Finite Group Databasebeta

← #13161468

Subgroups — #13161468

Order 1728.

486 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 75 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#11723647C212yesmaximal, complemented
2C3×#63626C212yesmaximal, complemented
3C3×C16×C3⋊D3C211yesmaximal, characteristic, complemented
4#11722067C212yesmaximal, complemented
5#10590234—31nomaximal
6D3×D48—32nomaximal
7C4⋊#1113—22no—
8C3×#3745C2214yescomplemented
9C3×C8×C3⋊D3C2211yescharacteristic
10#88810—22no—
11C33×C16C2211yescharacteristic, complemented
12C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
13#63797—32no—
14D3×D24—34no—
15#63626D312yescomplemented
16C16×C3⋊D3D311yescharacteristic, complemented
17C3×D48—32no—
18C3×D48—32no—
19C3×D48—62no—
20C3×D3×C16—32no—
21C3×D3×C16—61no—
22#63546—34no—
23#63545—32no—
24C2×#1113—42no—
25C3×C4⋊(C3⋊D3)—24no—
26C3×C4×C3⋊D3D411yescharacteristic
27#6295—42no—
28C33×C8C2311yescharacteristic
29C3×C3⋊(C3⋊C8)D411yescharacteristic
30D3×D16—92no—
31C2×D48—91no—
32#3802—62no—
33D3×D12—64no—
34#3745C2×D314yescomplemented
35C8×C3⋊D3C2×D311yescharacteristic
36C3×D24—34no—
37C3×D24—34no—
38C3×D24—64no—
39C3×D3×C8—32no—
40C3×D3×C8—61no—
41#3714—64no—
42#3713—62no—
43C32×C16—21no—
44C32×C16—22no—
45C32×C16—41no—
46C32×C16C2×D312yescomplemented
47C32×C16C2×D311yescharacteristic, complemented
48C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
49C3×C3⋊C16—32no—
50C3×C3⋊C16—61no—
51C2×C3×C3⋊D3—44no—
52C2×C3×C3⋊D3D811yescharacteristic
53#1113—84no—
54C33×C4C2×D411yescharacteristic
55C3×C3⋊Dic3D811yescharacteristic
56D3×D8—94no—
57C2×D24—92no—
58C3×D16—92no—
59C2×C3×C16—91no—
60#826—94no—
61D48—32no—
62D48—32no—
63D48—62no—
64D48—92no—
65D3×C16—32no—
66D3×C16—61no—
67C2×D32—124no—
68C2×D32—122no—
69C4⋊(C3⋊D3)—24no—
70C4×C3⋊D3D1211yescharacteristic
71C3×D12—64no—
72C3×D12—64no—
73C3×D12—124no—
74C3×C4×D3—32no—
75C3×C4×D3—61no—
76#626—124no—
77#625—122no—
78C32×C8—21no—
79C32×C8—22no—
80C32×C8—41no—
81C32×C8C22×D311yescharacteristic
82C32×C8C22×D312yes—
83C3⋊(C3⋊C8)D1211yescharacteristic
84C3×C3⋊C8—32no—
85C3×C3⋊C8—61no—
86C2×D16—271noSylow
87C2×C33C2×D811yescharacteristic
88C3×C3⋊D3—84no—
89C3×C3⋊D3D1612yescomplemented
90D3×D4—184no—
91C2×D12—182no—
92C3×D8—94no—
93C2×C3×C8—91no—
94#213—184no—
95D24—34no—
96D24—34no—
97D24—64no—
98D24—182no—
99D3×C8—32no—
100D3×C8—61no—
101C3×C16—21no—
102C3×C16—22no—
103C3×C16—41no—
104C3×C16—91no—
105C3×C16D3212yescomplemented
106C3×C16D3211yescharacteristic, complemented
107C3⋊C16—32no—
108C3⋊C16—61no—
109C2×C3⋊D3—44no—
110C2×C3⋊D3D2411yescharacteristic
111C2×C3×D3—32no—
112C2×C3×D3—61no—
113C2×C3×D3—124no—
114C2×C3×D3—124no—
115C2×C3×D3—244no—
116D32—248no—
117D32—244no—
118C32×C4—22no—
119C32×C4—21no—
120C32×C4—41no—
121C32×C4C2×D1211yescharacteristic
122C32×C4D3×D412yes—
123C3⋊Dic3D2411yescharacteristic
124C3×Dic3—32no—
125C3×Dic3—61no—
126C2×D8—272no—
127D16—92no—
128D16—272no—
129C2×C16—91no—
130C33C2×D1611yescharacteristic, complemented, Sylow
131C22×D3—364no—
132C22×D3—362no—
133C3×D4—184no—
134C2×C3×C4—91no—
135#67—364no—
136D12—64no—
137D12—64no—
138D12—124no—
139D12—362no—
140C4×D3—32no—
141C4×D3—61no—
142C3×C8—21no—
143C3×C8—22no—
144C3×C8—41no—
145C3×C8—91no—
146C3×C8C2×D3212yes—
147C3×C8C2×D3211yescharacteristic
148C3⋊C8—32no—
149C3⋊C8—61no—
150C2×C32—22no—
151C2×C32—21no—
152C2×C32—41no—
153C2×C32C2×D2411yescharacteristic
154C2×C32D3×D812yes—
155C3⋊D3—84no—
156C3⋊D3D4812yescomplemented
157C3×D3—34no—
158C3×D3—62no—
159C3×D3—244no—
160C3×D3—244no—
161C3×D3—484no—
162C2×D4—542no—
163D8—94no—
164D8—542no—
165C2×C8—91no—
166C16—91no—
167C16#111311yescharacteristic, complemented
168C22×C3—91no—
169C22×C3—364no—
170C2×D3—32no—
171C2×D3—61no—
172C2×D3—124no—
173C2×D3—124no—
174C2×D3—244no—
175C2×D3—728no—
176C2×D3—724no—
177C3×C4—21no—
178C3×C4—22no—
179C3×C4—41no—
180C3×C4—91no—
181C3×C4D3×D1212yes—
182C3×C4#380211yescharacteristic
183Dic3—32no—
184Dic3—61no—
185C32—22no—
186C32—21no—
187C32—41no—
188C32C2×D4811yescharacteristic, complemented
189C32D3×D1612yescomplemented
190C23—1082no—
191D4—184no—
192D4—1082no—
193C2×C4—91no—
194C8—91no—
195C8C2×#111311yescharacteristic
196C2×C3—21no—
197C2×C3—22no—
198C2×C3—41no—
199C2×C3—92no—
200C2×C3—724no—
201C2×C3D3×D2412yes—
202C2×C3#6379711yescharacteristic
203D3—34no—
204D3—62no—
205D3—244no—
206D3—244no—
207D3—484no—
208C22—91no—
209C22—364no—
210C22—2164no—
211C4—91no—
212C4C4⋊#111311yescharacteristic
213C3—21no—
214C3—22no—
215C3—41no—
216C3D3×D4812yescomplemented
217C3#1059023411yescharacteristic, complemented
218C2—92no—
219C2—724no—
220C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
hypercenterC16#1113
hypocenterC33C2×D16
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. #11723647C2semidirect
2. C3×#63626C2semidirect
3. C3×C16×C3⋊D3C2semidirect
4. #11722067C2semidirect
8. C3×#3745C22semidirect
11. C33×C16C22semidirect
12. C3×C3⋊(C3⋊C16)C22semidirect
15. #63626D3semidirect
16. C16×C3⋊D3D3semidirect
34. #3745C2×D3semidirect
46. C32×C16C2×D3semidirect
47. C32×C16C2×D3semidirect
48. C3⋊(C3⋊C16)C2×D3semidirect
89. C3×C3⋊D3D16semidirect
105. C3×C16D32semidirect
106. C3×C16D32semidirect
130. C33C2×D16semidirect
156. C3⋊D3D48semidirect
167. C16#1113semidirect
188. C32C2×D48semidirect
189. C32D3×D16semidirect
216. C3D3×D48semidirect
217. C3#10590234semidirect

Maximal subgroups

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

subgroupindex
#117236472
C3×#636262
C3×C16×C3⋊D32
#117220672
#105902343
D3×D483