Finite Group Databasebeta

← #10589412⋊C3 #13135639

Subgroups — #10589412⋊C3 #13135639

Order 1728.

292 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 86 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic
2C4⋊((C3⋊(C3⋊C8))⋊C3)C212yesmaximal
3#10589412C311yesmaximal, characteristic, complemented
4C3×#4546—31nomaximal
5C2×Q8×H(3,𝔽3)C412yes—
6C4⋊C4×H(3,𝔽3)—22no—
7C42×H(3,𝔽3)C2211yescharacteristic
8C2×(C3⋊(C3⋊C8))⋊C3—22no—
9C32×C4⋊Q8—21no—
10C32×C4⋊Q8D311yescharacteristic
11C32×C4⋊Q8C2×C311yescharacteristic
12C4⋊(C3⋊(C3⋊C8))C2×C312yes—
13C3×C4⋊(C3⋊C8)—32no—
14Q8×H(3,𝔽3)—42no—
15C2×C4×H(3,𝔽3)—22no—
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17C2×C4×H(3,𝔽3)D412yes—
18(C3⋊(C3⋊C8))⋊C3—42no—
19C3×#333—91no—
20#4546—31no—
21#4546—91no—
22C2×C32×Q8—22no—
23C2×C32×Q8Dic312yes—
24C2×C32×Q8C3×C412yes—
25C32×C4⋊C4—24no—
26C32×C4⋊C4—22no—
27C32×C4⋊C4—22no—
28C32×C42—21no—
29C32×C42C2×D311yescharacteristic
30C32×C42C22×C311yescharacteristic
31C2×C3⋊(C3⋊C8)—22no—
32C2×C3×C3⋊C8—62no—
33C22×H(3,𝔽3)C22⋊C411yescharacteristic
34C4×H(3,𝔽3)—21no—
35C4×H(3,𝔽3)—42no—
36C4×H(3,𝔽3)SD1612yes—
37C4×H(3,𝔽3)Q1612yes—
38C3×C4⋊Q8—31no—
39C3×C4⋊Q8—31no—
40C3×C4⋊Q8—61no—
41C3×C4⋊Q8C3×D311yescharacteristic
42C3×C4⋊C8—92no—
43C4⋊(C3⋊C8)—32no—
44C4⋊(C3⋊C8)—92no—
45C32×Q8—42no—
46C32×Q8—42no—
47C32×Q8—44no—
48C2×C32×C4—24no—
49C2×C32×C4—22no—
50C2×C32×C4—22no—
51C2×C32×C4—21no—
52C2×C32×C4—22no—
53C2×C32×C4C2×Dic311yescharacteristic
54C2×C32×C4#6712yes—
55C2×C32×C4C2×C3×C411yescharacteristic
56C2×C32×C4C3×D412yes—
57C3⋊(C3⋊C8)—42no—
58C3×C3⋊C8—122no—
59#333—91noSylow
60C2×H(3,𝔽3)#10011yescharacteristic
61C2×H(3,𝔽3)#10312yes—
62C2×C3×Q8—32no—
63C2×C3×Q8—32no—
64C2×C3×Q8—62no—
65C2×C3×Q8C3×Dic312yes—
66C2×C3×C8—182no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—62no—
69C3×C4⋊C4—62no—
70C3×C4⋊C4—64no—
71C3×C42—31no—
72C3×C42—31no—
73C3×C42—61no—
74C3×C42C2×C3×D311yescharacteristic
75C2×C3⋊C8—62no—
76C2×C3⋊C8—182no—
77C22×C32—21no—
78C22×C32#21711yescharacteristic
79C22×C32C3×C22⋊C411yescharacteristic
80C32×C4—22no—
81C32×C4—21no—
82C32×C4—21no—
83C32×C4—22no—
84C32×C4—22no—
85C32×C4—44no—
86C32×C4—42no—
87C32×C4—42no—
88C32×C4#21512yes—
89C32×C4#21612yes—
90C32×C4C3×SD1612yes—
91C32×C4C3×Q1612yes—
92C4⋊Q8(C3⋊D3)⋊C311yescharacteristic
93C4⋊C8—92no—
94H(3,𝔽3)#33311yescharacteristic, complemented, Sylow
95C3×Q8—42no—
96C3×Q8—122no—
97C3×Q8—122no—
98C3×Q8—124no—
99C2×C3×C4—22no—
100C2×C3×C4—31no—
101C2×C3×C4—31no—
102C2×C3×C4—32no—
103C2×C3×C4—32no—
104C2×C3×C4—62no—
105C2×C3×C4—62no—
106C2×C3×C4—64no—
107C2×C3×C4—61no—
108C2×C3×C4—62no—
109C2×C3×C4C2×C3×Dic311yescharacteristic
110C2×C3×C4C3×#6712yes—
111C3×C8—362no—
112C3⋊C8—122no—
113C3⋊C8—362no—
114C2×C32—21no—
115C2×C32—22no—
116C2×C32#83311yescharacteristic
117C2×C32#83512yes—
118C2×C32C3×#10011yescharacteristic
119C2×C32C3×#10312yes—
120C2×Q8(C3⋊Dic3)⋊C312yes—
121C2×C8—182no—
122C4⋊C4—22no—
123C42C2×(C3⋊D3)⋊C311yescharacteristic
124C22×C3—31no—
125C22×C3—31no—
126C22×C3—61no—
127C22×C3C3×#21711yescharacteristic
128C3×C4—21no—
129C3×C4—32no—
130C3×C4—32no—
131C3×C4—32no—
132C3×C4—32no—
133C3×C4—42no—
134C3×C4—61no—
135C3×C4—61no—
136C3×C4—62no—
137C3×C4—62no—
138C3×C4—62no—
139C3×C4—122no—
140C3×C4—122no—
141C3×C4—124no—
142C3×C4C3×#21512yes—
143C3×C4C3×#21612yes—
144C32—21no—
145C32#454611yescharacteristic, complemented
146C32C3×#33311yescharacteristic, complemented
147Q8—42no—
148C2×C4—22no—
149C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
150C2×C4#638⋊C312yes—
151C8—362no—
152C2×C3—31no—
153C2×C3—31no—
154C2×C3—32no—
155C2×C3—32no—
156C2×C3—61no—
157C2×C3—62no—
158C2×C3C3×#83311yescharacteristic
159C2×C3C3×#83512yes—
160C22#3757⋊C311yescharacteristic
161C4—21no—
162C4—42no—
163C4#3756⋊C312yes—
164C4#3755⋊C312yes—
165C3—31no—
166C3—31no—
167C3—61no—
168C3C3×#454611yescharacteristic
169C2#63660⋊C311yescharacteristic
170C2#63662⋊C312yes—
subgroupG/H
centerC22#3757⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC3×C42C2×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
socleC22×C3C3×#217
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC32×C42C22×C3
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#333
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #10589412C3semidirect
94. H(3,𝔽3)#333semidirect
145. C32#4546semidirect
146. C32C3×#333semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105894123
C3×#45463