Finite Group Databasebeta

← #13144724

Subgroups — #13144724

Order 1728.

660 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 113 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
2#11721861C211yesmaximal, characteristic, complemented
3D4×#1084C211yesmaximal, characteristic
4C2×#88546C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720547C211yesmaximal, characteristic, complemented
8C3×#5162—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084—21no—
12C2×#6229C2211yescharacteristic
13SD16×H(3,𝔽3)—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88552—21no—
16#88546—22no—
17C4⋊#1084C2211yescharacteristic
18#88540—21no—
19#88540—21no—
20C4×#1084C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C2×C32×SD16D314yes—
23C3×#944—34no—
24C3×D4×Dic3—34no—
25C2×C3×#214—34no—
26C3×#835—34no—
27C3×#816—34no—
28C3×Dic3⋊C8—34no—
29C23×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084—42no—
39C2×#1084D411yescharacteristic
40C2×#1084D411yescharacteristic
41#6229—21no—
42#6229—21no—
43C8×H(3,𝔽3)—21no—
44#6180—21no—
45C3×#461—91no—
46#5162—34no—
47C2×C32×Q8C2×D314yes—
48C2×C32×D4C2×D314yes—
49C22×C3×Dic3—64no—
50C2×C3×Dic6—34no—
51C32×SD16—28no—
52C2×C32×C8C2×D314yes—
53C3×#217—64no—
54C3×#214—68no—
55C3×C4⋊Dic3—34no—
56C3×Dic3⋊C4—64no—
57C3×Dic3⋊C4—64no—
58C3×C4×Dic3—34no—
59C2×C3×C3⋊C8—34no—
60C22×H(3,𝔽3)—22no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)#4111yescharacteristic
67#1084—21no—
68#1084—41no—
69#1084—41no—
70#1084SD1612yescomplemented
71C2×C3×SD16—34no—
72C2×C3×SD16—91no—
73C2×C3×SD16C3⋊D311yescharacteristic
74C3×C4⋊Q8—91no—
75C3×C4×D4—91no—
76#944—34no—
77D4×Dic3—34no—
78C2×#214—34no—
79C3×C4⋊C8—91no—
80C3×#103—91no—
81C3×#103—91no—
82#835—34no—
83#816—34no—
84Dic3⋊C8—34no—
85C23×C32—24no—
86C32×Q8—24no—
87C32×Q8—24no—
88C32×D4—24no—
89C32×D4#6718yes—
90C2×C32×C4—24no—
91C2×C32×C4C22×D314yes—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—34no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—128no—
97C3×Dic6—64no—
98C3×Dic6—64no—
99C32×C8—24no—
100C3×C3⋊C8—64no—
101#461—91noSylow
102C2×H(3,𝔽3)—42no—
103C2×H(3,𝔽3)#12111yescharacteristic
104C2×H(3,𝔽3)C2×SD1611yescharacteristic
105C2×H(3,𝔽3)#13711yescharacteristic
106C2×C3×Q8—34no—
107C2×C3×Q8—91no—
108C2×C3×Q8C2×C3⋊D311yescharacteristic
109C2×C3×D4—34no—
110C2×C3×D4C2×C3⋊D311yescharacteristic
111C22×C3×C4—181no—
112C22×Dic3—64no—
113C2×Dic6—34no—
114C3×SD16—22no—
115C3×SD16—68no—
116C3×SD16—182no—
117C2×C3×C8—34no—
118C2×C3×C8—91no—
119C2×C3×C8C2×C3⋊D311yescharacteristic
120C3×C4⋊C4—91no—
121C3×C4⋊C4—181no—
122C3×C4⋊C4—181no—
123C3×C22⋊C4—181no—
124C3×C42—91no—
125#217—64no—
126#214—68no—
127C4⋊Dic3—34no—
128Dic3⋊C4—64no—
129Dic3⋊C4—64no—
130C4×Dic3—34no—
131C2×C3⋊C8—34no—
132C22×C32—28no—
133C22×C32—44no—
134C22×C32—44no—
135C22×C32D3×D414yes—
136C32×C4—44no—
137C32×C4#23714yes—
138C32×C4C2×#6714yes—
139C3×Dic3—38no—
140C3×Dic3—64no—
141C3×Dic3—124no—
142C3×Dic3—124no—
143C2×SD16—91no—
144C2×SD16#27211yescharacteristic
145C4⋊Q8—91no—
146C4×D4—91no—
147C4⋊C8—91no—
148#103—91no—
149#103—91no—
150H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
151C23×C3—21no—
152C23×C3—64no—
153C3×Q8—21no—
154C3×Q8—21no—
155C3×Q8—64no—
156C3×Q8—64no—
157C3×Q8—181no—
158C3×Q8—181no—
159C3×D4—21no—
160C3×D4—38no—
161C3×D4—64no—
162C3×D4#63812yes—
163C2×C3×C4—21no—
164C2×C3×C4—34no—
165C2×C3×C4—64no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—181no—
169C2×C3×C4—181no—
170C2×C3×C4—362no—
171C2×C3×C4C22×C3⋊D311yescharacteristic
172C2×Dic3—34no—
173C2×Dic3—34no—
174C2×Dic3—64no—
175C2×Dic3—64no—
176C2×Dic3—128no—
177Dic6—64no—
178Dic6—64no—
179C3×C8—21no—
180C3×C8—64no—
181C3×C8—181no—
182C3⋊C8—64no—
183C2×C32—48no—
184C2×C32D3×SD1614yes—
185C2×C32#91514yes—
186C2×C32#93914yes—
187C2×Q8—91no—
188C2×Q8C2×#27211yescharacteristic
189C2×D4C2×#27211yescharacteristic
190C22×C4—181no—
191SD16—22no—
192SD16—182no—
193C2×C8—91no—
194C2×C8C2×#27211yescharacteristic
195C4⋊C4—91no—
196C4⋊C4—181no—
197C4⋊C4—181no—
198C22⋊C4—181no—
199C42—91no—
200C22×C3—22no—
201C22×C3—34no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—68no—
205C22×C3—124no—
206C22×C3—124no—
207C22×C3D4×C3⋊D311yescharacteristic
208C3×C4—34no—
209C3×C4—34no—
210C3×C4—41no—
211C3×C4—92no—
212C3×C4—124no—
213C3×C4—181no—
214C3×C4—361no—
215C3×C4—361no—
216C3×C4#383011yescharacteristic
217C3×C4C2×#63811yescharacteristic
218Dic3—38no—
219Dic3—64no—
220Dic3—124no—
221Dic3—124no—
222C32#516214yescomplemented
223C23—21no—
224Q8—21no—
225Q8—21no—
226Q8—181no—
227Q8—181no—
228D4—21no—
229D4#623512yes—
230C2×C4—21no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—181no—
234C2×C4—181no—
235C2×C4—362no—
236C2×C4C22×#27211yescharacteristic
237C8—21no—
238C8—181no—
239C2×C3—34no—
240C2×C3—34no—
241C2×C3—34no—
242C2×C3—42no—
243C2×C3—128no—
244C2×C3SD16×C3⋊D311yescharacteristic
245C2×C3#6412411yescharacteristic
246C2×C3#6414811yescharacteristic
247C22—22no—
248C22—41no—
249C22—41no—
250C22D4×#27211yescharacteristic
251C4—41no—
252C4—92no—
253C4—181no—
254C4—361no—
255C4—361no—
256C4#8874411yescharacteristic
257C4C2×#623511yescharacteristic
258C3—34no—
259C3#1059286111yescharacteristic
260C2—42no—
261C2SD16×#27211yescharacteristic
262C2#1172180111yescharacteristic
263C2#1172184911yescharacteristic
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 radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×C3×SD16C3⋊D3
hypocenterH(3,𝔽3)#461
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
2. #11721861C2semidirect
5. #11720583C2semidirect
6. #11720549C2semidirect
7. #11720547C2semidirect
70. #1084SD16semidirect
150. H(3,𝔽3)#461semidirect
222. C32#5162semidirect

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
D4×#10842
C2×#885462
#117205832
#117205492
#117205472
C3×#51623