Finite Group Databasebeta

← C2×#11724470 #13181405

Subgroups — C2×#11724470 #13181405

Order 1728.

1606 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 199 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#11724470C214yesmaximal, direct product factor
3C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
4C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
5C2×#64266—312nomaximal
6C2×#64159—31nomaximal
7C2×C4×C3⋊(C3⋊D3)—41nomaximal
8C32×#41⋊C3C2212yescomplemented
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
11C3⋊(C3⋊#226)C2212yescomplemented
12C2×C32×#41D311yescharacteristic, complemented
13C2×#3828—31no—
14C2×C3×#41⋊C3D3112yescomplemented
15#64266—348no—
16C2×C3⋊GL(2,𝔽3)—312no—
17C2×C3⋊#226—312no—
18#64159—34no—
19#64159—34no—
20C2×#3756—31no—
21C2×#3755—31no—
22C2×#3754—31no—
23C2×#3753—31no—
24C22×C3⋊(C3⋊C8)—31no—
25C22×C3⋊(C3⋊D3)—41no—
26C2×C33×C4—41no—
27C2×C3⋊(C3⋊Dic3)—41no—
28C4×C3⋊(C3⋊D3)—44no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30C2×#985—99no—
31C2×#950—94no—
32C32×#41—32no—
33C32×#41C2×D312yescomplemented
34C2×C32×Q8C2×D311yescharacteristic
35C2×C32×D4—31no—
36C22×C32×C4—31no—
37C2×#638—61no—
38#3828—34no—
39#3828—62no—
40C2×C4⋊(C3⋊D3)—31no—
41C2×C4×C3⋊D3—61no—
42C2×C4×C3⋊D3—1212no—
43C2×#634—31no—
44C3×#41⋊C3C2×D3124yescomplemented
45C2×C3×SL(2,𝔽3)C2×D3112yes—
46C3⋊GL(2,𝔽3)—324no—
47C3⋊#226—324no—
48#3756—32no—
49#3756—32no—
50#3755—32no—
51#3755—32no—
52#3754—34no—
53#3753—34no—
54C2×C3⋊(C3⋊C8)—31no—
55C2×C3⋊(C3⋊C8)—31no—
56C2×C3⋊(C3⋊C8)—34no—
57C22×C33—41no—
58C2×C3⋊(C3⋊D3)—42no—
59C2×C3⋊(C3⋊D3)—44no—
60C33×C4—42no—
61C3⋊(C3⋊Dic3)—42no—
62C2×C3×#41C3⋊D314yescomplemented
63C2×#235—94no—
64C2×#41⋊C3C3⋊D319yescomplemented
65#985—936no—
66C2×GL(2,𝔽3)—99no—
67C2×#226—99no—
68#950—916no—
69#950—916no—
70C2×#216—94no—
71C2×#215—94no—
72C2×#214—94no—
73C2×#213—94no—
74C22×C3⋊C8—94no—
75C23×C32—31no—
76C22×C3⋊D3—61no—
77C22×C3⋊D3—1212no—
78C32×Q8—31no—
79C32×Q8C22×D311yescharacteristic
80C32×D4—32no—
81C32×D4—61no—
82C2×C32×C4—31no—
83C2×C32×C4—32no—
84C2×C32×C4—32no—
85C2×C32×C4—412no—
86C2×C32×C4S411yescharacteristic
87#638—64no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—1212no—
90C4⋊(C3⋊D3)—32no—
91C4⋊(C3⋊D3)—61no—
92C4×C3⋊D3—64no—
93C4×C3⋊D3—1248no—
94#634—32no—
95#634—61no—
96C3×SL(2,𝔽3)C22×D3112yes—
97C3⋊(C3⋊C8)—32no—
98C3⋊(C3⋊C8)—32no—
99C2×#135—271noSylow
100C2×C33—42no—
101C2×C33—41no—
102C3⋊(C3⋊D3)—44no—
103C3×#41—38no—
104C3×#41C2×C3⋊D318yescomplemented
105C2×C3×Q8C2×C3⋊D314yes—
106C2×C3×D4—34no—
107C22×C3×C4—34no—
108C2×#67—184no—
109#235—916no—
110#235—188no—
111C2×D12—94no—
112C2×C4×D3—184no—
113C2×C4×D3—369no—
114C2×Dic6—94no—
115#41⋊C3C2×C3⋊D3118yescomplemented
116C2×SL(2,𝔽3)C2×C3⋊D319yes—
117GL(2,𝔽3)—918no—
118#226—918no—
119#216—98no—
120#216—98no—
121#215—98no—
122#215—98no—
123#214—916no—
124#213—916no—
125C2×C3⋊C8—94no—
126C2×C3⋊C8—94no—
127C2×C3⋊C8—916no—
128C22×C32—32no—
129C22×C32—412no—
130C22×C32—62no—
131C22×C32C2×S411yescharacteristic
132C2×C3⋊D3—62no—
133C2×C3⋊D3—1224no—
134C2×C3⋊D3—1248no—
135C2×C3⋊D3—122no—
136C32×C4—31no—
137C32×C4—31no—
138C32×C4—424no—
139C32×C4C2×S412yes—
140C3⋊Dic3—62no—
141C3⋊Dic3—1224no—
142C2×#41—271no—
143C2×#41C3⋊(C3⋊D3)11yescharacteristic, complemented
144#135—274no—
145#135—274no—
146C2×Q16—271no—
147C2×SD16—271no—
148C2×SD16—271no—
149C2×D8—271no—
150C22×C8—271no—
151C33—41noSylow
152C23×C3—34no—
153C22×D3—184no—
154C22×D3—369no—
155C3×Q8—34no—
156C3×Q8C22×C3⋊D314yes—
157C3×D4—38no—
158C3×D4—64no—
159C2×C3×C4—34no—
160C2×C3×C4—38no—
161C2×C3×C4—38no—
162C2×C3×C4—49no—
163C2×C3×C4C3⋊S414yes—
164#67—1816no—
165C2×Dic3—184no—
166C2×Dic3—369no—
167D12—98no—
168D12—184no—
169C4×D3—1816no—
170C4×D3—3636no—
171Dic6—98no—
172Dic6—184no—
173SL(2,𝔽3)C22×C3⋊D319yes—
174C3⋊C8—98no—
175C3⋊C8—98no—
176C2×C32—424no—
177C2×C32—412no—
178C2×C32—62no—
179C2×C32#98512yescomplemented
180C2×C32C22×S411yescharacteristic
181C3⋊D3—1248no—
182C3⋊D3—122no—
183#41—32no—
184#41—274no—
185#41—542no—
186#41C2×C3⋊(C3⋊D3)12yescomplemented
187C2×Q8—271no—
188C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
189C2×D4—31no—
190C2×D4—271no—
191C2×D4—541no—
192C22×C4—31no—
193C22×C4—541no—
194Q16—272no—
195Q16—272no—
196SD16—274no—
197SD16—272no—
198SD16—272no—
199D8—274no—
200C2×C8—271no—
201C2×C8—271no—
202C2×C8—274no—
203C22×C3—38no—
204C22×C3—49no—
205C22×C3—68no—
206C22×C3C2×C3⋊S414yes—
207C2×D3—188no—
208C2×D3—3618no—
209C2×D3—3636no—
210C2×D3—368no—
211C3×C4—34no—
212C3×C4—34no—
213C3×C4—418no—
214C3×C4C2×C3⋊S418yes—
215Dic3—188no—
216Dic3—3618no—
217C32—412no—
218C32C2×#98511yescharacteristic, complemented
219C23—31no—
220C23—541no—
221Q8—31no—
222Q8—272no—
223Q8—541no—
224Q8C22×C3⋊(C3⋊D3)11yescharacteristic
225D4—32no—
226D4—61no—
227D4—272no—
228D4—541no—
229D4—544no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—541no—
234C2×C4—544no—
235C2×C4C3⋊(C3⋊S4)11yescharacteristic
236C8—272no—
237C8—272no—
238C2×C3—418no—
239C2×C3—49no—
240C2×C3—68no—
241C2×C3#6426618yescomplemented
242C2×C3C22×C3⋊S414yes—
243D3—3636no—
244D3—368no—
245C22—32no—
246C22—62no—
247C22—542no—
248C22—1082no—
249C22C2×C3⋊(C3⋊S4)11yescharacteristic
250C4—31no—
251C4—31no—
252C4—542no—
253C4C2×C3⋊(C3⋊S4)12yes—
254C3—49no—
255C3C2×#6426614yescomplemented
256C2—62no—
257C2—1082no—
258C2#1172447012yesdirect product factor
259C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC2×C32×#41D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. #11724470C2direct (N × H)
3. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
4. C2×C3⋊(C3⋊#226)C2semidirect
8. C32×#41⋊C3C22semidirect
10. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
11. C3⋊(C3⋊#226)C22semidirect
12. C2×C32×#41D3semidirect
14. C2×C3×#41⋊C3D3semidirect
33. C32×#41C2×D3semidirect
44. C3×#41⋊C3C2×D3semidirect
62. C2×C3×#41C3⋊D3semidirect
64. C2×#41⋊C3C3⋊D3semidirect
104. C3×#41C2×C3⋊D3semidirect
115. #41⋊C3C2×C3⋊D3semidirect
143. C2×#41C3⋊(C3⋊D3)semidirect
179. C2×C32#985semidirect
186. #41C2×C3⋊(C3⋊D3)semidirect
218. C32C2×#985semidirect
241. C2×C3#64266semidirect
255. C3C2×#64266semidirect
258. C2#11724470direct (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
#117244702
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C3⋊(C3⋊#226)2
C2×#642663
C2×#641593
C2×C4×C3⋊(C3⋊D3)4