Finite Group Databasebeta

← #27157118

Subgroups — #27157118

Order 12416.

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

normal subgroups only

This group has 240 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#20261753C211yesmaximal, characteristic
2C4×C4⋊#87604C211yesmaximal, characteristic
3#20261589C212yesmaximal
4#20261588C212yesmaximal
5#20261289C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C2×C4×#87604—21no—
8#15690877—21no—
9#15690867C2212yes—
10C4×#11715897—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690809C411yescharacteristic, complemented
13#15690808C411yescharacteristic, complemented
14#15690800—21no—
15#15690790C2211yescharacteristic
16#15690788—21no—
17#15690786C2211yescharacteristic
18C4×Dic194—21no—
19C4×Dic194C411yescharacteristic, complemented
20#15690767—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4⋊#87604—24no—
26C4×#87604—42no—
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97C2311yescharacteristic
32C4⋊Dic97—21no—
33C4⋊Dic97C2×C411yescharacteristic
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97C2×C411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C2×Q8×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C42×C97—21no—
49C42×C97C2×C411yescharacteristic
50C22×D97C2×D411yescharacteristic
51C2×#87604—24no—
52C2×#87604—42no—
53C2×#87604—42no—
54Q8×C97—41no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C4×D97—22no—
66C4×D97—41no—
67C4×D97SD1612yes—
68C4×D97Q1612yes—
69Dic194—42no—
70Dic194—41no—
71#11715897—42no—
72C22×C97C2×C22⋊C411yescharacteristic
73C2×D97—21no—
74C2×D97—21no—
75C2×D97C2×SD1611yescharacteristic
76C2×D97C2×Q1611yescharacteristic
77#87604—44no—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—41no—
81C4×C97—41no—
82C4×C97#10312yes—
83Dic97—21no—
84Dic97—41no—
85Dic97—41no—
86Dic97#10312yes—
87Dic97#10412yes—
88C2×C97#41011yescharacteristic
89C2×C97C2×#10311yescharacteristic
90C2×C97C2×#10411yescharacteristic
91D97—21no—
92D97—21no—
93#1503—971nomaximal, Sylow
94C97#150311yescharacteristic, complemented, Sylow
95#533—971no—
96#431—971no—
97C4×C4⋊C4—971no—
98#328—972no—
99#328—972no—
100C4⋊Q8—971no—
101C4⋊Q8—971no—
102#125—1941no—
103C22⋊Q8—971no—
104C22⋊Q8—971no—
105C22⋊Q8—1941no—
106C4×Q8—971no—
107C4×Q8—1941no—
108C4×Q8#8760411yescharacteristic, complemented
109#117—1941no—
110C2×C4⋊C4—1941no—
111C2×C42—971no—
112C2×C42—1941no—
113C4⋊C8—1941no—
114C22⋊C8—972no—
115C4×C8—1941no—
116#95—972no—
117C2×Q8—971no—
118C2×Q8—1941no—
119C2×Q8C2×#8760411yescharacteristic
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—1942no—
124C2×C8—1942no—
125C4⋊C4—21no—
126C4⋊C4—971no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4—1944no—
133C4⋊C4—1941no—
134C4⋊C4C2×#8760411yescharacteristic
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C42—21no—
138C42—971no—
139C42—971no—
140C42—971no—
141C42—3882no—
142C42—1941no—
143C42C2×#8760411yescharacteristic
144C23—971no—
145Q8—41no—
146Q8—3881no—
147Q8—3882no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—972no—
153C2×C4—972no—
154C2×C4—971no—
155C2×C4—3881no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4—1942no—
159C2×C4—1944no—
160C2×C4—1941no—
161C2×C4—1941no—
162C2×C4C22⋊#8760411yescharacteristic
163C2×C4C22⋊#8760411yescharacteristic
164C2×C4C22×#8760411yescharacteristic
165C8—3882no—
166C22—971no—
167C22—971no—
168C22—1941no—
169C22—1941no—
170C22C2×C22⋊#8760411yescharacteristic
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—972no—
176C4—972no—
177C4—3884no—
178C4—3881no—
179C4—3881no—
180C4—1941no—
181C4#1569087312yes—
182C2—1941no—
183C2—1941no—
184C2#2026171411yescharacteristic
185C2C2×#1569087311yescharacteristic
186C2D97⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4×Q8#87604
hypocenterC97#1503
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
12. #15690809C4semidirect
13. #15690808C4semidirect
19. C4×Dic194C4semidirect
22. C4×Q8×C97C4semidirect
94. C97#1503semidirect
108. C4×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617532
C4×C4⋊#876042
#202615892
#202615882
#202612892
#150397