Finite Group Databasebeta

← #27157022

Subgroups — #27157022

Order 12416.

344 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 94 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#20261736C211yesmaximal, characteristic, complemented
2C4×#13095519C211yesmaximal, characteristic, complemented
3#20261583C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5C4×D4×D97C211yesmaximal, characteristic
6C2×#13095519—21no—
7C2×#13095519—21no—
8#15690877—21no—
9C4⋊#11715897C2212yescomplemented
10C4×#11715897—21no—
11C4×#11715897C2212yescomplemented
12C2×D4×D97C411yescharacteristic
13C22×C4×D97—21no—
14C4×D388C411yescharacteristic
15D4×Dic97C2211yescharacteristic
16#15690772C2211yescharacteristic
17C4×C22⋊D97—21no—
18#15690770—21no—
19C4⋊C4×D97C411yescharacteristic
20C22⋊C4×D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic
23#13095519—24no—
24#13095519—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C23×D97—21no—
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30D4×D97—22no—
31D4×D97C814yescomplemented
32C22×Dic97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—42no—
37C2×C4×D97—42no—
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41C4⋊Dic97C2×C411yescharacteristic
42#13095490—21no—
43C22⋊Dic97—21no—
44#13095488—21no—
45#13095488—21no—
46C4×Dic97D411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97C2311yescharacteristic
49C2×D4×C97C2×C411yescharacteristic
50C22×C4×C97—21no—
51C4⋊C4×C97C2×C411yescharacteristic
52C22⋊C4×C97—21no—
53C42×C97C2×C411yescharacteristic
54C23×C97—21no—
55C22×D97—21no—
56C22×D97—24no—
57C22×D97—42no—
58C22×D97—42no—
59C22×D97C22⋊C411yescharacteristic
60D4×C97—21no—
61D4×C97C2×C812yescomplemented
62C2×C4×C97—21no—
63C2×C4×C97—42no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C22⋊D97—24no—
68C2×Dic97—21no—
69C2×Dic97—41no—
70C2×Dic97—41no—
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73C2×Dic97C2×D411yescharacteristic
74D388—21no—
75D388C2×C812yescomplemented
76C4×D97—22no—
77C4×D97—42no—
78C4×D97—42no—
79C4×D97—41no—
80C4×D97—41no—
81C4×D97C2×C812yes—
82C4×D97M1612yes—
83C4×D97D812yes—
84C4×D97SD1612yes—
85#11715897—24no—
86#11715897—42no—
87C22×C97—22no—
88C22×C97—41no—
89C22×C97—41no—
90C22×C97C2×C22⋊C411yescharacteristic
91C2×D97—22no—
92C2×D97—41no—
93C2×D97—41no—
94C2×D97—48no—
95C2×D97C22⋊C812yes—
96C2×D97#10212yes—
97C2×D97#10412yes—
98C4×C97—21no—
99C4×C97—41no—
100C4×C97#10212yes—
101C4×C97C22×C811yescharacteristic
102C4×C97C2×M1611yescharacteristic
103Dic97—21no—
104Dic97—41no—
105Dic97C22⋊C812yes—
106Dic97C2×D811yescharacteristic
107Dic97C2×SD1611yescharacteristic
108C2×C97—42no—
109C2×C97C2×C22⋊C811yescharacteristic
110C2×C97C2×#10211yescharacteristic
111C2×C97C2×#10411yescharacteristic
112D97—42no—
113D97D4⋊C814yescomplemented
114C2×D4⋊C8—971nomaximal, Sylow
115C97C2×D4⋊C811yescharacteristic, complemented, Sylow
116C2×C4×D4—971no—
117C2×C4⋊C8—971no—
118C2×C4×C8—971no—
119D4⋊C8—972no—
120D4⋊C8—972no—
121C22×D4—971no—
122C23×C4—1941no—
123C22×C8—1941no—
124C22×C8—1941no—
125C4×D4—971no—
126C4×D4—971no—
127C4×D4—971no—
128C4×D4—1941no—
129C4×D4—1941no—
130C4×D4#8760411yescharacteristic
131C2×C4⋊C4—971no—
132C2×C22⋊C4—1941no—
133C2×C42—971no—
134C4⋊C8—972no—
135C4⋊C8—1941no—
136C4×C8—972no—
137C4×C8—1941no—
138C24—1941no—
139C2×D4—974no—
140C2×D4—971no—
141C2×D4—1942no—
142C2×D4—1941no—
143C2×D4C2×#8760411yescharacteristic
144C22×C4—21no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—3882no—
149C22×C4—3882no—
150C22×C4—1941no—
151C22×C4—1941no—
152C22×C4—1941no—
153C22×C4—1941no—
154C2×C8—3882no—
155C2×C8—1942no—
156C2×C8—1942no—
157C2×C8—1944no—
158C4⋊C4—971no—
159C4⋊C4—1941no—
160C4⋊C4C2×#8760411yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—1941no—
163C22⋊C4—1941no—
164C22⋊C4—1941no—
165C42—971no—
166C42—971no—
167C42—971no—
168C42C2×#8760411yescharacteristic
169C23—21no—
170C23—971no—
171C23—3882no—
172C23—3882no—
173C23—1944no—
174C23—1941no—
175D4—21no—
176D4—972no—
177D4—1944no—
178D4—1941no—
179D4#1309551912yescomplemented
180C2×C4—21no—
181C2×C4—42no—
182C2×C4—971no—
183C2×C4—971no—
184C2×C4—972no—
185C2×C4—972no—
186C2×C4—972no—
187C2×C4—972no—
188C2×C4—971no—
189C2×C4—3881no—
190C2×C4—3881no—
191C2×C4—3882no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—3882no—
195C2×C4—1942no—
196C2×C4—1941no—
197C2×C4C22⋊#8760411yescharacteristic
198C2×C4C22⋊#8760411yescharacteristic
199C2×C4C22×#8760411yescharacteristic
200C8—3882no—
201C8—1944no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—972no—
206C22—972no—
207C22—972no—
208C22—3888no—
209C22—3881no—
210C22—3881no—
211C22—1942no—
212C22C2×C22⋊#8760411yescharacteristic
213C4—21no—
214C4—41no—
215C4—972no—
216C4—971no—
217C4—971no—
218C4—3881no—
219C4—1941no—
220C4#1569087212yes—
221C4C2×#1309551911yescharacteristic
222C4D97⋊M1611yescharacteristic
223C2—42no—
224C2—974no—
225C2—3882no—
226C2#2026170311yescharacteristic
227C2C2×#1569087211yescharacteristic
228C2D97⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC4×C97C22×C8
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC4×Dic97C23
hypercenterC4×D4#87604
hypocenterC97C2×D4⋊C8
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261736C2semidirect
2. C4×#13095519C2semidirect
3. #20261583C2semidirect
4. D4⋊#11715897C2semidirect
9. C4⋊#11715897C22semidirect
11. C4×#11715897C22semidirect
31. D4×D97C8semidirect
61. D4×C97C2×C8semidirect
75. D388C2×C8semidirect
113. D97D4⋊C8semidirect
115. C97C2×D4⋊C8semidirect
179. D4#13095519semidirect

Maximal subgroups

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

subgroupindex
#202617362
C4×#130955192
#202615832
D4⋊#117158972
C4×D4×D972
C2×D4⋊C897