Finite Group Databasebeta

← #27140260

Subgroups — #27140260

Order 12416.

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

normal subgroups only

This group has 242 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#20261271C211yesmaximal, characteristic
2#20260789C211yesmaximal, characteristic, complemented
3#20260586C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260470C211yesmaximal, characteristic
6#20260468C211yesmaximal, characteristic
7#328×C97C211yesmaximal, characteristic, complemented
8#15690805—21no—
9#15690801C411yescharacteristic, complemented
10#15690799C411yescharacteristic, complemented
11#15690794—21no—
12#15690784C2211yescharacteristic
13Q8×Dic97C411yescharacteristic, complemented
14#15690772C411yescharacteristic, complemented
15#15690770—21no—
16#15690768—21no—
17C2×#13095490—21no—
18C2×C4×Dic97—21no—
19C2×C4×Dic97C2211yescharacteristic
20#15690739—21no—
21C22⋊#11715895C2211yescharacteristic
22#15690720—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C2×D388C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32C2×C4×D97—21no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—24no—
38#13095490—21no—
39#13095488—21no—
40#13095488—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97—21no—
44C4×Dic97—42no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×Q8×C97C2×C411yescharacteristic
50C22×C4×C97—21no—
51C22×C4×C97C2311yescharacteristic
52C2×C8×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C22⋊C4×C97—21no—
56C23×C97C2×D411yescharacteristic
57C22×D97—21no—
58Q8×C97—41no—
59C2×C4×C97—22no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C22⋊D97—41no—
68C22⋊D97—41no—
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—21no—
74C2×Dic97—41no—
75C2×Dic97—41no—
76C2×Dic97C22⋊C411yescharacteristic
77C2×Dic97C22⋊C411yescharacteristic
78C2×Dic97C22⋊C411yescharacteristic
79C2×Dic97C22⋊C411yescharacteristic
80D388—41no—
81C4×D97—42no—
82C8×C97—41no—
83#11715895—41no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97C2×C22⋊C411yescharacteristic
87C22×C97AGL(1,ℤ/8)11yescharacteristic
88C22×C97#13711yescharacteristic
89C2×D97—41no—
90C2×D97—41no—
91C2×D97—41no—
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—42no—
95C4×C97—41no—
96C4×C97—41no—
97Dic97—21no—
98Dic97—21no—
99Dic97—21no—
100Dic97—42no—
101Dic97—41no—
102Dic97#10412yes—
103C2×C97—21no—
104C2×C97—21no—
105C2×C97#41011yescharacteristic
106C2×C97#41711yescharacteristic
107C2×C97C2×#10411yescharacteristic
108D97—81no—
109#1509—971nomaximal, Sylow
110C97#150911yescharacteristic, complemented, Sylow
111#529—971no—
112#432—971no—
113C4×C4⋊C4—971no—
114#328—971no—
115#328D9711yescharacteristic, complemented
116#327—971no—
117#327—971no—
118#126—1941no—
119#125—971no—
120#124—971no—
121#123—1941no—
122C22⋊Q8C2×D9711yescharacteristic
123#121—971no—
124C4×Q8—971no—
125C4×D4—971no—
126C4×D4—1941no—
127#117—1941no—
128C2×C4⋊C4—1941no—
129C2×C42—971no—
130C2×C42—1941no—
131C4⋊C8—1941no—
132C22⋊C8—971no—
133C22⋊C8C2×D9711yescharacteristic
134#97—1941no—
135#95—971no—
136#95C2×D9711yescharacteristic
137C2×Q8C4×D9711yescharacteristic
138C2×D4—971no—
139C2×D4—1941no—
140C22×C4—21no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4C22×D9711yescharacteristic
146C2×C8—21no—
147C2×C8—1941no—
148C4⋊C4—21no—
149C4⋊C4—971no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1944no—
154C4⋊C4C4×D9711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C22⋊C4—1941no—
159C22⋊C4—1941no—
160C42—971no—
161C42—971no—
162C42—971no—
163C42—971no—
164C42—3882no—
165C42—1941no—
166C23—1941no—
167C23D4×D9711yescharacteristic
168Q8—41no—
169D4—3881no—
170D4—3881no—
171D4—3881no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—3882no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4C2×C4×D9711yescharacteristic
191C2×C4D4×D9711yescharacteristic
192C8—41no—
193C8—3881no—
194C22—21no—
195C22—21no—
196C22—3881no—
197C22—3881no—
198C22—3881no—
199C22C22⋊C4×D9711yescharacteristic
200C22#1569083711yescharacteristic
201C22#1569084211yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—972no—
208C4—3882no—
209C4—3881no—
210C4—1941no—
211C4—1941no—
212C4—1941no—
213C2—21no—
214C2—21no—
215C2—7761no—
216C2#2026070711yescharacteristic
217C2#2026073511yescharacteristic
218C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690784C22
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97#1509
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20260789C2semidirect
3. #20260586C2semidirect
4. #20260472C2semidirect
7. #328×C97C2semidirect
9. #15690801C4semidirect
10. #15690799C4semidirect
13. Q8×Dic97C4semidirect
14. #15690772C4semidirect
110. C97#1509semidirect
115. #328D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
#202607892
#202605862
#202604722
#202604702
#202604682
#328×C972
#150997