Finite Group Databasebeta

← C2×#11724491 #13181467

Subgroups — C2×#11724491 #13181467

Order 1728.

1352 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1352 conjugacy classes of subgroups — too many to draw. Only its 141 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×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2#11724491C214yesmaximal, direct product factor
3C2×C3⋊(C3⋊(#41⋊C3))C212yesmaximal, complemented
4C2×#64366C311yesmaximal, characteristic, complemented
5C2×C3⋊#994—34nomaximal
6C2×C3×#634—41nomaximal
7C32×#41⋊C3C2212yescomplemented
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C3⋊(C3⋊(#41⋊C3))C2214yescomplemented
10C2×C32×#41C2×C311yescharacteristic
11#64366—34no—
12#64366C2×C314yescomplemented
13C2×#3832C2×C312yes—
14C2×D4×C3⋊D3—32no—
15C2×#3828—31no—
16C22×C4⋊(C3⋊D3)—31no—
17C2×C3×#41⋊C3—24no—
18C2×C3×#41⋊C3D314yescomplemented
19C3⋊#994—316no—
20C2×C3⋊(#41⋊C3)—38no—
21C2×C33×C4—41no—
22C2×C3×C3⋊Dic3—42no—
23C3×#634—44no—
24C32×SL(2,𝔽3)C2311yescharacteristic, complemented
25C2×#1009—34no—
26C2×#994—91no—
27C23×C3⋊D3—32no—
28C32×#41—32no—
29C32×#41C22×C312yes—
30C2×C32×Q8C22×C311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×#638—32no—
34#3832—34no—
35#3832C22×C314yes—
36D4×C3⋊D3—38no—
37D4×C3⋊D3—38no—
38#3828—34no—
39#3828—34no—
40C2×C4⋊(C3⋊D3)—34no—
41C2×C4⋊(C3⋊D3)—32no—
42C2×C4⋊(C3⋊D3)—31no—
43C2×C4⋊(C3⋊D3)—34no—
44C2×C4×C3⋊D3—32no—
45C2×#634A411yescharacteristic
46C2×C3×Dic6—124no—
47C3×#41⋊C3—28no—
48C3×#41⋊C3C2×D318yescomplemented
49C2×C3×SL(2,𝔽3)—24no—
50C2×C3×SL(2,𝔽3)C2×D314yescomplemented
51C3⋊(#41⋊C3)—316no—
52C22×C33—41no—
53C33×C4—42no—
54C3×C3⋊Dic3—44no—
55C2×C3×#41C3×D314yes—
56#1009—316no—
57#1009—916no—
58C2×#239—38no—
59C2×D3×D4—98no—
60C2×#235—94no—
61C22×D12—94no—
62#994—94no—
63C2×#41⋊C3—24no—
64C2×#41⋊C3—92no—
65C2×#41⋊C3C3⋊D311yescharacteristic, complemented
66C23×C32—31no—
67C22×C3⋊D3—32no—
68C22×C3⋊D3—38no—
69C22×C3⋊D3—124no—
70C32×Q8—31no—
71C32×Q8C23×C311yescharacteristic
72C32×D4—32no—
73C32×D4—32no—
74C2×C32×C4—31no—
75C2×C32×C4—32no—
76C2×C32×C4—32no—
77C2×C32×C4—44no—
78C2×C32×C4—84no—
79C2×C32×C4C2×A411yescharacteristic
80#638—38no—
81C2×C3⋊Dic3C2×A412yes—
82C4⋊(C3⋊D3)—34no—
83C4⋊(C3⋊D3)—34no—
84C4⋊(C3⋊D3)—34no—
85C4×C3⋊D3—34no—
86C4×C3⋊D3—34no—
87#634C2×A414yes—
88C2×C3×Dic3—128no—
89C3×Dic6—1216no—
90C3×SL(2,𝔽3)—24no—
91C3×SL(2,𝔽3)C22×D314yescomplemented
92C2×H(5,𝔽2)—91noSylow
93C2×C33—42no—
94C2×C33—41no—
95C23×D3—98no—
96C3×#41—38no—
97C3×#41C2×C3×D318yes—
98C2×C3×Q8—361no—
99C2×C3×Q8C2×C3×D314yes—
100C2×C3×D4—34no—
101C22×C3×C4—34no—
102C2×#67—98no—
103#239—316no—
104#239—916no—
105D3×D4—932no—
106D3×D4—932no—
107#235—916no—
108#235—916no—
109C2×D12—98no—
110C2×D12—916no—
111C2×D12—94no—
112C2×D12—916no—
113C2×C4×D3—98no—
114C2×Dic6—34no—
115#41⋊C3—28no—
116#41⋊C3—94no—
117#41⋊C3C2×C3⋊D312yescomplemented
118C2×SL(2,𝔽3)—24no—
119C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic, complemented
120C22×C32—32no—
121C22×C32—44no—
122C22×C32—62no—
123C22×C32—84no—
124C22×C32C22×A411yescharacteristic
125C2×C3⋊D3—34no—
126C2×C3⋊D3—64no—
127C2×C3⋊D3—128no—
128C32×C4—31no—
129C32×C4—31no—
130C32×C4—48no—
131C32×C4—88no—
132C32×C4C22×A412yes—
133C3⋊Dic3C22×A414yes—
134C3×Dic3—1216no—
135H(5,𝔽2)—94no—
136H(5,𝔽2)—274no—
137C2×#41—92no—
138C2×#41—271no—
139C2×#41C3×C3⋊D311yescharacteristic
140C22×D4—271no—
141C22×D4—272no—
142C33—41noSylow
143C23×C3—34no—
144C22×D3—98no—
145C22×D3—932no—
146C22×D3—3616no—
147C3×Q8—34no—
148C3×Q8—364no—
149C3×Q8C22×C3×D314yes—
150C3×D4—38no—
151C3×D4—38no—
152C2×C3×C4—34no—
153C2×C3×C4—38no—
154C2×C3×C4—38no—
155C2×C3×C4—41no—
156C2×C3×C4—84no—
157C2×C3×C4—362no—
158C2×C3×C4D3×A414yes—
159#67—932no—
160C2×Dic3—38no—
161D12—916no—
162D12—916no—
163D12—916no—
164C4×D3—916no—
165C4×D3—916no—
166Dic6—316no—
167SL(2,𝔽3)—24no—
168SL(2,𝔽3)C22×C3⋊D311yescharacteristic, complemented
169C2×C32—48no—
170C2×C32—44no—
171C2×C32—62no—
172C2×C32—88no—
173C2×C32—84no—
174C2×C32#99412yescomplemented
175C2×C32C23×A411yescharacteristic
176C3⋊D3—64no—
177C24—272no—
178#41—32no—
179#41—94no—
180#41—274no—
181#41—274no—
182#41—274no—
183#41C2×C3×C3⋊D312yes—
184C2×Q8—91no—
185C2×Q8C2×C3×C3⋊D311yescharacteristic
186C2×D4—31no—
187C2×D4—272no—
188C2×D4—274no—
189C2×D4—274no—
190C2×D4—272no—
191C2×D4—278no—
192C2×D4—278no—
193C2×D4—271no—
194C22×C4—31no—
195C22×C4—272no—
196C22×C3—38no—
197C22×C3—41no—
198C22×C3—68no—
199C22×C3—84no—
200C22×C3C2×D3×A414yes—
201C2×D3—916no—
202C2×D3—1816no—
203C2×D3—3632no—
204C3×C4—34no—
205C3×C4—34no—
206C3×C4—42no—
207C3×C4—88no—
208C3×C4—364no—
209C3×C4C2×D3×A418yes—
210Dic3—316no—
211C32—44no—
212C32—84no—
213C32C2×#99411yescharacteristic, complemented
214C23—31no—
215C23—278no—
216C23—272no—
217C23—1084no—
218Q8—31no—
219Q8—94no—
220Q8C22×C3×C3⋊D311yescharacteristic
221D4—32no—
222D4—32no—
223D4—274no—
224D4—274no—
225D4—278no—
226D4—274no—
227C2×C4—31no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—92no—
231C2×C4—274no—
232C2×C4—274no—
233C2×C4A4×C3⋊D311yescharacteristic
234C2×C3—41no—
235C2×C3—42no—
236C2×C3—68no—
237C2×C3—84no—
238C2×C3—88no—
239C2×C3C3⋊#99418yescomplemented
240C2×C3C22×D3×A414yes—
241D3—1816no—
242C22—32no—
243C22—62no—
244C22—274no—
245C22—544no—
246C22—1088no—
247C22C2×A4×C3⋊D311yescharacteristic
248C4—31no—
249C4—31no—
250C4—94no—
251C4C2×A4×C3⋊D312yes—
252C3—41no—
253C3—84no—
254C3C2×C3⋊#99414yescomplemented
255C2—62no—
256C2—544no—
257C2#1172449112yesdirect product factor
258C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC22C2×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
hypercenterC2×C4A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. #11724491C2direct (N × H)
3. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
4. C2×#64366C3semidirect
7. C32×#41⋊C3C22semidirect
8. C2×C32×SL(2,𝔽3)C22semidirect
9. C3⋊(C3⋊(#41⋊C3))C22semidirect
12. #64366C2×C3semidirect
18. C2×C3×#41⋊C3D3semidirect
24. C32×SL(2,𝔽3)C23semidirect
48. C3×#41⋊C3C2×D3semidirect
50. C2×C3×SL(2,𝔽3)C2×D3semidirect
65. C2×#41⋊C3C3⋊D3semidirect
91. C3×SL(2,𝔽3)C22×D3semidirect
117. #41⋊C3C2×C3⋊D3semidirect
119. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
168. SL(2,𝔽3)C22×C3⋊D3semidirect
174. C2×C32#994semidirect
213. C32C2×#994semidirect
239. C2×C3C3⋊#994semidirect
254. C3C2×C3⋊#994semidirect
257. C2#11724491direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
#117244912
C2×C3⋊(C3⋊(#41⋊C3))2
C2×#643663
C2×C3⋊#9943
C2×C3×#6344