Finite Group Databasebeta

← #27140172

Subgroups — #27140172

Order 12416.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#13095510C211yesmaximal, characteristic
2#20260757C211yesmaximal, characteristic, complemented
3C4×#13095494C211yesmaximal, characteristic, complemented
4#20260467C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095510C411yescharacteristic
9C2×#13095494—21no—
10C2×#13095494—21no—
11C4×Dic194C2211yescharacteristic
12#15690775C411yescharacteristic
13D4×Dic97C411yescharacteristic
14C4×C22⋊D97—21no—
15#15690770—21no—
16#15690768C411yescharacteristic
17#15690766—21no—
18C2×C4×Dic97—21no—
19C42×D97C2211yescharacteristic
20C4⋊#11715895C2211yescharacteristic
21#15690739—21no—
22#15690720—21no—
23C4×#11715895C2211yescharacteristic
24C4×D4×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27#13095510—22no—
28#13095510—22no—
29C2×Dic194C2×C411yescharacteristic
30C2×C22⋊D97—21no—
31C22×Dic97—21no—
32C22×Dic97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36#13095494—24no—
37#13095494—42no—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C4⋊Dic97C2×C411yescharacteristic
41#13095490—21no—
42#13095490—21no—
43C22⋊Dic97—21no—
44#13095488—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C4×Dic97C2×C411yescharacteristic
49C4×Dic97D411yescharacteristic
50C4×Dic97D411yescharacteristic
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C2×C8×C97—21no—
54C2×C8×C97—21no—
55C4⋊C4×C97C2×C411yescharacteristic
56C22⋊C4×C97—21no—
57C42×C97C2311yescharacteristic
58C23×C97—21no—
59C22×D97C22⋊C411yescharacteristic
60D4×C97—21no—
61D4×C97M1612yescomplemented
62C2×C4×C97—21no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C22⋊D97—24no—
69C2×Dic97—24no—
70C2×Dic97—21no—
71C2×Dic97—21no—
72C2×Dic97—42no—
73C2×Dic97C22⋊C411yescharacteristic
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C22⋊C411yescharacteristic
76C4×D97—21no—
77C4×D97—21no—
78C4×D97—21no—
79C4×D97—21no—
80C4×D97—22no—
81Dic194—21no—
82Dic194M1612yescomplemented
83C8×C97—22no—
84C8×C97—41no—
85#11715895—22no—
86#11715895—41no—
87C22×C97—22no—
88C22×C97—41no—
89C22×C97—41no—
90C22×C97C2×C22⋊C411yescharacteristic
91C2×D97—21no—
92C2×D97—21no—
93C2×D97—21no—
94C4×C97—21no—
95C4×C97—41no—
96C4×C97C2×M1611yescharacteristic
97C4×C97C2×M1611yescharacteristic
98C4×C97AGL(1,ℤ/8)11yescharacteristic
99C4×C97#13711yescharacteristic
100Dic97—22no—
101Dic97—21no—
102Dic97—21no—
103Dic97—41no—
104Dic97#10412yes—
105C2×C97—42no—
106C2×C97#40711yescharacteristic
107C2×C97#41711yescharacteristic
108C2×C97C2×#10411yescharacteristic
109D97—41no—
110#1489—971nomaximal, Sylow
111C97#148911yescharacteristic, complemented, Sylow
112C4×#41—971no—
113#423—971no—
114C4×M16—971no—
115Q8⋊C8—971no—
116Q8⋊C8—971no—
117D4⋊C8—971no—
118D4⋊C8D9711yescharacteristic, complemented
119C2×#41—971no—
120C2×M16—1941no—
121C2×M16—1941no—
122C4×Q8—971no—
123C4×Q8—971no—
124C4×D4—971no—
125C4×D4—1941no—
126C4×D4—1941no—
127C4×D4C2×D9711yescharacteristic
128#117—971no—
129#117—1941no—
130C2×C42—971no—
131C2×C42—1941no—
132C4⋊C8—971no—
133C4⋊C8—1941no—
134C4⋊C8C2×D9711yescharacteristic
135#97—1941no—
136C4×C8—971no—
137C4×C8C2×D9711yescharacteristic
138#41—1942no—
139#41—1942no—
140C2×Q8—971no—
141C2×D4—1941no—
142C2×D4C4×D9711yescharacteristic
143C22×C4—21no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—1941no—
148C22×C4—1941no—
149M16—3882no—
150M16—1944no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—1941no—
154C2×C8—1941no—
155C4⋊C4—971no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C4⋊C4C4×D9711yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C42—971no—
163C42—971no—
164C42—971no—
165C42—1941no—
166C42—1941no—
167C42—1941no—
168C42C22×D9711yescharacteristic
169C23—21no—
170C23—971no—
171Q8—972no—
172Q8—1941no—
173D4—21no—
174D4—1944no—
175D4#1309549412yescomplemented
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—3882no—
183C2×C4—1942no—
184C2×C4—1941no—
185C2×C4—1944no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4—1941no—
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4D4×D9711yescharacteristic
194C8—22no—
195C8—41no—
196C8—3881no—
197C8—1942no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—1941no—
202C22—1941no—
203C22—1941no—
204C22C22⋊C4×D9711yescharacteristic
205C4—21no—
206C4—41no—
207C4—972no—
208C4—3881no—
209C4—1941no—
210C4—1942no—
211C4—1941no—
212C4M16×D9711yescharacteristic
213C4C2×#1309549411yescharacteristic
214C4#1569083411yescharacteristic
215C4#1569083911yescharacteristic
216C2—42no—
217C2—3881no—
218C2#2026069711yescharacteristic
219C2#2026073311yescharacteristic
220C2#104×D9711yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095510C4
cosocleC42×C97C23
hypercenterD4⋊C8D97
hypocenterC97#1489
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260757C2semidirect
3. C4×#13095494C2semidirect
4. #20260467C2semidirect
5. #20260465C2semidirect
6. D4⋊#11715895C2semidirect
7. D4⋊C8×C97C2semidirect
61. D4×C97M16semidirect
82. Dic194M16semidirect
111. C97#1489semidirect
118. D4⋊C8D97semidirect
175. D4#13095494semidirect

Maximal subgroups

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

subgroupindex
C4×#130955102
#202607572
C4×#130954942
#202604672
#202604652
D4⋊#117158952
D4⋊C8×C972
#148997