Finite Group Databasebeta

← #13143992

Subgroups — #13143992

Order 1728.

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

normal subgroups only

This group has 660 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, complemented
2C2×#88549C211yesmaximal, characteristic
3C2×#88528C211yesmaximal, characteristic
4#11721761C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4818—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C2×#6229C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88549—22no—
16#88542—21no—
17#88540—21no—
18#88540—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88528—22no—
22C3×#944—34no—
23C2×C3×#215—34no—
24C2×C3×#204—34no—
25C3×#893—34no—
26C32×#103D314yescomplemented
27C3×Dic3⋊C8—34no—
28C3×#810—34no—
29C22×#272—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272D412yes—
40C4×#272—42no—
41#6229—21no—
42#6229—21no—
43C8×H(3,𝔽3)—21no—
44#6180—21no—
45C3×#461—91no—
46#4818—34no—
47C2×C32×Q8C2×D314yes—
48C2×C3×D12—34no—
49C2×C3×C4×D3—64no—
50C2×C3×Dic6—34no—
51C2×C32×C8C2×D314yes—
52C32×C4⋊C4C2×D314yes—
53C3×#215—68no—
54C3×(C2×D3)⋊C4—64no—
55C3×Dic3⋊C4—64no—
56C3×Dic3⋊C4—64no—
57C3×C4×Dic3—34no—
58C2×C3×C3⋊C8—34no—
59C3×#204—68no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C2×#272—22no—
62C2×#272—41no—
63C2×#272—41no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)—41no—
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—21no—
69#1084—41no—
70#1084SD1612yescomplemented
71C2×C3×SD16—91no—
72C2×C3×SD16—91no—
73C3×C4⋊Q8—91no—
74C3×C4×D4—91no—
75#944—34no—
76C2×#215—34no—
77C2×#204—34no—
78#893—34no—
79C3×C4⋊C8—91no—
80C3×#103—34no—
81C3×#103—91no—
82C3×#103C3⋊D311yescharacteristic
83Dic3⋊C8—34no—
84#810—34no—
85C22×C3×D3—64no—
86C32×Q8—24no—
87C32×Q8—24no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4C22×D314yes—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—64no—
94C3×D12—38no—
95C3×D12—64no—
96C3×C4×D3—128no—
97C3×Dic6—64no—
98C3×Dic6—64no—
99C32×C8—24no—
100C3×C3⋊C8—64no—
101#461—91noSylow
102C2×H(3,𝔽3)#12111yescharacteristic
103C2×H(3,𝔽3)C2×SD1611yescharacteristic
104C2×H(3,𝔽3)#13711yescharacteristic
105#272—42no—
106C2×C3×Q8—34no—
107C2×C3×Q8—91no—
108C2×C3×Q8C2×C3⋊D311yescharacteristic
109C2×C3×D4—91no—
110C22×C3×C4—181no—
111C2×D12—34no—
112C2×C4×D3—64no—
113C2×Dic6—34no—
114C3×SD16—182no—
115C3×SD16—182no—
116C2×C3×C8—34no—
117C2×C3×C8—91no—
118C2×C3×C8C2×C3⋊D311yescharacteristic
119C3×C4⋊C4—34no—
120C3×C4⋊C4—181no—
121C3×C4⋊C4—181no—
122C3×C4⋊C4C2×C3⋊D311yescharacteristic
123C3×C22⋊C4—181no—
124C3×C42—91no—
125#215—68no—
126(C2×D3)⋊C4—64no—
127Dic3⋊C4—64no—
128Dic3⋊C4—64no—
129C4×Dic3—34no—
130C2×C3⋊C8—34no—
131#204—68no—
132C22×C32D3×D414yes—
133C2×C3×D3—68no—
134C2×C3×D3—124no—
135C2×C3×D3—124no—
136C32×C4—44no—
137C32×C4—44no—
138C32×C4#23514yes—
139C32×C4D3×D414yes—
140C3×Dic3—38no—
141C3×Dic3—64no—
142C3×Dic3—124no—
143C2×SD16—91no—
144C2×SD16—91no—
145C4⋊Q8—91no—
146C4×D4—91no—
147C4⋊C8—91no—
148#103—91no—
149#103#27211yescharacteristic, complemented
150H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
151C23×C3—181no—
152C22×D3—64no—
153C3×Q8—21no—
154C3×Q8—21no—
155C3×Q8—64no—
156C3×Q8—64no—
157C3×Q8—181no—
158C3×Q8—181no—
159C3×D4—92no—
160C3×D4—181no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—34no—
164C2×C3×C4—64no—
165C2×C3×C4—64no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—181no—
169C2×C3×C4—362no—
170C2×C3×C4C22×C3⋊D311yescharacteristic
171C2×Dic3—34no—
172C2×Dic3—34no—
173C2×Dic3—64no—
174D12—38no—
175D12—64no—
176C4×D3—128no—
177Dic6—64no—
178Dic6—64no—
179C3×C8—21no—
180C3×C8—64no—
181C3×C8—181no—
182C3⋊C8—64no—
183C2×C32#88414yes—
184C2×C32D3×SD1614yes—
185C2×C32#91814yes—
186C3×D3—128no—
187C2×Q8—91no—
188C2×Q8C2×#27211yescharacteristic
189C2×D4—91no—
190C22×C4—181no—
191SD16—182no—
192SD16—182no—
193C2×C8—91no—
194C2×C8C2×#27211yescharacteristic
195C4⋊C4—181no—
196C4⋊C4—181no—
197C4⋊C4C2×#27211yescharacteristic
198C22⋊C4—181no—
199C42—91no—
200C22×C3—34no—
201C22×C3—182no—
202C22×C3—361no—
203C22×C3—361no—
204C22×C3D4×C3⋊D311yescharacteristic
205C2×D3—68no—
206C2×D3—124no—
207C2×D3—124no—
208C3×C4—34no—
209C3×C4—34no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—92no—
213C3×C4—124no—
214C3×C4—124no—
215C3×C4—181no—
216C3×C4—361no—
217C3×C4#382811yescharacteristic
218C3×C4D4×C3⋊D311yescharacteristic
219Dic3—38no—
220Dic3—64no—
221Dic3—124no—
222C32#481814yescomplemented
223C23—181no—
224Q8—21no—
225Q8—21no—
226Q8—181no—
227Q8—181no—
228D4—92no—
229D4—181no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—181no—
235C2×C4—362no—
236C2×C4C22×#27211yescharacteristic
237C8—21no—
238C8—181no—
239C2×C3—34no—
240C2×C3—34no—
241C2×C3—34no—
242C2×C3—362no—
243C2×C3#6409311yescharacteristic
244C2×C3SD16×C3⋊D311yescharacteristic
245C2×C3#6412711yescharacteristic
246D3—128no—
247C22—182no—
248C22—361no—
249C22—361no—
250C22D4×#27211yescharacteristic
251C4—41no—
252C4—41no—
253C4—92no—
254C4—181no—
255C4—361no—
256C4#8874011yescharacteristic
257C4D4×#27211yescharacteristic
258C3—34no—
259C3#1059251711yescharacteristic
260C2—362no—
261C2#1172174011yescharacteristic
262C2SD16×#27211yescharacteristic
263C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#461
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
1. #11721861C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #11720547C2semidirect
7. #11720532C2semidirect
26. C32×#103D3semidirect
70. #1084SD16semidirect
149. #103#272semidirect
150. H(3,𝔽3)#461semidirect
222. C32#4818semidirect

Maximal subgroups

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

subgroupindex
#117218612
C2×#885492
C2×#885282
#117217612
H(3,𝔽3)×#1032
#117205472
#117205322
C3×#48183