Finite Group Databasebeta

← #13165960

Subgroups — #13165960

Order 1728.

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

normal subgroups only

This group has 568 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
1C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C32×#226C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593210—34nomaximal
5#10592885—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)—21no—
8C32×#226—21no—
9#88803—41nomaximal
10C2×Q8×C3⋊D3D311yescharacteristic
11C2×D3×SL(2,𝔽3)—34no—
12C2×C3×#226D314yescomplemented
13C2×C32×Q16—31no—
14#64155—31no—
15C2×#3756—31no—
16C3⋊#859—34no—
17C3⋊#859—64no—
18#63662—31no—
19#63647—31no—
20#63643—31no—
21C22×C3×C3⋊D3—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C2×C32×Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic
25#5419—91no—
26#5186—94no—
27C2×C32×Q8—31no—
28C2×C32×Q8C2×D311yescharacteristic
29Q8×C3⋊D3—21no—
30Q8×C3⋊D3—61no—
31C2×C4×C3⋊D3—31no—
32C2×#634—31no—
33C2×C3×SL(2,𝔽3)—24no—
34C2×C3×SL(2,𝔽3)C2×D314yes—
35D3×SL(2,𝔽3)—64no—
36C3×#226—24no—
37C32×Q16—61no—
38C32×Q16—61no—
39C2×C32×C8—31no—
40#3756—62no—
41#3752—61no—
42C4⋊(C3⋊Dic3)—31no—
43#3750—61no—
44C2×C3⋊(C3⋊C8)—31no—
45(C2×D3)⋊Dic3—124no—
46C22×C33—41no—
47C2×C3×C3⋊D3—42no—
48C2×C3×C3⋊D3—81no—
49C2×C3×C3⋊D3—81no—
50C3⋊(C3⋊Dic3)—81no—
51C32×Dic3—81no—
52C2×D3×Q8—34no—
53C22×SL(2,𝔽3)—91no—
54C2×#226C3⋊D311yescharacteristic, complemented
55C2×C3×Q16—34no—
56#946—94no—
57C2×#216—94no—
58#859—91no—
59#859—184no—
60#835—94no—
61(C2×D3)⋊C8—94no—
62#816—94no—
63C22×C3⋊D3S411yescharacteristic
64C22×C3×D3—124no—
65C32×Q8—31no—
66C32×Q8—61no—
67C32×Q8—61no—
68C32×Q8#6711yescharacteristic
69C2×C32×C4—31no—
70C2×C32×C4—61no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—61no—
73C2×C3⋊Dic3—124no—
74C2×C3⋊Dic3—244no—
75C4×C3⋊D3—61no—
76C4×C3⋊D3—61no—
77#634—32no—
78#634—61no—
79C2×C3×Dic3—44no—
80C3×SL(2,𝔽3)—24no—
81C3×SL(2,𝔽3)#6714yes—
82C32×C8—61no—
83C3⋊(C3⋊C8)—61no—
84#451—271noSylow
85C2×C33—41no—
86C2×C33—41no—
87C2×C33—41no—
88C3×C3⋊D3—82no—
89C2×C3×Q8—34no—
90C2×C3×Q8D3214yes—
91D3×Q8—64no—
92D3×Q8—184no—
93C2×C4×D3—94no—
94C2×Dic6—94no—
95C2×SL(2,𝔽3)—24no—
96C2×SL(2,𝔽3)—181no—
97C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
98#226—21no—
99C3×Q16—64no—
100C3×Q16—64no—
101C2×C3×C8—34no—
102#217—361no—
103#216—188no—
104(C2×D3)⋊C4—184no—
105C4⋊Dic3—94no—
106Dic3⋊C4—184no—
107C2×C3⋊C8—94no—
108C22×C32—44no—
109C22×C32—84no—
110C22×C32C2×S411yescharacteristic
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C2×C3⋊D3#22612yescomplemented
114C2×C3×D3—128no—
115C2×C3×D3—244no—
116C2×C3×D3—244no—
117C32×C4—31no—
118C32×C4—31no—
119C32×C4—121no—
120C3⋊Dic3—61no—
121C3⋊Dic3—121no—
122C3⋊Dic3—244no—
123C3⋊Dic3—248no—
124C3×Dic3—84no—
125C22×Q8—91no—
126C2×Q16—31no—
127C2×Q16—271no—
128C22⋊Q8—271no—
129#103—271no—
130#103—271no—
131C22⋊C8—271no—
132C33—41noSylow
133C23×C3—361no—
134C22×D3—34no—
135C3×Q8—34no—
136C3×Q8—64no—
137C3×Q8—64no—
138C3×Q8#62514yes—
139C2×C3×C4—34no—
140C2×C3×C4—64no—
141C2×Dic3—41no—
142C2×Dic3—94no—
143C2×Dic3—184no—
144C2×Dic3—361no—
145C2×Dic3—724no—
146C4×D3—184no—
147C4×D3—184no—
148Dic6—98no—
149Dic6—184no—
150SL(2,𝔽3)—24no—
151SL(2,𝔽3)#63811yescharacteristic
152C3×C8—64no—
153C3⋊C8—184no—
154C2×C32—44no—
155C2×C32—44no—
156C2×C32—44no—
157C2×C32—84no—
158C2×C32—84no—
159C2×C32—84no—
160C2×C32C2×#22611yescharacteristic
161C2×C32#98311yescharacteristic
162C2×C32GL(2,ℤ/4)11yescharacteristic
163C3⋊D3—22no—
164C3×D3—248no—
165C2×Q8—31no—
166C2×Q8—181no—
167C2×Q8—271no—
168C2×Q8—541no—
169C2×Q8D3×C3⋊D311yescharacteristic
170C22×C4—271no—
171Q16—61no—
172Q16—61no—
173Q16—542no—
174C2×C8—31no—
175C2×C8—271no—
176C4⋊C4—271no—
177C4⋊C4—541no—
178C22⋊C4—541no—
179C22×C3—41no—
180C22×C3—84no—
181C22×C3—362no—
182C22×C3—721no—
183C22×C3—721no—
184C22×C3D3×S414yes—
185C2×D3—38no—
186C2×D3—64no—
187C2×D3—64no—
188C3×C4—34no—
189C3×C4—34no—
190C3×C4—124no—
191Dic3—81no—
192Dic3—184no—
193Dic3—364no—
194Dic3—721no—
195Dic3—728no—
196C32—44no—
197C32—84no—
198C32#541911yescharacteristic, complemented
199C23—91no—
200Q8—31no—
201Q8—61no—
202Q8—61no—
203Q8—272no—
204Q8—541no—
205Q8#629011yescharacteristic
206C2×C4—31no—
207C2×C4—61no—
208C2×C4—271no—
209C2×C4—541no—
210C2×C4—541no—
211C2×C4—541no—
212C8—61no—
213C8—541no—
214C2×C3—41no—
215C2×C3—41no—
216C2×C3—41no—
217C2×C3—84no—
218C2×C3—84no—
219C2×C3—84no—
220C2×C3—722no—
221C2×C3D3×#22614yes—
222C2×C3#6420214yes—
223C2×C3#6420914yes—
224D3—68no—
225C22—92no—
226C22—181no—
227C22—181no—
228C22C3⋊D3×S411yescharacteristic
229C4—31no—
230C4—31no—
231C4—121no—
232C4—541no—
233C4—1081no—
234C3—41no—
235C3—84no—
236C3#1059321014yescomplemented
237C2—182no—
238C2C3⋊D3×#22611yescharacteristic
239C2#1172418211yescharacteristic
240C2#1172418911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
hypocenterC32×SL(2,𝔽3)D4
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
2. C2×C32×#226C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
12. C2×C3×#226D3semidirect
54. C2×#226C3⋊D3semidirect
113. C2×C3⋊D3#226semidirect
198. C32#5419semidirect
236. C3#10593210semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C2×C32×#2262
C3⋊(C3⋊#859)2
#105932103
#105928853
#888034