Finite Group Databasebeta

← #13165962

Subgroups — #13165962

Order 1728.

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

normal subgroups only

This group has 478 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#10593212—34nomaximal
5#10592490—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×#6293—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#63644—31no—
17#63641—31no—
18#63635—31no—
19C2×C3⋊(C3⋊Dic3)—41no—
20C2×C3×C3⋊Dic3—41no—
21C2×C32×Dic3—41no—
22C32×SL(2,𝔽3)Q811yescharacteristic, complemented
23#6293—44no—
24#5386—91no—
25#4791—94no—
26C2×C32×Q8C2×D311yescharacteristic
27C2×#634—61no—
28C2×C3×SL(2,𝔽3)—24no—
29C2×C3×SL(2,𝔽3)C2×D314yes—
30C2×#627—124no—
31C2×C32×C8—31no—
32C32×C4⋊C4—31no—
33C4⋊(C3⋊Dic3)—31no—
34C4⋊(C3⋊Dic3)—31no—
35#3750—61no—
36C4×C3⋊Dic3—31no—
37C2×C3⋊(C3⋊C8)—31no—
38C22×C33—41no—
39C3⋊(C3⋊Dic3)—42no—
40C3×C3⋊Dic3—42no—
41C32×Dic3—42no—
42Q8×Dic3—34no—
43#888—94no—
44C4×SL(2,𝔽3)—91no—
45#859—91no—
46#859—184no—
47#859C3⋊D311yescharacteristic
48C3×#103—34no—
49#835—94no—
50#817—94no—
51Dic3⋊C8—94no—
52#808—94no—
53C32×Q8—31no—
54C32×Q8Dic611yescharacteristic, complemented
55C2×C32×C4—31no—
56C2×C32×C4—61no—
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—61no—
59C2×C3⋊Dic3—124no—
60C2×C3⋊Dic3—244no—
61C2×C3⋊Dic3S411yescharacteristic
62#634—122no—
63C2×C3×Dic3—44no—
64C2×C3×Dic3—124no—
65C3×SL(2,𝔽3)—24no—
66C3×SL(2,𝔽3)Dic614yescomplemented
67#627—1216no—
68C32×C8—61no—
69C3⋊(C3⋊C8)—61no—
70#475—271noSylow
71C2×C33—41no—
72C2×C33—41no—
73C2×C33—41no—
74C2×C3×Q8D3214yes—
75C2×Dic6—184no—
76C2×Dic6—361no—
77C2×SL(2,𝔽3)—24no—
78C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
79C2×C3×C8—34no—
80C3×C4⋊C4—34no—
81C4⋊Dic3—94no—
82C4⋊Dic3—94no—
83Dic3⋊C4—184no—
84C4×Dic3—94no—
85C2×C3⋊C8—94no—
86C22×C32—44no—
87C22×C32—84no—
88C22×C32C2×S411yescharacteristic
89C32×C4—31no—
90C32×C4—31no—
91C32×C4—121no—
92C3⋊Dic3—61no—
93C3⋊Dic3—121no—
94C3⋊Dic3—128no—
95C3⋊Dic3—248no—
96C3⋊Dic3GL(2,𝔽3)12yes—
97C3×Dic3—48no—
98C3×Dic3—128no—
99C4⋊Q8—271no—
100C4×Q8—91no—
101#106—271no—
102#106—271no—
103C4⋊C8—271no—
104#103—31no—
105#103—271no—
106C33—41noSylow
107C3×Q8—34no—
108C3×Q8#62714yescomplemented
109C2×C3×C4—34no—
110C2×C3×C4—64no—
111C2×C3×C4—361no—
112C2×Dic3—34no—
113C2×Dic3—41no—
114C2×Dic3—94no—
115C2×Dic3—184no—
116C2×Dic3—361no—
117C2×Dic3—724no—
118Dic6—368no—
119Dic6—364no—
120SL(2,𝔽3)—24no—
121SL(2,𝔽3)#63411yescharacteristic, complemented
122C3×C8—64no—
123C3⋊C8—184no—
124C2×C32—44no—
125C2×C32—44no—
126C2×C32—44no—
127C2×C32—84no—
128C2×C32—84no—
129C2×C32—84no—
130C2×C32A4⋊Q811yescharacteristic
131C2×C32C2×GL(2,𝔽3)11yescharacteristic
132C2×C32#98411yescharacteristic
133C2×Q8—541no—
134C2×Q8D3×C3⋊D311yescharacteristic
135C2×C8—31no—
136C2×C8—271no—
137C4⋊C4—31no—
138C4⋊C4—271no—
139C4⋊C4—271no—
140C4⋊C4—541no—
141C42—271no—
142C22×C3—41no—
143C22×C3—84no—
144C22×C3D3×S414yes—
145C3×C4—34no—
146C3×C4—34no—
147C3×C4—124no—
148C3×C4—362no—
149Dic3—38no—
150Dic3—42no—
151Dic3—184no—
152Dic3—362no—
153Dic3—364no—
154Dic3—728no—
155C32—44no—
156C32—84no—
157C32#538611yescharacteristic, complemented
158Q8—31no—
159Q8—1082no—
160Q8#629311yescharacteristic, complemented
161C2×C4—31no—
162C2×C4—61no—
163C2×C4—91no—
164C2×C4—271no—
165C2×C4—541no—
166C8—61no—
167C8—541no—
168C2×C3—41no—
169C2×C3—41no—
170C2×C3—41no—
171C2×C3—84no—
172C2×C3—84no—
173C2×C3—84no—
174C2×C3#6419614yes—
175C2×C3D3×GL(2,𝔽3)14yes—
176C2×C3#6420414yes—
177C22C3⋊D3×S411yescharacteristic
178C4—31no—
179C4—31no—
180C4—92no—
181C4—121no—
182C4—541no—
183C4—1081no—
184C3—41no—
185C3—84no—
186C3#1059321214yescomplemented
187C2#1172417611yescharacteristic
188C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
189C2#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
54. C32×Q8Dic6semidirect
66. C3×SL(2,𝔽3)Dic6semidirect
108. C3×Q8#627semidirect
121. SL(2,𝔽3)#634semidirect
157. C32#5386semidirect
160. Q8#6293semidirect
186. C3#10593212semidirect

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
#105932123
#105924903
C2×#62934