Finite Group Databasebeta

← #26126909

Subgroups — #26126909

Order 12032.

444 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 87 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#20247705C211yesmaximal, characteristic
2C4×#13093941C211yesmaximal, characteristic
3#20241416C214yesmaximal, complemented
4C47×#1465C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic
6C2×#13093941—21no—
7C4×#10625329—21no—
8#15689062C2212yes—
9#15689059C412yes—
10#15689021—21no—
11#15689003—21no—
12#15688995—21no—
13#15688983—21no—
14#15688983C411yescharacteristic
15#15688892C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C23×Dic47—21no—
21C2×#10625329—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—22no—
30#13093941—24no—
31#13093935—41no—
32C4×C47⋊C8—41no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95C2×C412yes—
36C22×D4×C47D411yescharacteristic
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C2×M16×C47—41no—
40C4×D4×C47—41no—
41C4×D4×C47—41no—
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C22⋊C8×C47—22no—
47C22⋊C8×C47—41no—
48C24×C47C22⋊C411yescharacteristic
49C24×C47C22⋊C411yescharacteristic
50C2×D4×C47—22no—
51C2×D4×C47—22no—
52C2×D4×C47—41no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—42no—
58C22×C4×C47—42no—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—42no—
67C22×Dic47—82no—
68M16×C47—82no—
69C2×C8×C47—42no—
70C4⋊C4×C47—41no—
71C22⋊C4×C47—41no—
72C22⋊C4×C47—41no—
73C42×C47—21no—
74C42×C47—21no—
75C22⋊Dic47—42no—
76C22⋊Dic47—82no—
77C4⋊Dic47—42no—
78C4⋊Dic47—41no—
79C4⋊Dic47—41no—
80C4⋊Dic47—42no—
81Dic47⋊C4—82no—
82#10625329—24no—
83#10625329—42no—
84C2×C47⋊C8—42no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47#10412yes—
95C23×C47C2×C22⋊C411yescharacteristic
96D4×C47—44no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—81no—
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47#9911yescharacteristic
112C2×C4×C47#13712yes—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—84no—
120C8×C47—82no—
121C47⋊C8—44no—
122#11790—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—22no—
126C22×C47—42no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47—41no—
130C22×C47—41no—
131C22×C47—41no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47—81no—
135C22×C47C2×#9911yescharacteristic
136C22×C47#41911yescharacteristic
137C22×C47C2×#10411yescharacteristic
138C4×C47—21no—
139C4×C47—22no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47—41no—
143C4×C47#35212yes—
144Dic47—82no—
145Dic47—82no—
146#2487—471no—
147C4×#100—471no—
148#1465D4711yescharacteristic
149#1353—474no—
150C2×C47—21no—
151C2×C47—21no—
152C2×C47—42no—
153C2×C47—81no—
154C2×C47#150811yescharacteristic
155C2×C47C2×#35211yescharacteristic
156C2×C47#213411yescharacteristic
157C2×C4×D4C2×D4711yescharacteristic
158C22×C4⋊C4—471no—
159C2×#100—941no—
160#407—21no—
161C4×M16—941no—
162#400—472no—
163#399—472no—
164#389—941no—
165#386—941no—
166#385—941no—
167#380—471no—
168#380—941no—
169#343—472no—
170#324C2×D4712yes—
171#323—21no—
172C47#1179011yescharacteristic, complemented, Sylow
173C22×D4D18811yescharacteristic
174C23×C4—941no—
175C23×C4C4×D4711yescharacteristic
176C23×C4C22⋊D4711yescharacteristic
177C2×M16—41no—
178C2×M16—942no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—472no—
182C2×C4⋊C4—942no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4—941no—
185C2×C4⋊C4—1881no—
186C2×C4⋊C4—1881no—
187C2×C4⋊C4C22×D4711yescharacteristic
188C2×C22⋊C4—942no—
189C2×C22⋊C4—941no—
190C2×C22⋊C4C4×D4711yescharacteristic
191C2×C22⋊C4C22⋊D4711yescharacteristic
192C2×C42D18811yescharacteristic
193#100—944no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196#97—1881no—
197C4×C8—1881no—
198#95—472no—
199#95—942no—
200#95—942no—
201C24(C2×D47)⋊C411yescharacteristic
202C24(C2×D47)⋊C411yescharacteristic
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—942no—
213C22×C4—942no—
214C22×C4—3762no—
215C22×C4—1882no—
216C22×C4(C2×D47)⋊C411yescharacteristic
217C22×C4(C2×D47)⋊C411yescharacteristic
218C22×C4C2×C4×D4711yescharacteristic
219C22×C4C2×D18811yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221M16—82no—
222M16—944no—
223M16—1882no—
224C2×C8—42no—
225C2×C8—1882no—
226C4⋊C4—41no—
227C4⋊C4—3762no—
228C4⋊C4—1882no—
229C4⋊C4—1882no—
230C4⋊C4—1881no—
231C4⋊C4—1881no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—3762no—
235C22⋊C4—1882no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1309395012yes—
248C23C2×(C2×D47)⋊C411yescharacteristic
249D4—44no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—3764no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4—1882no—
269C2×C4—1882no—
270C2×C4#1309393811yescharacteristic
271C2×C4#1309393811yescharacteristic
272C2×C4#1309405712yes—
273C8—82no—
274C8—1884no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22C2×#1309393811yescharacteristic
288C22#1568913311yescharacteristic
289C22C2×#1309395011yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—3762no—
296C4—3762no—
297C4#1568891212yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2024190911yescharacteristic
303C2C2×#1568891211yescharacteristic
304C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1465C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#1465D47
hypocenterC47#11790
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20241416C2semidirect
172. C47#11790semidirect

Maximal subgroups

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

subgroupindex
#202477052
C4×#130939412
#202414162
C47×#14652
#1179047