Finite Group Databasebeta

← #13165952

Subgroups — #13165952

Order 1728.

442 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 67 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
1SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic
3C32×#859C211yesmaximal, characteristic
4#10593202—34nomaximal
5#10592492—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7#88807—41nomaximal
8Q8×C3⋊Dic3D311yescharacteristic
9#64097—31no—
10Dic3×SL(2,𝔽3)—34no—
11C3⋊#859—34no—
12C3⋊#859—64no—
13C3×#859D314yes—
14C32×#103—31no—
15#63662—31no—
16#63645—31no—
17#63641—31no—
18#63634—31no—
19C2×C3⋊(C3⋊Dic3)—41no—
20C2×C3×C3⋊Dic3—41no—
21C2×C32×Dic3—41no—
22C32×SL(2,𝔽3)Q811yescharacteristic, complemented
23#5391—91no—
24#4793—94no—
25C2×C32×Q8C2×D311yescharacteristic
26C2×#634—61no—
27C2×C3×SL(2,𝔽3)—24no—
28C2×C3×SL(2,𝔽3)C2×D314yes—
29C2×C32×C8—31no—
30C32×C4⋊C4—31no—
31C4⋊(C3⋊Dic3)—31no—
32C4⋊(C3⋊Dic3)—31no—
33#3750—61no—
34C4×C3⋊Dic3—31no—
35C2×C3⋊(C3⋊C8)—31no—
36Dic3⋊Dic3—124no—
37C22×C33—41no—
38C3⋊(C3⋊Dic3)—81no—
39C3×C3⋊Dic3—42no—
40C32×Dic3—81no—
41Q8×Dic3—34no—
42#888—94no—
43C4×SL(2,𝔽3)—91no—
44#859—91no—
45#859—184no—
46#859C3⋊D311yescharacteristic
47C3×#103—34no—
48#835—94no—
49#818—94no—
50Dic3⋊C8—94no—
51#807—94no—
52C32×Q8—31no—
53C32×Q8Dic611yescharacteristic
54C2×C32×C4—31no—
55C2×C32×C4—61no—
56C2×C3⋊Dic3—31no—
57C2×C3⋊Dic3—61no—
58C2×C3⋊Dic3—124no—
59C2×C3⋊Dic3—244no—
60C2×C3⋊Dic3S411yescharacteristic
61#634—122no—
62C2×C3×Dic3—44no—
63C2×C3×Dic3—124no—
64C3×SL(2,𝔽3)—24no—
65C3×SL(2,𝔽3)Dic614yescomplemented
66C32×C8—61no—
67C3⋊(C3⋊C8)—61no—
68#477—271noSylow
69C2×C33—41no—
70C2×C33—41no—
71C2×C33—41no—
72C2×C3×Q8D3214yes—
73C2×Dic6—184no—
74C2×SL(2,𝔽3)—24no—
75C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
76C2×C3×C8—34no—
77C3×C4⋊C4—34no—
78C4⋊Dic3—94no—
79C4⋊Dic3—94no—
80C4⋊Dic3—361no—
81Dic3⋊C4—184no—
82C4×Dic3—94no—
83C2×C3⋊C8—94no—
84C22×C32—44no—
85C22×C32—84no—
86C22×C32C2×S411yescharacteristic
87C32×C4—31no—
88C32×C4—31no—
89C32×C4—121no—
90C3⋊Dic3—61no—
91C3⋊Dic3—121no—
92C3⋊Dic3—244no—
93C3⋊Dic3—248no—
94C3⋊Dic3#22612yes—
95C3×Dic3—84no—
96C3×Dic3—128no—
97C4⋊Q8—271no—
98C4×Q8—91no—
99#107—271no—
100#107—271no—
101C4⋊C8—271no—
102#103—31no—
103#103—271no—
104C33—41noSylow
105C3×Q8—34no—
106C3×Q8#62714yes—
107C2×C3×C4—34no—
108C2×C3×C4—64no—
109C2×C3×C4—361no—
110C2×Dic3—34no—
111C2×Dic3—41no—
112C2×Dic3—94no—
113C2×Dic3—184no—
114C2×Dic3—361no—
115C2×Dic3—724no—
116Dic6—368no—
117SL(2,𝔽3)—24no—
118SL(2,𝔽3)#63411yescharacteristic, complemented
119C3×C8—64no—
120C3⋊C8—184no—
121C2×C32—44no—
122C2×C32—44no—
123C2×C32—44no—
124C2×C32—84no—
125C2×C32—84no—
126C2×C32—84no—
127C2×C32A4⋊Q811yescharacteristic
128C2×C32C2×#22611yescharacteristic
129C2×C32#98611yescharacteristic
130C2×Q8—541no—
131C2×Q8D3×C3⋊D311yescharacteristic
132C2×C8—31no—
133C2×C8—271no—
134C4⋊C4—31no—
135C4⋊C4—271no—
136C4⋊C4—271no—
137C4⋊C4—541no—
138C42—271no—
139C22×C3—41no—
140C22×C3—84no—
141C22×C3D3×S414yes—
142C3×C4—34no—
143C3×C4—34no—
144C3×C4—124no—
145C3×C4—362no—
146Dic3—38no—
147Dic3—81no—
148Dic3—184no—
149Dic3—364no—
150Dic3—721no—
151Dic3—728no—
152C32—44no—
153C32—84no—
154C32#539111yescharacteristic, complemented
155Q8—31no—
156Q8—1082no—
157Q8#629311yescharacteristic
158C2×C4—31no—
159C2×C4—61no—
160C2×C4—91no—
161C2×C4—271no—
162C2×C4—541no—
163C8—61no—
164C8—541no—
165C2×C3—41no—
166C2×C3—41no—
167C2×C3—41no—
168C2×C3—84no—
169C2×C3—84no—
170C2×C3—84no—
171C2×C3#6419914yes—
172C2×C3D3×#22614yes—
173C2×C3#6420414yes—
174C22C3⋊D3×S411yescharacteristic
175C4—31no—
176C4—31no—
177C4—92no—
178C4—121no—
179C4—541no—
180C4—1081no—
181C3—41no—
182C3—84no—
183C3#1059320214yescomplemented
184C2#1172417911yescharacteristic
185C2C3⋊D3×#22611yescharacteristic
186C2#1172418411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, prime radical, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)Q8
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
22. C32×SL(2,𝔽3)Q8semidirect
65. C3×SL(2,𝔽3)Dic6semidirect
118. SL(2,𝔽3)#634semidirect
154. C32#5391semidirect
183. C3#10593202semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
C3⋊(C3⋊#859)2
C32×#8592
#105932023
#105924923
#888074