Finite Group Databasebeta

← #13165979

Subgroups — #13165979

Order 1728.

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

normal subgroups only

This group has 488 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, complemented
2C32×#986C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593229—34nomaximal
5#10592900—31nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)—21no—
8C32×GL(2,𝔽3)—21no—
9#88789—41nomaximal
10#64367D311yescharacteristic, complemented
11#1010⋊C3—34no—
12C3×#986D314yescomplemented
13C32×AGL(1,ℤ/8)—31no—
14#64145—31no—
15#64141—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×D12—41no—
23C32×SL(2,𝔽3)D411yescharacteristic
24C3⋊(C3⋊(C3⋊C8))—41no—
25#5431—91no—
26#5201—94no—
27C32×#41C2×D311yescharacteristic
28C2×C32×D4—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×GL(2,𝔽3)—24no—
37C32×SD16—61no—
38C32×D8—61no—
39C32×M16—31no—
40#3757—62no—
41#3754—61no—
42#3753—61no—
43C4×C3⋊Dic3—31no—
44#3748—31no—
45#3713—124no—
46C2×C3×C3⋊D3—81no—
47C2×C32×D3—81no—
48C33×C4—41no—
49#1010—34no—
502−1+4⋊C3—91no—
51#986C3⋊D311yescharacteristic, complemented
52C3×AGL(1,ℤ/8)—34no—
53#936—94no—
54#932—94no—
55GU(2,𝔽3)—91no—
56GU(2,𝔽3)—184no—
57#837—94no—
58#825—94no—
59#824—94no—
60C23×C32—61no—
61C32×Q8#6711yescharacteristic
62C32×D4—31no—
63C32×D4—61no—
64C32×D4—61no—
65C2×C32×C4—31no—
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—44no—
74C3×D12—124no—
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#454—271noSylow
82C2×C33—41no—
83C3×C3⋊D3—161no—
84C32×D3—161no—
85C3×#41D3214yes—
86C2×C3×D4—34no—
87D3×Q8—64no—
88#237—184no—
89#235—94no—
90C2×Dic6—94no—
91#41⋊C3—24no—
92#41⋊C3C2×C3⋊D311yescharacteristic
93C2×SL(2,𝔽3)—181no—
94GL(2,𝔽3)—21no—
95C3×SD16—64no—
96C3×D8—64no—
97C3×M16—34no—
98#217—188no—
99#214—184no—
100#213—184no—
101#213—361no—
102C4×Dic3—94no—
103#208—94no—
104C22×C32—31no—
105C22×C32—122no—
106C22×C32—121no—
107C2×C3⋊D3—21no—
108C2×C3×D3—84no—
109C2×C3×D3—244no—
110C32×C4—31no—
111C32×C4—44no—
112C32×C4—84no—
113C32×C4C2×S411yescharacteristic
114C3⋊Dic3—61no—
115C3⋊Dic3—121no—
1162−1+4—91no—
117AGL(1,ℤ/8)—31no—
118AGL(1,ℤ/8)—271no—
119#124—271no—
120#104—271no—
121#104—271no—
122#101—271no—
123C33—41noSylow
124C23×C3—64no—
125C3×Q8#62514yes—
126C3×D4—34no—
127C3×D4—64no—
128C3×D4—64no—
129C3×D4—361no—
130C2×C3×C4—34no—
131#67—184no—
132C2×Dic3—98no—
133C2×Dic3—184no—
134D12—34no—
135D12—41no—
136C4×D3—184no—
137Dic6—94no—
138Dic6—184no—
139SL(2,𝔽3)—24no—
140SL(2,𝔽3)#63811yescharacteristic
141C3×C8—64no—
142C3⋊C8—184no—
143C3⋊C8—361no—
144C3⋊C8—724no—
145C2×C32—44no—
146C2×C32—61no—
147C2×C32—84no—
148C2×C32—241no—
149C2×C32GL(2,ℤ/4)11yescharacteristic
150C3⋊D3—41no—
151C3×D3—164no—
152C3×D3—484no—
153#41—271no—
154#41—541no—
155#41D3×C3⋊D311yescharacteristic
156C2×Q8—181no—
157C2×Q8—271no—
158C2×D4—31no—
159SD16—61no—
160SD16—541no—
161D8—61no—
162D8—541no—
163M16—31no—
164M16—271no—
165C22⋊C4—542no—
166C42—271no—
167C22×C3—34no—
168C22×C3—128no—
169C22×C3—124no—
170C22×C3—721no—
171C2×D3—64no—
172C2×D3—81no—
173C3×C4—34no—
174C3×C4—41no—
175C3×C4—84no—
176C3×C4D3×S414yes—
177Dic3—184no—
178Dic3—364no—
179C32—44no—
180C32—84no—
181C32#543111yescharacteristic, complemented
182C23—61no—
183Q8—271no—
184Q8—541no—
185Q8#629011yescharacteristic
186D4—31no—
187D4—61no—
188D4—61no—
189D4—91no—
190D4—541no—
191C2×C4—31no—
192C2×C4—272no—
193C2×C4—541no—
194C2×C4—541no—
195C8—61no—
196C8—541no—
197C2×C3—41no—
198C2×C3—64no—
199C2×C3—84no—
200C2×C3—244no—
201C2×C3—1441no—
202C2×C3#6420914yes—
203D3—124no—
204D3—161no—
205C22—31no—
206C22—122no—
207C22—121no—
208C22—181no—
209C4—31no—
210C4—541no—
211C4—1081no—
212C4C3⋊D3×S411yescharacteristic
213C3—41no—
214C3—84no—
215C3#1059322914yescomplemented
216C2—61no—
217C2—241no—
218C2—361no—
219C2#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. #11724493C2semidirect
2. C32×#986C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
10. #64367D3semidirect
12. C3×#986D3semidirect
51. #986C3⋊D3semidirect
181. C32#5431semidirect
215. C3#10593229semidirect

Maximal subgroups

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

subgroupindex
#117244932
C32×#9862
C3⋊(C3⋊GU(2,𝔽3))2
#105932293
#105929003
#887894