Finite Group Databasebeta

← #26133112

Subgroups — #26133112

Order 12032.

492 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15688920C211yesmaximal, characteristic
2#20243792C211yesmaximal, characteristic
3#20243602C212yesmaximal, complemented
4#20241515C212yesmaximal, complemented
5C47×#2372C211yesmaximal, characteristic
6#15689601C2211yescharacteristic
7#15689163—22no—
8#15689128—21no—
9C2×#13093943C2211yescharacteristic
10C2×#13093943C2212yes—
11#15688920C414yescomplemented
12#15688883—21no—
13#15688883—22no—
14C2×#124×C47C2211yescharacteristic
15C47×#391C2212yes—
16C47×#388—22no—
17C47×#386—21no—
18C4×C22⋊C4×C47—22no—
19C47×#376—21no—
20C2×#10625329—22no—
21C2×#10625329—21no—
22Q8×Dic47—42no—
23#13093983—21no—
24#13093983—22no—
25#13093957—42no—
26#13093955—42no—
27#13093945—42no—
28#13093943—22no—
29#13093943—22no—
30#13093943C2×C414yescomplemented
31C22×Q8×C47C2311yescharacteristic
32C22×D4×C47D411yescharacteristic
33C23×C4×C47—21no—
34#124×C47—22no—
35#124×C47C2×C412yes—
36#124×C47Q812yes—
37C2×C4⋊C4×C47D412yes—
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—21no—
41C2×C42×C47—22no—
42C2×C42×C47D411yescharacteristic
43C2×C42×C47D412yes—
44#95×C47—21no—
45#95×C47—21no—
46#95×C47—22no—
47#95×C47—22no—
48C24×C47—21no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47C22×C411yescharacteristic
52C2×Q8×C47C2×Q811yescharacteristic
53C2×Q8×C47#4112yes—
54C2×D4×C47—21no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47C4⋊C412yes—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—44no—
62C22×C4×C47—81no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65C4⋊C4×C47—42no—
66C22⋊C4×C47—24no—
67C22⋊C4×C47—24no—
68C22⋊C4×C47—44no—
69C22⋊C4×C47—81no—
70C42×C47—21no—
71C42×C47—44no—
72C42×C47—42no—
73C42×C47C4⋊C412yes—
74C22⋊Dic47—41no—
75C4⋊Dic47—22no—
76C4⋊Dic47—22no—
77C4⋊Dic47—22no—
78C4×Dic47—42no—
79C4×Dic47—41no—
80#10625329—24no—
81#10625329—42no—
82#10625329—41no—
83#10625329—41no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—41no—
86C23×C47—22no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47#12011yescharacteristic
92Q8×C47—41no—
93Q8×C47—41no—
94Q8×C47—42no—
95D4×C47—42no—
96D4×C47—81no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—24no—
102C2×C4×C47—22no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—84no—
110C2×C4×C47C2×C4⋊C411yescharacteristic
111C2×C4×C47C4×D412yes—
112C2×C4×C47C22⋊Q812yes—
113C2×C4×C47#12311yescharacteristic
114C2×Dic47—42no—
115C47⋊C8—44no—
116C47⋊C8—81no—
117#12972—471nomaximal, Sylow
118C22×C47—22no—
119C22×C47—44no—
120C22×C47—42no—
121C22×C47—81no—
122C22×C47—81no—
123C22×C47—82no—
124C22×C47#38511yescharacteristic
125C22×C47B(4,𝔽2)11yescharacteristic
126C22×C47#45811yescharacteristic
127C4×C47—22no—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—22no—
131C4×C47—42no—
132C4×C47—44no—
133C4×C47—82no—
134Dic47—82no—
135#2372D4711yescharacteristic
136C2×#355—471no—
137#1888—471no—
138#1857—472no—
139#1394—472no—
140C2×C47—21no—
141C2×C47—21no—
142C2×C47—82no—
143C2×C47#186711yescharacteristic
144C2×C47#220412yes—
145C2×#124C2×D4711yescharacteristic
146#519—471no—
147#428—942no—
148#418—941no—
149C2×#101—472no—
150C2×#101—471no—
151#391C2×D4712yes—
152#388—22no—
153#386—21no—
154C4×C22⋊C4—22no—
155#376—21no—
156#355—474no—
157#339—942no—
158#339—941no—
159C47#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D4711yescharacteristic
161C22×D4C22⋊D4711yescharacteristic
162C23×C4—21no—
163C2×M16—942no—
164C2×M16—941no—
165#124—22no—
166#124Dic9412yes—
167#124C4×D4712yes—
168C4×Q8—1882no—
169#117—941no—
170#117—942no—
171C2×C4⋊C4C22⋊D4712yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C42—22no—
176C2×C42C22⋊D4711yescharacteristic
177C2×C42C22⋊D4712yes—
178#107—1882no—
179#106—1882no—
180#102—1882no—
181#101—474no—
182#101—942no—
183#101—942no—
184#95—21no—
185#95—21no—
186#95—22no—
187#95—22no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic9411yescharacteristic
192C2×Q8C2×C4×D4711yescharacteristic
193C2×Q8#1062535012yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4Dic47⋊C412yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4C2×C22⋊D4711yescharacteristic
204C22×C4C2×C22⋊D4712yes—
205M16—944no—
206M16—1881no—
207M16—1881no—
208M16—1882no—
209C2×C8—1881no—
210C2×C8—1882no—
211C4⋊C4—42no—
212C4⋊C4—942no—
213C4⋊C4—942no—
214C4⋊C4—942no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—81no—
219C22⋊C4—1881no—
220C42—21no—
221C42—42no—
222C42—44no—
223C42—1882no—
224C42—1881no—
225C42Dic47⋊C412yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1309399311yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—1882no—
251C2×C4C2×Dic47⋊C411yescharacteristic
252C2×C4C4×C22⋊D4712yes—
253C2×C4#1309400312yes—
254C2×C4#1309400711yescharacteristic
255C8—3761no—
256C8—1884no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1568899611yescharacteristic
264C22#1568931011yescharacteristic
265C22#1568931311yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—22no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—3762no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2024366911yescharacteristic
278C2#2024541412yes—
subgroupG/H
centerC22#15689310
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#2372C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
cosocleC22×Q8×C47C23
hypercenter#2372D47
hypocenterC47#12972
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2372

Complements

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

NH ≅ G/Ntype
3. #20243602C2semidirect
4. #20241515C2semidirect
11. #15688920C4semidirect
30. #13093943C2×C4semidirect
159. C47#12972semidirect

Maximal subgroups

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

subgroupindex
C2×#156889202
#202437922
#202436022
#202415152
C47×#23722
#1297247