Finite Group Databasebeta

← #26186026

Subgroups — #26186026

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252349C212yesmaximal, complemented
2C4×#13094046C211yesmaximal, characteristic
3#20242284C212yesmaximal, complemented
4#20242277C214yesmaximal, complemented
5#20242272C212yesmaximal, complemented
6#20241768C212yesmaximal, complemented
7C2×C8×C47⋊C8C211yesmaximal, characteristic, complemented
8C47×#1564C211yesmaximal, characteristic, complemented
9C2×#13094046C411yescharacteristic
10C4×#10625350C2212yes—
11#15689250C2214yes—
12C4×Dic188C411yescharacteristic
13C4×Dic188C2211yescharacteristic
14C4×#10625325C412yes—
15C4×#10625325C2212yes—
16C4×D376C411yescharacteristic
17C4×D376C2211yescharacteristic
18#15689200C2214yes—
19#15689191—24no—
20#15689174—22no—
21#15689161C411yescharacteristic
22#15689120C412yes—
23C4×#10625324—22no—
24C2×C4×C47⋊C8C2211yescharacteristic
25C2×C4×C47⋊C8C2211yescharacteristic
26#15688853C2214yes—
27#15688850C2214yes—
28C8×C47⋊C8C2212yes—
29C8×C47⋊C8C2212yes—
30C47×#431C2212yes—
31C2×C4×C8×C47C2211yescharacteristic
32C47×#335C2212yes—
33C47×#334C2212yes—
34C2×#10625350C2×C412yes—
35#13094046—24no—
36C2×Dic188C2×C411yescharacteristic
37C2×#10625325C2×C412yes—
38C2×D376C2×C411yescharacteristic
39C2×#10625324—24no—
40C22×C47⋊C8—21no—
41C22×C47⋊C8D412yes—
42C4×Dic94C2×C412yes—
43C4×Dic94C2312yes—
44C4×D188C2×C412yes—
45C4×D188C2312yes—
46C4×C22⋊D47—22no—
47C4×C22⋊D47—22no—
48#13093983C2×C412yes—
49#13093981—22no—
50C42×D47—22no—
51#13093958C2×C411yescharacteristic
52#13093958C2×C411yescharacteristic
53#13093956C2×C412yes—
54Dic47⋊C8—24no—
55#13093949C2×C412yes—
56#13093949C2×C412yes—
57#13093946C2×C412yes—
58#13093946C2×C412yes—
59(C2×D47)⋊C8—24no—
60#13093934—24no—
61C4×C47⋊C8—21no—
62C4×C47⋊C8D411yescharacteristic
63C4×C47⋊C8D411yescharacteristic
64C4×C47⋊C8C2312yes—
65C4×C47⋊C8C2312yes—
66C22×C8×C47C2×C411yescharacteristic
67C22×C8×C47C2×C411yescharacteristic
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47C2314yes—
70C22⋊C8×C47—22no—
71C4×C8×C47—22no—
72C4×C8×C47C2×C411yescharacteristic
73C4×C8×C47C2×C411yescharacteristic
74C4×C8×C47C2311yescharacteristic
75C4×C8×C47C2311yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C2×C22⋊D47—22no—
80#10625350—24no—
81#10625350—24no—
82C2×D188C22×C412yes—
83C2×C4×D47—22no—
84C2×C4×D47—22no—
85C2×C4×D47—22no—
86C2×Dic94C22×C412yes—
87C2×C8×C47—22no—
88C2×C8×C47—24no—
89C2×C8×C47M1614yescomplemented
90C2×C8×C47C22×C411yescharacteristic
91C2×C8×C47C22×C411yescharacteristic
92C2×C8×C47C22×C411yescharacteristic
93C2×C8×C47C22×C411yescharacteristic
94C42×C47C22×C411yescharacteristic
95C42×C47C2×D411yescharacteristic
96C42×C47#4111yescharacteristic
97C42×C47C2411yescharacteristic
98C22⋊Dic47—22no—
99(C2×D47)⋊C4—22no—
100(C2×D47)⋊C4—22no—
101C4⋊Dic47C22×C412yes—
102C4⋊Dic47C22×C412yes—
103Dic47⋊C4—22no—
104Dic47⋊C4—22no—
105C4×Dic47—22no—
106C4×Dic47—22no—
107C4×Dic47—22no—
108C2×C47⋊C8—24no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—24no—
111C2×C47⋊C8C2×D412yes—
112C2×C47⋊C8C2×D412yes—
113Dic188—22no—
114D376—22no—
115#10625325—24no—
116#10625324—48no—
117C23×C47C4×D411yescharacteristic
118C22×D47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47C4×D411yescharacteristic
122C2×C4×C47C4×D411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47C2×M1612yes—
125C2×C4×C47#13112yes—
126C2×C4×C47C23×C411yescharacteristic
127C2×C4×C47C22×D411yescharacteristic
128C2×C4×C47C2×#4111yescharacteristic
129C22⋊D47—44no—
130C2×Dic47—22no—
131C2×Dic47—22no—
132C2×Dic47—22no—
133D188—22no—
134D188—22no—
135C4×D47—44no—
136C4×D47—44no—
137C4×D47—44no—
138Dic94—22no—
139Dic94—22no—
140C8×C47—21no—
141C8×C47—21no—
142C8×C47—44no—
143C8×C47C2×M1612yes—
144C8×C47C2×M1612yes—
145C47⋊C8—24no—
146C47⋊C8—22no—
147C47⋊C8#13514yescomplemented
148#17886—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47#43612yes—
152C22×C47C2×C4×D411yescharacteristic
153C2×D47—42no—
154C2×D47—42no—
155C2×D47—42no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47#43612yes—
159C4×C47#44312yes—
160C4×C47C22×M1611yescharacteristic
161C4×C47C2×#13111yescharacteristic
162C4×C47C2×#13512yes—
163Dic47—42no—
164Dic47—42no—
165Dic47—42no—
166#2965—472no—
167C4×#135—471no—
168#1591—472no—
169#1590—474no—
170#1589—472no—
171#1564D4711yescharacteristic, complemented
172#1476—472no—
173C2×C82—471no—
174C2×C47—22no—
175C2×C47C2×#43611yescharacteristic
176C2×C47C4×#13511yescharacteristic
177C2×C47C2×#44311yescharacteristic
178D47—82no—
179C2×#135—471no—
180C4×#41—472no—
181#446—474no—
182C4×Q16—471no—
183C4×Q16—471no—
184C4×SD16—472no—
185C4×SD16—472no—
186C4×D8—471no—
187C4×D8—471no—
188#436—474no—
189#435—944no—
190#431—942no—
191#431C2×D4712yes—
192#427—471no—
193#416—472no—
194C4×M16—942no—
195C2×C4×C8—471no—
196C2×C4×C8—471no—
197C2×C4×C8C2×D4711yescharacteristic
198#335C2×D4712yes—
199#334C2×D4712yes—
200Q8⋊C8—474no—
201D4⋊C8—474no—
202C82—472no—
203C82—472no—
204C47#1788611yescharacteristic, complemented, Sylow
205C2×#41—472no—
206#135—944no—
207C2×Q16—471no—
208C2×SD16—472no—
209C2×D8—471no—
210C2×M16—944no—
211C22×C8—472no—
212C22×C8—941no—
213C22×C8C4×D4711yescharacteristic
214C22×C8C4×D4711yescharacteristic
215C4×Q8—472no—
216C4×Q8—472no—
217C4×D4—472no—
218C4×D4—472no—
219C4×D4—942no—
220C4×D4—942no—
221#117—472no—
222#117—942no—
223C2×C42—942no—
224C2×C42C22×D4711yescharacteristic
225#107—471no—
226#107—471no—
227#106—472no—
228C4⋊C8—944no—
229C4⋊C8C22×D4714yes—
230#103—472no—
231#103—472no—
232#102—472no—
233#102—472no—
234C22⋊C8—22no—
235C22⋊C8—944no—
236#97—944no—
237C4×C8—22no—
238C4×C8—471no—
239C4×C8—471no—
240C4×C8—472no—
241C4×C8—472no—
242C4×C8—941no—
243C4×C8C4×D4711yescharacteristic
244C4×C8C4×D4711yescharacteristic
245C4×C8C22×D4711yescharacteristic
246C4×C8C22×D4711yescharacteristic
247#41—944no—
248#41—944no—
249C2×Q8—472no—
250C2×D4—472no—
251C2×D4—942no—
252C22×C4—942no—
253C22×C4—942no—
254C22×C4—942no—
255C22×C4C2×C4×D4711yescharacteristic
256C22×C4D4×D4711yescharacteristic
257C22×C4#1062535411yescharacteristic
258Q16—942no—
259SD16—944no—
260D8—942no—
261M16—1888no—
262C2×C8—22no—
263C2×C8—24no—
264C2×C8—472no—
265C2×C8—472no—
266C2×C8—944no—
267C2×C8—944no—
268C2×C8—942no—
269C2×C8#1062532414yescomplemented
270C2×C8C2×C4×D4711yescharacteristic
271C2×C8C2×C4×D4711yescharacteristic
272C2×C8C2×C4×D4711yescharacteristic
273C2×C8C2×C4×D4711yescharacteristic
274C4⋊C4—472no—
275C4⋊C4—472no—
276C4⋊C4—942no—
277C4⋊C4—942no—
278C22⋊C4—942no—
279C22⋊C4—942no—
280C22⋊C4—942no—
281C42—942no—
282C42—942no—
283C42—942no—
284C42C2×C4×D4711yescharacteristic
285C42D4×D4711yescharacteristic
286C42#1062535411yescharacteristic
287C42C23×D4711yescharacteristic
288C23—942no—
289C23#1309398611yescharacteristic
290Q8—942no—
291Q8—942no—
292D4—942no—
293D4—942no—
294D4—1884no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—942no—
298C2×C4—942no—
299C2×C4—942no—
300C2×C4—1884no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4#1309398611yescharacteristic
304C2×C4#1309398611yescharacteristic
305C2×C4#1309398611yescharacteristic
306C2×C4C2×#1062532412yes—
307C2×C4#1309403212yes—
308C2×C4C22×C4×D4711yescharacteristic
309C2×C4C2×D4×D4711yescharacteristic
310C2×C4C2×#1062535411yescharacteristic
311C8—21no—
312C8—21no—
313C8—44no—
314C8—474no—
315C8—944no—
316C8—942no—
317C8C2×#1062532412yes—
318C8C2×#1062532412yes—
319C22—21no—
320C22—21no—
321C22—1882no—
322C22—1882no—
323C22—1882no—
324C22#1568920012yes—
325C22C2×#1309398611yescharacteristic
326C4—21no—
327C4—21no—
328C4—1882no—
329C4—1882no—
330C4—1882no—
331C4#1568920012yes—
332C4#1568923912yes—
333C4C22×#1062532411yescharacteristic
334C4C2×#1309403211yescharacteristic
335C4#135×D4712yes—
336C2—22no—
337C2—3762no—
338C2C2×#1568920011yescharacteristic
339C2#2025215911yescharacteristic
340C2C2×#1568923911yescharacteristic
subgroupG/H
centerC42D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1564C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094046C4
cosocleC42×C47C24
hypercenter#1564D47
hypocenterC47#17886
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C42 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #20252349C2semidirect
3. #20242284C2semidirect
4. #20242277C2semidirect
5. #20242272C2semidirect
6. #20241768C2semidirect
7. C2×C8×C47⋊C8C2semidirect
8. C47×#1564C2semidirect
89. C2×C8×C47M16semidirect
147. C47⋊C8#135semidirect
171. #1564D47semidirect
204. C47#17886semidirect
269. C2×C8#10625324semidirect

Maximal subgroups

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

subgroupindex
#202523492
C4×#130940462
#202422842
#202422772
#202422722
#202417682
C2×C8×C47⋊C82
C47×#15642
#1788647