Finite Group Databasebeta

← #13161472

Subgroups — #13161472

Order 1728.

426 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 71 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#11723654C212yesmaximal, complemented
2C3×#63627C212yesmaximal, complemented
3C3×C16×C3⋊D3C211yesmaximal, characteristic, complemented
4#11722069C211yesmaximal, characteristic, complemented
5#11722067C211yesmaximal, characteristic, complemented
6#10590245—31nomaximal
7#10590242—32nomaximal
8#89041—22no—
9C3×#3746C2212yes—
10C3×#3745C2212yes—
11C3×C8×C3⋊D3C2211yescharacteristic
12#88811—22no—
13C33×C16C2211yescharacteristic
14C3×C3⋊(C3⋊C16)C2211yescharacteristic
15#63810—32no—
16#63808—32no—
17#63807—32no—
18#63627D312yescomplemented
19C16×C3⋊D3D311yescharacteristic, complemented
20C3×#800—32no—
21C3×#800—32no—
22C3×#800—62no—
23C3×D3×C16—32no—
24C3×D3×C16—61no—
25#63551—32no—
26#63550—31no—
27#63546—32no—
28#63545—31no—
29C3×C4⋊(C3⋊D3)—22no—
30C3×C4×C3⋊D3D411yescharacteristic
31C3×#634—22no—
32#6295—42no—
33#6294—42no—
34C33×C8C2311yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36#4916—92no—
37#4904—91no—
38#3799—62no—
39#3796—62no—
40#3795—62no—
41#3746C2×D312yes—
42#3745C2×D312yes—
43C8×C3⋊D3C2×D311yescharacteristic
44C3×Dic12—32no—
45C3×Dic12—32no—
46C3×Dic12—62no—
47C3×D24—32no—
48C3×D24—32no—
49C3×D24—62no—
50C3×D3×C8—32no—
51C3×D3×C8—61no—
52#3717—62no—
53#3716—62no—
54#3715—62no—
55C32×C16—21no—
56C32×C16—22no—
57C32×C16—41no—
58C32×C16C2×D312yes—
59C32×C16C2×D311yescharacteristic
60C3⋊(C3⋊C16)C2×D311yescharacteristic
61C3×C3⋊C16—32no—
62C3×C3⋊C16—61no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3D811yescharacteristic
65C33×C4C2×D411yescharacteristic
66C3×C3⋊Dic3—42no—
67C3×C3⋊Dic3D811yescharacteristic
68#919—92no—
69#912—92no—
70#904—92no—
71C3×SD32—92no—
72C2×C3×C16—91no—
73#829—92no—
74#826—92no—
75Dic24—91no—
76#800—32no—
77#800—32no—
78#800—62no—
79D48—91no—
80D3×C16—32no—
81D3×C16—61no—
82C4⋊(C3⋊D3)—22no—
83C4×C3⋊D3D1211yescharacteristic
84#634—22no—
85C3×D12—62no—
86C3×D12—62no—
87C3×D12—122no—
88C3×C4×D3—32no—
89C3×C4×D3—61no—
90C3×Dic6—62no—
91C3×Dic6—62no—
92C3×Dic6—122no—
93#626—122no—
94#626—122no—
95#625—122no—
96#624—122no—
97D3×Dic3—122no—
98D3×Dic3—122no—
99C32×C8—21no—
100C32×C8—22no—
101C32×C8—41no—
102C32×C8C22×D311yescharacteristic
103C32×C8C22×D312yes—
104C3⋊(C3⋊C8)D1211yescharacteristic
105C3×C3⋊C8—32no—
106C3×C3⋊C8—61no—
107#508—271noSylow
108C2×C33C2×D811yescharacteristic
109C3×C3⋊D3—21no—
110C3×C3⋊D3—82no—
111#239—182no—
112#237—182no—
113#235—182no—
114C3×Q16—92no—
115C3×D8—92no—
116C2×C3×C8—91no—
117#215—182no—
118#214—182no—
119Dic12—32no—
120Dic12—32no—
121Dic12—62no—
122D24—32no—
123D24—32no—
124D24—62no—
125#204—182no—
126D3×C8—32no—
127D3×C8—61no—
128C3×C16—21no—
129C3×C16—22no—
130C3×C16—41no—
131C3×C16—91no—
132C3×C16D3212yes—
133C3×C16D3211yescharacteristic
134C3⋊C16—32no—
135C3⋊C16—61no—
136C2×C3⋊D3—42no—
137C2×C3⋊D3D2411yescharacteristic
138C2×C3×D3—32no—
139C2×C3×D3—61no—
140C2×C3×D3—122no—
141C2×C3×D3—122no—
142C2×C3×D3—242no—
143C32×C4—22no—
144C32×C4—21no—
145C32×C4—41no—
146C32×C4C2×D1211yescharacteristic
147C32×C4D3×D412yes—
148C3⋊Dic3—42no—
149C3⋊Dic3D2411yescharacteristic
150C3×Dic3—32no—
151C3×Dic3—61no—
152C3×Dic3—122no—
153C3×Dic3—122no—
154C3×Dic3—242no—
155#135—272no—
156Q32—271no—
157SD32—92no—
158D16—271no—
159C2×C16—91no—
160C33#50811yescharacteristic, complemented, Sylow
161C3×Q8—182no—
162C3×D4—182no—
163C2×C3×C4—91no—
164#67—362no—
165#67—362no—
166C2×Dic3—362no—
167D12—62no—
168D12—62no—
169D12—122no—
170D12—362no—
171C4×D3—32no—
172C4×D3—61no—
173C4×D3—362no—
174C4×D3—362no—
175Dic6—62no—
176Dic6—62no—
177Dic6—122no—
178C3×C8—21no—
179C3×C8—22no—
180C3×C8—41no—
181C3×C8—91no—
182C3×C8C2×D3212yes—
183C3×C8C2×D3211yescharacteristic
184C3⋊C8—32no—
185C3⋊C8—61no—
186C2×C32—22no—
187C2×C32—21no—
188C2×C32—41no—
189C2×C32C2×D2411yescharacteristic
190C2×C32D3×D812yes—
191C3⋊D3—21no—
192C3⋊D3—82no—
193C3×D3—62no—
194C3×D3—62no—
195C3×D3—242no—
196C3×D3—242no—
197C3×D3—482no—
198#41—542no—
199Q16—92no—
200SD16—542no—
201D8—92no—
202C2×C8—91no—
203C16—91no—
204C16#111311yescharacteristic
205C22×C3—91no—
206C22×C3—362no—
207C2×D3—32no—
208C2×D3—61no—
209C2×D3—122no—
210C2×D3—122no—
211C2×D3—242no—
212C3×C4—21no—
213C3×C4—22no—
214C3×C4—41no—
215C3×C4—91no—
216C3×C4—362no—
217C3×C4D3×D1212yes—
218C3×C4#380211yescharacteristic
219Dic3—32no—
220Dic3—61no—
221Dic3—122no—
222Dic3—122no—
223Dic3—242no—
224C32—22no—
225C32—21no—
226C32—41no—
227C32#490411yescharacteristic, complemented
228C32#491612yescomplemented
229Q8—182no—
230D4—182no—
231D4—1082no—
232C2×C4—91no—
233C2×C4—1082no—
234C8—91no—
235C8C2×#111311yescharacteristic
236C2×C3—21no—
237C2×C3—22no—
238C2×C3—41no—
239C2×C3—181no—
240C2×C3—722no—
241C2×C3D3×D2412yes—
242C2×C3#6379711yescharacteristic
243D3—62no—
244D3—62no—
245D3—242no—
246D3—242no—
247D3—482no—
248C22—91no—
249C22—362no—
250C4—91no—
251C4—362no—
252C4C4⋊#111311yescharacteristic
253C3—21no—
254C3—22no—
255C3—41no—
256C3#1059024212yescomplemented
257C3#1059024511yescharacteristic, complemented
258C2—181no—
259C2—722no—
260C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
hypercenterC16#1113
hypocenterC33#508
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
1. #11723654C2semidirect
2. C3×#63627C2semidirect
3. C3×C16×C3⋊D3C2semidirect
4. #11722069C2semidirect
5. #11722067C2semidirect
18. #63627D3semidirect
19. C16×C3⋊D3D3semidirect
160. C33#508semidirect
227. C32#4904semidirect
228. C32#4916semidirect
256. C3#10590242semidirect
257. C3#10590245semidirect

Maximal subgroups

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

subgroupindex
#117236542
C3×#636272
C3×C16×C3⋊D32
#117220692
#117220672
#105902453
#105902423