Finite Group Databasebeta

← C22×#88949 #13181319

Subgroups — C22×#88949 #13181319

Order 1728.

6210 conjugacy classes of subgroups; only the 1132 normal subgroups are shown.

normal subgroups only

This group has 6210 conjugacy classes of subgroups — too many to draw. Only its 1132 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
1C23×#1113C211yesmaximal, characteristic
2C22×Dic3×C3⋊D3C213yesmaximal, complemented
3C22×C3×#624C213yesmaximal, complemented
4C2×#88949C2124yesmaximal, direct product factor
5C22×C4×D32—33nomaximal
6C23×C3×C3⋊D3C2213yes—
7C22×#1113C4116yescomplemented
8C22×#1113C22112yes—
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
10C22×C32×Dic3C2213yescomplemented
11C2×Dic3×C3⋊D3C22136yescomplemented
12C2×C3×#624C22136yescomplemented
13#88949C22164yesdirect product factor
14C23×D32—33no—
15C22×C4×C3⋊D3—33no—
16C22×C3×C4×D3—36no—
17C22×#624D313yescomplemented
18C22×D3×Dic3—36no—
19C22×D3×Dic3—63no—
20C2×C4×D32—372no—
21C23×C33C2311yescharacteristic
22C22×C3×C3⋊D3C2×C4124yescomplemented
23C22×C3×C3⋊D3C23118yes—
24C2×#1113C2×C4196yescomplemented
25C2×#1113C23116yes—
26C2×C3⋊(C3⋊Dic3)C2316yescomplemented
27C2×C32×Dic3C23118yescomplemented
28Dic3×C3⋊D3C23148yescomplemented
29C3×#624C23148yescomplemented
30C23×C4×D3—93no—
31C23×C3⋊D3C2×D313yes—
32C23×C3×D3—36no—
33C23×C3×D3—63no—
34C22×D32—348no—
35C22×D32—336no—
36C22×C32×C4—33no—
37C22×C3⋊Dic3—33no—
38C22×C3⋊Dic3—63no—
39C22×C3⋊Dic3—121no—
40C2×C4×C3⋊D3—336no—
41C22×C3×Dic3—23no—
42C22×C3×Dic3C2×D316yescomplemented
43C2×C3×C4×D3—372no—
44C2×#624C2×D3136yescomplemented
45C2×D3×Dic3—372no—
46C2×D3×Dic3—636no—
47C4×D32—3192no—
48C22×C33C22×C414yescomplemented
49C22×C33C2413yes—
50C2×C3×C3⋊D3C22×C4172yescomplemented
51C2×C3×C3⋊D3C24112yes—
52#1113C22×C4164yescomplemented
53C3⋊(C3⋊Dic3)C2414yescomplemented
54C32×Dic3C24112yescomplemented
55C24×D3—93no—
56C23×C3×C4—93no—
57C23×Dic3—93no—
58C22×C4×D3—36no—
59C22×C4×D3—972no—
60C22×C4×D3—96no—
61C22×C4×D3—183no—
62C23×C32—23no—
63C23×C32—41no—
64C23×C32C22×D313yes—
65C22×C3⋊D3C4×D3124yescomplemented
66C22×C3⋊D3C22×D3118yes—
67C22×C3×D3—348no—
68C22×C3×D3—336no—
69C22×C3×D3—624no—
70C22×C3×D3—618no—
71C2×D32—3288no—
72C2×D32—348no—
73C2×C32×C4—318no—
74C2×C3⋊Dic3—318no—
75C2×C3⋊Dic3—618no—
76C2×C3⋊Dic3—126no—
77C4×C3⋊D3—348no—
78C2×C3×Dic3—218no—
79C2×C3×Dic3C22×D3136yescomplemented
80C3×C4×D3—396no—
81#624C22×D3148yescomplemented
82D3×Dic3—396no—
83D3×Dic3—648no—
84C24×C4—271noSylow
85C2×C33C23×C416yescomplemented
86C2×C33C2511yescharacteristic
87C3×C3⋊D3C23×C4124yescomplemented
88C24×C3—93no—
89C23×D3—36no—
90C23×D3—63no—
91C23×D3—936no—
92C23×D3—948no—
93C22×C3×C4—36no—
94C22×C3×C4—63no—
95C22×C3×C4—936no—
96C22×Dic3—93no—
97C22×Dic3—936no—
98C22×Dic3—183no—
99C22×Dic3—361no—
100C22×Dic3D3213yescomplemented
101C2×C4×D3—372no—
102C2×C4×D3—9192no—
103C2×C4×D3—972no—
104C2×C4×D3—1836no—
105C22×C32—29no—
106C22×C32—212no—
107C22×C32—44no—
108C22×C32—43no—
109C22×C32C2×C4×D3112yescomplemented
110C22×C32C23×D319yes—
111C2×C3⋊D3C2×C4×D3172yescomplemented
112C2×C3⋊D3C23×D3112yes—
113C2×C3×D3—3144no—
114C2×C3×D3—324no—
115C2×C3×D3—672no—
116C2×C3×D3—612no—
117D32—3192no—
118C32×C4—312no—
119C3⋊Dic3—312no—
120C3⋊Dic3—612no—
121C3⋊Dic3—124no—
122C3×Dic3—212no—
123C3×Dic3C23×D3124yescomplemented
124C25—271no—
125C23×C4—93no—
126C23×C4—273no—
127C23×C4—2724no—
128C33C24×C411yescharacteristic, complemented, Sylow
129C23×C3—23no—
130C23×C3—41no—
131C23×C3—918no—
132C23×C3—924no—
133C23×C3C2×D3213yes—
134C22×D3—336no—
135C22×D3—348no—
136C22×D3—624no—
137C22×D3—618no—
138C22×D3—948no—
139C22×D3—9288no—
140C2×C3×C4—336no—
141C2×C3×C4—618no—
142C2×C3×C4—948no—
143C2×Dic3—918no—
144C2×Dic3—948no—
145C2×Dic3—1818no—
146C2×Dic3—366no—
147C2×Dic3C2×D32118yescomplemented
148C4×D3—396no—
149C4×D3—996no—
150C4×D3—1848no—
151C2×C32—218no—
152C2×C32—23no—
153C2×C32—46no—
154C2×C32—41no—
155C2×C32C22×C4×D3118yescomplemented
156C2×C32C24×D313yes—
157C3⋊D3C22×C4×D3124yescomplemented
158C3×D3—348no—
159C3×D3—624no—
160C24—93no—
161C24—2712no—
162C24—2716no—
163C22×C4—33no—
164C22×C4—936no—
165C22×C4—2736no—
166C22×C4—2764no—
167C22×C4—271no—
168C22×C3—29no—
169C22×C3—212no—
170C22×C3—43no—
171C22×C3—44no—
172C22×C3—972no—
173C22×C3—912no—
174C22×C3C4×D32112yescomplemented
175C22×C3C22×D3219yes—
176C2×D3—3144no—
177C2×D3—324no—
178C2×D3—612no—
179C2×D3—672no—
180C2×D3—9192no—
181C3×C4—324no—
182C3×C4—612no—
183Dic3—912no—
184Dic3—1812no—
185Dic3—364no—
186Dic3C22×D32112yescomplemented
187C32—23no—
188C32—41no—
189C32C23×C4×D313yescomplemented
190C23—918no—
191C23—924no—
192C23—2716no—
193C23—2796no—
194C23D3311yescharacteristic
195C2×C4—318no—
196C2×C4—948no—
197C2×C4—2748no—
198C2×C4—276no—
199C2×C3—218no—
200C2×C3—23no—
201C2×C3—46no—
202C2×C3—41no—
203C2×C3—924no—
204C2×C3C2×C4×D32118yescomplemented
205C2×C3C23×D3213yes—
206D3—348no—
207D3—624no—
208C22—972no—
209C22—912no—
210C22—2764no—
211C22#8894914yesdirect product factor
212C22C2×D3313yes—
213C4—312no—
214C4—274no—
215C3—23no—
216C3—41no—
217C3C22×C4×D3213yescomplemented
218C2—924no—
219C2C2×#8894916yesdirect product factor
220C2C22×D3311yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, hypocenterC33C24×C4
FrattiniC2C22×D33
Fitting, socleC23×C33C23
prime radicalC23×#1113C2
cosocleC2×C33C25
LCSG ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×C3⋊D3C2semidirect
3. C22×C3×#624C2semidirect
4. C2×#88949C2direct (N × H)
7. C22×#1113C4semidirect
9. C22×C3⋊(C3⋊Dic3)C22semidirect
10. C22×C32×Dic3C22semidirect
11. C2×Dic3×C3⋊D3C22semidirect
12. C2×C3×#624C22semidirect
13. #88949C22direct (N × H)
17. C22×#624D3semidirect
22. C22×C3×C3⋊D3C2×C4semidirect
24. C2×#1113C2×C4semidirect
26. C2×C3⋊(C3⋊Dic3)C23semidirect
27. C2×C32×Dic3C23semidirect
28. Dic3×C3⋊D3C23semidirect
29. C3×#624C23semidirect
42. C22×C3×Dic3C2×D3semidirect
44. C2×#624C2×D3semidirect
48. C22×C33C22×C4semidirect
50. C2×C3×C3⋊D3C22×C4semidirect
52. #1113C22×C4semidirect
53. C3⋊(C3⋊Dic3)C24semidirect
54. C32×Dic3C24semidirect
65. C22×C3⋊D3C4×D3semidirect
79. C2×C3×Dic3C22×D3semidirect
81. #624C22×D3semidirect
85. C2×C33C23×C4semidirect
87. C3×C3⋊D3C23×C4semidirect
100. C22×Dic3D32semidirect
109. C22×C32C2×C4×D3semidirect
111. C2×C3⋊D3C2×C4×D3semidirect
123. C3×Dic3C23×D3semidirect
128. C33C24×C4semidirect
147. C2×Dic3C2×D32semidirect
155. C2×C32C22×C4×D3semidirect
157. C3⋊D3C22×C4×D3semidirect
174. C22×C3C4×D32semidirect
186. Dic3C22×D32semidirect
189. C32C23×C4×D3semidirect
204. C2×C3C2×C4×D32semidirect
211. C22#88949direct (N × H)
217. C3C22×C4×D32semidirect
219. C2C2×#88949direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11132
C22×Dic3×C3⋊D32
C22×C3×#6242
C2×#889492
C22×C4×D323