Finite Group Databasebeta

← #27157644

Subgroups — #27157644

Order 12416.

306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261706C211yesmaximal, characteristic
2C4×C22⋊#87604C211yesmaximal, characteristic
3#20261605C212yesmaximal, complemented
4#20261605C212yesmaximal, complemented
5#117×D97C211yesmaximal, characteristic
6C2×#13095521C2212yes—
7C2×C22⋊#87604—21no—
8C2×C4×#87604C2212yes—
9#15690867—21no—
10C22⋊#11715897—21no—
11(C2×#87604)⋊C4—21no—
12C22×C4×D97C2211yescharacteristic
13#15690769C411yescharacteristic, complemented
14#15690768C2211yescharacteristic
15#15690768C2211yescharacteristic
16#15690767—21no—
17#15690766—21no—
18C4⋊C4×D97C411yescharacteristic, complemented
19C4⋊C4×D97C411yescharacteristic, complemented
20C22⋊C4×D97—21no—
21C42×D97—21no—
22#117×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095521—24no—
25C2×#11715897—22no—
26C22⋊#87604—24no—
27C4×#87604—24no—
28C23×D97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C2×C4×D97—24no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97C2311yescharacteristic
37C4⋊Dic97C2×C411yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40#13095490—21no—
41C22⋊Dic97—21no—
42#13095488—21no—
43#13095488—21no—
44C4×Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C22×C4×C97C2×C411yescharacteristic
48C4⋊C4×C97C2×C411yescharacteristic
49C4⋊C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C42×C97—21no—
52C23×C97C22⋊C411yescharacteristic
53C22×D97—22no—
54C22×D97—22no—
55C22×D97—41no—
56C22×D97C22⋊C411yescharacteristic
57C22×D97C2×D411yescharacteristic
58C2×#87604—24no—
59C2×#87604—42no—
60C2×#87604—42no—
61C2×C4×C97—22no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22×C411yescharacteristic
66C2×Dic97—22no—
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C2×D411yescharacteristic
71C4×D97—22no—
72C4×D97—22no—
73C4×D97—22no—
74C4×D97—42no—
75C4×D97—42no—
76C4×D97#4114yes—
77#11715897—42no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97C2×C22⋊C411yescharacteristic
82C2×D97—21no—
83C2×D97—21no—
84C2×D97—21no—
85C2×D97—21no—
86C2×D97—44no—
87C2×D97#10412yes—
88C2×D97#12312yes—
89#87604—44no—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97—41no—
93C4×C97#11712yes—
94Dic97—21no—
95Dic97—41no—
96Dic97—41no—
97Dic97#12312yes—
98C2×C97—41no—
99C2×C97#38111yescharacteristic
100C2×C97C2×#10411yescharacteristic
101C2×C97#42111yescharacteristic
102D97—22no—
103D97—22no—
104#1826—971nomaximal, Sylow
105C97#182611yescharacteristic, complemented, Sylow
106C2×#117—971no—
107#407—971no—
108C4×C22⋊C4—971no—
109#339—972no—
110#339—972no—
111C23×C4—971no—
112C2×M16—972no—
113#117—971no—
114#117—971no—
115#117—971no—
116#117—1941no—
117#117—1941no—
118#117#8760411yescharacteristic, complemented
119C2×C4⋊C4—971no—
120C2×C4⋊C4—971no—
121C2×C22⋊C4—1941no—
122C2×C22⋊C4—1941no—
123C2×C42—972no—
124C2×C42—1941no—
125C22⋊C8—1941no—
126C22⋊C8—1941no—
127#95—1941no—
128C24—971no—
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—971no—
134C22×C4—1944no—
135C22×C4—1941no—
136C22×C4—1941no—
137C22×C4—1942no—
138C22×C4C2×#8760411yescharacteristic
139M16—1944no—
140C2×C8—1942no—
141C4⋊C4—971no—
142C4⋊C4—971no—
143C4⋊C4—1941no—
144C4⋊C4—1941no—
145C4⋊C4C2×#8760411yescharacteristic
146C4⋊C4C2×#8760411yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—1941no—
149C22⋊C4—1941no—
150C22⋊C4—1944no—
151C22⋊C4—1941no—
152C42—21no—
153C42—1941no—
154C42—1941no—
155C42—1944no—
156C42—1941no—
157C23—971no—
158C23—971no—
159C23—3881no—
160C23—1942no—
161C23—1942no—
162C23C22⋊#8760411yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—971no—
167C2×C4—974no—
168C2×C4—971no—
169C2×C4—3882no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—3882no—
173C2×C4—1942no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1944no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4—1942no—
180C2×C4C22⋊#8760411yescharacteristic
181C2×C4C22×#8760411yescharacteristic
182C8—3882no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—972no—
187C22—972no—
188C22—3884no—
189C22—1941no—
190C22—1941no—
191C22—1941no—
192C22—1941no—
193C22C2×C22⋊#8760411yescharacteristic
194C4—21no—
195C4—41no—
196C4—41no—
197C4—972no—
198C4—3884no—
199C4—3881no—
200C4—3881no—
201C4—1941no—
202C4#1569088912yes—
203C2—41no—
204C2—1942no—
205C2—1942no—
206C2#2026167811yescharacteristic
207C2D97⋊#10411yescharacteristic
208C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenter#117#87604
hypocenterC97#1826
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20261605C2semidirect
4. #20261605C2semidirect
13. #15690769C4semidirect
18. C4⋊C4×D97C4semidirect
19. C4⋊C4×D97C4semidirect
22. #117×C97C4semidirect
105. C97#1826semidirect
118. #117#87604semidirect

Maximal subgroups

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

subgroupindex
#202617062
C4×C22⋊#876042
#202616052
#117×D972
#182697