Finite Group Databasebeta

← C2×#11722814 #13170492

Subgroups — C2×#11722814 #13170492

Order 1728.

1602 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 324 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×C4×#1113C211yesmaximal, characteristic
2C2×C3⋊D3×C3⋊C8C213yesmaximal, complemented
3C2×C3×#3710C213yesmaximal, complemented
4#11722814C218yesmaximal, direct product factor
5C2×D32×C8—33nomaximal
6C22×#1113C411yescharacteristic
7C2×C3×C4×C3⋊D3C2213yes—
8C2×#6294C413yes—
9C4×#1113C414yes—
10C4×#1113C2214yes—
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
12C2×C32×C3⋊C8C2213yescomplemented
13C3⋊D3×C3⋊C8C22112yescomplemented
14C3×#3710C22112yescomplemented
15C2×C4×D32—33no—
16C2×C8×C3⋊D3—33no—
17C2×C3×D3×C8—36no—
18C2×#3710D313yescomplemented
19C2×D3×C3⋊C8—36no—
20C2×D3×C3⋊C8—63no—
21D32×C8—324no—
22C22×C3×C3⋊D3C2×C413yes—
23C2×#1113C818yescomplemented
24C2×#1113C2×C414yes—
25C2×C33×C4C2311yescharacteristic
26C2×C3×C3⋊Dic3C2×C413yes—
27C3×C4×C3⋊D3C2×C416yes—
28C3×C4×C3⋊D3C2316yes—
29#6294C2×C4112yes—
30C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
31C32×C3⋊C8C2316yescomplemented
32C22×D3×C8—93no—
33C22×D32—33no—
34C2×C4×C3⋊D3C2×D313yes—
35C2×C3×C4×D3—36no—
36C2×C3×C4×D3—63no—
37C2×#624—33no—
38C2×D3×Dic3—36no—
39C4×D32—312no—
40C4×D32—312no—
41C2×C32×C8—33no—
42C2×C3⋊(C3⋊C8)—33no—
43C2×C3⋊(C3⋊C8)—63no—
44C2×C3⋊(C3⋊C8)—121no—
45C8×C3⋊D3—312no—
46C2×C3×C3⋊C8—23no—
47C2×C3×C3⋊C8C2×D316yescomplemented
48C3×D3×C8—324no—
49#3710C2×D3112yescomplemented
50D3×C3⋊C8—324no—
51D3×C3⋊C8—612no—
52C22×C33C22×C411yescharacteristic
53C2×C3×C3⋊D3C2×C8112yescomplemented
54C2×C3×C3⋊D3C22×C416yes—
55#1113C2×C8116yescomplemented
56C33×C4C22×C411yescharacteristic
57C33×C4C2411yescharacteristic
58C3×C3⋊Dic3C22×C416yes—
59C22×C4×D3—93no—
60C22×C3×C8—93no—
61C22×C3⋊C8—93no—
62C2×D3×C8—36no—
63C2×D3×C8—924no—
64C2×D3×C8—96no—
65C2×D3×C8—183no—
66C22×C3⋊D3C4×D313yes—
67C22×C3×D3—36no—
68C22×C3×D3—63no—
69C2×D32—324no—
70C2×D32—312no—
71C2×C32×C4—23no—
72C2×C32×C4—41no—
73C2×C32×C4C22×D313yes—
74C2×C3⋊Dic3C4×D313yes—
75C4×C3⋊D3C4×D316yes—
76C4×C3⋊D3C22×D316yes—
77C2×C3×Dic3—36no—
78C2×C3×Dic3—63no—
79C3×C4×D3—312no—
80C3×C4×D3—312no—
81C3×C4×D3—66no—
82C3×C4×D3—66no—
83#624—312no—
84D3×Dic3—324no—
85C32×C8—36no—
86C3⋊(C3⋊C8)—36no—
87C3⋊(C3⋊C8)—66no—
88C3⋊(C3⋊C8)—122no—
89C3×C3⋊C8—26no—
90C3×C3⋊C8C22×D3112yescomplemented
91C23×C8—271noSylow
92C2×C33C22×C812yescomplemented
93C2×C33C23×C411yescharacteristic
94C3×C3⋊D3C22×C8112yescomplemented
95C23×D3—93no—
96C22×C3×C4—93no—
97C22×Dic3—93no—
98C2×C4×D3—36no—
99C2×C4×D3—63no—
100C2×C4×D3—912no—
101C2×C4×D3—912no—
102C2×C4×D3—96no—
103C2×C3×C8—36no—
104C2×C3×C8—63no—
105C2×C3×C8—912no—
106C2×C3⋊C8—93no—
107C2×C3⋊C8—912no—
108C2×C3⋊C8—183no—
109C2×C3⋊C8—361no—
110C2×C3⋊C8D3213yescomplemented
111D3×C8—324no—
112D3×C8—924no—
113D3×C8—1812no—
114C22×C32—23no—
115C22×C32—41no—
116C22×C32C2×C4×D313yes—
117C2×C3⋊D3D3×C8112yescomplemented
118C2×C3⋊D3C2×C4×D316yes—
119C2×C3×D3—324no—
120C2×C3×D3—312no—
121C2×C3×D3—612no—
122C2×C3×D3—66no—
123D32—348no—
124C32×C4—23no—
125C32×C4—23no—
126C32×C4—41no—
127C32×C4—41no—
128C32×C4C2×C4×D313yes—
129C32×C4C23×D313yes—
130C3⋊Dic3C2×C4×D316yes—
131C3×Dic3—312no—
132C3×Dic3—66no—
133C23×C4—271no—
134C22×C8—93no—
135C22×C8—273no—
136C22×C8—278no—
137C33C23×C811yescharacteristic, complemented, Sylow
138C23×C3—93no—
139C22×D3—36no—
140C22×D3—63no—
141C22×D3—912no—
142C22×D3—924no—
143C2×C3×C4—23no—
144C2×C3×C4—41no—
145C2×C3×C4—93no—
146C2×C3×C4—96no—
147C2×C3×C4—96no—
148C2×C3×C4C2×D3213yes—
149C2×Dic3—36no—
150C2×Dic3—63no—
151C2×Dic3—912no—
152C4×D3—312no—
153C4×D3—312no—
154C4×D3—66no—
155C4×D3—66no—
156C4×D3—924no—
157C3×C8—312no—
158C3×C8—66no—
159C3⋊C8—96no—
160C3⋊C8—186no—
161C3⋊C8—362no—
162C3⋊C8C2×D3216yescomplemented
163C2×C32—26no—
164C2×C32—23no—
165C2×C32—42no—
166C2×C32—41no—
167C2×C32C2×D3×C816yescomplemented
168C2×C32C22×C4×D313yes—
169C3⋊D3C2×D3×C8112yescomplemented
170C3×D3—324no—
171C3×D3—612no—
172C24—271no—
173C22×C4—93no—
174C22×C4—274no—
175C22×C4—274no—
176C22×C4—273no—
177C2×C8—33no—
178C2×C8—912no—
179C2×C8—2712no—
180C2×C8—271no—
181C22×C3—23no—
182C22×C3—41no—
183C22×C3—912no—
184C22×C3—96no—
185C22×C3C4×D3213yes—
186C2×D3—324no—
187C2×D3—312no—
188C2×D3—66no—
189C2×D3—612no—
190C2×D3—948no—
191C3×C4—23no—
192C3×C4—23no—
193C3×C4—41no—
194C3×C4—41no—
195C3×C4—96no—
196C3×C4C4×D3213yes—
197C3×C4C22×D3213yes—
198Dic3—312no—
199Dic3—66no—
200C32—23no—
201C32—41no—
202C32C22×D3×C813yescomplemented
203C23—93no—
204C23—274no—
205C23—278no—
206C2×C4—93no—
207C2×C4—96no—
208C2×C4—96no—
209C2×C4—2712no—
210C2×C4D3311yescharacteristic
211C8—36no—
212C8—272no—
213C2×C3—26no—
214C2×C3—23no—
215C2×C3—42no—
216C2×C3—41no—
217C2×C3—912no—
218C2×C3D32×C816yescomplemented
219C2×C3C2×C4×D3213yes—
220D3—324no—
221D3—612no—
222C22—912no—
223C22—96no—
224C22—2716no—
225C22#8894911yescharacteristic
226C4—96no—
227C4#8894911yescharacteristic
228C4C2×D3311yescharacteristic
229C3—23no—
230C3—41no—
231C3C2×D32×C813yescomplemented
232C2—912no—
233C2#1172281412yesdirect product factor
234C2C2×#8894911yescharacteristic
subgroupG/H
center, hypercenterC2×C4D33
commutator, hypocenterC33C23×C8
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×#1113C4
cosocleC33×C4C24
LCSG ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×C3⋊C8C2semidirect
3. C2×C3×#3710C2semidirect
4. #11722814C2direct (N × H)
11. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
12. C2×C32×C3⋊C8C22semidirect
13. C3⋊D3×C3⋊C8C22semidirect
14. C3×#3710C22semidirect
18. C2×#3710D3semidirect
23. C2×#1113C8semidirect
30. C3⋊(C3⋊(C3⋊C8))C23semidirect
31. C32×C3⋊C8C23semidirect
47. C2×C3×C3⋊C8C2×D3semidirect
49. #3710C2×D3semidirect
53. C2×C3×C3⋊D3C2×C8semidirect
55. #1113C2×C8semidirect
90. C3×C3⋊C8C22×D3semidirect
92. C2×C33C22×C8semidirect
94. C3×C3⋊D3C22×C8semidirect
110. C2×C3⋊C8D32semidirect
117. C2×C3⋊D3D3×C8semidirect
137. C33C23×C8semidirect
162. C3⋊C8C2×D32semidirect
167. C2×C32C2×D3×C8semidirect
169. C3⋊D3C2×D3×C8semidirect
202. C32C22×D3×C8semidirect
218. C2×C3D32×C8semidirect
231. C3C2×D32×C8semidirect
233. C2#11722814direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×C3⋊D3×C3⋊C82
C2×C3×#37102
#117228142
C2×D32×C83