Finite Group Databasebeta

← C2×#11724493 #13181469

Subgroups — C2×#11724493 #13181469

Order 1728.

1328 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1328 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#11724493C214yesmaximal, direct product factor
3C2×C3⋊D3×SL(2,𝔽3)C212yesmaximal, complemented
4C2×#64367C311yesmaximal, characteristic, complemented
5C2×#1010⋊C3—34nomaximal
6C2×C3×C4⋊(C3⋊D3)—41nomaximal
7C32×#41⋊C3C2212yescomplemented
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3⋊D3×SL(2,𝔽3)C2214yescomplemented
10C2×C32×#41C2×C311yescharacteristic, complemented
11#64367—34no—
12#64367C2×C314yescomplemented
13C2×Q8×C3⋊D3C2×C312yescomplemented
14C2×#3830—32no—
15C2×#3828—31no—
16C22×#634—31no—
17C2×C3×#41⋊C3—24no—
18C2×C3×#41⋊C3D314yescomplemented
19#1010⋊C3—316no—
20C2×D3×SL(2,𝔽3)—38no—
21C22×C3×C3⋊D3—42no—
22C2×C33×C4—41no—
23C3×C4⋊(C3⋊D3)—44no—
24C32×SL(2,𝔽3)C2311yescharacteristic
25C2×#1010—34no—
26C2×2−1+4⋊C3—91no—
27C32×#41—32no—
28C32×#41C22×C312yescomplemented
29C2×C32×Q8C22×C311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×#638—32no—
33C22×C3⋊Dic3—32no—
34Q8×C3⋊D3—34no—
35Q8×C3⋊D3C22×C314yescomplemented
36#3830—38no—
37#3830—38no—
38#3828—34no—
39#3828—34no—
40C2×C4⋊(C3⋊D3)A411yescharacteristic
41C2×C4×C3⋊D3—32no—
42C2×#634—34no—
43C2×#634—31no—
44C2×#634—32no—
45C2×#634—34no—
46C2×C3×D12—124no—
47C3×#41⋊C3—28no—
48C3×#41⋊C3C2×D318yescomplemented
49C2×C3×SL(2,𝔽3)—24no—
50C2×C3×SL(2,𝔽3)C2×D314yes—
51D3×SL(2,𝔽3)—316no—
52C22×C33—41no—
53C2×C3×C3⋊D3—44no—
54C2×C3×C3⋊D3—84no—
55C33×C4—42no—
56C2×C3×#41C3×D314yescomplemented
57#1010—316no—
58#1010—916no—
59C2×D3×Q8—38no—
60C2×#237—98no—
61C2×#235—94no—
62C22×Dic6—94no—
632−1+4⋊C3—94no—
64C2×#41⋊C3—24no—
65C2×#41⋊C3C3⋊D311yescharacteristic, complemented
66C22×SL(2,𝔽3)—92no—
67C23×C32—31no—
68C22×C3⋊D3C2×A412yes—
69C22×C3×D3—128no—
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⋊Dic3—38no—
82C2×C3⋊Dic3—32no—
83C4⋊(C3⋊D3)C2×A414yes—
84C4×C3⋊D3—34no—
85C4×C3⋊D3—34no—
86#634—34no—
87#634—34no—
88#634—34no—
89C3×D12—1216no—
90C3×SL(2,𝔽3)—24no—
91C3×SL(2,𝔽3)C22×D314yes—
92C2×2−1+4—91noSylow
93C2×C33—42no—
94C2×C33—41no—
95C3×C3⋊D3—84no—
96C3×#41—38no—
97C3×#41C2×C3×D318yescomplemented
98C2×C3×Q8C2×C3×D314yes—
99C2×C3×D4—34no—
100C2×C3×D4—361no—
101C22×C3×C4—34no—
102C2×#67—98no—
103C22×Dic3—98no—
104D3×Q8—316no—
105D3×Q8—916no—
106#237—932no—
107#237—932no—
108#235—916no—
109#235—916no—
110C2×D12—34no—
111C2×C4×D3—98no—
112C2×Dic6—916no—
113C2×Dic6—94no—
114C2×Dic6—98no—
115C2×Dic6—916no—
116#41⋊C3—28no—
117#41⋊C3C2×C3⋊D312yescomplemented
118C2×SL(2,𝔽3)—24no—
119C2×SL(2,𝔽3)—94no—
120C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
121C22×C32—32no—
122C22×C32—44no—
123C22×C32—62no—
124C22×C32—84no—
125C22×C32C22×A411yescharacteristic
126C2×C3⋊D3—24no—
127C2×C3⋊D3C22×A414yes—
128C2×C3×D3—1216no—
129C2×C3×D3—2416no—
130C32×C4—31no—
131C32×C4—31no—
132C32×C4—48no—
133C32×C4—88no—
134C32×C4C22×A412yes—
135C3⋊Dic3—34no—
1362−1+4—94no—
1372−1+4—274no—
138C2×#41—271no—
139C2×#41—272no—
140C2×#41C3×C3⋊D311yescharacteristic, complemented
141C22×Q8—92no—
142C22×Q8—271no—
143C33—41noSylow
144C23×C3—34no—
145C23×C3—362no—
146C22×D3—38no—
147C3×Q8—34no—
148C3×Q8C22×C3×D314yes—
149C3×D4—38no—
150C3×D4—38no—
151C3×D4—364no—
152C2×C3×C4—34no—
153C2×C3×C4—38no—
154C2×C3×C4—38no—
155C2×C3×C4—41no—
156C2×C3×C4—84no—
157C2×C3×C4D3×A414yes—
158#67—932no—
159C2×Dic3—932no—
160C2×Dic3—98no—
161D12—316no—
162C4×D3—916no—
163C4×D3—916no—
164Dic6—916no—
165Dic6—916no—
166Dic6—916no—
167SL(2,𝔽3)—24no—
168SL(2,𝔽3)C22×C3⋊D311yescharacteristic
169C2×C32—48no—
170C2×C32—44no—
171C2×C32—62no—
172C2×C32—88no—
173C2×C32—84no—
174C2×C322−1+4⋊C312yescomplemented
175C2×C32C23×A411yescharacteristic
176C3⋊D3—24no—
177C3×D3—2416no—
178#41—32no—
179#41—274no—
180#41—274no—
181#41—278no—
182#41—278no—
183#41C2×C3×C3⋊D312yescomplemented
184C2×Q8—94no—
185C2×Q8—274no—
186C2×Q8—274no—
187C2×Q8—272no—
188C2×Q8—271no—
189C2×Q8—274no—
190C2×Q8C2×C3×C3⋊D311yescharacteristic
191C2×D4—31no—
192C2×D4—91no—
193C2×D4—272no—
194C22×C4—31no—
195C22×C4—272no—
196C22×C4—272no—
197C22×C3—38no—
198C22×C3—41no—
199C22×C3—68no—
200C22×C3—84no—
201C22×C3—364no—
202C22×C3—724no—
203C22×C3C2×D3×A414yes—
204C2×D3—316no—
205C2×D3—616no—
206C3×C4—34no—
207C3×C4—34no—
208C3×C4—42no—
209C3×C4—88no—
210C3×C4C2×D3×A418yes—
211Dic3—916no—
212C32—44no—
213C32—84no—
214C32C2×2−1+4⋊C311yescharacteristic, complemented
215C23—31no—
216C23—92no—
217Q8—31no—
218Q8—274no—
219Q8—274no—
220Q8—274no—
221Q8C22×C3×C3⋊D311yescharacteristic
222D4—32no—
223D4—32no—
224D4—94no—
225D4—278no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—274no—
230C2×C4—274no—
231C2×C4—278no—
232C2×C4—272no—
233C2×C4A4×C3⋊D311yescharacteristic
234C2×C3—41no—
235C2×C3—42no—
236C2×C3—68no—
237C2×C3—84no—
238C2×C3—88no—
239C2×C3—724no—
240C2×C3#1010⋊C318yescomplemented
241C2×C3C22×D3×A414yes—
242D3—616no—
243C22—32no—
244C22—62no—
245C22—94no—
246C22—184no—
247C22C2×A4×C3⋊D311yescharacteristic
248C4—31no—
249C4—31no—
250C4—274no—
251C4C2×A4×C3⋊D312yes—
252C3—41no—
253C3—84no—
254C3C2×#1010⋊C314yescomplemented
255C2—62no—
256C2—184no—
257C2#1172449312yesdirect 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. #11724493C2direct (N × H)
3. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
4. C2×#64367C3semidirect
7. C32×#41⋊C3C22semidirect
9. C3⋊D3×SL(2,𝔽3)C22semidirect
10. C2×C32×#41C2×C3semidirect
12. #64367C2×C3semidirect
13. C2×Q8×C3⋊D3C2×C3semidirect
18. C2×C3×#41⋊C3D3semidirect
28. C32×#41C22×C3semidirect
35. Q8×C3⋊D3C22×C3semidirect
48. C3×#41⋊C3C2×D3semidirect
56. C2×C3×#41C3×D3semidirect
65. C2×#41⋊C3C3⋊D3semidirect
97. C3×#41C2×C3×D3semidirect
117. #41⋊C3C2×C3⋊D3semidirect
140. C2×#41C3×C3⋊D3semidirect
174. C2×C322−1+4⋊C3semidirect
183. #41C2×C3×C3⋊D3semidirect
214. C32C2×2−1+4⋊C3semidirect
240. C2×C3#1010⋊C3semidirect
254. C3C2×#1010⋊C3semidirect
257. C2#11724493direct (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
#117244932
C2×C3⋊D3×SL(2,𝔽3)2
C2×#643673
C2×#1010⋊C33
C2×C3×C4⋊(C3⋊D3)4