Finite Group Databasebeta

← #10593661

Subgroups — #10593661

Order 576.

216 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 33 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
1C2×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊#859C211yesmaximal, characteristic, complemented
4#5427—33nomaximal
5#5238—31nomaximal
6C3×#41⋊C3—21no—
7C2×C3×SL(2,𝔽3)C2211yescharacteristic
8C3⋊GL(2,𝔽3)—21no—
9#3752—41nomaximal
10C2×C3×#41D311yescharacteristic, complemented
11C2×#41⋊C3D313yescomplemented
12C2×GL(2,𝔽3)—33no—
13C2×#215—31no—
14C2×#213—31no—
15#930—31no—
16#859—33no—
17#835—31no—
18#832—31no—
19#830—31no—
20C22×C3⋊D3—41no—
21C2×C32×C4—41no—
22C2×C3⋊Dic3—41no—
23C3×SL(2,𝔽3)D411yescharacteristic, complemented
24#449—91noSylow
25C3×#41—21no—
26C3×#41—61no—
27C2×C3×Q8C2×D311yescharacteristic
28C2×C3×D4—31no—
29C22×C3×C4—31no—
30C2×#67—61no—
31C2×D12—31no—
32#41⋊C3—23no—
33C2×SL(2,𝔽3)C2×D313yes—
34GL(2,𝔽3)—63no—
35#215—61no—
36#215—61no—
37#213—62no—
38(C2×D3)⋊C4—61no—
39(C2×D3)⋊C4—123no—
40C4⋊Dic3—31no—
41C2×C3⋊C8—31no—
42C2×C3⋊C8—31no—
43C22×C32—41no—
44C2×C3⋊D3—42no—
45C2×C3⋊D3—81no—
46C2×C3⋊D3—81no—
47C32×C4—81no—
48C3⋊Dic3—81no—
49C2×#41C3⋊D311yescharacteristic, complemented
50C2×SD16—91no—
51C2×D8—91no—
52#121—91no—
53#103—91no—
54#102—91no—
55C22⋊C8—91no—
56C23×C3—31no—
57C22×D3—61no—
58C22×D3—123no—
59C3×Q8—31no—
60C3×Q8#6711yescharacteristic
61C3×D4—32no—
62C3×D4—61no—
63C2×C3×C4—31no—
64C2×C3×C4—43no—
65C2×C3×C4—61no—
66C2×C3×C4—61no—
67C2×C3×C4S411yescharacteristic
68#67—122no—
69C2×Dic3—61no—
70C2×Dic3—123no—
71D12—61no—
72D12—61no—
73SL(2,𝔽3)#6713yescomplemented
74C3⋊C8—61no—
75C3⋊C8—61no—
76C2×C32—41no—
77C2×C32—41no—
78C2×C32—41no—
79C3⋊D3—82no—
80#41—21no—
81#41—61no—
82C2×Q8C2×C3⋊D311yescharacteristic
83C2×D4—31no—
84C2×D4—91no—
85C2×D4—181no—
86C22×C4—31no—
87SD16—181no—
88SD16—181no—
89D8—182no—
90C2×C8—91no—
91C2×C8—91no—
92C4⋊C4—91no—
93C22⋊C4—181no—
94C22×C3—43no—
95C22×C3—61no—
96C22×C3—61no—
97C22×C3—61no—
98C22×C3C2×S411yescharacteristic
99C2×D3—126no—
100C2×D3—121no—
101C2×D3—121no—
102C2×D3—121no—
103C2×D3—243no—
104C2×D3—243no—
105C3×C4—21no—
106C3×C4—31no—
107C3×C4—31no—
108C3×C4—83no—
109Dic3—121no—
110Dic3—243no—
111C32—41noSylow
112C23—31no—
113C23—181no—
114Q8—31no—
115Q8#63811yescharacteristic
116D4—32no—
117D4—61no—
118D4—181no—
119D4—181no—
120D4—362no—
121C2×C4—31no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—181no—
125C2×C4C3⋊S411yescharacteristic
126C8—181no—
127C8—181no—
128C2×C3—43no—
129C2×C3—43no—
130C2×C3—43no—
131C2×C3—121no—
132C2×C3#98511yescharacteristic
133C2×C3#98611yescharacteristic
134C2×C3GL(2,ℤ/4)11yescharacteristic
135D3—246no—
136D3—241no—
137C22—61no—
138C22—61no—
139C22—61no—
140C22—361no—
141C22—361no—
142C22—361no—
143C22C2×C3⋊S411yescharacteristic
144C4—21no—
145C4—31no—
146C4—31no—
147C4—361no—
148C3—43no—
149C3#542711yescharacteristic, complemented
150C2—121no—
151C2—721no—
152C2#6426611yescharacteristic
153C2#6426711yescharacteristic
154C2#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22
FittingC2×C3×#41D3
socleC22×C3C2×S4
hypercenterC2×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C3C2semidirect
2. C2×C3⋊GL(2,𝔽3)C2semidirect
3. C3⋊#859C2semidirect
10. C2×C3×#41D3semidirect
11. C2×#41⋊C3D3semidirect
23. C3×SL(2,𝔽3)D4semidirect
49. C2×#41C3⋊D3semidirect
73. SL(2,𝔽3)#67semidirect
149. C3#5427semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C32
C2×C3⋊GL(2,𝔽3)2
C3⋊#8592
#54273
#52383
#37524