Finite Group Databasebeta

← #13143998

Subgroups — #13143998

Order 1728.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721861C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic
3C2×#88530C211yesmaximal, characteristic
4#11721755C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic
6#11720547C211yesmaximal, characteristic
7#11720532C211yesmaximal, characteristic
8C3×#4795—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88550—22no—
15#88540—21no—
16#88540—21no—
17#88540—21no—
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88530—22no—
22C3×#944—34no—
23C2×C3×#216—34no—
24C2×C3×Dic12—34no—
25C3×#887—34no—
26C32×#103D314yes—
27C3×Dic3⋊C8—34no—
28C3×#810—34no—
29Q8×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38#6229—21no—
39#6229—21no—
40#6229—21no—
41#6229D412yes—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44C3×#462—91no—
45#4795—34no—
46C2×C32×Q8C2×D314yes—
47C2×C3×Dic6—34no—
48C2×C3×Dic6—34no—
49C2×C32×C8C2×D314yes—
50C32×C4⋊C4C2×D314yes—
51C3×#216—68no—
52C3×Dic3⋊C4—64no—
53C3×Dic3⋊C4—64no—
54C3×Dic3⋊C4—64no—
55C3×C4×Dic3—34no—
56C3×C4×Dic3—64no—
57C2×C3×C3⋊C8—34no—
58C3×Dic12—68no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C4×H(3,𝔽3)—41no—
61C4×H(3,𝔽3)—41no—
62C4×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—22no—
65#1084—21no—
66#1084—41no—
67#1084Q1612yescomplemented
68C2×C3×Q16—91no—
69C2×C3×Q16—91no—
70C3×C4⋊Q8—91no—
71C3×C4×Q8—91no—
72#944—34no—
73C2×#216—34no—
74C2×Dic12—34no—
75#887—34no—
76C3×C4⋊C8—91no—
77C3×#103—34no—
78C3×#103—91no—
79C3×#103C3⋊D311yescharacteristic
80Dic3⋊C8—34no—
81#810—34no—
82C32×Q8—24no—
83C32×Q8—24no—
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4C22×D314yes—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—64no—
91C3×Dic6—38no—
92C3×Dic6—64no—
93C3×Dic6—64no—
94C3×Dic6—64no—
95C32×C8—24no—
96C3×C3⋊C8—64no—
97#462—91noSylow
98C2×H(3,𝔽3)#12111yescharacteristic
99C2×H(3,𝔽3)C2×Q1611yescharacteristic
100C2×H(3,𝔽3)#13711yescharacteristic
101C2×C3×Q8—34no—
102C2×C3×Q8—91no—
103C2×C3×Q8—91no—
104C2×C3×Q8C2×C3⋊D311yescharacteristic
105C2×Dic6—34no—
106C2×Dic6—34no—
107C3×Q16—182no—
108C3×Q16—182no—
109C2×C3×C8—34no—
110C2×C3×C8—91no—
111C2×C3×C8C2×C3⋊D311yescharacteristic
112C3×C4⋊C4—34no—
113C3×C4⋊C4—181no—
114C3×C4⋊C4—181no—
115C3×C4⋊C4—181no—
116C3×C4⋊C4C2×C3⋊D311yescharacteristic
117C3×C42—91no—
118C3×C42—181no—
119#216—68no—
120Dic3⋊C4—64no—
121Dic3⋊C4—64no—
122Dic3⋊C4—64no—
123C4×Dic3—34no—
124C4×Dic3—64no—
125C2×C3⋊C8—34no—
126Dic12—68no—
127C22×C32D3×D414yes—
128C32×C4—44no—
129C32×C4—44no—
130C32×C4#23514yes—
131C32×C4D3×D414yes—
132C3×Dic3—38no—
133C3×Dic3—68no—
134C3×Dic3—64no—
135C3×Dic3—124no—
136C2×Q16—91no—
137C2×Q16—91no—
138C4⋊Q8—91no—
139C4×Q8—91no—
140C4⋊C8—91no—
141#103—91no—
142#103#27211yescharacteristic
143H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
144C3×Q8—21no—
145C3×Q8—21no—
146C3×Q8—64no—
147C3×Q8—64no—
148C3×Q8—92no—
149C3×Q8—181no—
150C3×Q8—181no—
151C3×Q8—181no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—34no—
155C2×C3×C4—64no—
156C2×C3×C4—64no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—181no—
160C2×C3×C4—181no—
161C2×C3×C4C22×C3⋊D311yescharacteristic
162C2×Dic3—34no—
163C2×Dic3—34no—
164C2×Dic3—64no—
165C2×Dic3—64no—
166Dic6—38no—
167Dic6—64no—
168Dic6—64no—
169Dic6—64no—
170C3×C8—21no—
171C3×C8—64no—
172C3×C8—181no—
173C3⋊C8—64no—
174C2×C32#88414yes—
175C2×C32#91514yes—
176C2×C32D3×Q1614yes—
177C2×Q8—91no—
178C2×Q8—91no—
179C2×Q8C2×#27211yescharacteristic
180Q16—182no—
181Q16—182no—
182C2×C8—91no—
183C2×C8C2×#27211yescharacteristic
184C4⋊C4—181no—
185C4⋊C4—181no—
186C4⋊C4—181no—
187C4⋊C4C2×#27211yescharacteristic
188C42—91no—
189C42—181no—
190C22×C3—34no—
191C22×C3D4×C3⋊D311yescharacteristic
192C3×C4—34no—
193C3×C4—34no—
194C3×C4—41no—
195C3×C4—41no—
196C3×C4—92no—
197C3×C4—124no—
198C3×C4—124no—
199C3×C4—182no—
200C3×C4—181no—
201C3×C4—361no—
202C3×C4#382811yescharacteristic
203C3×C4D4×C3⋊D311yescharacteristic
204Dic3—38no—
205Dic3—68no—
206Dic3—64no—
207Dic3—124no—
208C32#479514yescomplemented
209Q8—21no—
210Q8—21no—
211Q8—92no—
212Q8—181no—
213Q8—181no—
214Q8—181no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—91no—
218C2×C4—91no—
219C2×C4—181no—
220C2×C4—181no—
221C2×C4C22×#27211yescharacteristic
222C8—21no—
223C8—181no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3#6409311yescharacteristic
228C2×C3#6412411yescharacteristic
229C2×C3Q16×C3⋊D311yescharacteristic
230C22D4×#27211yescharacteristic
231C4—41no—
232C4—41no—
233C4—92no—
234C4—182no—
235C4—181no—
236C4—361no—
237C4#8874011yescharacteristic
238C4D4×#27211yescharacteristic
239C3—34no—
240C3#1059249411yescharacteristic
241C2#1172174011yescharacteristic
242C2#1172180111yescharacteristic
243C2Q16×#27211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
67. #1084Q16semidirect
143. H(3,𝔽3)#462semidirect
208. C32#4795semidirect

Maximal subgroups

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

subgroupindex
#117218612
C2×#885502
C2×#885302
#117217552
H(3,𝔽3)×#1032
#117205472
#117205322
C3×#47953