Finite Group Databasebeta

← #26192538

Subgroups — #26192538

Order 12032.

1004 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#15689853C211yesmaximal, characteristic, complemented
2#20253506C212yesmaximal
3#20242526C214yesmaximal, complemented
4#20242464C214yesmaximal, complemented
5#20242114C211yesmaximal, characteristic, complemented
6#20241836C212yesmaximal, complemented
7C47×#3152C211yesmaximal, characteristic
8#15689853—24no—
9#15689853—22no—
10#15689853C2214yes—
11C2×#13094023—22no—
12C2×#13094022—22no—
13C2×C4×Dic94C2212yes—
14C2×#13093989—22no—
15#15689434—22no—
16#15689415C2214yes—
17#15689320C2214yes—
18#15689261—22no—
19#15689223—24no—
20#15689180C2212yes—
21#15688872C2214yes—
22#15688855C2218yes—
23C2×#137×C47—22no—
24C47×#532C2211yescharacteristic
25C2×C4⋊Q8×C47C2211yescharacteristic
26C47×#497—22no—
27C47×#494C2212yes—
28C47×#492C2212yes—
29C4×M16×C47C2211yescharacteristic
30C22×Dic94—22no—
31C22×Dic94D412yes—
32C2×#10625335—24no—
33#13094023—28no—
34#13094023—28no—
35#13094022—28no—
36#13094022—28no—
37C4×Dic94—22no—
38C4×Dic94—22no—
39C4×Dic94D414yes—
40C4×Dic94C2314yes—
41#13093989—28no—
42#13093989—28no—
43C2×C4⋊Dic47D412yes—
44C2×Dic47⋊C4—24no—
45C2×Dic47⋊C4—22no—
46C2×C4×Dic47—22no—
47C2×C4×Dic47—22no—
48#13093957—24no—
49C4⋊(C47⋊C8)C2314yes—
50#13093947—24no—
51#13093945—24no—
52C22⋊(C47⋊C8)—22no—
53#13093935—22no—
54C2×#41×C47—21no—
55C22×Q8×C47D412yes—
56#137×C47—48no—
57C2×Q16×C47—24no—
58C2×SD16×C47—24no—
59C2×M16×C47—21no—
60C4⋊Q8×C47—22no—
61C4⋊Q8×C47D412yes—
62C4⋊Q8×C47C2311yescharacteristic
63C4⋊Q8×C47C2312yes—
64#127×C47C2311yescharacteristic
65#124×C47—21no—
66#121×C47—22no—
67C4×D4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C42×C47C2311yescharacteristic
70#97×C47—21no—
71C4×C8×C47C2312yes—
72#41×C47—44no—
73C2×Q8×C47—22no—
74C2×Q8×C47—21no—
75C2×Q8×C47—44no—
76C2×Q8×C47C2×D414yes—
77C2×D4×C47—22no—
78C2×D4×C47—21no—
79C2×D4×C47—41no—
80C22×C4×C47—22no—
81C22×C4×C47—41no—
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2×D411yescharacteristic
84C22×Dic47—22no—
85C22×Dic47—22no—
86C2×Dic94—22no—
87C2×Dic94—28no—
88C2×Dic94—28no—
89C2×Dic94—84no—
90C2×Dic94C2×D414yes—
91Q16×C47—48no—
92SD16×C47—48no—
93M16×C47—24no—
94C2×C8×C47—22no—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47—22no—
98C22⋊C4×C47—42no—
99C42×C47C2×D412yes—
100C42×C47C2×D411yescharacteristic
101C42×C47C2411yescharacteristic
102#10625335—48no—
103C4⋊Dic47—22no—
104C4⋊Dic47C2×D414yes—
105Dic47⋊C4—216no—
106Dic47⋊C4—24no—
107Dic47⋊C4—24no—
108C4×Dic47—28no—
109C4×Dic47—24no—
110C4×Dic47—24no—
111C2×C47⋊C8—24no—
112C23×C47—41no—
113C23×C47#12011yescharacteristic
114Q8×C47—44no—
115Q8×C47—42no—
116Q8×C47—82no—
117D4×C47—42no—
118D4×C47—44no—
119D4×C47—82no—
120C2×C4×C47—22no—
121C2×C4×C47—24no—
122C2×C4×C47—24no—
123C2×C4×C47—44no—
124C2×C4×C47—41no—
125C2×C4×C47—82no—
126C2×C4×C47#12012yes—
127C2×C4×C47#12011yescharacteristic
128C2×C4×C47C22×D412yes—
129C2×C4×C47C22×D411yescharacteristic
130C2×Dic47—28no—
131C2×Dic47—24no—
132C2×Dic47—24no—
133C2×Dic47—44no—
134Dic94—28no—
135Dic94—24no—
136Dic94—88no—
137Dic94#13718yescomplemented
138C8×C47—24no—
139C47⋊C8—44no—
140#18546—471nomaximal, Sylow
141C22×C47—22no—
142C22×C47—82no—
143C22×C47—81no—
144C22×C47#45312yes—
145C22×C47C2×#12011yescharacteristic
146C4×C47—22no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—44no—
150C4×C47—81no—
151C4×C47C2×#13714yes—
152Dic47—28no—
153Dic47—44no—
154C2×Q82—471no—
155#3152D4711yescharacteristic
156#3112—472no—
157#1633—474no—
158#1623—474no—
159#1558—471no—
160#1491—472no—
161C2×C47—22no—
162C2×C47—161no—
163C2×C47#300412yes—
164C2×C47C2×#45311yescharacteristic
165C2×#137—22no—
166Q82—474no—
167Q82—944no—
168Q82—942no—
169#532C2×D4711yescharacteristic
170C2×C4⋊Q8—942no—
171C2×C4⋊Q8—942no—
172C2×C4⋊Q8C2×D4711yescharacteristic
173C2×C4×Q8—472no—
174C2×C4×Q8—942no—
175#497—22no—
176#494C2×D4712yes—
177#492C2×D4712yes—
178#480—942no—
179#477—474no—
180#460—474no—
181#448—942no—
182#440—944no—
183#432—472no—
184C4×M16C2×D4711yescharacteristic
185#333—474no—
186Q8⋊C8—478no—
187C47#1854611yescharacteristic, complemented, Sylow
188C2×#41—21no—
189C22×Q8—472no—
190C22×Q8—942no—
191C22×Q8C22⋊D4712yes—
192#137—48no—
193C2×Q16—24no—
194C2×SD16—24no—
195C2×SD16—944no—
196C2×M16—21no—
197C4⋊Q8—22no—
198C4⋊Q8—948no—
199C4⋊Q8—948no—
200C4⋊Q8—948no—
201C4⋊Q8—948no—
202C4⋊Q8C22⋊D4712yes—
203C4⋊Q8C22×D4711yescharacteristic
204C4⋊Q8C22×D4712yes—
205#127C22×D4711yescharacteristic
206#124—21no—
207#121—22no—
208C4×Q8—474no—
209C4×Q8—474no—
210C4×Q8—948no—
211C4×Q8—948no—
212C4×Q8—942no—
213C4×Q8—942no—
214C4×D4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—472no—
217C2×C4⋊C4—944no—
218C2×C4⋊C4—942no—
219C2×C42—942no—
220C2×C42—942no—
221C2×C42C22×D4711yescharacteristic
222#107—944no—
223C4⋊C8—474no—
224#103—944no—
225#102—944no—
226C22⋊C8—942no—
227#97—21no—
228#97—942no—
229C4×C8C22×D4712yes—
230#41—44no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—44no—
234C2×Q8—474no—
235C2×Q8—948no—
236C2×Q8—948no—
237C2×Q8—942no—
238C2×Q8—3764no—
239C2×Q8C2×C22⋊D4714yes—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—942no—
246C22×C4—942no—
247C22×C4D4×D4712yes—
248C22×C4C2×C22⋊D4711yescharacteristic
249Q16—48no—
250SD16—48no—
251SD16—1888no—
252M16—24no—
253C2×C8—22no—
254C2×C8—944no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—474no—
259C4⋊C4—9416no—
260C4⋊C4—944no—
261C4⋊C4—942no—
262C4⋊C4—944no—
263C22⋊C4—42no—
264C42—948no—
265C42—944no—
266C42—944no—
267C42D4×D4712yes—
268C42C2×C22⋊D4711yescharacteristic
269C42C23×D4711yescharacteristic
270C23—41no—
271C23#1309399211yescharacteristic
272Q8—44no—
273Q8—42no—
274Q8—82no—
275Q8—478no—
276Q8—948no—
277Q8—944no—
278Q8—3768no—
279D4—42no—
280D4—44no—
281D4—82no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—948no—
289C2×C4—944no—
290C2×C4—944no—
291C2×C4—1884no—
292C2×C4#1309399212yes—
293C2×C4#1309399211yescharacteristic
294C2×C4C2×D4×D4712yes—
295C2×C4C22×C22⋊D4711yescharacteristic
296C8—24no—
297C8—1884no—
298C22—22no—
299C22—81no—
300C22—82no—
301C22#1568929012yes—
302C22C2×#1309399211yescharacteristic
303C4—21no—
304C4—21no—
305C4—22no—
306C4—44no—
307C4—81no—
308C4—948no—
309C4—1884no—
310C4#137×D4714yes—
311C2—22no—
312C2—161no—
313C2#2025262012yes—
314C2C2×#1568929011yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3152C2
socleC22×C47C2×#120
prime radicalC47×#532C22
hypercenter#3152D47
hypocenterC47#18546
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#15689853C2semidirect
3. #20242526C2semidirect
4. #20242464C2semidirect
5. #20242114C2semidirect
6. #20241836C2semidirect
137. Dic94#137semidirect
187. C47#18546semidirect

Maximal subgroups

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

subgroupindex
C2×#156898532
#202535062
#202425262
#202424642
#202421142
#202418362
C47×#31522
#1854647