Finite Group Databasebeta

← #27158144

Subgroups — #27158144

Order 12416.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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#20261781C211yesmaximal, characteristic
2#20261775C211yesmaximal, characteristic, complemented
3#20261660C211yesmaximal, characteristic, complemented
4#20261654C211yesmaximal, characteristic, complemented
5#20261652C211yesmaximal, characteristic
6#20261626C211yesmaximal, characteristic, complemented
7#20261177C211yesmaximal, characteristic
8#15690898—21no—
9Q8×#87604—21no—
10#15690885C2211yescharacteristic
11#15690880C2211yescharacteristic
12#15690878C2211yescharacteristic
13#15690875—21no—
14#15690873—21no—
15#15690864C2211yescharacteristic
16#15690833C2211yescharacteristic
17Q16×D97C2211yescharacteristic
18#15690756C411yescharacteristic, complemented
19#15690755C411yescharacteristic, complemented
20#15690753C411yescharacteristic, complemented
21#15690749C2211yescharacteristic
22Q32×C97C411yescharacteristic, complemented
23#13095521—21no—
24#13095521—41no—
25#13095519—41no—
26#13095519D411yescharacteristic
27C4⋊#87604—21no—
28C4⋊#87604—41no—
29C4×#87604—41no—
30C4×#87604D411yescharacteristic
31#13095514—21no—
32#13095512—21no—
33Q8×D97—21no—
34Dic388C2×C411yescharacteristic
35#13095501—21no—
36#13095498—21no—
37D776C2×C411yescharacteristic
38C8×D97C2311yescharacteristic
39#13095486C2×C411yescharacteristic
40Q16×C97C2×C411yescharacteristic
41Q16×C97C2×C411yescharacteristic
42C16×C97C2×C411yescharacteristic
43C2×#87604—21no—
44C2×#87604—41no—
45Q8×C97—21no—
46Q8×C97—21no—
47D388—21no—
48D388—41no—
49C4×D97—41no—
50C4×D97—41no—
51C4×D97C2×D411yescharacteristic
52Dic194—21no—
53Dic194—41no—
54#11715897—21no—
55#11715897—41no—
56C8×C97C22×C411yescharacteristic
57#11715895#4111yescharacteristic
58C2×D97—41no—
59C2×D97AGL(1,ℤ/8)11yescharacteristic
60#87604—41no—
61#87604—81no—
62C4×C97—41no—
63C4×C97—41no—
64C4×C97C4×D411yescharacteristic
65Dic97—41no—
66Dic97AGL(1,ℤ/8)11yescharacteristic
67C2×C97#44211yescharacteristic
68D97—21no—
69D97—81no—
70#2180—971nomaximal, Sylow
71C97#218011yescharacteristic, complemented, Sylow
72#510—971no—
73#444—971no—
74#441—971no—
75#365—971no—
76#362—971no—
77#360—971no—
78#347—971no—
79#135—971no—
80C2×Q16—971no—
81#131—1941no—
82C4×Q8—1941no—
83Q32—971no—
84Q32#8760411yescharacteristic, complemented
85SD32—971no—
86SD32—971no—
87M32—971no—
88M32—971no—
89#108—971no—
90#106—971no—
91#104—1941no—
92#103—1941no—
93#97—971no—
94#41—1941no—
95C2×Q8—1941no—
96Q16—971no—
97Q16—1941no—
98Q16C2×#8760411yescharacteristic
99Q16C2×#8760411yescharacteristic
100SD16—1941no—
101D8—971no—
102M16—3881no—
103M16—1941no—
104C2×C8—971no—
105C2×C8—971no—
106C2×C8—3881no—
107C4⋊C4—3881no—
108C4⋊C4—1941no—
109C42—971no—
110C42—3881no—
111C16—971no—
112C16—1941no—
113C16C2×#8760411yescharacteristic
114Q8—21no—
115Q8—21no—
116Q8—3881no—
117Q8—1941no—
118D4—3881no—
119D4—1941no—
120C2×C4—971no—
121C2×C4—3881no—
122C2×C4—3881no—
123C2×C4—3881no—
124C2×C4—1941no—
125C8—971no—
126C8—3881no—
127C8—1941no—
128C8C22×#8760411yescharacteristic
129C22—971no—
130C22—3881no—
131C4—41no—
132C4—41no—
133C4—7761no—
134C4—971no—
135C4—3881no—
136C4—3881no—
137C4D4×#8760411yescharacteristic
138C2—7761no—
139C2—1941no—
140C2#2026177611yescharacteristic
subgroupG/H
centerC2#20261776
commutatorC8×C97C22×C4
FrattiniC8C22×#87604
FittingQ32×C97C4
socleC2×C97#442
prime radical#15690833C22
cosocleC8×D97C23
hypercenterQ32#87604
hypocenterC97#2180
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32

Complements

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

NH ≅ G/Ntype
2. #20261775C2semidirect
3. #20261660C2semidirect
4. #20261654C2semidirect
6. #20261626C2semidirect
18. #15690756C4semidirect
19. #15690755C4semidirect
20. #15690753C4semidirect
22. Q32×C97C4semidirect
71. C97#2180semidirect
84. Q32#87604semidirect

Maximal subgroups

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

subgroupindex
#202617812
#202617752
#202616602
#202616542
#202616522
#202616262
#202611772
#218097