Finite Group Databasebeta

← #27157245

Subgroups — #27157245

Order 12416.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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#20261752C211yesmaximal, characteristic
2#20261738C211yesmaximal, characteristic
3#20261594C214yesmaximal, complemented
4C2×#15690809C211yesmaximal, characteristic
5C2×#13095520—21no—
6#15690877C2212yes—
7#15690877C2212yes—
8C4⋊#11715897—21no—
9C22⋊#11715897—21no—
10C4×#11715897—21no—
11C22×Dic194—21no—
12#15690809—21no—
13#15690809—21no—
14#15690809C412yes—
15#15690809C2212yes—
16C2×C4⋊Dic97C411yescharacteristic
17C2×#13095490—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4⋊C4×C97C411yescharacteristic
20#13095520—42no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×Dic194—22no—
24C2×Dic194—22no—
25C2×Dic194—42no—
26C2×Dic194—42no—
27C22×Dic97—21no—
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C412yes—
32#13095490—22no—
33#13095490—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97C2311yescharacteristic
38C22×C4×C97—21no—
39C22×C4×C97C2×C411yescharacteristic
40C4⋊C4×C97—21no—
41C4⋊C4×C97C2×C412yes—
42C23×C97C22⋊C411yescharacteristic
43C2×C4×C97—22no—
44C2×C4×C97—22no—
45C2×C4×C97—42no—
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—22no—
49C2×Dic97—22no—
50C2×Dic97—42no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97D812yes—
54C2×Dic97SD1612yes—
55C2×Dic97C2×D411yescharacteristic
56C2×Dic97C2×D411yescharacteristic
57Dic194—44no—
58Dic194—42no—
59Dic194—42no—
60#11715897—42no—
61#11715897—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97#10212yes—
65C22×C97C2×C22⋊C411yescharacteristic
66C4×C97—21no—
67C4×C97—42no—
68C4×C97#10112yes—
69Dic97—21no—
70Dic97—42no—
71Dic97#10212yes—
72Dic97C2×D811yescharacteristic
73Dic97C2×SD1611yescharacteristic
74Dic97#13712yes—
75C2×C97—22no—
76C2×C97C2×#10111yescharacteristic
77C2×C97C2×#10211yescharacteristic
78C2×C97#41911yescharacteristic
79#1546—971nomaximal, Sylow
80C97#154611yescharacteristic, complemented, Sylow
81C2×C4⋊Q8—971no—
82#431—971no—
83#423—971no—
84#332—974no—
85C22×Q8—1941no—
86C2×M16—1941no—
87C4⋊Q8—972no—
88C4⋊Q8—972no—
89C4⋊Q8—1941no—
90C4⋊Q8—1941no—
91C2×C4⋊C4—971no—
92C2×C4⋊C4—1941no—
93C2×C4⋊C4#8760411yescharacteristic
94C2×C42—971no—
95C4⋊C8—972no—
96C4⋊C8—972no—
97C4⋊C8—1941no—
98C22⋊C8—1941no—
99C4×C8—1941no—
100C2×Q8—3882no—
101C2×Q8—3882no—
102C2×Q8—1942no—
103C2×Q8—1942no—
104C22×C4—21no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4—1941no—
108C22×C4C2×#8760411yescharacteristic
109M16—3882no—
110C2×C8—1942no—
111C2×C8—1942no—
112C4⋊C4—21no—
113C4⋊C4—972no—
114C4⋊C4—1942no—
115C4⋊C4—1942no—
116C4⋊C4—1941no—
117C4⋊C4C2×#8760412yes—
118C42—971no—
119C42—971no—
120C42—971no—
121C42—971no—
122C23C22⋊#8760411yescharacteristic
123Q8—3882no—
124Q8—3884no—
125Q8—3882no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—42no—
129C2×C4—972no—
130C2×C4—972no—
131C2×C4—971no—
132C2×C4—971no—
133C2×C4—971no—
134C2×C4—971no—
135C2×C4—3882no—
136C2×C4—1942no—
137C2×C4—1942no—
138C2×C4C22⋊#8760411yescharacteristic
139C2×C4C22×#8760411yescharacteristic
140C8—3882no—
141C8—3882no—
142C22—21no—
143C22—21no—
144C22#1569087212yes—
145C22C2×C22⋊#8760411yescharacteristic
146C4—21no—
147C4—42no—
148C4—971no—
149C4—972no—
150C4—971no—
151C4—972no—
152C4—3882no—
153C4—1941no—
154C4#1569087112yes—
155C2—22no—
156C2C2×#1569087111yescharacteristic
157C2C2×#1569087211yescharacteristic
158C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C22⋊C4
cosocleC4×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1546
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261594C2semidirect
80. C97#1546semidirect

Maximal subgroups

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

subgroupindex
#202617522
#202617382
#202615942
C2×#156908092
#154697