Finite Group Databasebeta

← C3⋊D3×#859 #13165966

Subgroups — C3⋊D3×#859 #13165966

Order 1728.

670 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 79 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×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4D3×#859—34nomaximal
5C3⋊D3×#103—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C2×Dic3×C3⋊D3—41nomaximal
8C3⋊D3×SL(2,𝔽3)C412yescomplemented
9C2×Q8×C3⋊D3D311yescharacteristic
10C2×D3×SL(2,𝔽3)—34no—
11C2×C8×C3⋊D3—31no—
12C4⋊C4×C3⋊D3—31no—
13C3⋊#859—34no—
14C3⋊#859—64no—
15C3×#859D314yescomplemented
16C32×#103—31no—
17#63662—31no—
18#63643—31no—
19#63637—31no—
20C22×C3×C3⋊D3—41no—
21C2×C3⋊(C3⋊Dic3)—41no—
22C2×C32×Dic3—41no—
23C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24Dic3×C3⋊D3—44no—
25C2×#859—91no—
26D3×#103—94no—
27C2×C32×Q8C2×D311yescharacteristic
28Q8×C3⋊D3—32no—
29Q8×C3⋊D3Dic312yescomplemented
30C2×C4×C3⋊D3—31no—
31C2×C4×C3⋊D3—61no—
32C2×#634—31no—
33C2×C3×SL(2,𝔽3)—24no—
34C2×C3×SL(2,𝔽3)C2×D314yes—
35C2×D3×Dic3—124no—
36D3×SL(2,𝔽3)—38no—
37C2×C32×C8—31no—
38C32×C4⋊C4—31no—
39C4⋊(C3⋊Dic3)—31no—
40#3750—61no—
41C2×C3⋊(C3⋊C8)—31no—
42C8×C3⋊D3—62no—
43C22×C33—41no—
44C2×C3×C3⋊D3—42no—
45C2×C3×C3⋊D3—42no—
46C2×C3×C3⋊D3—42no—
47C3⋊(C3⋊Dic3)—42no—
48C32×Dic3—42no—
49C2×D3×Q8—34no—
50C22×SL(2,𝔽3)—91no—
51C2×D3×C8—94no—
52D3×C4⋊C4—94no—
53#859—91no—
54#859—184no—
55#859C3⋊D311yescharacteristic, direct product factor
56C3×#103—34no—
57#835—94no—
58#816—94no—
59#810—94no—
60C22×C3⋊D3S411yescharacteristic
61C22×C3×D3—124no—
62C32×Q8—31no—
63C32×Q8C2×Dic311yescharacteristic, complemented
64C2×C32×C4—31no—
65C2×C32×C4—61no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—124no—
69C2×C3⋊Dic3—244no—
70C4×C3⋊D3—32no—
71C4×C3⋊D3—32no—
72C4×C3⋊D3—122no—
73#634—32no—
74#634—61no—
75C2×C3×Dic3—44no—
76C3×SL(2,𝔽3)—24no—
77C3×SL(2,𝔽3)C4×D314yescomplemented
78D3×Dic3—1216no—
79C32×C8—61no—
80C3⋊(C3⋊C8)—61no—
81C2×#103—271noSylow
82C2×C33—41no—
83C2×C33—41no—
84C2×C33—41no—
85C3×C3⋊D3—44no—
86C2×C3×Q8D3214yes—
87C22×Dic3—361no—
88D3×Q8—38no—
89D3×Q8—98no—
90C2×C4×D3—94no—
91C2×C4×D3—184no—
92C2×Dic6—94no—
93C2×SL(2,𝔽3)—24no—
94C2×SL(2,𝔽3)—92no—
95C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
96C2×C3×C8—34no—
97C3×C4⋊C4—34no—
98C4⋊Dic3—94no—
99Dic3⋊C4—184no—
100C2×C3⋊C8—94no—
101D3×C8—188no—
102C22×C32—44no—
103C22×C32—84no—
104C22×C32C2×S411yescharacteristic
105C2×C3⋊D3#22612yes—
106C2×C3⋊D3GL(2,𝔽3)12yes—
107C2×C3⋊D3SL(2,ℤ/4)12yes—
108C2×C3×D3—128no—
109C2×C3×D3—128no—
110C2×C3×D3—128no—
111C32×C4—31no—
112C32×C4—31no—
113C32×C4—121no—
114C3⋊Dic3—32no—
115C3⋊Dic3—121no—
116C3⋊Dic3—128no—
117C3⋊Dic3—248no—
118C3×Dic3—48no—
119C22×Q8—91no—
120C22×C8—271no—
121C2×C4⋊C4—271no—
122#103—31no—
123#103—271no—
124#103—271no—
125#103—271no—
126C33—41noSylow
127C23×C3—361no—
128C22×D3—34no—
129C3×Q8—34no—
130C3×Q8D3×Dic314yescomplemented
131C2×C3×C4—34no—
132C2×C3×C4—64no—
133C2×Dic3—41no—
134C2×Dic3—94no—
135C2×Dic3—184no—
136C2×Dic3—361no—
137C2×Dic3—364no—
138C2×Dic3—724no—
139C4×D3—98no—
140C4×D3—98no—
141C4×D3—368no—
142Dic6—98no—
143Dic6—184no—
144SL(2,𝔽3)—24no—
145SL(2,𝔽3)C4×C3⋊D311yescharacteristic, complemented
146C3×C8—64no—
147C3⋊C8—184no—
148C2×C32—44no—
149C2×C32—44no—
150C2×C32—44no—
151C2×C32—84no—
152C2×C32—84no—
153C2×C32—84no—
154C2×C32C2×#22611yescharacteristic
155C2×C32C2×GL(2,𝔽3)11yescharacteristic
156C2×C32C2×SL(2,ℤ/4)11yescharacteristic
157C3⋊D3#85914yesdirect product factor
158C3×D3—1216no—
159C2×Q8—92no—
160C2×Q8—271no—
161C2×Q8—272no—
162C2×Q8D3×C3⋊D311yescharacteristic
163C22×C4—271no—
164C22×C4—541no—
165C2×C8—31no—
166C2×C8—271no—
167C2×C8—542no—
168C4⋊C4—31no—
169C4⋊C4—271no—
170C4⋊C4—541no—
171C22×C3—41no—
172C22×C3—84no—
173C22×C3—362no—
174C22×C3—362no—
175C22×C3—362no—
176C22×C3D3×S414yes—
177C2×D3—38no—
178C2×D3—38no—
179C2×D3—38no—
180C3×C4—34no—
181C3×C4—34no—
182C3×C4—124no—
183Dic3—42no—
184Dic3—98no—
185Dic3—362no—
186Dic3—364no—
187Dic3—728no—
188C32—44no—
189C32—84no—
190C32C2×#85911yescharacteristic, complemented
191C23—91no—
192Q8—31no—
193Q8—272no—
194Q8—541no—
195Q8Dic3×C3⋊D311yescharacteristic, complemented
196C2×C4—31no—
197C2×C4—61no—
198C2×C4—272no—
199C2×C4—272no—
200C2×C4—271no—
201C2×C4—541no—
202C2×C4—1082no—
203C8—61no—
204C8—541no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—84no—
209C2×C3—84no—
210C2×C3—84no—
211C2×C3—364no—
212C2×C3D3×#22614yes—
213C2×C3D3×GL(2,𝔽3)14yes—
214C2×C3D3×SL(2,ℤ/4)14yes—
215D3—316no—
216C22—92no—
217C22—92no—
218C22—92no—
219C22C3⋊D3×S411yescharacteristic
220C4—31no—
221C4—31no—
222C4—121no—
223C4—272no—
224C4—1081no—
225C3—41no—
226C3—84no—
227C3D3×#85914yescomplemented
228C2—94no—
229C2C3⋊D3×#22611yescharacteristic
230C2C3⋊D3×GL(2,𝔽3)11yescharacteristic
231C2C3⋊D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊#859)C2semidirect
3. C32×#859C2semidirect
8. C3⋊D3×SL(2,𝔽3)C4semidirect
15. C3×#859D3semidirect
23. C32×SL(2,𝔽3)C2×C4semidirect
29. Q8×C3⋊D3Dic3semidirect
55. #859C3⋊D3direct (N × H)
63. C32×Q8C2×Dic3semidirect
77. C3×SL(2,𝔽3)C4×D3semidirect
130. C3×Q8D3×Dic3semidirect
145. SL(2,𝔽3)C4×C3⋊D3semidirect
157. C3⋊D3#859direct (N × H)
190. C32C2×#859semidirect
195. Q8Dic3×C3⋊D3semidirect
227. C3D3×#859semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C32×#8592
D3×#8593
C3⋊D3×#1033
C2×Dic3×C3⋊D34