Finite Group Databasebeta

← #26192540

Subgroups — #26192540

Order 12032.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253845C211yesmaximal, characteristic
2#20253510C212yesmaximal
3#20242528C214yesmaximal, complemented
4#20242466C214yesmaximal, complemented
5#20242114C211yesmaximal, characteristic, complemented
6#20241835C212yesmaximal, complemented
7C2×Q82×C47C211yesmaximal, characteristic, complemented
8C2×#13094055—22no—
9#15689673C2211yescharacteristic
10#15689515—22no—
11#15689502C2212yes—
12#15689497C2212yes—
13#15689436—22no—
14#15689418C2214yes—
15#15689323C2214yes—
16#15689262—22no—
17#15689221—24no—
18#15689180C2212yes—
19C4×#10625329C2211yescharacteristic
20#15688872C2214yes—
21Q8⋊(C47⋊C8)C2218yes—
22Q82×C47—22no—
23Q82×C47—24no—
24Q82×C47C2214yes—
25C2×C4⋊Q8×C47—22no—
26C2×C4⋊Q8×C47—22no—
27C2×C4⋊Q8×C47C2211yescharacteristic
28C2×C4×Q8×C47—22no—
29C2×C4×Q8×C47C2212yes—
30C2×#10625350—21no—
31#13094055—48no—
32C2×#10625337—24no—
33C2×#10625336—24no—
34C2×#10625336—24no—
35C2×#10625329—21no—
36#13094024C2311yescharacteristic
37#13094021C2311yescharacteristic
38#13094012—21no—
39#13093998—22no—
40C4×D188—22no—
41#13093957—24no—
42C4⋊(C47⋊C8)C2314yes—
43#13093948—24no—
44#13093944—24no—
45C22⋊(C47⋊C8)—22no—
46#13093935—22no—
47#13093935—21no—
48C4×C47⋊C8C2312yes—
49C22×Q8×C47—22no—
50C22×Q8×C47D412yes—
51C22×Q8×C47D412yes—
52C4⋊Q8×C47—28no—
53C4⋊Q8×C47—28no—
54C4⋊Q8×C47—28no—
55C4⋊Q8×C47—28no—
56C4⋊Q8×C47—22no—
57C4⋊Q8×C47D412yes—
58C4⋊Q8×C47C2312yes—
59C4×Q8×C47—28no—
60C4×Q8×C47—28no—
61C4×Q8×C47—22no—
62C4×Q8×C47—22no—
63C4×Q8×C47D414yes—
64C4×Q8×C47C2314yes—
65C2×C4⋊C4×C47—24no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47D412yes—
69C2×C42×C47—22no—
70C2×C42×C47—22no—
71C2×C42×C47C2311yescharacteristic
72C2×Q8×C47—28no—
73C2×Q8×C47—28no—
74C2×Q8×C47—22no—
75C2×Q8×C47—22no—
76C2×Q8×C47—44no—
77C2×Q8×C47—84no—
78C2×Q8×C47C2×D414yes—
79C2×Q8×C47C2×D414yes—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D411yescharacteristic
85C2×C22⋊D47—41no—
86#10625350—44no—
87C2×D188—22no—
88C2×D188—21no—
89C2×C4×D47—41no—
90C2×Dic94—21no—
91C4⋊C4×C47—216no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—24no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47C2×D414yes—
98C42×C47—28no—
99C42×C47—24no—
100C42×C47—24no—
101C42×C47C2×D411yescharacteristic
102C42×C47C2×D412yes—
103C42×C47C2411yescharacteristic
104#10625337—48no—
105#10625336—48no—
106#10625336—48no—
107(C2×D47)⋊C4—42no—
108C4⋊Dic47—22no—
109#10625329—24no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—24no—
112C23×C47#12011yescharacteristic
113C22×D47—41no—
114Q8×C47—28no—
115Q8×C47—24no—
116Q8×C47—44no—
117Q8×C47—82no—
118Q8×C47—88no—
119Q8×C47#13718yescomplemented
120C2×C4×C47—28no—
121C2×C4×C47—24no—
122C2×C4×C47—22no—
123C2×C4×C47—24no—
124C2×C4×C47—24no—
125C2×C4×C47—24no—
126C2×C4×C47—44no—
127C2×C4×C47—44no—
128C2×C4×C47#12012yes—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47C22×D411yescharacteristic
131C2×C4×C47C22×D412yes—
132C22⋊D47—82no—
133C2×Dic47—41no—
134D188—42no—
135D188—44no—
136C4×D47—82no—
137Dic94—42no—
138C47⋊C8—24no—
139C47⋊C8—44no—
140#18546—471nomaximal, Sylow
141C22×C47—22no—
142C22×C47#45312yes—
143C22×C47C2×#12011yescharacteristic
144C2×D47—82no—
145C2×D47—81no—
146C4×C47—28no—
147C4×C47—22no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—44no—
151C4×C47—44no—
152C4×C47C2×#13714yes—
153Dic47—81no—
154C2×Q82D4711yescharacteristic, complemented
155#3152—471no—
156#3112—472no—
157#1633—474no—
158#1623—474no—
159#1558—471no—
160#1491—472no—
161C2×C47—22no—
162C2×C47#300412yes—
163C2×C47C2×#45311yescharacteristic
164D47—161no—
165C2×#137—942no—
166Q82—22no—
167Q82—24no—
168Q82C2×D4714yes—
169#532—471no—
170C2×C4⋊Q8—22no—
171C2×C4⋊Q8—22no—
172C2×C4⋊Q8C2×D4711yescharacteristic
173C2×C4×Q8—22no—
174C2×C4×Q8C2×D4712yes—
175#497—942no—
176#494—472no—
177#492—472no—
178#480—942no—
179#477—474no—
180#460—474no—
181#448—942no—
182#440—944no—
183#432—472no—
184C4×M16—471no—
185#333—474no—
186Q8⋊C8—478no—
187C47#1854611yescharacteristic, complemented, Sylow
188C2×#41—941no—
189C22×Q8—22no—
190C22×Q8C22⋊D4712yes—
191C22×Q8C22⋊D4712yes—
192#137—1888no—
193C2×Q16—944no—
194C2×SD16—944no—
195C2×SD16—944no—
196C2×M16—941no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—471no—
203C4⋊Q8C22⋊D4712yes—
204C4⋊Q8C22×D4712yes—
205#127—471no—
206#124—941no—
207#121—942no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊D4714yes—
213C4×Q8C22×D4714yes—
214C4×D4—942no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C22⋊D4712yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×D4711yescharacteristic
222#107—944no—
223C4⋊C8—474no—
224#103—944no—
225#102—944no—
226C22⋊C8—942no—
227#97—941no—
228#97—942no—
229C4×C8—472no—
230#41—1884no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—941no—
238C2×Q8C2×C22⋊D4714yes—
239C2×Q8C2×C22⋊D4714yes—
240C2×D4—942no—
241C2×D4—941no—
242C2×D4—1881no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—1881no—
247C22×C4C2×C22⋊D4712yes—
248C22×C4C2×C22⋊D4711yescharacteristic
249Q16—1888no—
250SD16—1888no—
251SD16—1888no—
252M16—944no—
253C2×C8—942no—
254C2×C8—944no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—942no—
262C4⋊C4C2×C22⋊D4714yes—
263C22⋊C4—1882no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D4712yes—
268C42C2×C22⋊D4711yescharacteristic
269C42C23×D4711yescharacteristic
270C23—1881no—
271C23#1309399311yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—1882no—
278Q8#1309405518yescomplemented
279D4—3762no—
280D4—1882no—
281D4—1884no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—3762no—
291C2×C4—1881no—
292C2×C4#1309399312yes—
293C2×C4#1309399311yescharacteristic
294C2×C4C22×C22⋊D4711yescharacteristic
295C2×C4C22×C22⋊D4712yes—
296C8—944no—
297C8—1884no—
298C22—22no—
299C22—3761no—
300C22—3762no—
301C22#1568929112yes—
302C22C2×#1309399311yescharacteristic
303C4—21no—
304C4—21no—
305C4—28no—
306C4—22no—
307C4—44no—
308C4—44no—
309C4—3761no—
310C4C2×#1309405514yes—
311C2—22no—
312C2—7521no—
313C2#2025262212yes—
314C2C2×#1568929111yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×Q82×C47C2
socleC22×C47C2×#120
prime radical#15689673C22
hypercenterC2×Q82D47
hypocenterC47#18546
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #20242528C2semidirect
4. #20242466C2semidirect
5. #20242114C2semidirect
6. #20241835C2semidirect
7. C2×Q82×C47C2semidirect
119. Q8×C47#137semidirect
154. C2×Q82D47semidirect
187. C47#18546semidirect
278. Q8#13094055semidirect

Maximal subgroups

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

subgroupindex
#202538452
#202535102
#202425282
#202424662
#202421142
#202418352
C2×Q82×C472
#1854647