Finite Group Databasebeta

← C3⋊D3×#994 #13181471

Subgroups — C3⋊D3×#994 #13181471

Order 1728.

1336 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C32×#994C211yesmaximal, characteristic, complemented
2C3⋊D3×#41⋊C3C213yesmaximal, complemented
3#11724491C213yesmaximal, complemented
4C3⋊D3×H(5,𝔽2)C311yesmaximal, characteristic, complemented
5D3×#994—34nomaximal
6C3×Q8×C3⋊D3—41nomaximal
7C32×#41⋊C3C2213yescomplemented
8C3⋊D3×SL(2,𝔽3)C2211yescharacteristic, complemented
9C3⋊(C3⋊(#41⋊C3))C2213yescomplemented
10C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
11#64366—31no—
12#64366C2×C313yescomplemented
13#41×C3⋊D3—31no—
14#41×C3⋊D3C2×C313yes—
15#64361—33no—
16C2×D4×C3⋊D3—33no—
17C3×#994—24no—
18C3×#994D314yescomplemented
19D3×#41⋊C3—312no—
20C3⋊#994—312no—
21C33×Q8—41no—
22C3×C4×C3⋊D3—43no—
23C3×#634—43no—
24C32×SL(2,𝔽3)C2311yescharacteristic, complemented
25D3×H(5,𝔽2)—34no—
26C2×#994—91no—
27C23×C3⋊D3—32no—
28C32×#41—31no—
29C32×#41C22×C313yes—
30C2×C32×D4—33no—
31C2×#638—36no—
32#3832—31no—
33#3832C22×C313yes—
34Q8×C3⋊D3A411yescharacteristic
35Q8×C3⋊D3C22×C311yescharacteristic
36#3830—33no—
37#3830—33no—
38D4×C3⋊D3—36no—
39D4×C3⋊D3—33no—
40D4×C3⋊D3—36no—
41D4×C3⋊D3—33no—
42#3828—33no—
43#3828—33no—
44C2×C4⋊(C3⋊D3)—33no—
45C2×C4×C3⋊D3—33no—
46C3×D3×Q8—124no—
47C3×#41⋊C3—212no—
48C3×#41⋊C3C2×D3112yescomplemented
49D3×SL(2,𝔽3)—34no—
50C3⋊(#41⋊C3)—312no—
51C2×C3×C3⋊D3—41no—
52C33×C4—43no—
53C3×C3⋊Dic3—43no—
54C3×H(5,𝔽2)C3×D314yescomplemented
55#1009—312no—
56#1009—94no—
57D3×#41—312no—
58D3×#41—94no—
59#1004—912no—
60C2×D3×D4—912no—
61#994—24no—
62#994—93no—
63#994C3⋊D311yescharacteristic, direct product factor
64C2×#41⋊C3—93no—
65C23×C32—32no—
66C22×C3⋊D3—36no—
67C22×C3⋊D3—33no—
68C22×C3⋊D3—124no—
69C32×Q8—44no—
70C32×Q8—84no—
71C32×Q8C2×A411yescharacteristic
72C32×Q8C23×C311yescharacteristic
73C32×D4—33no—
74C32×D4—33no—
75C2×C32×C4—33no—
76#638—36no—
77#638—36no—
78C2×C3⋊Dic3—33no—
79C4⋊(C3⋊D3)—33no—
80C4⋊(C3⋊D3)—33no—
81C4×C3⋊D3—31no—
82C4×C3⋊D3—33no—
83C4×C3⋊D3—33no—
84C4×C3⋊D3C2×A413yes—
85#634—31no—
86#634C2×A413yes—
87C3×C4×D3—1212no—
88C3×Dic6—1212no—
89C3×SL(2,𝔽3)—24no—
90C3×SL(2,𝔽3)C22×D314yescomplemented
91C2×H(5,𝔽2)—91noSylow
92C2×C33—41no—
93C3×C3⋊D3—42no—
94C23×D3—98no—
95C3×#41—34no—
96C3×#41C2×C3×D3112yes—
97C2×C3×Q8—361no—
98C2×C3×D4—312no—
99C2×#67—924no—
100#239—312no—
101#239—94no—
102D3×Q8—34no—
103D3×Q8—34no—
104#237—912no—
105#237—912no—
106D3×D4—912no—
107D3×D4—912no—
108D3×D4—924no—
109D3×D4—924no—
110#235—912no—
111#235—912no—
112C2×D12—912no—
113C2×C4×D3—912no—
114#41⋊C3—212no—
115#41⋊C3—93no—
116#41⋊C3C2×C3⋊D313yescomplemented
117C2×SL(2,𝔽3)—91no—
118C22×C32—33no—
119C22×C32—122no—
120C2×C3⋊D3—33no—
121C2×C3⋊D3—66no—
122C2×C3⋊D3—126no—
123C2×C3⋊D3C22×A411yescharacteristic
124C2×C3×D3—124no—
125C32×C4—31no—
126C32×C4—412no—
127C32×C4—812no—
128C32×C4C22×A413yes—
129C3⋊Dic3—31no—
130C3⋊Dic3C22×A413yes—
131C3×Dic3—1212no—
132H(5,𝔽2)—93no—
133H(5,𝔽2)—273no—
134H(5,𝔽2)—271no—
135H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
136C2×#41—93no—
137C2×#41—271no—
138C22×D4—273no—
139C33—41noSylow
140C23×C3—38no—
141C22×D3—912no—
142C22×D3—924no—
143C22×D3—3616no—
144C3×Q8—41no—
145C3×Q8—84no—
146C3×Q8—363no—
147C3×Q8D3×A414yes—
148C3×Q8C22×C3×D314yes—
149C3×D4—312no—
150C3×D4—312no—
151C2×C3×C4—312no—
152C2×C3×C4—363no—
153#67—924no—
154#67—924no—
155C2×Dic3—912no—
156D12—912no—
157D12—912no—
158C4×D3—312no—
159C4×D3—912no—
160C4×D3—94no—
161C4×D3—912no—
162Dic6—312no—
163Dic6—94no—
164SL(2,𝔽3)—24no—
165SL(2,𝔽3)C22×C3⋊D311yescharacteristic, complemented
166C2×C32—44no—
167C2×C32—63no—
168C2×C32—84no—
169C2×C32C23×A411yescharacteristic
170C3⋊D3—63no—
171C3⋊D3#99412yesdirect product factor
172C3×D3—128no—
173C24—272no—
174#41—31no—
175#41—93no—
176#41—273no—
177#41—273no—
178#41—273no—
179#41—271no—
180#41—273no—
181#41C2×C3×C3⋊D313yes—
182C2×Q8—91no—
183C2×Q8—91no—
184C2×D4—33no—
185C2×D4—273no—
186C2×D4—273no—
187C2×D4—273no—
188C2×D4—276no—
189C2×D4—276no—
190C2×D4—276no—
191C22×C4—273no—
192C22×C3—312no—
193C22×C3—128no—
194C22×C3—361no—
195C2×D3—34no—
196C2×D3—912no—
197C2×D3—1824no—
198C2×D3—3624no—
199C3×C4—34no—
200C3×C4—43no—
201C3×C4—812no—
202C3×C4—363no—
203C3×C4C2×D3×A4112yes—
204Dic3—312no—
205Dic3—94no—
206C32—44no—
207C32—84no—
208C32C2×#99411yescharacteristic, complemented
209C23—32no—
210C23—273no—
211C23—276no—
212C23—1084no—
213Q8—93no—
214Q8—271no—
215Q8A4×C3⋊D311yescharacteristic
216Q8C22×C3×C3⋊D311yescharacteristic
217D4—33no—
218D4—33no—
219D4—273no—
220D4—276no—
221D4—276no—
222D4—273no—
223C2×C4—33no—
224C2×C4—93no—
225C2×C4—273no—
226C2×C4—271no—
227C2×C4—273no—
228C2×C4—273no—
229C2×C3—41no—
230C2×C3—612no—
231C2×C3—84no—
232C2×C3—362no—
233C2×C3C22×D3×A414yes—
234D3—38no—
235D3—1812no—
236C22—33no—
237C22—91no—
238C22—122no—
239C22—273no—
240C22—546no—
241C22—1086no—
242C4—31no—
243C4—93no—
244C4—271no—
245C4C2×A4×C3⋊D313yes—
246C3—41no—
247C3—84no—
248C3D3×#99414yescomplemented
249C2—63no—
250C2—92no—
251C2—543no—
252C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FittingC32×H(5,𝔽2)C2×C3
socleC2×C32C23×A4
hypercenterQ8A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×#994C2semidirect
2. C3⋊D3×#41⋊C3C2semidirect
3. #11724491C2semidirect
4. C3⋊D3×H(5,𝔽2)C3semidirect
7. C32×#41⋊C3C22semidirect
8. C3⋊D3×SL(2,𝔽3)C22semidirect
9. C3⋊(C3⋊(#41⋊C3))C22semidirect
10. C32×H(5,𝔽2)C2×C3semidirect
12. #64366C2×C3semidirect
18. C3×#994D3semidirect
24. C32×SL(2,𝔽3)C23semidirect
48. C3×#41⋊C3C2×D3semidirect
54. C3×H(5,𝔽2)C3×D3semidirect
63. #994C3⋊D3direct (N × H)
90. C3×SL(2,𝔽3)C22×D3semidirect
116. #41⋊C3C2×C3⋊D3semidirect
135. H(5,𝔽2)C3×C3⋊D3semidirect
165. SL(2,𝔽3)C22×C3⋊D3semidirect
171. C3⋊D3#994direct (N × H)
208. C32C2×#994semidirect
248. C3D3×#994semidirect

Maximal subgroups

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

subgroupindex
C32×#9942
C3⋊D3×#41⋊C32
#117244912
C3⋊D3×H(5,𝔽2)3
D3×#9943
C3×Q8×C3⋊D34