Finite Group Databasebeta

← #27158031

Subgroups — #27158031

Order 12416.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#15690871C212yesmaximal, complemented
2#20261645C212yesmaximal, complemented
3#20261643C212yesmaximal, complemented
4#20261306C211yesmaximal, characteristic
5C2×#13095520—22no—
6#15690871—22no—
7#15690871C2214yes—
8C22×Dic194C2211yescharacteristic
9#15690808C411yescharacteristic
10#15690808C2211yescharacteristic
11#15690797C411yescharacteristic
12#15690797C2211yescharacteristic
13#15690791—22no—
14#15690789—22no—
15#15690769C411yescharacteristic
16C2×#41×C97C411yescharacteristic
17#13095520—24no—
18#13095520—42no—
19C2×#11715897—42no—
20C2×Dic194—42no—
21C2×Dic194C2×C411yescharacteristic
22C2×Dic194D412yes—
23C2×Dic194C2311yescharacteristic
24C22×Dic97D412yes—
25C4⋊Dic97—21no—
26#13095490—22no—
27#13095490—22no—
28C22⋊Dic97—21no—
29C22⋊Dic97—22no—
30C22⋊Dic97—22no—
31C4×Dic97C2×C411yescharacteristic
32C4×Dic97C2×C411yescharacteristic
33#41×C97—42no—
34C2×Q8×C97C2×C411yescharacteristic
35C2×D4×C97—21no—
36C2×D4×C97C2×C411yescharacteristic
37C22×C4×C97—21no—
38C22×C4×C97C2×C411yescharacteristic
39C23×C97—21no—
40C23×C97C22⋊C411yescharacteristic
41Q8×C97—41no—
42D4×C97—41no—
43D4×C97—42no—
44C2×C4×C97—22no—
45C2×C4×C97—21no—
46C2×C4×C97—42no—
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22×C411yescharacteristic
49C2×Dic97—22no—
50C2×Dic97—22no—
51C2×Dic97—21no—
52C2×Dic97—21no—
53C2×Dic97C22⋊C412yes—
54C2×Dic97C2×D412yes—
55Dic194—41no—
56Dic194—42no—
57Dic194—41no—
58#11715897—44no—
59C22×C97—41no—
60C22×C97—42no—
61C22×C97—41no—
62C22×C97#9912yes—
63C22×C97C2×C22⋊C411yescharacteristic
64C4×C97—21no—
65C4×C97—21no—
66C4×C97—41no—
67Dic97—22no—
68Dic97—22no—
69Dic97—41no—
70Dic97—41no—
71C2×C97—21no—
72C2×C97—22no—
73C2×C97—81no—
74C2×C97C2×#9911yescharacteristic
75#2133—971nomaximal, Sylow
76C97#213311yescharacteristic, complemented, Sylow
77#536—971no—
78C2×#101—972no—
79#356—972no—
80#355—972no—
81C2×#41#8760411yescharacteristic
82C22×Q8—971no—
83C2×M16—1942no—
84C4⋊Q8—971no—
85C4⋊Q8—971no—
86#124—971no—
87#124—971no—
88#123—1942no—
89C22⋊Q8—1942no—
90#117—971no—
91#101—974no—
92#101—1942no—
93#41—42no—
94C2×Q8—972no—
95C2×Q8—971no—
96C2×Q8—971no—
97C2×Q8—3882no—
98C2×Q8C2×#8760411yescharacteristic
99C2×D4—21no—
100C2×D4C2×#8760411yescharacteristic
101C22×C4—21no—
102C22×C4—972no—
103C22×C4C2×#8760411yescharacteristic
104M16—3882no—
105M16—1944no—
106C2×C8—3882no—
107C4⋊C4—1942no—
108C4⋊C4—1942no—
109C4⋊C4—1941no—
110C22⋊C4—1942no—
111C22⋊C4—1942no—
112C22⋊C4—1941no—
113C42—971no—
114C42—971no—
115C23—21no—
116C23C22⋊#8760411yescharacteristic
117Q8—41no—
118Q8—3882no—
119Q8—3881no—
120Q8—3881no—
121D4—41no—
122D4—42no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—42no—
126C2×C4—972no—
127C2×C4—972no—
128C2×C4—1941no—
129C2×C4—1941no—
130C2×C4—1942no—
131C2×C4—1942no—
132C2×C4C22⋊#8760411yescharacteristic
133C2×C4C22×#8760411yescharacteristic
134C8—3884no—
135C22—41no—
136C22—42no—
137C22—41no—
138C22#1569086812yes—
139C22C2×C22⋊#8760411yescharacteristic
140C4—21no—
141C4—21no—
142C4—41no—
143C4—3881no—
144C4—3881no—
145C4—1942no—
146C4—1942no—
147C2—21no—
148C2—22no—
149C2—81no—
150C2C2×#1569086811yescharacteristic
subgroupG/H
centerC2C2×#15690868
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting, prime radicalC2×#41×C97C4
socleC2×C97C2×#99
cosocleC2×Dic194C23
hypercenterC2×#41#87604
hypocenterC97#2133
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15690871C2semidirect
2. #20261645C2semidirect
3. #20261643C2semidirect
76. C97#2133semidirect

Maximal subgroups

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

subgroupindex
C2×#156908712
#202616452
#202616432
#202613062
#213397