Finite Group Databasebeta

← #27143042

Subgroups — #27143042

Order 12416.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261535C211yesmaximal, characteristic
2#20260732C212yesmaximal, complemented
3#20260708C212yesmaximal, complemented
4#20260492C211yesmaximal, characteristic, complemented
5#424×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2212yes—
7#15690843—22no—
8#15690838—22no—
9#15690832—22no—
10#15690769C2212yes—
11#15690735—22no—
12#15690732—22no—
13#15690724—24no—
14C2×M16×C97C2211yescharacteristic
15C22×C8×C97C2211yescharacteristic
16#117×C97C2211yescharacteristic
17C4⋊C8×C97—22no—
18#13095511—22no—
19#13095511—22no—
20#13095511—22no—
21#13095511—22no—
22C2×Dic194D412yes—
23C2×D388D412yes—
24C2×C22⋊D97—22no—
25C2×C4×D97—22no—
26Dic388—22no—
27#13095500—24no—
28D776—22no—
29C4⋊Dic97—22no—
30C22⋊Dic97—22no—
31C4×Dic97—22no—
32C22×C4×C97C2311yescharacteristic
33M16×C97—22no—
34C2×C8×C97—21no—
35C2×C8×C97—22no—
36C2×C8×C97D412yes—
37C4⋊C4×C97—21no—
38C22⋊C4×C97D412yes—
39C42×C97—21no—
40C23×C97C2×D411yescharacteristic
41C22×D97—22no—
42C2×C4×C97—22no—
43C2×C4×C97C2×D412yes—
44C2×C4×C97C2×D411yescharacteristic
45C2×C4×C97#4111yescharacteristic
46C2×C4×C97#4112yes—
47C22⋊D97—42no—
48C22⋊D97—42no—
49C2×Dic97—22no—
50C2×Dic97—42no—
51D388—22no—
52D388—22no—
53C4×D97—42no—
54C4×D97—42no—
55Dic194—22no—
56Dic194—22no—
57C8×C97—22no—
58C8×C97—41no—
59C22×C97—41no—
60C22×C97#12111yescharacteristic
61C22×C97#12112yes—
62C2×D97—42no—
63C2×D97—82no—
64C4×C97—42no—
65C4×C97#12011yescharacteristic
66C4×C97#12312yes—
67C4×C97#12411yescharacteristic
68Dic97—42no—
69Dic97—82no—
70C2×C97—22no—
71C2×C97—21no—
72C2×C97#39411yescharacteristic
73D97—82no—
74#2053—971nomaximal, Sylow
75C97#205311yescharacteristic, complemented, Sylow
76#575—971no—
77#424D9711yescharacteristic, complemented
78#416—972no—
79#410—972no—
80H(3,ℤ/4)—971no—
81C2×#41—972no—
82#137—1942no—
83AGL(1,ℤ/8)—1942no—
84#135—1942no—
85C2×M16C2×D9711yescharacteristic
86C22×C8C2×D9711yescharacteristic
87#117—972no—
88#117C2×D9711yescharacteristic
89C4⋊C8—22no—
90#103—1942no—
91#102—1942no—
92#99—1944no—
93#41—1942no—
94#41—1942no—
95#41—1942no—
96#41—1942no—
97C2×Q8—972no—
98C2×D4—972no—
99C2×D4—1942no—
100C22×C4—1942no—
101C22×C4C22×D9711yescharacteristic
102Q16—1942no—
103SD16—1944no—
104D8—1942no—
105M16—22no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8C22⋊D9712yes—
109C4⋊C4—21no—
110C4⋊C4—1942no—
111C22⋊C4—1942no—
112C22⋊C4D38812yes—
113C42—21no—
114C42—1942no—
115C23—1942no—
116C23C2×D38811yescharacteristic
117Q8—1942no—
118Q8—1942no—
119D4—3882no—
120D4—3882no—
121D4—1942no—
122D4—1942no—
123C2×C4—22no—
124C2×C4—3882no—
125C2×C4—3882no—
126C2×C4—3882no—
127C2×C4—1942no—
128C2×C4D4×D9712yes—
129C2×C4C2×C22⋊D9711yescharacteristic
130C2×C4#1309551111yescharacteristic
131C2×C4#1309551212yes—
132C8—22no—
133C8—41no—
134C22—41no—
135C22—7762no—
136C22—3882no—
137C22#1569078411yescharacteristic
138C22#1569078512yes—
139C4—42no—
140C4—7762no—
141C4—3882no—
142C4#1569077811yescharacteristic
143C4#1569079512yes—
144C4#1569079911yescharacteristic
145C2—21no—
146C2—22no—
147C2—7762no—
148C2#2026065211yescharacteristic
subgroupG/H
centerC4#15690799
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#424×C97C2
socleC2×C97#394
prime radical#20261535C2
hypercenter#424D97
hypocenterC97#2053
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #20260732C2semidirect
3. #20260708C2semidirect
4. #20260492C2semidirect
5. #424×C97C2semidirect
75. C97#2053semidirect
77. #424D97semidirect

Maximal subgroups

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

subgroupindex
#202615352
#202607322
#202607082
#202604922
#424×C972
#205397