Finite Group Databasebeta

← #13165981

Subgroups — #13165981

Order 1728.

470 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 53 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#11724493C211yesmaximal, characteristic
2C32×#984C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593231—34nomaximal
5#10592904—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)—21no—
8C32×#226—21no—
9#88793—41nomaximal
10#64367D311yescharacteristic
11#1010⋊C3—34no—
12C3×#984D314yescomplemented
13C32×#137—31no—
14#64153—31no—
15#64151—31no—
16C3⋊GU(2,𝔽3)—34no—
17C3⋊GU(2,𝔽3)—64no—
18#63664—31no—
19#63652—31no—
20#63651—31no—
21C3×C4⋊(C3⋊D3)—41no—
22C32×Dic6—41no—
23C32×SL(2,𝔽3)D411yescharacteristic
24C3⋊(C3⋊(C3⋊C8))—41no—
25#5430—91no—
26#5205—94no—
27C32×#41C2×D311yescharacteristic
28C2×C32×Q8—31no—
29Q8×C3⋊D3—21no—
30#3830—61no—
31#3828—31no—
32C2×#634—31no—
33C3×#41⋊C3—24no—
34C3×#41⋊C3C2×D314yes—
35D3×SL(2,𝔽3)—64no—
36C3×#226—24no—
37C32×Q16—61no—
38C32×SD16—61no—
39C32×M16—31no—
40#3756—61no—
41#3755—61no—
42#3750—62no—
43C4×C3⋊Dic3—31no—
44#3748—31no—
45#3715—124no—
46C2×C3×C3⋊D3—81no—
47C33×C4—41no—
48C32×Dic3—81no—
49#1010—34no—
502−1+4⋊C3—91no—
51#984C3⋊D311yescharacteristic, complemented
52C3×#137—34no—
53#944—94no—
54#942—94no—
55GU(2,𝔽3)—91no—
56GU(2,𝔽3)—184no—
57#837—94no—
58#825—94no—
59#824—94no—
60C32×Q8—61no—
61C32×Q8—61no—
62C32×Q8#6711yescharacteristic
63C32×D4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—61no—
66#638—61no—
67C2×C3⋊Dic3—32no—
68C2×C3⋊Dic3—61no—
69C4⋊(C3⋊D3)S411yescharacteristic
70C4×C3⋊D3—61no—
71#634—31no—
72#634—61no—
73C3×D12—124no—
74C3×Dic6—44no—
75C3×SL(2,𝔽3)—24no—
76C3×SL(2,𝔽3)#6714yes—
77C32×C8—61no—
78C3⋊(C3⋊C8)—61no—
79C3⋊(C3⋊C8)—124no—
80C3⋊(C3⋊C8)—244no—
81#456—271noSylow
82C2×C33—41no—
83C3×C3⋊D3—161no—
84C3×#41D3214yes—
85C2×C3×Q8—34no—
86D3×Q8—64no—
87#237—184no—
88#235—94no—
89C2×Dic6—94no—
90#41⋊C3—24no—
91#41⋊C3C2×C3⋊D311yescharacteristic
92C2×SL(2,𝔽3)—181no—
93#226—21no—
94C3×Q16—64no—
95C3×SD16—64no—
96C3×M16—34no—
97#216—184no—
98#215—184no—
99#214—361no—
100Dic3⋊C4—188no—
101C4×Dic3—94no—
102#208—94no—
103C22×C32—31no—
104C2×C3⋊D3—21no—
105C2×C3×D3—244no—
106C32×C4—31no—
107C32×C4—44no—
108C32×C4—84no—
109C32×C4—121no—
110C32×C4C2×S411yescharacteristic
111C3⋊Dic3—62no—
112C3⋊Dic3—61no—
113C3×Dic3—84no—
1142−1+4—91no—
115#137—31no—
116#137—271no—
117C4⋊Q8—271no—
118#104—271no—
119#104—271no—
120#101—271no—
121C33—41noSylow
122C3×Q8—64no—
123C3×Q8—64no—
124C3×Q8#62514yes—
125C3×D4—34no—
126C3×D4—361no—
127C2×C3×C4—34no—
128C2×C3×C4—64no—
129#67—184no—
130C2×Dic3—98no—
131C2×Dic3—184no—
132D12—34no—
133C4×D3—184no—
134Dic6—41no—
135Dic6—94no—
136Dic6—184no—
137SL(2,𝔽3)—24no—
138SL(2,𝔽3)#63811yescharacteristic
139C3×C8—64no—
140C3⋊C8—184no—
141C3⋊C8—361no—
142C3⋊C8—724no—
143C2×C32—44no—
144C2×C32—61no—
145C2×C32—84no—
146C2×C32GL(2,ℤ/4)11yescharacteristic
147C3⋊D3—41no—
148C3×D3—484no—
149#41—271no—
150#41—541no—
151#41D3×C3⋊D311yescharacteristic
152C2×Q8—31no—
153C2×Q8—181no—
154C2×Q8—271no—
155Q16—61no—
156Q16—541no—
157SD16—61no—
158SD16—541no—
159M16—31no—
160M16—271no—
161C4⋊C4—542no—
162C42—271no—
163C22×C3—34no—
164C22×C3—721no—
165C2×D3—64no—
166C3×C4—34no—
167C3×C4—41no—
168C3×C4—84no—
169C3×C4—124no—
170C3×C4D3×S414yes—
171Dic3—81no—
172Dic3—188no—
173Dic3—184no—
174C32—44no—
175C32—84no—
176C32#543011yescharacteristic, complemented
177Q8—61no—
178Q8—61no—
179Q8—271no—
180Q8—541no—
181Q8#629011yescharacteristic
182D4—31no—
183D4—91no—
184D4—541no—
185C2×C4—31no—
186C2×C4—61no—
187C2×C4—272no—
188C2×C4—541no—
189C2×C4—541no—
190C8—61no—
191C8—541no—
192C2×C3—41no—
193C2×C3—64no—
194C2×C3—84no—
195C2×C3—1441no—
196C2×C3#6420914yes—
197D3—124no—
198C22—31no—
199C22—181no—
200C4—31no—
201C4—121no—
202C4—542no—
203C4—541no—
204C4C3⋊D3×S411yescharacteristic
205C3—41no—
206C3—84no—
207C3#1059323114yescomplemented
208C2—61no—
209C2—361no—
210C2#1172418911yescharacteristic
subgroupG/H
centerC2#11724189
commutator, cosocleC32×#41⋊C3C22
Frattini, hypercenterC4C3⋊D3×S4
FittingC32×#41C2×D3
socleC2×C32GL(2,ℤ/4)
prime radical#11724493C2
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C32×#984C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
12. C3×#984D3semidirect
51. #984C3⋊D3semidirect
176. C32#5430semidirect
207. C3#10593231semidirect

Maximal subgroups

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

subgroupindex
#117244932
C32×#9842
C3⋊(C3⋊GU(2,𝔽3))2
#105932313
#105929043
#887934