Finite Group Databasebeta

← #27157274

Subgroups — #27157274

Order 12416.

254 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261740C212yesmaximal, complemented
2#20261595C212yesmaximal, complemented
3#20261593C212yesmaximal, complemented
4#20261284C211yesmaximal, characteristic
5C2×#13095521—22no—
6#15690877—22no—
7#15690877C2214yes—
8C2×#13095512C411yescharacteristic
9C2×#13095510C411yescharacteristic
10#15690808C2211yescharacteristic
11#15690806C2211yescharacteristic
12#15690798C411yescharacteristic
13#15690783—22no—
14#15690770—22no—
15C42×D97C2211yescharacteristic
16#124×C97C411yescharacteristic
17#13095521—44no—
18C2×#11715897—24no—
19#13095512—42no—
20#13095510—42no—
21C2×Dic194C2×C411yescharacteristic
22C2×D388—21no—
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25C2×C22⋊D97—22no—
26C22×Dic97—21no—
27C2×C4×D97—21no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D412yes—
30#13095490—22no—
31#13095488—22no—
32C4×Dic97D412yes—
33C4×Dic97C2311yescharacteristic
34C2×Q8×C97C2×C411yescharacteristic
35C2×D4×C97C2×C411yescharacteristic
36C22⋊C4×C97—22no—
37C42×C97C2×C411yescharacteristic
38C23×C97—21no—
39C22×D97—21no—
40C22×D97C22⋊C411yescharacteristic
41Q8×C97—41no—
42D4×C97—41no—
43C2×C4×C97—21no—
44C2×C4×C97C22⋊C412yes—
45C2×C4×C97C22×C411yescharacteristic
46C22⋊D97—42no—
47C22⋊D97—42no—
48C22⋊D97—42no—
49C2×Dic97—21no—
50C2×Dic97—42no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C2×D412yes—
53D388—41no—
54D388—42no—
55C4×D97—22no—
56C4×D97—22no—
57C4×D97—22no—
58C4×D97—42no—
59Dic194—41no—
60#11715897—44no—
61C22×C97—41no—
62C22×C97—42no—
63C22×C97C2×C22⋊C411yescharacteristic
64C2×D97—21no—
65C2×D97—22no—
66C2×D97—41no—
67C2×D97—42no—
68C4×C97—21no—
69C4×C97—22no—
70C4×C97—41no—
71Dic97—41no—
72Dic97#10012yes—
73Dic97AGL(1,ℤ/8)12yes—
74Dic97#13712yes—
75C2×C97—81no—
76C2×C97C2×#10011yescharacteristic
77C2×C97#41712yes—
78D97—41no—
79D97—81no—
80#1560—971nomaximal, Sylow
81C97#156011yescharacteristic, complemented, Sylow
82#532—971no—
83#423—972no—
84#333—972no—
85#331—972no—
86C2×#41—971no—
87C2×#41—971no—
88C2×M16—1942no—
89C4⋊Q8—971no—
90#127—971no—
91#124—971no—
92#124#8760411yescharacteristic
93#121—1942no—
94C4×D4—1942no—
95C2×C42—971no—
96C4⋊C8—974no—
97C4⋊C8—1942no—
98#41—3882no—
99#41—3882no—
100C2×Q8—971no—
101C2×Q8C2×#8760411yescharacteristic
102C2×D4—971no—
103C2×D4—1941no—
104C2×D4—1942no—
105C2×D4—1941no—
106C2×D4C2×#8760411yescharacteristic
107C22×C4—972no—
108C22×C4—971no—
109C22×C4—1941no—
110C22×C4—1941no—
111M16—3884no—
112C2×C8—1944no—
113C4⋊C4—1942no—
114C22⋊C4—22no—
115C22⋊C4—1942no—
116C42—972no—
117C42—971no—
118C42C2×#8760411yescharacteristic
119C23—21no—
120C23—971no—
121C23—1941no—
122Q8—41no—
123Q8—3881no—
124D4—41no—
125D4—3881no—
126D4—3882no—
127D4—3882no—
128D4—3882no—
129D4—3882no—
130C2×C4—21no—
131C2×C4—972no—
132C2×C4—971no—
133C2×C4—3882no—
134C2×C4—3882no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4—1942no—
138C2×C4—1941no—
139C2×C4C22⋊#8760412yes—
140C2×C4C22×#8760411yescharacteristic
141C8—3884no—
142C22—41no—
143C22—42no—
144C22—3881no—
145C22—3882no—
146C22—1941no—
147C22—1942no—
148C22C2×C22⋊#8760411yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—972no—
153C4—972no—
154C4—972no—
155C4—3881no—
156C2—81no—
157C2—7761no—
158C2—3881no—
159C2D97⋊#10011yescharacteristic
160C2#2026172612yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261284C2
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97#1560
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261740C2semidirect
2. #20261595C2semidirect
3. #20261593C2semidirect
81. C97#1560semidirect

Maximal subgroups

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

subgroupindex
#202617402
#202615952
#202615932
#202612842
#156097