Finite Group Databasebeta

← #27157643

Subgroups — #27157643

Order 12416.

306 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261605C214yesmaximal, complemented
4C2×#15690768C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C22⋊#87604—21no—
7C2×C4×#87604C2212yes—
8#15690867—21no—
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4—21no—
11C22×C4×D97C2211yescharacteristic
12#15690768—21no—
13#15690768—21no—
14#15690768C412yescomplemented
15#15690768C2212yes—
16C2×C4⋊Dic97C411yescharacteristic, complemented
17C2×#13095488—21no—
18C2×C4×Dic97—21no—
19C2×C4⋊C4×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095521—24no—
22C2×#11715897—22no—
23C22⋊#87604—24no—
24C4×#87604—24no—
25C23×D97D411yescharacteristic
26C22×Dic97—21no—
27C22×Dic97D411yescharacteristic
28C2×C4×D97—24no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×C4×D97C2311yescharacteristic
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C412yes—
35#13095488—22no—
36#13095488—22no—
37C4×Dic97—22no—
38C4×Dic97—22no—
39C22×C4×C97—21no—
40C22×C4×C97C2×C411yescharacteristic
41C4⋊C4×C97—21no—
42C4⋊C4×C97C2×C412yes—
43C23×C97C22⋊C411yescharacteristic
44C22×D97—22no—
45C22×D97—22no—
46C22×D97—41no—
47C22×D97C22⋊C411yescharacteristic
48C22×D97C2×D411yescharacteristic
49C2×#87604—24no—
50C2×#87604—42no—
51C2×#87604—42no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—42no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×Dic97—22no—
58C2×Dic97—22no—
59C2×Dic97—42no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C2×D411yescharacteristic
62C4×D97—22no—
63C4×D97—22no—
64C4×D97—22no—
65C4×D97#4114yes—
66#11715897—42no—
67C22×C97—21no—
68C22×C97—21no—
69C22×C97#10412yes—
70C22×C97C2×C22⋊C411yescharacteristic
71C2×D97—21no—
72C2×D97—21no—
73C2×D97—21no—
74C2×D97—21no—
75C2×D97—21no—
76C2×D97—44no—
77C2×D97#12312yes—
78#87604—44no—
79C4×C97—21no—
80C4×C97—42no—
81C4×C97#11712yes—
82Dic97—21no—
83Dic97—42no—
84Dic97#12312yes—
85C2×C97—22no—
86C2×C97#38111yescharacteristic
87C2×C97C2×#10411yescharacteristic
88C2×C97#42111yescharacteristic
89D97—22no—
90D97—41no—
91#1826—971nomaximal, Sylow
92C97#182611yescharacteristic, complemented, Sylow
93C2×#117—971no—
94#407—971no—
95C4×C22⋊C4—971no—
96#339—974no—
97C23×C4—971no—
98C2×M16—972no—
99#117—972no—
100#117—972no—
101#117—1941no—
102#117—1941no—
103C2×C4⋊C4—971no—
104C2×C4⋊C4#8760411yescharacteristic, complemented
105C2×C22⋊C4—1941no—
106C2×C22⋊C4—1941no—
107C2×C42—972no—
108C2×C42—1941no—
109C22⋊C8—1941no—
110C22⋊C8—1941no—
111#95—1941no—
112C24—971no—
113C22×C4—21no—
114C22×C4—971no—
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—1944no—
120C22×C4—1942no—
121C22×C4—1941no—
122C22×C4C2×#8760411yescharacteristic
123M16—1944no—
124C2×C8—1942no—
125C4⋊C4—21no—
126C4⋊C4—972no—
127C4⋊C4—1941no—
128C4⋊C4C2×#8760412yes—
129C22⋊C4—1942no—
130C22⋊C4—1942no—
131C22⋊C4—1944no—
132C42—1942no—
133C42—1942no—
134C42—1944no—
135C23—971no—
136C23—971no—
137C23—3881no—
138C23—1942no—
139C23—1942no—
140C23C22⋊#8760411yescharacteristic
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—971no—
145C2×C4—974no—
146C2×C4—971no—
147C2×C4—3882no—
148C2×C4—3882no—
149C2×C4—3882no—
150C2×C4—1942no—
151C2×C4—1942no—
152C2×C4—1942no—
153C2×C4—1944no—
154C2×C4—1942no—
155C2×C4—1942no—
156C2×C4C22⋊#8760411yescharacteristic
157C2×C4C22×#8760411yescharacteristic
158C8—3882no—
159C22—21no—
160C22—21no—
161C22—972no—
162C22—3884no—
163C22—1941no—
164C22—1941no—
165C22—1941no—
166C22—1941no—
167C22—1941no—
168C22#1569087512yes—
169C22C2×C22⋊#8760411yescharacteristic
170C4—21no—
171C4—42no—
172C4—972no—
173C4—3884no—
174C4—3882no—
175C4—1941no—
176C4#1569088912yes—
177C2—22no—
178C2—3881no—
179C2—1942no—
180C2#2026167811yescharacteristic
181C2C2×#1569087511yescharacteristic
182C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1826
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261605C2semidirect
14. #15690768C4semidirect
16. C2×C4⋊Dic97C4semidirect
19. C2×C4⋊C4×C97C4semidirect
92. C97#1826semidirect
104. C2×C4⋊C4#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
C2×#156907682
#182697