Finite Group Databasebeta

← #26371028

Subgroups — #26371028

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253919C211yesmaximal, characteristic, complemented
2#20245564C214yesmaximal, complemented
3#20245559C214yesmaximal, complemented
4#20245162C211yesmaximal, characteristic, complemented
5C2×#15688947C211yesmaximal, characteristic, complemented
6#20245154C211yesmaximal, characteristic, complemented
7#20245124C211yesmaximal, characteristic
8C2×#15688933C211yesmaximal, characteristic
9C47×#2957C211yesmaximal, characteristic
10#15689751—21no—
11C2×#13093964C2211yescharacteristic
12C2×#13093964C2211yescharacteristic
13#15689532—22no—
14#15689525C2212yescomplemented
15#15689521C2212yescomplemented
16#15689165—21no—
17#15689161C2211yescharacteristic
18C2×#13093956C2211yescharacteristic
19#15688947C2212yescomplemented
20#15688947C2214yescomplemented
21#15688947C2212yescomplemented
22#15688944C2214yescomplemented
23#15688933C2212yes—
24#15688933C2214yes—
25#15688933C2212yes—
26C2×#135×C47C2211yescharacteristic
27C47×#568—21no—
28C47×#444—22no—
29C47×#443C2214yes—
30C2×#108×C47C2211yescharacteristic
31C47×#421—21no—
32C47×#405C2211yescharacteristic
33#13094024—22no—
34#13094023—41no—
35#13094016—22no—
36#13094015—41no—
37#13094001—42no—
38#13093983—21no—
39C2×C4⋊Dic47—21no—
40#13093964—21no—
41#13093964—21no—
42#13093964C2312yes—
43#13093964C2312yes—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46#13093958—22no—
47#13093958—21no—
48#13093956—22no—
49#13093956—21no—
50#13093956C2311yescharacteristic
51#13093956C2312yes—
52#13093956C2311yescharacteristic
53C2×#41×C47—21no—
54#135×C47—21no—
55#135×C47—21no—
56#135×C47Q812yescomplemented
57#135×C47C2312yes—
58C2×Q16×C47Q811yescharacteristic, complemented
59C2×SD16×C47—21no—
60C2×D8×C47Q811yescharacteristic, complemented
61#131×C47—24no—
62#131×C47—42no—
63C2×M16×C47—21no—
64C2×M16×C47—42no—
65C2×M16×C47D411yescharacteristic
66C22×C8×C47C2311yescharacteristic
67#117×C47D411yescharacteristic
68#108×C47—21no—
69#108×C47C2312yes—
70#104×C47—24no—
71#97×C47D412yes—
72C4×C8×C47C2312yes—
73#41×C47—22no—
74#41×C47—22no—
75#41×C47—41no—
76#41×C47—41no—
77C2×Q8×C47—21no—
78C2×D4×C47—21no—
79C2×D4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47C2×D411yescharacteristic
82C22×Dic47—41no—
83C2×Dic94—42no—
84Q16×C47—21no—
85Q16×C47C2×Q812yes—
86SD16×C47—22no—
87SD16×C47—22no—
88D8×C47—21no—
89D8×C47C2×Q812yes—
90M16×C47—22no—
91M16×C47—41no—
92M16×C47—44no—
93M16×C47SD1612yes—
94M16×C47SD1612yes—
95C2×C8×C47—41no—
96C2×C8×C47—44no—
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D412yes—
99C2×C8×C47C2×Q811yescharacteristic
100C2×C8×C47C2×Q811yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C2×C8×C47C2411yescharacteristic
104C4⋊C4×C47—21no—
105C22⋊C4×C47—21no—
106C42×C47C2×D412yes—
107C22⋊Dic47—41no—
108C4⋊Dic47—22no—
109C4⋊Dic47—21no—
110C4⋊Dic47—21no—
111C4⋊Dic47—42no—
112C4⋊Dic47—41no—
113Dic47⋊C4—42no—
114Dic47⋊C4—42no—
115C4×Dic47—41no—
116C47⋊C16—22no—
117C47⋊C16—22no—
118C23×C47—41no—
119C23×C47AGL(1,ℤ/8)11yescharacteristic
120Q8×C47—22no—
121Q8×C47—41no—
122D4×C47—22no—
123D4×C47—42no—
124D4×C47—42no—
125D4×C47—41no—
126C2×C4×C47—22no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47C22⋊Q811yescharacteristic
131C2×C4×C47C22⋊Q811yescharacteristic
132C2×C4×C47C2×SD1611yescharacteristic
133C2×C4×C47C2×SD1611yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47C22×D411yescharacteristic
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—42no—
139C2×Dic47—161no—
140Dic94—162no—
141C8×C47—42no—
142C8×C47C22⋊Q811yescharacteristic
143C8×C47C22⋊Q811yescharacteristic
144C8×C47C2×SD1614yes—
145C8×C47C22×Q811yescharacteristic
146C8×C47C2×#4111yescharacteristic
147#33638—471nomaximal, Sylow
148C22×C47—41no—
149C22×C47—42no—
150C22×C47—161no—
151C22×C47#47611yescharacteristic
152C22×C47#47611yescharacteristic
153C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
154C4×C47—42no—
155C4×C47—42no—
156C4×C47#47611yescharacteristic
157C4×C47#47611yescharacteristic
158C4×C47C2×C22⋊Q811yescharacteristic
159C4×C47C22×SD1611yescharacteristic
160Dic47—81no—
161Dic47—82no—
162Dic47—81no—
163#3163—471no—
164#2957D4711yescharacteristic
165#2231—474no—
166#2230—474no—
167#2155—471no—
168C2×#365—471no—
169#2153—471no—
170#2145—471no—
171C2×#360—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—21no—
175C2×C47—82no—
176C2×C47C2×#47611yescharacteristic
177C2×#135C2×D4711yescharacteristic
178#568—21no—
179#544—941no—
180C2×M32—471no—
181C2×M32—471no—
182#501—942no—
183#499—472no—
184#498—472no—
185#444—22no—
186#443C2×D4714yes—
187C2×#108C2×D4711yescharacteristic
188#428—941no—
189#427—471no—
190C2×#106—471no—
191#421—21no—
192#405C2×D4711yescharacteristic
193#365—474no—
194#365—472no—
195#365—472no—
196#364—474no—
197#360—474no—
198#360—472no—
199#360—472no—
200C47#3363811yescharacteristic, complemented, Sylow
201C2×#41—21no—
202#135—21no—
203#135—21no—
204#135Dic9412yescomplemented
205#135C22×D4712yes—
206C2×Q16Dic9411yescharacteristic, complemented
207C2×SD16—21no—
208C2×D8Dic9411yescharacteristic, complemented
209#131—24no—
210#131—42no—
211C2×M16—21no—
212C2×M16—42no—
213C2×M16C22⋊D4711yescharacteristic
214C22×C8C22×D4711yescharacteristic
215C4⋊Q8—942no—
216C4⋊Q8—1881no—
217#125—942no—
218#125—1881no—
219C22⋊Q8—1882no—
220#117—941no—
221#117C22⋊D4711yescharacteristic
222C2×C4⋊C4—941no—
223M32—472no—
224M32—472no—
225M32—941no—
226M32—941no—
227C2×C16—941no—
228C2×C16—941no—
229#108—21no—
230#108C22×D4712yes—
231#107—941no—
232#107—942no—
233#106—472no—
234#106—471no—
235#106—471no—
236#106—941no—
237#106—942no—
238#104—24no—
239#97C22⋊D4712yes—
240C4×C8C22×D4712yes—
241#41—22no—
242#41—22no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—1882no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—1881no—
251C22×C4C2×C22⋊D4711yescharacteristic
252Q16—21no—
253Q16C2×Dic9412yes—
254SD16—22no—
255SD16—22no—
256D8—21no—
257D8C2×Dic9412yes—
258M16—22no—
259M16—41no—
260M16—44no—
261M16#1062533512yes—
262M16#1062533512yes—
263C2×C8—41no—
264C2×C8—44no—
265C2×C8C2×Dic9411yescharacteristic
266C2×C8C2×Dic9411yescharacteristic
267C2×C8#1062535011yescharacteristic
268C2×C8#1062535011yescharacteristic
269C2×C8C2×C22⋊D4711yescharacteristic
270C2×C8C2×C22⋊D4712yes—
271C2×C8C23×D4711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—942no—
274C4⋊C4—941no—
275C4⋊C4—941no—
276C4⋊C4—1882no—
277C4⋊C4—1882no—
278C4⋊C4—1882no—
279C4⋊C4—1881no—
280C22⋊C4—21no—
281C22⋊C4—1881no—
282C42—1881no—
283C42C2×C22⋊D4712yes—
284C16—942no—
285C16—942no—
286C23—41no—
287C23#1309404911yescharacteristic
288Q8—22no—
289Q8—41no—
290Q8—7522no—
291D4—22no—
292D4—42no—
293D4—42no—
294D4—41no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—1881no—
300C2×C4—1881no—
301C2×C4—1882no—
302C2×C4—7521no—
303C2×C4#1309400311yescharacteristic
304C2×C4#1309400311yescharacteristic
305C2×C4C2×#1062533511yescharacteristic
306C2×C4C2×#1062533511yescharacteristic
307C2×C4#1309404911yescharacteristic
308C2×C4C22×C22⋊D4711yescharacteristic
309C8—42no—
310C8#1309400311yescharacteristic
311C8#1309400311yescharacteristic
312C8C2×#1062533514yes—
313C8C22×Dic9411yescharacteristic
314C8C2×#1062535011yescharacteristic
315C22—41no—
316C22—42no—
317C22—161no—
318C22#1568941311yescharacteristic
319C22#1568941311yescharacteristic
320C22C2×#1309404911yescharacteristic
321C4—42no—
322C4—42no—
323C4—3762no—
324C4—3761no—
325C4—3761no—
326C4#1568941311yescharacteristic
327C4#1568941311yescharacteristic
328C4C2×#1309400311yescharacteristic
329C4C22×#1062533511yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—82no—
334C2C2×#1568941311yescharacteristic
subgroupG/H
centerC2C2×#15689413
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2957C2
socleC2×C47C2×#476
prime radicalC2×#135×C47C22
hypercenter#2957D47
hypocenterC47#33638
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. #20253919C2semidirect
2. #20245564C2semidirect
3. #20245559C2semidirect
4. #20245162C2semidirect
5. C2×#15688947C2semidirect
6. #20245154C2semidirect
14. #15689525C22semidirect
15. #15689521C22semidirect
19. #15688947C22semidirect
20. #15688947C22semidirect
21. #15688947C22semidirect
22. #15688944C22semidirect
56. #135×C47Q8semidirect
58. C2×Q16×C47Q8semidirect
60. C2×D8×C47Q8semidirect
200. C47#33638semidirect
204. #135Dic94semidirect
206. C2×Q16Dic94semidirect
208. C2×D8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539192
#202455642
#202455592
#202451622
C2×#156889472
#202451542
#202451242
C2×#156889332
C47×#29572
#3363847