Finite Group Databasebeta

← #27158039

Subgroups — #27158039

Order 12416.

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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#20261722C212yesmaximal
2#20261646C212yesmaximal, complemented
3#20261645C212yesmaximal, complemented
4#20261305C211yesmaximal, characteristic
5C2×#13095520—22no—
6#15690871C2214yes—
7#15690870—22no—
8C2×#13095510C2211yescharacteristic
9C2×Q8×D97C411yescharacteristic
10#15690810C2211yescharacteristic
11#15690808C2211yescharacteristic
12#15690797C411yescharacteristic
13#15690792—22no—
14#15690787—22no—
15#15690767C411yescharacteristic
16#124×C97C411yescharacteristic
17#13095520—24no—
18#13095520—42no—
19C2×#11715897—42no—
20#13095510—42no—
21C2×Dic194—21no—
22C2×Dic194C2×C411yescharacteristic
23C2×Dic194C2311yescharacteristic
24Q8×D97—42no—
25C2×C22⋊D97D412yes—
26C22×Dic97D412yes—
27C2×C4×D97—21no—
28C2×C4×D97C2×C411yescharacteristic
29C4⋊Dic97—22no—
30#13095490—21no—
31#13095490—22no—
32C22⋊Dic97—22no—
33#13095488—21no—
34C4×Dic97C2×C411yescharacteristic
35C2×Q8×C97C2×C411yescharacteristic
36C2×D4×C97C2×C411yescharacteristic
37C22⋊C4×C97—22no—
38C42×C97C2×C411yescharacteristic
39C23×C97C22⋊C412yes—
40C22×D97C22⋊C411yescharacteristic
41Q8×C97—41no—
42D4×C97—41no—
43C2×C4×C97—21no—
44C2×C4×C97—21no—
45C2×C4×C97C22×C411yescharacteristic
46C22⋊D97—42no—
47C2×Dic97—22no—
48C2×Dic97—22no—
49C2×Dic97—21no—
50C2×Dic97—21no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C2×D412yes—
53C4×D97—22no—
54C4×D97—42no—
55Dic194—41no—
56Dic194—42no—
57Dic194—41no—
58#11715897—44no—
59C22×C97—22no—
60C22×C97—42no—
61C22×C97C2×C22⋊C411yescharacteristic
62C2×D97—21no—
63C2×D97—41no—
64C4×C97—21no—
65C4×C97—41no—
66C4×C97—41no—
67Dic97—22no—
68Dic97—41no—
69Dic97—41no—
70Dic97#9912yes—
71C2×C97—21no—
72C2×C97—42no—
73C2×C97C2×#9911yescharacteristic
74D97—41no—
75#2135—971nomaximal, Sylow
76C97#213511yescharacteristic, complemented, Sylow
77#536—971no—
78#413—972no—
79#356—972no—
80#356—972no—
81C2×#41—971no—
82C22×Q8—971no—
83C2×M16—1942no—
84C4⋊Q8—971no—
85C4⋊Q8—971no—
86#124—971no—
87#124#8760411yescharacteristic
88#123—1942no—
89C22⋊Q8—1942no—
90#117—971no—
91#101—974no—
92#100—1942no—
93#41—3882no—
94C2×Q8—971no—
95C2×Q8—971no—
96C2×Q8—3882no—
97C2×Q8—1941no—
98C2×Q8C2×#8760411yescharacteristic
99C2×D4—972no—
100C2×D4C2×#8760411yescharacteristic
101C22×C4—972no—
102C22×C4—971no—
103C22×C4—1941no—
104M16—3882no—
105M16—1944no—
106C2×C8—3882no—
107C4⋊C4—1941no—
108C4⋊C4—1942no—
109C4⋊C4—1942no—
110C22⋊C4—22no—
111C22⋊C4—1941no—
112C22⋊C4—1942no—
113C42—971no—
114C42C2×#8760411yescharacteristic
115C23—971no—
116C23C22⋊#8760412yes—
117Q8—41no—
118Q8—3881no—
119Q8—3881no—
120Q8—3882no—
121D4—41no—
122D4—3882no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—971no—
126C2×C4—972no—
127C2×C4—3882no—
128C2×C4—1942no—
129C2×C4—1942no—
130C2×C4—1942no—
131C2×C4—1941no—
132C2×C4—1941no—
133C2×C4C22×#8760411yescharacteristic
134C8—3884no—
135C22—22no—
136C22—42no—
137C22—3881no—
138C22—1941no—
139C22C2×C22⋊#8760411yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—972no—
144C4—3881no—
145C4—3881no—
146C4—1942no—
147C2—21no—
148C2—42no—
149C2—3881no—
150C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC2×C97C2×#99
prime radicalC2×#13095510C22
cosocleC2×Dic194C23
hypercenter#124#87604
hypocenterC97#2135
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20261646C2semidirect
3. #20261645C2semidirect
76. C97#2135semidirect

Maximal subgroups

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

subgroupindex
#202617222
#202616462
#202616452
#202613052
#213597