Finite Group Databasebeta

← #27157879

Subgroups — #27157879

Order 12416.

292 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 76 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#20261738C211yesmaximal, characteristic
2C2×#15690871C214yesmaximal
3C4×#13095520C211yesmaximal, characteristic
4C2×#15690809C211yesmaximal, characteristic
5C2×#13095520C2212yes—
6C2×#13095520C2212yes—
7#15690877—21no—
8C4⋊#11715897—21no—
9#15690871—28no—
10#15690865—21no—
11C4×#11715897—21no—
12C22×Dic194C2212yes—
13#15690809—21no—
14#15690809—21no—
15#15690809—21no—
16#15690809—21no—
17C2×C4⋊Dic97C411yescharacteristic
18C2×#13095490C412yes—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4⋊C4×C97C411yescharacteristic
21#13095520—22no—
22#13095520—24no—
23#13095520D414yes—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194—22no—
27C2×Dic194—22no—
28C2×Dic194—24no—
29C2×Dic194—24no—
30C22×Dic97C2×C412yes—
31C22×Dic97C2×C411yescharacteristic
32C22×Dic97C2311yescharacteristic
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35#13095490—22no—
36#13095490—22no—
37C4×Dic97D411yescharacteristic
38C4×Dic97D411yescharacteristic
39C4×Dic97D411yescharacteristic
40C4×Dic97D411yescharacteristic
41C22×C4×C97C2×C412yes—
42C22×C4×C97C2×C411yescharacteristic
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C23×C97C22×C411yescharacteristic
46C2×C4×C97—22no—
47C2×C4×C97—22no—
48C2×C4×C97—24no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×Dic97—22no—
52C2×Dic97—22no—
53C2×Dic97—24no—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C2×D411yescharacteristic
57C2×Dic97C2×D412yes—
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97#4112yes—
60Dic194—44no—
61Dic194—42no—
62Dic194—42no—
63#11715897—24no—
64#11715897—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97C4×D412yes—
69C4×C97—21no—
70C4×C97—42no—
71C4×C97#10112yescomplemented
72Dic97—21no—
73Dic97—42no—
74Dic97#10112yescomplemented
75Dic97#12112yes—
76Dic97#12411yescharacteristic
77Dic97#12711yescharacteristic
78C2×C97—22no—
79C2×C97#39011yescharacteristic
80C2×C97C2×#10111yescharacteristic
81C2×C97C2×#10111yescharacteristic
82#1966—971nomaximal, Sylow
83C97#196611yescharacteristic, complemented, Sylow
84C2×C4⋊Q8—971no—
85#423—971no—
86C2×#101—974no—
87C4×M16—971no—
88C22×Q8—972no—
89C2×M16—972no—
90C2×M16—972no—
91C4⋊Q8—1941no—
92C4⋊Q8—1941no—
93C4⋊Q8—1941no—
94C4⋊Q8—1941no—
95C2×C4⋊C4—972no—
96C2×C4⋊C4—971no—
97C2×C4⋊C4#8760411yescharacteristic
98C2×C42—971no—
99C4⋊C8—1941no—
100C4⋊C8—1941no—
101#101—1948no—
102#97—1941no—
103C4×C8—1941no—
104C2×Q8—1944no—
105C2×Q8—1942no—
106C2×Q8—1942no—
107C2×Q8—1944no—
108C22×C4—971no—
109C22×C4—971no—
110C22×C4—972no—
111C22×C4C2×#8760412yes—
112C22×C4C2×#8760411yescharacteristic
113M16—974no—
114M16—1944no—
115M16—1942no—
116C2×C8—1942no—
117C2×C8—1942no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—1942no—
121C4⋊C4—1942no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C42—971no—
125C42—971no—
126C42—971no—
127C42—971no—
128C23C22×#8760411yescharacteristic
129Q8—3882no—
130Q8—3884no—
131Q8—3882no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—971no—
136C2×C4—971no—
137C2×C4—971no—
138C2×C4—972no—
139C2×C4—972no—
140C2×C4—971no—
141C2×C4—1942no—
142C2×C4—1942no—
143C2×C4—1944no—
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C22⋊#8760411yescharacteristic
146C8—3882no—
147C8—1944no—
148C22—21no—
149C22—21no—
150C22C2×C22⋊#8760411yescharacteristic
151C22D4×#8760412yes—
152C4—21no—
153C4—42no—
154C4—971no—
155C4—972no—
156C4—971no—
157C4—972no—
158C4—3882no—
159C4—1941no—
160C4#1569087112yescomplemented
161C2—22no—
162C2#2026168711yescharacteristic
163C2C2×#1569087111yescharacteristic
164C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutator, prime radicalC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1966
LCSG ▷ C23×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
71. C4×C97#101semidirect
74. Dic97#101semidirect
83. C97#1966semidirect
160. C4#15690871semidirect

Maximal subgroups

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

subgroupindex
#202617382
C2×#156908712
C4×#130955202
C2×#156908092
#196697