Finite Group Databasebeta

← #27158136

Subgroups — #27158136

Order 12416.

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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#20261779C211yesmaximal, characteristic, complemented
2#20261777C211yesmaximal, characteristic, complemented
3#20261664C211yesmaximal, characteristic, complemented
4#20261650C211yesmaximal, characteristic, complemented
5#20261649C211yesmaximal, characteristic, complemented
6C16×#87604C211yesmaximal, characteristic, complemented
7#20261166C211yesmaximal, characteristic
8#15690898—21no—
9#15690897—21no—
10#15690883C2211yescharacteristic
11#15690881C2211yescharacteristic
12#15690881C2211yescharacteristic
13#15690875—21no—
14#15690874—21no—
15C8×#87604C2211yescharacteristic
16#15690833C2211yescharacteristic
17#15690829C2211yescharacteristic
18#15690756C411yescharacteristic, complemented
19#15690755C411yescharacteristic, complemented
20#15690751C411yescharacteristic, complemented
21C16×D97C2211yescharacteristic
22SD32×C97C411yescharacteristic, complemented
23#13095521—21no—
24#13095521—21no—
25#13095521—41no—
26#13095520—41no—
27#13095519—41no—
28#13095519D411yescharacteristic
29C2×#11715897—41no—
30C4×#87604D411yescharacteristic
31#13095514—21no—
32#13095512—21no—
33#13095510—21no—
34Dic388C2×C411yescharacteristic
35#13095499—21no—
36#13095498—21no—
37D776C2×C411yescharacteristic
38C8×D97C2311yescharacteristic
39#13095486C2×C411yescharacteristic
40Q16×C97C2×C411yescharacteristic
41D8×C97C2×C411yescharacteristic
42C16×C97C2×C411yescharacteristic
43C2×#87604—21no—
44Q8×C97—21no—
45D4×C97—21no—
46C22⋊D97—41no—
47C2×Dic97—41no—
48D388—21no—
49D388—41no—
50C4×D97—41no—
51C4×D97C2×D411yescharacteristic
52Dic194—21no—
53#11715897—41no—
54#11715897—41no—
55#11715897D811yescharacteristic
56#11715897D811yescharacteristic
57C8×C97C22×C411yescharacteristic
58#11715895#4111yescharacteristic
59C22×C97—41no—
60C2×D97—41no—
61C2×D97#13511yescharacteristic
62#87604—22no—
63C4×C97—41no—
64C4×C97C4×D411yescharacteristic
65Dic97—41no—
66Dic97C2×D811yescharacteristic
67C2×C97—81no—
68C2×C97C4×D811yescharacteristic
69D97—21no—
70D97—81no—
71#2178—971nomaximal, Sylow
72C97#217811yescharacteristic, complemented, Sylow
73#508—971no—
74#443—971no—
75#443—971no—
76#368—971no—
77#359—971no—
78#359—971no—
79C4×C16—971no—
80#135—971no—
81#135—971no—
82#131—1941no—
83#131—1941no—
84Q32—971no—
85SD32—971no—
86SD32#8760411yescharacteristic, complemented
87D16—971no—
88C2×C16—971no—
89C2×C16—971no—
90#108—971no—
91#108—971no—
92#104—1941no—
93#104—1941no—
94C4×C8—971no—
95#41—1941no—
96#41—1941no—
97Q16—971no—
98Q16C2×#8760411yescharacteristic
99SD16—1941no—
100SD16—1941no—
101D8—971no—
102D8C2×#8760411yescharacteristic
103M16—3881no—
104M16—3881no—
105M16—1941no—
106M16—1941no—
107C2×C8—971no—
108C2×C8—971no—
109C2×C8—3881no—
110C2×C8—3881no—
111C42—971no—
112C16—971no—
113C16—1941no—
114C16C2×#8760411yescharacteristic
115Q8—21no—
116Q8—1941no—
117D4—21no—
118D4—3881no—
119D4—3881no—
120D4—1941no—
121C2×C4—971no—
122C2×C4—3881no—
123C2×C4—3881no—
124C2×C4—1941no—
125C8—971no—
126C8—971no—
127C8—971no—
128C8—3881no—
129C8—3881no—
130C8C22×#8760411yescharacteristic
131C22—41no—
132C22—971no—
133C22—3881no—
134C4—41no—
135C4—971no—
136C4—3881no—
137C4—1942no—
138C4D4×#8760411yescharacteristic
139C2—81no—
140C2—7761no—
141C2—1941no—
142C2D8×#8760411yescharacteristic
subgroupG/H
centerC2D8×#87604
commutatorC8×C97C22×C4
FrattiniC8C22×#87604
FittingSD32×C97C4
socleC2×C97C4×D8
prime radical#20261166C2
cosocleC8×D97C23
hypercenterSD32#87604
hypocenterC97#2178
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32

Complements

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

NH ≅ G/Ntype
1. #20261779C2semidirect
2. #20261777C2semidirect
3. #20261664C2semidirect
4. #20261650C2semidirect
5. #20261649C2semidirect
6. C16×#87604C2semidirect
18. #15690756C4semidirect
19. #15690755C4semidirect
20. #15690751C4semidirect
22. SD32×C97C4semidirect
72. C97#2178semidirect
86. SD32#87604semidirect

Maximal subgroups

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

subgroupindex
#202617792
#202617772
#202616642
#202616502
#202616492
C16×#876042
#202611662
#217897