Finite Group Databasebeta

← #13165983

Subgroups — #13165983

Order 1728.

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

normal subgroups only

This group has 524 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#11724491C211yesmaximal, characteristic, complemented
2C32×#986C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593233—34nomaximal
5#10592897—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C3⋊(C3⋊(#41⋊C3))—21no—
8C32×GL(2,𝔽3)—21no—
9#88792—41nomaximal
10#64366D311yescharacteristic, complemented
11C3⋊#994—34no—
12C3×#986D314yescomplemented
13C32×AGL(1,ℤ/8)—31no—
14#64149—31no—
15#64141—31no—
16C3⋊GU(2,𝔽3)—34no—
17C3⋊GU(2,𝔽3)—64no—
18#63664—31no—
19#63652—31no—
20#63650—31no—
21C3×#634—41no—
22C32×D12—41no—
23C32×SL(2,𝔽3)D411yescharacteristic, complemented
24C3⋊(C3⋊(C3⋊C8))—41no—
25#5429—91no—
26#5198—94no—
27C32×#41C2×D311yescharacteristic
28C2×C32×D4—31no—
29C2×#638—62no—
30#3832—21no—
31D4×C3⋊D3—61no—
32#3828—31no—
33C2×C4⋊(C3⋊D3)—31no—
34C3×#41⋊C3—24no—
35C3×#41⋊C3C2×D314yes—
36C3⋊(#41⋊C3)—64no—
37C3×GL(2,𝔽3)—24no—
38C32×SD16—61no—
39C32×D8—61no—
40C32×M16—31no—
41#3754—61no—
42#3753—61no—
43C4×C3⋊Dic3—31no—
44#3748—31no—
45#3715—124no—
46C2×C32×D3—81no—
47C33×C4—41no—
48C3×C3⋊Dic3—81no—
49#1009—34no—
50#994—91no—
51#986C3⋊D311yescharacteristic, complemented
52C3×AGL(1,ℤ/8)—34no—
53#940—94no—
54#932—94no—
55GU(2,𝔽3)—91no—
56GU(2,𝔽3)—184no—
57#837—94no—
58#825—94no—
59#823—94no—
60C23×C32—61no—
61C22×C3⋊D3—32no—
62C32×Q8#6711yescharacteristic
63C32×D4—31no—
64C32×D4—61no—
65C32×D4—61no—
66C2×C32×C4—31no—
67#638—61no—
68#638—124no—
69C2×C3⋊Dic3—61no—
70C4⋊(C3⋊D3)—31no—
71C4⋊(C3⋊D3)—61no—
72C4×C3⋊D3—61no—
73#634S411yescharacteristic
74C3×D12—44no—
75C3×Dic6—124no—
76C3×SL(2,𝔽3)—24no—
77C3×SL(2,𝔽3)#6714yescomplemented
78C32×C8—61no—
79C3⋊(C3⋊C8)—61no—
80C3⋊(C3⋊C8)—124no—
81C3⋊(C3⋊C8)—244no—
82#453—271noSylow
83C2×C33—41no—
84C32×D3—161no—
85C3×#41D3214yes—
86C2×C3×D4—34no—
87C2×#67—188no—
88#239—64no—
89D3×D4—184no—
90#235—94no—
91C2×D12—94no—
92#41⋊C3—24no—
93#41⋊C3—181no—
94#41⋊C3C2×C3⋊D311yescharacteristic
95GL(2,𝔽3)—21no—
96C3×SD16—64no—
97C3×D8—64no—
98C3×M16—34no—
99#215—361no—
100#214—184no—
101#213—184no—
102C4×Dic3—94no—
103#208—94no—
104C22×C32—31no—
105C22×C32—122no—
106C22×C32—121no—
107C2×C3⋊D3—61no—
108C2×C3⋊D3—122no—
109C2×C3×D3—84no—
110C32×C4—31no—
111C32×C4—44no—
112C32×C4—84no—
113C32×C4C2×S411yescharacteristic
114C3⋊Dic3—21no—
115C3⋊Dic3—62no—
116C3×Dic3—244no—
117H(5,𝔽2)—91no—
118AGL(1,ℤ/8)—31no—
119AGL(1,ℤ/8)—271no—
120#127—271no—
121#104—271no—
122#104—271no—
123#100—271no—
124C33—41noSylow
125C23×C3—64no—
126C22×D3—98no—
127C3×Q8—361no—
128C3×Q8#62514yes—
129C3×D4—34no—
130C3×D4—64no—
131C3×D4—64no—
132C2×C3×C4—34no—
133#67—184no—
134#67—3616no—
135C2×Dic3—184no—
136D12—41no—
137D12—94no—
138D12—184no—
139C4×D3—184no—
140Dic6—34no—
141SL(2,𝔽3)—24no—
142SL(2,𝔽3)#63811yescharacteristic, complemented
143C3×C8—64no—
144C3⋊C8—184no—
145C3⋊C8—361no—
146C3⋊C8—724no—
147C2×C32—44no—
148C2×C32—61no—
149C2×C32—84no—
150C2×C32—241no—
151C2×C32GL(2,ℤ/4)11yescharacteristic
152C3⋊D3—121no—
153C3×D3—164no—
154#41—181no—
155#41—271no—
156#41D3×C3⋊D311yescharacteristic
157C2×D4—31no—
158C2×D4—271no—
159C2×D4—541no—
160C2×D4—542no—
161SD16—61no—
162SD16—541no—
163D8—61no—
164D8—541no—
165M16—31no—
166M16—271no—
167C42—271no—
168C22×C3—34no—
169C22×C3—128no—
170C22×C3—124no—
171C2×D3—81no—
172C2×D3—184no—
173C2×D3—368no—
174C3×C4—34no—
175C3×C4—41no—
176C3×C4—84no—
177C3×C4—721no—
178C3×C4D3×S414yes—
179Dic3—64no—
180Dic3—188no—
181C32—44no—
182C32—84no—
183C32#542911yescharacteristic, complemented
184C23—61no—
185C23—272no—
186Q8—91no—
187Q8#629011yescharacteristic
188D4—31no—
189D4—61no—
190D4—61no—
191D4—271no—
192D4—541no—
193D4—541no—
194D4—1084no—
195C2×C4—31no—
196C2×C4—541no—
197C2×C4—541no—
198C8—61no—
199C8—541no—
200C2×C3—41no—
201C2×C3—64no—
202C2×C3—84no—
203C2×C3—244no—
204C2×C3#6420914yes—
205D3—161no—
206D3—364no—
207C22—31no—
208C22—122no—
209C22—121no—
210C22—541no—
211C22—1082no—
212C4—31no—
213C4—181no—
214C4—542no—
215C4C3⋊D3×S411yescharacteristic
216C3—41no—
217C3—84no—
218C3#1059323314yescomplemented
219C2—61no—
220C2—241no—
221C2—1081no—
222C2#1172418911yescharacteristic
subgroupG/H
centerC2#11724189
commutator, cosocleC32×#41⋊C3C22
Frattini, hypercenterC4C3⋊D3×S4
FittingC32×#41C2×D3
socleC2×C32GL(2,ℤ/4)
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
1. #11724491C2semidirect
2. C32×#986C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
10. #64366D3semidirect
12. C3×#986D3semidirect
23. C32×SL(2,𝔽3)D4semidirect
51. #986C3⋊D3semidirect
77. C3×SL(2,𝔽3)#67semidirect
142. SL(2,𝔽3)#638semidirect
183. C32#5429semidirect
218. C3#10593233semidirect

Maximal subgroups

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

subgroupindex
#117244912
C32×#9862
C3⋊(C3⋊GU(2,𝔽3))2
#105932333
#105928973
#887924