Finite Group Databasebeta

← #13138721

Subgroups — #13138721

Order 1728.

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

normal subgroups only

This group has 322 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
1#64113⋊C3C212yesmaximal, complemented
2C2×#88386C211yesmaximal, characteristic, complemented
3#11720250C211yesmaximal, characteristic, complemented
4#11720249C212yesmaximal, complemented
5#11720248C211yesmaximal, characteristic, complemented
6#10590632—32nomaximal
7#3828⋊C3—22no—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9#3745⋊C3C2212yes—
10#3744⋊C3—22no—
11#3746⋊C3C2212yes—
12#88386C2211yescharacteristic
13#88386C2211yescharacteristic
14#64113D312yescomplemented
15C3×#904—32no—
16C2×C3×C3⋊C16—32no—
17C2×C3×C3⋊C16—61no—
18#63551—32no—
19#63549—32no—
20#63548—32no—
21#63546—32no—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23#638⋊C3—42no—
24C4⋊((C3⋊D3)⋊C3)—22no—
25C4×(C3⋊D3)⋊C3—42no—
26#634⋊C3—22no—
27C8×H(3,𝔽3)D411yescharacteristic
28C8×H(3,𝔽3)C2311yescharacteristic
29#5194—92no—
30#4904—91no—
31#3828—22no—
32C3×#235—62no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D312yes—
35#3746C2×D312yes—
36#3745C2×D312yes—
37#3744—22no—
38C3×Dic12—32no—
39C3×D24—32no—
40C3×#204—62no—
41C3×C3⋊C16—32no—
42C3×C3⋊C16—32no—
43C3×C3⋊C16—61no—
44C3×C3⋊C16—61no—
45C22×H(3,𝔽3)D811yescharacteristic
46C2×(C3⋊D3)⋊C3—42no—
47C4×H(3,𝔽3)D811yescharacteristic
48C4×H(3,𝔽3)C2×D411yescharacteristic
49(C3⋊Dic3)⋊C3—42no—
50C3×#135—92no—
51#904—32no—
52#904—92no—
53C2×C3×C16—91no—
54#829—92no—
55#828—92no—
56#827—92no—
57#826—92no—
58C2×C3⋊C16—32no—
59C2×C3⋊C16—61no—
60Dic24—91no—
61#800—92no—
62D48—91no—
63C2×C32×C4—21no—
64C2×C32×C4#6712yes—
65#638—42no—
66C4⋊(C3⋊D3)—22no—
67C4×C3⋊D3—42no—
68#634—22no—
69C3×#67—122no—
70C3×D12—62no—
71C3×C4×D3—122no—
72C3×Dic6—62no—
73C32×C8—21no—
74C32×C8—21no—
75C32×C8#6712yes—
76C32×C8C22×D312yes—
77#508—271noSylow
78C2×H(3,𝔽3)—21no—
79C2×H(3,𝔽3)C2×D811yescharacteristic
80(C3⋊D3)⋊C3—82no—
81C3×#41—182no—
82#235—62no—
83#235—182no—
84C3×Q16—92no—
85C3×SD16—182no—
86C3×D8—92no—
87C2×C3×C8—32no—
88C2×C3×C8—61no—
89C2×C3×C8D3211yescharacteristic
90Dic12—32no—
91Dic12—92no—
92D24—32no—
93D24—92no—
94#204—62no—
95#204—182no—
96C3×C16—91no—
97C3×C16—91no—
98C3⋊C16—32no—
99C3⋊C16—32no—
100C3⋊C16—61no—
101C3⋊C16—61no—
102C22×C32—21no—
103C22×C32#21312yes—
104C2×C3⋊D3—42no—
105C2×C3×D3—122no—
106C32×C4—21no—
107C32×C4—21no—
108C32×C4#21312yes—
109C32×C4C2×#6712yes—
110C3⋊Dic3—42no—
111C3×Dic3—122no—
112#135—92no—
113Q32—271no—
114SD32—272no—
115D16—271no—
116C2×C16—91no—
117H(3,𝔽3)#50811yescharacteristic, complemented, Sylow
118C3×Q8—182no—
119C3×D4—182no—
120C3×D4—362no—
121C2×C3×C4—32no—
122C2×C3×C4—61no—
123C2×C3×C4—362no—
124C2×C3×C4#62511yescharacteristic
125#67—122no—
126#67—362no—
127D12—62no—
128D12—182no—
129C4×D3—122no—
130C4×D3—362no—
131Dic6—62no—
132Dic6—182no—
133C3×C8—32no—
134C3×C8—32no—
135C3×C8—61no—
136C3×C8—61no—
137C3×C8#62511yescharacteristic
138C3×C8C2×D3211yescharacteristic
139C2×C32—22no—
140C2×C32—22no—
141C2×C32—21no—
142C2×C32C2×#21312yes—
143C3⋊D3—82no—
144C3×D3—242no—
145#41—182no—
146Q16—92no—
147SD16—182no—
148D8—92no—
149C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
150C16—91no—
151C16—91no—
152C22×C3—32no—
153C22×C3—61no—
154C22×C3—362no—
155C22×C3#371311yescharacteristic
156C2×D3—122no—
157C2×D3—362no—
158C3×C4—32no—
159C3×C4—32no—
160C3×C4—61no—
161C3×C4—61no—
162C3×C4—362no—
163C3×C4#371311yescharacteristic
164C3×C4C2×#62511yescharacteristic
165Dic3—122no—
166Dic3—362no—
167C32—21no—
168C32#519412yescomplemented
169Q8—182no—
170D4—182no—
171D4—362no—
172C2×C4—362no—
173C2×C4#619811yescharacteristic
174C8#619811yescharacteristic
175C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
176C2×C3—21no—
177C2×C3—32no—
178C2×C3—62no—
179C2×C3—62no—
180C2×C3—61no—
181C2×C3—722no—
182C2×C3C2×#371311yescharacteristic
183D3—242no—
184D3—722no—
185C22—362no—
186C22#8843211yescharacteristic
187C4—362no—
188C4#8843211yescharacteristic
189C4C2×#619811yescharacteristic
190C3—32no—
191C3—61no—
192C3#1059062911yescharacteristic
193C2—21no—
194C2—722no—
195C2C2×#8843211yescharacteristic
subgroupG/H
centerC4#88432
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3713
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#508
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #64113⋊C3C2semidirect
2. C2×#88386C2semidirect
3. #11720250C2semidirect
4. #11720249C2semidirect
5. #11720248C2semidirect
14. #64113D3semidirect
117. H(3,𝔽3)#508semidirect
168. C32#5194semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
C2×#883862
#117202502
#117202492
#117202482
#105906323