Finite Group Databasebeta

← #13165969

Subgroups — #13165969

Order 1728.

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

normal subgroups only

This group has 440 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
1C32×#985C211yesmaximal, characteristic
2C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593219—34nomaximal
5#10592895—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C32×GL(2,𝔽3)C411yescharacteristic, complemented
8C32×#226C411yescharacteristic, complemented
9D3×C3⋊(C3⋊C8)—41nomaximal
10C3×#985D314yes—
11C32×#135—31no—
12#64157—31no—
13#64157D311yescharacteristic, complemented
14C3⋊(#131⋊C3)—34no—
15C3⋊GU(2,𝔽3)—34no—
16C3⋊GU(2,𝔽3)—64no—
17#63664—31no—
18#63664—31no—
19#63646—31no—
20C8×C3⋊Dic3—31no—
21C32×C4×D3—41no—
22C32×SL(2,𝔽3)C2×C411yescharacteristic
23C3⋊(C3⋊(C3⋊C8))—41no—
24C3×C3⋊(C3⋊C8)—41no—
25#5404—91no—
26#5196—94no—
27C32×#41—31no—
28C32×#41C2×D311yescharacteristic
29C3×#41⋊C3—24no—
30C3×#41⋊C3C2×D314yes—
31C3×GL(2,𝔽3)Dic314yescomplemented
32C3×#226Dic314yescomplemented
33C32×Q16—31no—
34C32×SD16—31no—
35C32×SD16—31no—
36C32×D8—31no—
37C2×C32×C8—31no—
38C4×C3⋊Dic3—31no—
39#3748—31no—
40#3748—31no—
41#3748—61no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—61no—
44D3×C3⋊C8—124no—
45C2×C32×D3—41no—
46C33×C4—41no—
47C32×Dic3—41no—
48#985C3⋊D311yescharacteristic
49C3×#135—34no—
50#948—34no—
51#948—94no—
52#131⋊C3—91no—
53GU(2,𝔽3)—91no—
54GU(2,𝔽3)—184no—
55#837—94no—
56#837—94no—
57#819—94no—
58C8×Dic3—94no—
59C32×Q8—31no—
60C32×Q8C4×D311yescharacteristic
61C32×D4—31no—
62C32×D4—31no—
63C32×D4—61no—
64C2×C32×C4—31no—
65C2×C32×C4—61no—
66C2×C3⋊Dic3—61no—
67C3×C4×D3—44no—
68C3×SL(2,𝔽3)—24no—
69C3×SL(2,𝔽3)C2×Dic314yes—
70C32×C8—31no—
71C32×C8—31no—
72C3⋊(C3⋊C8)—31no—
73C3⋊(C3⋊C8)—61no—
74C3⋊(C3⋊C8)—124no—
75C3⋊(C3⋊C8)—244no—
76C3⋊(C3⋊C8)S411yescharacteristic
77C3×C3⋊C8—124no—
78#443—271noSylow
79C2×C33—41no—
80C32×D3—42no—
81C3×#41—34no—
82C3×#41D3214yes—
83#41⋊C3—24no—
84#41⋊C3C2×C3⋊D311yescharacteristic
85GL(2,𝔽3)C3⋊Dic311yescharacteristic, complemented
86#226C3⋊Dic311yescharacteristic, complemented
87C3×Q16—34no—
88C3×SD16—34no—
89C3×SD16—34no—
90C3×D8—34no—
91C2×C3×C8—34no—
92C4×Dic3—94no—
93#208—94no—
94#208—94no—
95#208—184no—
96C2×C3⋊C8—94no—
97C2×C3⋊C8—184no—
98D3×C8—361no—
99C22×C32—31no—
100C22×C32—61no—
101C2×C3×D3—44no—
102C32×C4—31no—
103C32×C4—44no—
104C32×C4—61no—
105C32×C4—84no—
106C32×C4C2×S411yescharacteristic
107C3⋊Dic3—62no—
108C3×Dic3—44no—
109#135—31no—
110#131—91no—
111#131—271no—
112#108—271no—
113#104—271no—
114#104—271no—
115C4×C8—271no—
116C33—41noSylow
117C3×Q8—34no—
118C3×Q8D3×Dic314yes—
119C3×D4—34no—
120C3×D4—34no—
121C3×D4—64no—
122C2×C3×C4—34no—
123C2×C3×C4—64no—
124C2×Dic3—184no—
125C4×D3—41no—
126SL(2,𝔽3)—24no—
127SL(2,𝔽3)C2×C3⋊Dic311yescharacteristic
128C3×C8—34no—
129C3×C8—34no—
130C3×C8—361no—
131C3⋊C8—34no—
132C3⋊C8—94no—
133C3⋊C8—184no—
134C3⋊C8—361no—
135C3⋊C8—724no—
136C2×C32—44no—
137C2×C32—61no—
138C2×C32—84no—
139C2×C32—121no—
140C2×C32C4×S411yescharacteristic
141C3×D3—48no—
142#41—31no—
143#41D3×C3⋊D311yescharacteristic
144Q16—31no—
145SD16—31no—
146SD16—31no—
147D8—31no—
148M16—271no—
149M16—271no—
150M16—541no—
151C2×C8—31no—
152C2×C8—271no—
153C2×C8—541no—
154C42—271no—
155C22×C3—34no—
156C22×C3—64no—
157C2×D3—41no—
158C3×C4—34no—
159C3×C4—41no—
160C3×C4—64no—
161C3×C4—84no—
162C3×C4D3×S414yes—
163Dic3—41no—
164Dic3—188no—
165C32—44no—
166C32—84no—
167C32#540411yescharacteristic, complemented
168Q8—31no—
169Q8D3×C3⋊Dic311yescharacteristic
170D4—31no—
171D4—31no—
172D4—61no—
173C2×C4—31no—
174C2×C4—61no—
175C2×C4—541no—
176C8—31no—
177C8—31no—
178C8—91no—
179C8—271no—
180C8—541no—
181C2×C3—41no—
182C2×C3—64no—
183C2×C3—84no—
184C2×C3—124no—
185C2×C3Dic3×S414yes—
186D3—42no—
187C22—31no—
188C22—61no—
189C4—31no—
190C4—61no—
191C4—542no—
192C4C3⋊D3×S411yescharacteristic
193C3—41no—
194C3—84no—
195C3#1059321914yescomplemented
196C2—61no—
197C2—121no—
198C2S4×C3⋊Dic311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4C3⋊D3×S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC32×#41C2×D3
socleC2×C32C4×S4
prime radicalC32×#985C2
cosocleC32×#41⋊C3C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊(#131⋊C3))C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
7. C32×GL(2,𝔽3)C4semidirect
8. C32×#226C4semidirect
13. #64157D3semidirect
31. C3×GL(2,𝔽3)Dic3semidirect
32. C3×#226Dic3semidirect
85. GL(2,𝔽3)C3⋊Dic3semidirect
86. #226C3⋊Dic3semidirect
167. C32#5404semidirect
195. C3#10593219semidirect

Maximal subgroups

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

subgroupindex
C32×#9852
C3⋊(C3⋊(#131⋊C3))2
C3⋊(C3⋊GU(2,𝔽3))2
#105932193
#105928953
D3×C3⋊(C3⋊C8)4