Finite Group Databasebeta

← #27157216

Subgroups — #27157216

Order 12416.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261757C212yesmaximal, complemented
2#20261591C212yesmaximal, complemented
3#20261590C212yesmaximal, complemented
4#124×D97C211yesmaximal, characteristic
5C4⋊#11715897—22no—
6#15690867—22no—
7#15690865C2214yescomplemented
8C2×Q8×D97C411yescharacteristic
9C2×D4×D97C411yescharacteristic
10#15690799C2211yescharacteristic
11#15690798C411yescharacteristic
12#15690797C2211yescharacteristic
13#15690796—22no—
14C22⋊C4×D97—22no—
15C42×D97C2211yescharacteristic
16#124×C97C411yescharacteristic
17C2×#11715897—24no—
18C23×D97—21no—
19C2×Dic194—21no—
20C2×Dic194C2×C411yescharacteristic
21Q8×D97—42no—
22C2×D388C2×C411yescharacteristic
23C2×C22⋊D97—21no—
24D4×D97—42no—
25C2×C4×D97—21no—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D412yes—
28C22⋊Dic97—22no—
29#13095488—22no—
30#13095488—22no—
31C4×Dic97D412yes—
32C4×Dic97C2311yescharacteristic
33C2×Q8×C97C2×C411yescharacteristic
34C2×D4×C97C2×C411yescharacteristic
35C22⋊C4×C97—22no—
36C42×C97C2×C411yescharacteristic
37C23×C97—21no—
38C22×D97—21no—
39C22×D97—44no—
40C22×D97—42no—
41C22×D97C22⋊C411yescharacteristic
42Q8×C97—41no—
43D4×C97—41no—
44C2×C4×C97—21no—
45C2×C4×C97C22⋊C412yes—
46C2×C4×C97C22×C411yescharacteristic
47C22⋊D97—42no—
48C2×Dic97—21no—
49C2×Dic97C22⋊C411yescharacteristic
50C2×Dic97C2×D412yes—
51D388—41no—
52C4×D97—22no—
53C4×D97—42no—
54C4×D97—42no—
55Dic194—42no—
56Dic194—41no—
57#11715897—44no—
58C22×C97—41no—
59C22×C97—42no—
60C22×C97C2×C22⋊C411yescharacteristic
61C2×D97—22no—
62C2×D97—41no—
63C2×D97—42no—
64C2×D97—84no—
65C2×D97#10012yes—
66C4×C97—21no—
67C4×C97—22no—
68C4×C97—41no—
69Dic97—21no—
70Dic97—22no—
71Dic97—41no—
72C2×C97—81no—
73C2×C97C2×#10011yescharacteristic
74C2×C97#42112yes—
75D97—22no—
76D97—81no—
77#1533—971nomaximal, Sylow
78C97#153311yescharacteristic, complemented, Sylow
79C2×#124—971no—
80#432—972no—
81#329—972no—
82#329—972no—
83C22×Q8—971no—
84C22×D4—971no—
85#124—971no—
86#124—971no—
87#124—971no—
88#124—1942no—
89#124#8760411yescharacteristic
90C2×C22⋊C4—1942no—
91C2×C42—971no—
92C4⋊C8—1942no—
93C22⋊C8—1942no—
94#97—974no—
95C24—1941no—
96C2×Q8—971no—
97C2×Q8—3882no—
98C2×Q8—1941no—
99C2×Q8C2×#8760411yescharacteristic
100C2×D4—971no—
101C2×D4—3882no—
102C2×D4—1941no—
103C2×D4C2×#8760411yescharacteristic
104C22×C4—972no—
105C22×C4—971no—
106C22×C4—1941no—
107C2×C8—1944no—
108C22⋊C4—22no—
109C22⋊C4—1942no—
110C22⋊C4—1942no—
111C22⋊C4—1942no—
112C42—972no—
113C42—971no—
114C42C2×#8760411yescharacteristic
115C23—21no—
116C23—971no—
117C23—3882no—
118C23—3884no—
119C23—1941no—
120Q8—41no—
121Q8—3881no—
122Q8—3882no—
123D4—41no—
124D4—3881no—
125D4—3882no—
126C2×C4—21no—
127C2×C4—972no—
128C2×C4—971no—
129C2×C4—3882no—
130C2×C4—3882no—
131C2×C4—1942no—
132C2×C4—1941no—
133C2×C4C22⋊#8760412yes—
134C2×C4C22×#8760411yescharacteristic
135C8—3884no—
136C22—41no—
137C22—42no—
138C22—7764no—
139C22—972no—
140C22—3881no—
141C22—3882no—
142C22—1942no—
143C22C2×C22⋊#8760411yescharacteristic
144C4—21no—
145C4—22no—
146C4—41no—
147C4—3881no—
148C4—1941no—
149C4—1942no—
150C2—81no—
151C2—7761no—
152C2—1942no—
153C2D97⋊#10011yescharacteristic
154C2#2026173212yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97#1533
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. #20261757C2semidirect
2. #20261591C2semidirect
3. #20261590C2semidirect
7. #15690865C22semidirect
78. C97#1533semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202615912
#202615902
#124×D972
#153397