Finite Group Databasebeta

← #10593641⋊C3 #13146864

Subgroups — #10593641⋊C3 #13146864

Order 1728.

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

normal subgroups only

This group has 263 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
1#64267⋊C3C212yesmaximal, complemented
2#131⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#10593641C311yesmaximal, characteristic, complemented
4C3×#5142—31nomaximal
5#41⋊H(3,𝔽3)C2211yescharacteristic
6(C3⋊GL(2,𝔽3))⋊C3—22no—
7#3745⋊C3—41nomaximal
8#64267C2×C312yescomplemented
9C32×#131D311yescharacteristic, complemented
10C3×#949—32no—
11C2×C3×D24—31no—
12C3×#131⋊C3—21no—
13C3×#131⋊C3C2×C311yescharacteristic, complemented
14C3×#823—32no—
15C3×#819—31no—
16C4⋊((C3⋊D3)⋊C3)—42no—
17Q8⋊H(3,𝔽3)D411yescharacteristic
18C8×H(3,𝔽3)—41no—
19#5407—91no—
20C3×#472—91no—
21#5142—31no—
22C32×#41C2×D311yescharacteristic
23C2×C3×D12—32no—
24C3×#41⋊C3—21no—
25C3×#41⋊C3C22×C311yescharacteristic
26C3⋊GL(2,𝔽3)—22no—
27C32×M16—31no—
28C2×C32×C8—31no—
29#3745—41no—
30C3×#215—62no—
31C3×#213—62no—
32C3×#208—32no—
33C3×D24—61no—
34C3×D24—61no—
35C2×(C3⋊D3)⋊C3—82no—
36C4×H(3,𝔽3)—41no—
37#986—92no—
38C3×AGL(1,ℤ/8)—92no—
39C2×C3×D8—91no—
40C3×#131—31no—
41C3×#131C3×D311yescharacteristic
42#949—32no—
43C2×D24—31no—
44#131⋊C3—31no—
45#131⋊C3—61no—
46C3×#108—91no—
47C3×#100—92no—
48#823—32no—
49#819—31no—
50C22×C3×D3—62no—
51C32×Q8D1211yescharacteristic
52C32×D4—31no—
53C2×C32×C4—31no—
54C4⋊(C3⋊D3)—42no—
55C3×D12—62no—
56C3×D12—62no—
57C3×SL(2,𝔽3)—21no—
58C3×SL(2,𝔽3)C3×D411yescharacteristic
59C32×C8—31no—
60C32×C8—41no—
61C32×C8—81no—
62C32×C8S411yescharacteristic
63C3×C3⋊C8—62no—
64#472—91noSylow
65C2×H(3,𝔽3)—41no—
66(C3⋊D3)⋊C3—162no—
67C3×#41—31no—
68C3×#41C2×C3×D311yescharacteristic
69C2×C3×D4—92no—
70C2×D12—32no—
71#41⋊C3—31no—
72#41⋊C3—61no—
73GL(2,𝔽3)—182no—
74C3×SD16—182no—
75C3×D8—181no—
76C3×D8—181no—
77C3×D8—182no—
78C3×M16—31no—
79C3×M16—31no—
80C3×M16—61no—
81C3×M16—92no—
82C2×C3×C8—31no—
83C2×C3×C8—31no—
84C2×C3×C8—61no—
85#215—62no—
86#213—62no—
87#208—32no—
88D24—61no—
89D24—61no—
90D24—361no—
91C22×C32—31no—
92C2×C3⋊D3—82no—
93C2×C3×D3—122no—
94C2×C3×D3—242no—
95C32×C4—31no—
96C32×C4—41no—
97C32×C4—81no—
98C32×C4C2×S411yescharacteristic
99AGL(1,ℤ/8)—92no—
100C2×D8—91no—
101#131(C3⋊D3)⋊C311yescharacteristic, complemented
102#108—91no—
103#100—92no—
104H(3,𝔽3)—41noSylow
105C23×C3—182no—
106C22×D3—62no—
107C3×Q8—31no—
108C3×Q8C3×D1211yescharacteristic
109C3×D4—31no—
110C3×D4—31no—
111C3×D4—61no—
112C3×D4—182no—
113C3×D4—182no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—61no—
117D12—62no—
118D12—62no—
119D12—362no—
120SL(2,𝔽3)—31no—
121SL(2,𝔽3)—61no—
122C3×C8—31no—
123C3×C8—31no—
124C3×C8—31no—
125C3×C8—61no—
126C3×C8—121no—
127C3×C8—182no—
128C3×C8—241no—
129C3×C8C3×S411yescharacteristic
130C3⋊C8—62no—
131C2×C32—41no—
132C2×C32—61no—
133C2×C32—81no—
134C2×C32C4⋊S411yescharacteristic
135C3⋊D3—162no—
136C3×D3—242no—
137#41C2×(C3⋊D3)⋊C311yescharacteristic
138C2×D4—92no—
139SD16—182no—
140D8—181no—
141D8—182no—
142D8—181no—
143M16—31no—
144M16—92no—
145C2×C8—31no—
146C22×C3—31no—
147C22×C3—31no—
148C22×C3—61no—
149C22×C3—362no—
150C22×C3—722no—
151C2×D3—122no—
152C2×D3—242no—
153C2×D3—722no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—31no—
157C3×C4—61no—
158C3×C4—121no—
159C3×C4—241no—
160C3×C4C2×C3×S411yescharacteristic
161C32—41no—
162C32—81no—
163C32#540711yescharacteristic, complemented
164C23—182no—
165Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
166D4—31no—
167D4—182no—
168D4—182no—
169C2×C4—31no—
170C8—31no—
171C8—182no—
172C8(C3⋊S4)⋊C311yescharacteristic
173C2×C3—31no—
174C2×C3—61no—
175C2×C3—61no—
176C2×C3—121no—
177C2×C3—121no—
178C2×C3—241no—
179C2×C3—722no—
180C2×C3C3×C4⋊S411yescharacteristic
181D3—242no—
182D3—1442no—
183C22—31no—
184C22—362no—
185C22—722no—
186C4—31no—
187C4C2×(C3⋊S4)⋊C311yescharacteristic
188C3—31no—
189C3—121no—
190C3—241no—
191C3C3×#540711yescharacteristic
192C2—61no—
193C2—722no—
194C2C4⋊((C3⋊S4)⋊C3)11yescharacteristic
subgroupG/H
centerC2C4⋊((C3⋊S4)⋊C3)
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C3C3×C4⋊S4
hypercenterC8(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #64267⋊C3C2semidirect
2. #131⋊H(3,𝔽3)C2semidirect
3. #10593641C3semidirect
8. #64267C2×C3semidirect
9. C32×#131D3semidirect
13. C3×#131⋊C3C2×C3semidirect
101. #131(C3⋊D3)⋊C3semidirect
163. C32#5407semidirect

Maximal subgroups

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

subgroupindex
#64267⋊C32
#131⋊H(3,𝔽3)2
#105936413
C3×#51423
#3745⋊C34