Finite Group Databasebeta

← C2×#11722945 #13172277

Subgroups — C2×#11722945 #13172277

Order 1728.

1570 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 1570 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
1C22×#6296C211yesmaximal, characteristic
2C2×Dic3×C3⋊Dic3C213yesmaximal
3C2×C3×#3724C213yesmaximal
4#11722945C218yesmaximal, direct product factor
5C2×#63838—33nomaximal
6C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
7C22×C3×C3⋊Dic3C2213yes—
8C22×C32×Dic3C2213yes—
9C2×#6296C418yescomplemented
10C2×#6296C2214yes—
11Dic3×C3⋊Dic3C22112yes—
12C3×#3724C22112yes—
13C22×#627—33no—
14C2×C4×C3⋊Dic3—33no—
15C2×C3×Dic3⋊C4—36no—
16C2×#3724D313yes—
17C2×Dic32—36no—
18C2×Dic32—63no—
19#63838—324no—
20C23×C33C2311yescharacteristic
21C2×C3⋊(C3⋊Dic3)Q814yescomplemented
22C2×C3⋊(C3⋊Dic3)C2312yes—
23C2×C3×C3⋊Dic3C2×C4112yes—
24C2×C3×C3⋊Dic3C2316yes—
25C2×C32×Dic3—26no—
26C2×C32×Dic3C2316yes—
27#6296C2×C4116yescomplemented
28C2×#887—93no—
29C22×C32×C4—33no—
30C22×C3⋊Dic3—33no—
31C22×C3⋊Dic3—63no—
32C22×C3⋊Dic3—121no—
33C22×C3⋊Dic3C2×D313yes—
34C22×C3×Dic3—23no—
35C22×C3×Dic3—36no—
36C22×C3×Dic3—63no—
37C22×C3×Dic3C2×D316yes—
38C2×#627—324no—
39C2×#627—312no—
40C4×C3⋊Dic3—312no—
41C3×Dic3⋊C4—324no—
42#3724C2×D3112yes—
43Dic32—324no—
44Dic32—612no—
45C22×C33C22×C412yes—
46C22×C33C2×Q812yes—
47C22×C33#4112yes—
48C22×C33C2411yescharacteristic
49C3⋊(C3⋊Dic3)C2×Q814yescomplemented
50C3×C3⋊Dic3C22×C4112yes—
51C32×Dic3—26no—
52C22×Dic6—93no—
53C2×C3×C4⋊C4—93no—
54C2×Dic3⋊C4—36no—
55C2×C4×Dic3—96no—
56C2×C4×Dic3—93no—
57C2×C4×Dic3—183no—
58#887—924no—
59C23×C32—23no—
60C23×C32—41no—
61C23×C32C22×D313yes—
62C2×C32×C4—36no—
63C2×C32×C4—66no—
64C2×C3⋊Dic3—312no—
65C2×C3⋊Dic3—36no—
66C2×C3⋊Dic3—66no—
67C2×C3⋊Dic3—612no—
68C2×C3⋊Dic3—124no—
69C2×C3⋊Dic3—122no—
70C2×C3⋊Dic3C4×D3112yes—
71C2×C3⋊Dic3C22×D316yes—
72C2×C3×Dic3—212no—
73C2×C3×Dic3—212no—
74C2×C3×Dic3—26no—
75C2×C3×Dic3—324no—
76C2×C3×Dic3—312no—
77C2×C3×Dic3—66no—
78C2×C3×Dic3—612no—
79C2×C3×Dic3C22×D3112yes—
80#627—348no—
81C2×C4×Q8—271noSylow
82C2×C33C4×Q814yescomplemented
83C2×C33C23×C411yescharacteristic
84C2×C33C22×Q811yescharacteristic
85C2×C33C2×#4111yescharacteristic
86C22×C3×C4—36no—
87C22×C3×C4—63no—
88C22×C3×C4—93no—
89C22×Dic3—36no—
90C22×Dic3—63no—
91C22×Dic3—93no—
92C22×Dic3—183no—
93C22×Dic3—361no—
94C22×Dic3D3213yes—
95C2×Dic6—912no—
96C2×Dic6—924no—
97C3×C4⋊C4—912no—
98Dic3⋊C4—324no—
99C4×Dic3—924no—
100C4×Dic3—912no—
101C4×Dic3—1812no—
102C22×C32—23no—
103C22×C32—26no—
104C22×C32—26no—
105C22×C32—26no—
106C22×C32—42no—
107C22×C32—42no—
108C22×C32—42no—
109C22×C32—41no—
110C22×C32C2×C4×D316yes—
111C22×C32#23716yes—
112C22×C32D3×Q816yes—
113C22×C32C23×D313yes—
114C32×C4—66no—
115C3⋊Dic3—312no—
116C3⋊Dic3—612no—
117C3⋊Dic3—124no—
118C3⋊Dic3C2×C4×D3112yes—
119C3×Dic3—212no—
120C3×Dic3—212no—
121C3×Dic3—324no—
122C3×Dic3—612no—
123C22×Q8—271no—
124C4×Q8—278no—
125C2×C4⋊C4—93no—
126C2×C42—273no—
127C33C2×C4×Q811yescharacteristic, complemented, Sylow
128C23×C3—23no—
129C23×C3—41no—
130C23×C3C2×D3213yes—
131C2×C3×C4—312no—
132C2×C3×C4—66no—
133C2×C3×C4—612no—
134C2×C3×C4—612no—
135C2×C3×C4—96no—
136C2×C3×C4—912no—
137C2×Dic3—26no—
138C2×Dic3—312no—
139C2×Dic3—324no—
140C2×Dic3—612no—
141C2×Dic3—66no—
142C2×Dic3—96no—
143C2×Dic3—912no—
144C2×Dic3—186no—
145C2×Dic3—1812no—
146C2×Dic3—362no—
147C2×Dic3—364no—
148C2×Dic3C2×D3216yes—
149Dic6—948no—
150C2×C32—23no—
151C2×C32—23no—
152C2×C32—23no—
153C2×C32—212no—
154C2×C32—44no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—41no—
158C2×C32#887112yescomplemented
159C2×C32C22×C4×D313yes—
160C2×C32C2×#23713yes—
161C2×C32C2×D3×Q813yes—
162C2×Q8—274no—
163C2×Q8—278no—
164C22×C4—33no—
165C22×C4—93no—
166C22×C4—271no—
167C4⋊C4—912no—
168C42—2712no—
169C22×C3—23no—
170C22×C3—26no—
171C22×C3—26no—
172C22×C3—26no—
173C22×C3—41no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—42no—
177C22×C3#379716yes—
178C22×C3C4×D3216yes—
179C22×C3#380516yes—
180C22×C3C22×D3213yes—
181C3×C4—612no—
182C3×C4—612no—
183C3×C4—912no—
184Dic3—26no—
185Dic3—324no—
186Dic3—612no—
187Dic3—912no—
188Dic3—1812no—
189Dic3—364no—
190C32—23no—
191C32—41no—
192C32C2×#88713yescomplemented
193C23D3311yescharacteristic
194Q8—2716no—
195C2×C4—36no—
196C2×C4—66no—
197C2×C4—96no—
198C2×C4—912no—
199C2×C4—272no—
200C2×C4—274no—
201C2×C3—23no—
202C2×C3—23no—
203C2×C3—23no—
204C2×C3—212no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—44no—
209C2×C3#63838112yescomplemented
210C2×C3C2×#379713yes—
211C2×C3C2×C4×D3213yes—
212C2×C3C2×#380513yes—
213C22#8894912yes—
214C22#8895812yes—
215C22#8896312yes—
216C22C2×D3311yescharacteristic
217C4—66no—
218C4—912no—
219C4—274no—
220C3—23no—
221C3—41no—
222C3C2×#6383813yescomplemented
223C2#1172294514yesdirect product factor
224C2C2×#8894911yescharacteristic
225C2C2×#8895811yescharacteristic
226C2C2×#8896311yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socle, prime radicalC23×C33C23
cosocleC22×C33C24
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722945C2direct (N × H)
9. C2×#6296C4semidirect
21. C2×C3⋊(C3⋊Dic3)Q8semidirect
27. #6296C2×C4semidirect
49. C3⋊(C3⋊Dic3)C2×Q8semidirect
82. C2×C33C4×Q8semidirect
127. C33C2×C4×Q8semidirect
158. C2×C32#887semidirect
192. C32C2×#887semidirect
209. C2×C3#63838semidirect
222. C3C2×#63838semidirect
223. C2#11722945direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62962
C2×Dic3×C3⋊Dic32
C2×C3×#37242
#117229452
C2×#638383