Finite Group Databasebeta

← #27143375

Subgroups — #27143375

Order 12416.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261242C211yesmaximal, characteristic
2#20260719C211yesmaximal, characteristic, complemented
3#20260524C212yesmaximal, complemented
4#20260521C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C411yescharacteristic, complemented
7C2×#13095495—21no—
8#15690791C411yescharacteristic, complemented
9#15690791C2211yescharacteristic
10#15690786—21no—
11#15690781—21no—
12#15690778C411yescharacteristic, complemented
13#15690778C2211yescharacteristic
14C4×C22⋊D97—21no—
15C4×C22⋊D97—21no—
16#15690769C411yescharacteristic, complemented
17#15690729—21no—
18#15690727C2212yes—
19C2×#41×C97C2211yescharacteristic
20#117×C97—21no—
21#99×C97—21no—
22#99×C97C2212yes—
23C23×D97C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25C2×C22⋊D97—21no—
26C22×Dic97C2×C411yescharacteristic
27C2×C4×D97—21no—
28C2×C4×D97—21no—
29C2×C4×D97—41no—
30C2×C4×D97—41no—
31#13095495—22no—
32#13095495—41no—
33C2×#11715895—41no—
34C4⋊Dic97—21no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97C2×C411yescharacteristic
40C22⋊Dic97C2×C411yescharacteristic
41#13095488—21no—
42#13095488—21no—
43C4×Dic97—21no—
44#41×C97—41no—
45#41×C97—41no—
46C2×Q8×C97D411yescharacteristic
47C2×D4×C97C2×C411yescharacteristic
48C2×D4×C97D411yescharacteristic
49C2×D4×C97C2311yescharacteristic
50C22×C4×C97C2×C411yescharacteristic
51C22×C4×C97D411yescharacteristic
52C22×C4×C97D411yescharacteristic
53C4⋊C4×C97—41no—
54C22⋊C4×C97—22no—
55C42×C97—41no—
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C23×C97C2×D411yescharacteristic
59C22×D97—21no—
60C22×D97—42no—
61C22×D97—41no—
62Q8×C97—41no—
63D4×C97—41no—
64D4×C97—41no—
65D4×C97—41no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—42no—
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C22⋊C411yescharacteristic
75C2×C4×C97C22⋊C411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C22⋊D97—42no—
78C22⋊D97—42no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81C2×Dic97—21no—
82C2×Dic97—41no—
83C4×D97—42no—
84C4×D97—41no—
85C4×D97—41no—
86#11715895—42no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97C2×C22⋊C411yescharacteristic
94C2×D97—22no—
95C2×D97—41no—
96C2×D97—42no—
97C2×D97—41no—
98C2×D97—42no—
99C4×C97—21no—
100C4×C97—21no—
101C4×C97—21no—
102C4×C97—82no—
103C4×C97#9911yescharacteristic
104C4×C97#9911yescharacteristic
105Dic97—41no—
106Dic97—41no—
107Dic97—41no—
108C2×C97—21no—
109C2×C97—41no—
110C2×C97—41no—
111C2×C97—41no—
112C2×C97C2×#9911yescharacteristic
113D97—42no—
114#2120—971nomaximal, Sylow
115C97#212011yescharacteristic, complemented, Sylow
116#525—971no—
117#413—971no—
118#410D9711yescharacteristic, complemented
119#352—972no—
120AΣL(1,𝔽16)—972no—
121C2×#41C2×D9711yescharacteristic
122C23×C4—971no—
123C2×M16—1941no—
124#123—971no—
125#123—971no—
126C22⋊Q8—1941no—
127#121—1941no—
128#120—971no—
129#120—971no—
130C4×D4—1941no—
131C4×D4—1941no—
132#117—21no—
133#117—971no—
134#101—1941no—
135#100—972no—
136#99—21no—
137#99C2×D9712yes—
138C24—971no—
139#41—41no—
140#41—41no—
141C2×Q8D38811yescharacteristic
142C2×D4—1941no—
143C2×D4—1941no—
144C2×D4C4×D9711yescharacteristic
145C2×D4C22⋊D9711yescharacteristic
146C2×D4C22×D9711yescharacteristic
147C22×C4—971no—
148C22×C4—3881no—
149C22×C4—3881no—
150C22×C4—1941no—
151C22×C4—1941no—
152C22×C4C4×D9711yescharacteristic
153C22×C4D38811yescharacteristic
154C22×C4C22⋊D9711yescharacteristic
155M16—3881no—
156M16—1942no—
157C2×C8—3881no—
158C4⋊C4—41no—
159C4⋊C4—1941no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C22⋊C4—22no—
163C22⋊C4—971no—
164C22⋊C4—971no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C22⋊C4—1941no—
169C42—41no—
170C42—1941no—
171C23—3881no—
172C23—3882no—
173C23—1941no—
174C23#1309548811yescharacteristic
175C23C2×C4×D9711yescharacteristic
176C23C2×C22⋊D9711yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—3882no—
182D4—3882no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—3882no—
191C2×C4—3881no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—1941no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4#1309548811yescharacteristic
198C2×C4#1309548811yescharacteristic
199C2×C4#1309548811yescharacteristic
200C2×C4C2×D38811yescharacteristic
201C8—3882no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3881no—
209C22—3882no—
210C22—3882no—
211C22—3881no—
212C22—1942no—
213C22C2×#1309548811yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—3881no—
219C4—3881no—
220C4—3881no—
221C4#1569072411yescharacteristic
222C4#1569072411yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—3882no—
228C2C2×#1569072411yescharacteristic
subgroupG/H
centerC4#15690724
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#410×C97C2
socleC2×C97C2×#99
prime radical#20261242C2
cosocleC2×D4×C97C23
hypercenter#410D97
hypocenterC97#2120
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20260719C2semidirect
3. #20260524C2semidirect
4. #20260521C2semidirect
5. #410×C97C2semidirect
6. C22×C4×D97C4semidirect
8. #15690791C4semidirect
12. #15690778C4semidirect
16. #15690769C4semidirect
115. C97#2120semidirect
118. #410D97semidirect

Maximal subgroups

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

subgroupindex
#202612422
#202607192
#202605242
#202605212
#410×C972
#212097