
350C2#2024432111yescharacteristic
subgroupG/H
centerC23#13093995
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#486C2
socleC23×C47#121
prime radicalC2×#127×C47C22
cosocleC2×C42×C47C23
hypercenterC2×#486D47
hypocenterC47#9735
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #20244337C2semidirect
3. C8×C4⋊Dic47C2semidirect
4. C2×#15688874C2semidirect
5. C2×#15688872C2semidirect
22. C47×#486C4semidirect
186. #486Dic47semidirect
201. C47#9735semidirect

Maximal subgroups

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

subgroupindex
#202443372
#202443162
C8×C4⋊Dic472
C2×#156888742
C2×#156888722
C2×#156888672
C2×C47×#4862
#973547