Finite Group Databasebeta

← #26189410

Subgroups — #26189410

Order 12032.

890 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 187 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#20257198C211yesmaximal, characteristic
2#20253427C212yesmaximal
3#20242429C214yesmaximal
4#20242361C214yesmaximal
5#20241951C212yesmaximal
6#20241712C211yesmaximal, characteristic
7C47×#2581C211yesmaximal, characteristic
8Q8×Dic94—22no—
9#15689852—22no—
10#15689849—24no—
11#15689754—21no—
12#15689754C2214yes—
13#15689680—22no—
14#15689680C2212yes—
15#15689601—21no—
16#15689458C2214yes—
17#15689418—22no—
18#15689333—22no—
19#15689279C2214yes—
20C4×#10625337—24no—
21#15689176C2212yes—
22#15688859C2218yes—
23#15688849C2214yes—
24C2×C4×Q8×C47C2211yescharacteristic
25C2×C4×Q8×C47C2211yescharacteristic
26C47×#400C2212yes—
27C47×#395C2212yes—
28C47×#391—21no—
29C47×#389—21no—
30C47×#387—22no—
31C47×#382C2211yescharacteristic
32C2×#10625337—24no—
33#13094024—22no—
34#13094024—22no—
35#13094024D412yes—
36#13094024D412yes—
37#13094023—24no—
38#13094023—28no—
39#13094016—24no—
40#13094016—22no—
41#13094015—24no—
42#13094015—28no—
43#13094003—24no—
44#13094003—22no—
45#13094003C2314yes—
46Q8×Dic47—24no—
47Q8×Dic47—24no—
48C4×Dic94—24no—
49C4×Dic94—24no—
50C4×Dic94—22no—
51C4×Dic94D414yes—
52#13093983—22no—
53#13093983—24no—
54#13093957—24no—
55C4⋊(C47⋊C8)—22no—
56#13093948—24no—
57#13093947—24no—
58C22⋊(C47⋊C8)C2314yes—
59C4×C47⋊C8—22no—
60C22×Q8×C47D411yescharacteristic
61C22×Q8×C47C2311yescharacteristic
62C4×Q8×C47—22no—
63C4×Q8×C47—22no—
64C4×Q8×C47—22no—
65C4×Q8×C47—22no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C4⋊C4×C47C2311yescharacteristic
71C2×C4⋊C4×C47C2312yes—
72C2×C42×C47—21no—
73C2×C42×C47D412yes—
74C2×C42×C47C2311yescharacteristic
75#95×C47—21no—
76#95×C47—22no—
77#95×C47—22no—
78#95×C47—22no—
79#95×C47C2312yes—
80C2×Q8×C47—22no—
81C2×Q8×C47—22no—
82C2×Q8×C47—22no—
83C2×Q8×C47—42no—
84C2×Q8×C47Q1614yescomplemented
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—22no—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D412yes—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2411yescharacteristic
93C2×Dic94—22no—
94C2×Dic94—24no—
95C2×Dic94C2×D414yes—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—44no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47—42no—
103C42×C47—22no—
104C42×C47—22no—
105C42×C47—42no—
106C42×C47C2×D412yes—
107C42×C47C2×D412yes—
108C22⋊Dic47—24no—
109#10625337—48no—
110C4⋊Dic47—24no—
111C4⋊Dic47—24no—
112C4⋊Dic47—24no—
113C4⋊Dic47—22no—
114C4⋊Dic47—22no—
115C4⋊Dic47—22no—
116C4⋊Dic47C2×D414yes—
117Dic47⋊C4—28no—
118Dic47⋊C4—24no—
119Dic47⋊C4—24no—
120C4×Dic47—28no—
121C4×Dic47—24no—
122C2×C47⋊C8—24no—
123C23×C47C22×D411yescharacteristic
124Q8×C47—28no—
125Q8×C47—24no—
126Q8×C47—42no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—41no—
140C2×C4×C47—42no—
141C2×C4×C47#12011yescharacteristic
142C2×C4×C47#12011yescharacteristic
143C2×C4×C47#12012yes—
144C2×C4×C47C2×Q1614yes—
145C2×C4×C47C2×Q1612yes—
146C2×C4×C47#13712yes—
147C2×C4×C47C22×D412yes—
148C2×Dic47—24no—
149C2×Dic47—24no—
150Dic94—44no—
151Dic94—44no—
152Dic94—84no—
153C47⋊C8—44no—
154#18221—471nomaximal, Sylow
155C22×C47—21no—
156C22×C47—21no—
157C22×C47C2×#12011yescharacteristic
158C22×C47C22×Q1611yescharacteristic
159C22×C47C2×#13711yescharacteristic
160C4×C47—24no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—22no—
164C4×C47—42no—
165C4×C47—42no—
166C4×C47—82no—
167C4×C47—81no—
168C4×C47#45114yes—
169Dic47—44no—
170Dic47—44no—
171#3502—471no—
172#3104—472no—
173#2581D4711yescharacteristic
174#1617—474no—
175#1605—474no—
176#1518—472no—
177#1461—471no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47C2×#45111yescharacteristic
181C2×C47#302011yescharacteristic
182C2×C47#302511yescharacteristic
183Q82—942no—
184#557—944no—
185#557—942no—
186#544—474no—
187#544—941no—
188#533—472no—
189#533—942no—
190#519—941no—
191C2×C4×Q8C2×D4711yescharacteristic
192C2×C4×Q8C2×D4711yescharacteristic
193#484—474no—
194#477—942no—
195#462—942no—
196#451—474no—
197C4×Q16—944no—
198#431—472no—
199#400C2×D4712yes—
200#395C2×D4712yes—
201#391—21no—
202#389—21no—
203#387—22no—
204#382C2×D4711yescharacteristic
205#328—478no—
206#324—474no—
207C47#1822111yescharacteristic, complemented, Sylow
208C22×Q8C22⋊D4711yescharacteristic
209C22×Q8C22×D4711yescharacteristic
210C2×Q16—944no—
211C4⋊Q8—472no—
212C4⋊Q8—472no—
213C4⋊Q8—942no—
214C4⋊Q8—942no—
215C4⋊Q8—948no—
216C4⋊Q8—944no—
217#125—944no—
218#125—948no—
219#125—944no—
220#125—942no—
221C22⋊Q8—474no—
222C22⋊Q8—942no—
223C22⋊Q8—944no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—474no—
229C4×Q8—944no—
230C4×Q8—944no—
231C4×Q8—944no—
232C4×Q8—944no—
233C4×Q8—942no—
234#117—942no—
235#117—944no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22⋊D4711yescharacteristic
240C2×C4⋊C4C22×D4711yescharacteristic
241C2×C4⋊C4C22×D4712yes—
242C2×C42—21no—
243C2×C42C22⋊D4712yes—
244C2×C42C22×D4711yescharacteristic
245#107—944no—
246C4⋊C8—942no—
247#103—944no—
248#103—944no—
249C22⋊C8—474no—
250C4×C8—942no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22×D4712yes—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—42no—
260C2×Q8—474no—
261C2×Q8—944no—
262C2×Q8—942no—
263C2×Q8#1062533714yescomplemented
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4C2×C22⋊D4712yes—
268C22×C4C2×C22⋊D4712yes—
269C22×C4C2×C22⋊D4711yescharacteristic
270C22×C4C2×C22⋊D4711yescharacteristic
271C22×C4C23×D4711yescharacteristic
272Q16—1888no—
273C2×C8—944no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—474no—
282C4⋊C4—948no—
283C4⋊C4—944no—
284C4⋊C4—944no—
285C4⋊C4—944no—
286C4⋊C4—944no—
287C4⋊C4—942no—
288C4⋊C4—942no—
289C4⋊C4—942no—
290C4⋊C4—944no—
291C22⋊C4—944no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—948no—
296C42—944no—
297C42D4×D4712yes—
298C42D4×D4712yes—
299C23C22×C22⋊D4711yescharacteristic
300Q8—28no—
301Q8—24no—
302Q8—42no—
303Q8—3764no—
304Q8—1884no—
305Q8—1884no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—944no—
321C2×C4—944no—
322C2×C4#1309399211yescharacteristic
323C2×C4#1309399211yescharacteristic
324C2×C4#1309399212yes—
325C2×C4C2×#1062533714yes—
326C2×C4C2×#1062533712yes—
327C2×C4#1309405412yes—
328C2×C4C2×D4×D4712yes—
329C8—1884no—
330C22—21no—
331C22—21no—
332C22C2×#1309399211yescharacteristic
333C22C22×#1062533711yescharacteristic
334C22C2×#1309405411yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—24no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—81no—
343C4—1884no—
344C4—1884no—
345C4#1568927814yes—
346C2—21no—
347C2—21no—
348C2C2×#1568927811yescharacteristic
349C2#2025275111yescharacteristic
350C2#2025277911yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2581C2
socleC22×C47C2×#120
prime radicalC23×C47C22×D4
hypercenter#2581D47
hypocenterC47#18221
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
84. C2×Q8×C47Q16semidirect
207. C47#18221semidirect
263. C2×Q8#10625337semidirect

Maximal subgroups

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

subgroupindex
#202571982
#202534272
#202424292
#202423612
#202419512
#202417122
C47×#25812
#1822147