Finite Group Databasebeta

← #13144726

Subgroups — #13144726

Order 1728.

781 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 117 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721861C211yesmaximal, characteristic, complemented
3C2×#88549C211yesmaximal, characteristic, complemented
4#11721850C211yesmaximal, characteristic
5C2×#88546C211yesmaximal, characteristic, complemented
6C2×#88528C211yesmaximal, characteristic, complemented
7C8×#1084C211yesmaximal, characteristic, complemented
8C3×#5175—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—22no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#6229C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#88549—22no—
17#88546—22no—
18#88540—22no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88528—22no—
22C2×C32×SD16D314yescomplemented
23C3×#944—34no—
24C2×C3×#215—34no—
25C3×#940—34no—
26C2×C3×#214—34no—
27C2×C3×#204—34no—
28C3×C8×Dic3—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#6235—44no—
38C2×#1084—21no—
39C2×#1084D412yes—
40C4⋊#272—21no—
41C4⋊#272—21no—
42#6229—21no—
43#6229—21no—
44C8×H(3,𝔽3)D412yescomplemented
45#6180D412yescomplemented
46C3×#492—91no—
47#5175—34no—
48C2×C32×Q8C2×D314yes—
49C2×C32×D4C2×D314yes—
50C2×C3×#67—68no—
51C2×C3×D12—34no—
52C2×C3×Dic6—34no—
53C32×SD16—28no—
54C2×C32×C8C2×D314yes—
55C3×#215—68no—
56C3×#214—68no—
57C3×Dic3⋊C4—68no—
58C3×C4×Dic3—34no—
59C2×C3×C3⋊C8—34no—
60C3×#204—68no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—42no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—41no—
65C2×#272—42no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69#1084—41no—
70#1084SD1614yescomplemented
71C2×C3×SD16—34no—
72C2×C3×SD16—91no—
73C2×C3×SD16—91no—
74C2×C3×SD16—91no—
75C2×C3×SD16C3⋊D311yescharacteristic
76C3×C4⋊Q8—91no—
77C3×#127—91no—
78#944—34no—
79C2×#215—34no—
80#940—34no—
81C2×#214—34no—
82C2×#204—34no—
83C3×C4×C8—91no—
84C8×Dic3—34no—
85C23×C32—24no—
86C22×C3×D3—64no—
87C32×Q8—24no—
88C32×Q8—24no—
89C32×D4—24no—
90C32×D4—24no—
91C2×C32×C4—24no—
92C2×C32×C4C22×D314yes—
93C3×#67—1216no—
94C2×C3×Dic3—38no—
95C2×C3×Dic3—64no—
96C3×D12—64no—
97C3×D12—64no—
98C3×Dic6—64no—
99C3×Dic6—64no—
100C32×C8#6718yescomplemented
101C3×C3⋊C8—38no—
102#492—91noSylow
103C2×H(3,𝔽3)—81no—
104C2×H(3,𝔽3)#12711yescharacteristic
105C2×H(3,𝔽3)C2×SD1612yes—
106#272—81no—
107C2×C3×Q8—34no—
108C2×C3×Q8—91no—
109C2×C3×Q8C2×C3⋊D311yescharacteristic
110C2×C3×D4—34no—
111C2×C3×D4—91no—
112C2×C3×D4—182no—
113C2×C3×D4C2×C3⋊D311yescharacteristic
114C2×#67—68no—
115C2×D12—34no—
116C2×Dic6—34no—
117C3×SD16—22no—
118C3×SD16—68no—
119C3×SD16—182no—
120C3×SD16—182no—
121C3×SD16—182no—
122C2×C3×C8—34no—
123C2×C3×C8—91no—
124C2×C3×C8C2×C3⋊D311yescharacteristic
125C3×C4⋊C4—182no—
126C3×C42—91no—
127#215—68no—
128#214—68no—
129Dic3⋊C4—68no—
130C4×Dic3—34no—
131C2×C3⋊C8—34no—
132#204—68no—
133C22×C32—48no—
134C22×C32—44no—
135C22×C32D3×D414yes—
136C2×C3×D3—128no—
137C2×C3×D3—124no—
138C32×C4—44no—
139C32×C4D3×D414yes—
140C32×C4C2×#6714yes—
141C3×Dic3—316no—
142C3×Dic3—124no—
143C2×SD16—91no—
144C2×SD16—91no—
145C2×SD16—91no—
146C2×SD16#27211yescharacteristic, complemented
147C4⋊Q8—91no—
148#127—91no—
149C4×C8—91no—
150H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
151C23×C3—21no—
152C23×C3—64no—
153C23×C3—181no—
154C22×D3—64no—
155C3×Q8—21no—
156C3×Q8—21no—
157C3×Q8—64no—
158C3×Q8—64no—
159C3×Q8—181no—
160C3×Q8—181no—
161C3×D4—21no—
162C3×D4—21no—
163C3×D4—64no—
164C3×D4—64no—
165C3×D4—181no—
166C3×D4—181no—
167C3×D4—364no—
168C2×C3×C4—21no—
169C2×C3×C4—34no—
170C2×C3×C4—64no—
171C2×C3×C4—92no—
172C2×C3×C4—181no—
173C2×C3×C4C22×C3⋊D311yescharacteristic
174#67—1216no—
175C2×Dic3—38no—
176C2×Dic3—64no—
177D12—64no—
178D12—64no—
179Dic6—64no—
180Dic6—64no—
181C3×C8—38no—
182C3×C8—92no—
183C3×C8#63812yes—
184C3⋊C8—38no—
185C2×C32—84no—
186C2×C32D3×SD1618yes—
187C2×C32#94014yes—
188C3×D3—244no—
189C2×Q8—91no—
190C2×Q8C2×#27211yescharacteristic
191C2×D4—91no—
192C2×D4—182no—
193C2×D4C2×#27211yescharacteristic
194SD16—22no—
195SD16—182no—
196SD16—182no—
197SD16—182no—
198C2×C8—91no—
199C2×C8C2×#27211yescharacteristic
200C4⋊C4—182no—
201C42—91no—
202C22×C3—34no—
203C22×C3—42no—
204C22×C3—41no—
205C22×C3—128no—
206C22×C3—124no—
207C22×C3—362no—
208C22×C3—361no—
209C22×C3D4×C3⋊D311yescharacteristic
210C2×D3—124no—
211C2×D3—128no—
212C3×C4—34no—
213C3×C4—34no—
214C3×C4—41no—
215C3×C4—94no—
216C3×C4—124no—
217C3×C4—361no—
218C3×C4D4×C3⋊D311yescharacteristic
219C3×C4C2×#63811yescharacteristic
220Dic3—316no—
221Dic3—124no—
222C32#517514yescomplemented
223C23—21no—
224C23—181no—
225Q8—21no—
226Q8—21no—
227Q8—181no—
228Q8—181no—
229D4—21no—
230D4—21no—
231D4—181no—
232D4—181no—
233D4—364no—
234C2×C4—21no—
235C2×C4—92no—
236C2×C4—181no—
237C2×C4C22×#27211yescharacteristic
238C8—92no—
239C8#623512yescomplemented
240C2×C3—38no—
241C2×C3—34no—
242C2×C3—81no—
243C2×C3—244no—
244C2×C3—721no—
245C2×C3SD16×C3⋊D312yes—
246C2×C3#6414911yescharacteristic
247D3—244no—
248C22—42no—
249C22—41no—
250C22—362no—
251C22—361no—
252C22D4×#27211yescharacteristic
253C4—41no—
254C4—94no—
255C4—361no—
256C4D4×#27211yescharacteristic
257C4C2×#623511yescharacteristic
258C3—34no—
259C3#1059287411yescharacteristic
260C2—81no—
261C2—721no—
262C2SD16×#27212yes—
263C2#1172185011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×SD16×H(3,𝔽3)C2
prime radical#11721850C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#492
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. #11721861C2semidirect
3. C2×#88549C2semidirect
5. C2×#88546C2semidirect
6. C2×#88528C2semidirect
7. C8×#1084C2semidirect
22. C2×C32×SD16D3semidirect
44. C8×H(3,𝔽3)D4semidirect
45. #6180D4semidirect
70. #1084SD16semidirect
100. C32×C8#67semidirect
146. C2×SD16#272semidirect
150. H(3,𝔽3)#492semidirect
222. C32#5175semidirect
239. C8#6235semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
#117218612
C2×#885492
#117218502
C2×#885462
C2×#885282
C8×#10842
C3×#51753