Finite Group Databasebeta

← #27140242

Subgroups — #27140242

Order 12416.

272 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 63 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#20261280C211yesmaximal, characteristic
2#20260789C211yesmaximal, characteristic, complemented
3#20260580C211yesmaximal, characteristic, complemented
4#20260470C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic, complemented
6#20260468C211yesmaximal, characteristic, complemented
7#327×C97C211yesmaximal, characteristic, complemented
8C2×#13095510—21no—
9#15690809C411yescharacteristic, complemented
10#15690806C411yescharacteristic, complemented
11#15690796—21no—
12#15690784C2211yescharacteristic
13#15690782—21no—
14D4×Dic97C411yescharacteristic, complemented
15#15690772C411yescharacteristic, complemented
16#15690770—21no—
17C2×C4×Dic97—21no—
18C2×C4×Dic97C2211yescharacteristic
19#15690739—21no—
20C22⋊#11715895C2211yescharacteristic
21#15690720—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic
24#121×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27#13095510—42no—
28#13095510—42no—
29C2×Dic194—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C22×Dic97—21no—
35C22×Dic97—21no—
36C22×Dic97D411yescharacteristic
37C22×Dic97D411yescharacteristic
38C2×C4×D97—21no—
39C2×#11715895—21no—
40C4⋊Dic97C2×C411yescharacteristic
41#13095490—21no—
42C22⋊Dic97—21no—
43#13095488—21no—
44#13095488—21no—
45C4×Dic97—24no—
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97D411yescharacteristic
49C4×Dic97D411yescharacteristic
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54C2×C8×C97—21no—
55C4⋊C4×C97C2×C411yescharacteristic
56C22⋊C4×C97—21no—
57C23×C97—21no—
58C23×C97C2×D411yescharacteristic
59C22×D97—21no—
60D4×C97—41no—
61D4×C97—41no—
62D4×C97—41no—
63C2×C4×C97—22no—
64C2×C4×C97—21no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C22⋊D97—42no—
71C22⋊D97—41no—
72C22⋊D97—41no—
73C22⋊D97—42no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—21no—
79C2×Dic97—42no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97C22⋊C411yescharacteristic
83C2×Dic97C22⋊C411yescharacteristic
84C2×Dic97C22⋊C411yescharacteristic
85C2×Dic97C22⋊C411yescharacteristic
86D388—41no—
87C4×D97—42no—
88Dic194—42no—
89C8×C97—41no—
90#11715895—41no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97—41no—
96C22×C97C2×C22⋊C411yescharacteristic
97C22×C97AGL(1,ℤ/8)11yescharacteristic
98C22×C97AGL(1,ℤ/8)11yescharacteristic
99C2×D97—41no—
100C2×D97—41no—
101C2×D97—41no—
102C4×C97—21no—
103C4×C97—21no—
104C4×C97—42no—
105C4×C97—41no—
106Dic97—21no—
107Dic97—21no—
108Dic97—21no—
109Dic97—42no—
110Dic97—41no—
111Dic97#10412yes—
112C2×C97—21no—
113C2×C97—21no—
114C2×C97—81no—
115C2×C97#41011yescharacteristic
116C2×C97#41811yescharacteristic
117C2×C97C2×#10411yescharacteristic
118D97—81no—
119#1505—971nomaximal, Sylow
120C97#150511yescharacteristic, complemented, Sylow
121#532—971no—
122#432—971no—
123#376—971no—
124#327—971no—
125#327—971no—
126#327—971no—
127#327D9711yescharacteristic, complemented
128C2×#41—1941no—
129C4⋊Q8—971no—
130#127—971no—
131#124—1941no—
132#121—971no—
133#121—1941no—
134#121C2×D9711yescharacteristic
135C4×D4—971no—
136C4×D4—971no—
137C4×D4—1941no—
138C2×C42—971no—
139C2×C42—1941no—
140C4⋊C8—1941no—
141C22⋊C8—971no—
142C22⋊C8C2×D9711yescharacteristic
143#97—1941no—
144#95—971no—
145#95—1941no—
146#95C2×D9711yescharacteristic
147#41—3882no—
148#41—3882no—
149C2×Q8—1941no—
150C2×D4—21no—
151C2×D4—971no—
152C2×D4—1941no—
153C2×D4—1941no—
154C2×D4—1941no—
155C2×D4C4×D9711yescharacteristic
156C22×C4—21no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—1941no—
160C22×C4—1941no—
161C22×C4—1941no—
162C22×C4C22×D9711yescharacteristic
163C2×C8—21no—
164C2×C8—1941no—
165C4⋊C4—971no—
166C4⋊C4—1941no—
167C4⋊C4C4×D9711yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—1941no—
170C22⋊C4—1941no—
171C22⋊C4—1941no—
172C42—971no—
173C42—971no—
174C42—971no—
175C42—971no—
176C42—1944no—
177C23—21no—
178C23—1941no—
179C23D4×D9711yescharacteristic
180Q8—3882no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—3881no—
185D4—3882no—
186D4—3881no—
187D4—3881no—
188D4—3882no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—971no—
195C2×C4—971no—
196C2×C4—971no—
197C2×C4—971no—
198C2×C4—3882no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—3882no—
202C2×C4—1942no—
203C2×C4—1942no—
204C2×C4—1941no—
205C2×C4—1941no—
206C2×C4—1941no—
207C2×C4C2×C4×D9711yescharacteristic
208C2×C4D4×D9711yescharacteristic
209C8—41no—
210C8—3881no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—3881no—
217C22—3881no—
218C22—3881no—
219C22C22⋊C4×D9711yescharacteristic
220C22#1569083411yescharacteristic
221C22#1569083711yescharacteristic
222C4—21no—
223C4—21no—
224C4—42no—
225C4—41no—
226C4—972no—
227C4—3882no—
228C4—3881no—
229C4—1941no—
230C4—1941no—
231C4—1941no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—7761no—
236C2#2026070711yescharacteristic
237C2#2026073811yescharacteristic
238C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#327×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261280C2
cosocleC22×C4×C97C23
hypercenter#327D97
hypocenterC97#1505
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20260789C2semidirect
3. #20260580C2semidirect
4. #20260470C2semidirect
5. #20260469C2semidirect
6. #20260468C2semidirect
7. #327×C97C2semidirect
9. #15690809C4semidirect
10. #15690806C4semidirect
14. D4×Dic97C4semidirect
15. #15690772C4semidirect
120. C97#1505semidirect
127. #327D97semidirect

Maximal subgroups

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

subgroupindex
#202612802
#202607892
#202605802
#202604702
#202604692
#202604682
#327×C972
#150597