Finite Group Databasebeta

← C2×#11724471 #13181406

Subgroups — C2×#11724471 #13181406

Order 1728.

1702 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1702 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#11724471C214yesmaximal, direct product factor
3C2×C3⋊(C3⋊GL(2,𝔽3))C212yesmaximal, complemented
4C2×#64267—312nomaximal
5C2×#64158—31nomaximal
6C2×C4⋊(C3⋊(C3⋊D3))—41nomaximal
7C32×#41⋊C3C2212yescomplemented
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C3⋊(C3⋊GL(2,𝔽3))C2214yescomplemented
10C2×C32×#41D311yescharacteristic, complemented
11C22×C4⋊(C3⋊D3)—31no—
12C2×C3×#41⋊C3D3112yescomplemented
13#64267—348no—
14C2×C3⋊GL(2,𝔽3)—324no—
15#64158—34no—
16#64158—34no—
17C2×#3755—32no—
18C2×#3753—32no—
19C2×#3748—31no—
20C22×C3⋊(C3⋊D3)—42no—
21C2×C33×C4—41no—
22C4⋊(C3⋊(C3⋊D3))—44no—
23C32×SL(2,𝔽3)C2311yescharacteristic, complemented
24C2×#986—99no—
25C2×#949—94no—
26C23×C3⋊D3—61no—
27C32×#41—32no—
28C32×#41C2×D312yescomplemented
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C4⋊(C3⋊D3)—32no—
33C2×C4⋊(C3⋊D3)—34no—
34C2×C4⋊(C3⋊D3)—62no—
35C2×C4⋊(C3⋊D3)—61no—
36C2×C4⋊(C3⋊D3)—1212no—
37C3×#41⋊C3C2×D3124yescomplemented
38C2×C3×SL(2,𝔽3)C2×D3112yescomplemented
39C3⋊GL(2,𝔽3)—348no—
40#3755—34no—
41#3755—34no—
42#3753—38no—
43#3748—34no—
44C2×C3⋊(C3⋊C8)—32no—
45C22×C33—41no—
46C2×C3⋊(C3⋊D3)—44no—
47C2×C3⋊(C3⋊D3)—84no—
48C33×C4—42no—
49C2×C3×#41C3⋊D314yescomplemented
50C22×D12—94no—
51C2×#41⋊C3C3⋊D319yescomplemented
52#986—936no—
53C2×GL(2,𝔽3)—918no—
54#949—916no—
55#949—916no—
56C2×#215—98no—
57C2×#213—98no—
58C2×#208—94no—
59C23×C32—31no—
60C22×C3⋊D3—64no—
61C22×C3⋊D3—62no—
62C22×C3⋊D3—1224no—
63C22×C3⋊D3—124no—
64C32×Q8—31no—
65C32×Q8C22×D311yescharacteristic
66C32×D4—32no—
67C32×D4—61no—
68C2×C32×C4—31no—
69C2×C32×C4—32no—
70C2×C32×C4—32no—
71C2×C32×C4—412no—
72C2×C32×C4S411yescharacteristic
73C4⋊(C3⋊D3)—34no—
74C4⋊(C3⋊D3)—62no—
75C4⋊(C3⋊D3)—64no—
76C4⋊(C3⋊D3)—1248no—
77C3×SL(2,𝔽3)C22×D3112yescomplemented
78C3⋊(C3⋊C8)—34no—
79C2×AGL(1,ℤ/8)—271noSylow
80C2×C33—42no—
81C2×C33—41no—
82C3⋊(C3⋊D3)—84no—
83C23×D3—184no—
84C3×#41—38no—
85C3×#41C2×C3⋊D318yescomplemented
86C2×C3×Q8C2×C3⋊D314yes—
87C2×C3×D4—34no—
88C22×C3×C4—34no—
89C2×D12—98no—
90C2×D12—916no—
91C2×D12—188no—
92C2×D12—184no—
93C2×D12—369no—
94#41⋊C3C2×C3⋊D3118yescomplemented
95C2×SL(2,𝔽3)C2×C3⋊D319yescomplemented
96GL(2,𝔽3)—936no—
97#215—916no—
98#215—916no—
99#213—932no—
100#208—916no—
101C2×C3⋊C8—98no—
102C22×C32—32no—
103C22×C32—412no—
104C22×C32—62no—
105C22×C32C2×S411yescharacteristic
106C2×C3⋊D3—64no—
107C2×C3⋊D3—1248no—
108C2×C3⋊D3—128no—
109C2×C3⋊D3—124no—
110C2×C3⋊D3—2448no—
111C32×C4—31no—
112C32×C4—31no—
113C32×C4—424no—
114C32×C4C2×S412yes—
115C2×#41C3⋊(C3⋊D3)11yescharacteristic, complemented
116C22×D4—271no—
117AGL(1,ℤ/8)—274no—
118AGL(1,ℤ/8)—274no—
119C2×SD16—272no—
120C2×D8—272no—
121C2×M16—271no—
122C33—41noSylow
123C23×C3—34no—
124C22×D3—1816no—
125C22×D3—188no—
126C22×D3—3618no—
127C22×D3—3616no—
128C3×Q8—34no—
129C3×Q8C22×C3⋊D314yes—
130C3×D4—38no—
131C3×D4—64no—
132C2×C3×C4—34no—
133C2×C3×C4—38no—
134C2×C3×C4—38no—
135C2×C3×C4—49no—
136C2×C3×C4C3⋊S414yes—
137D12—916no—
138D12—188no—
139D12—1816no—
140D12—3636no—
141SL(2,𝔽3)C22×C3⋊D319yescomplemented
142C3⋊C8—916no—
143C2×C32—424no—
144C2×C32—412no—
145C2×C32—62no—
146C2×C32#98612yescomplemented
147C2×C32C22×S411yescharacteristic
148C3⋊D3—124no—
149C3⋊D3—2448no—
150C24—541no—
151#41—32no—
152#41C2×C3⋊(C3⋊D3)12yescomplemented
153C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
154C2×D4—31no—
155C2×D4—272no—
156C2×D4—274no—
157C2×D4—542no—
158C2×D4—541no—
159C22×C4—31no—
160SD16—274no—
161SD16—274no—
162D8—278no—
163M16—274no—
164C2×C8—272no—
165C22×C3—38no—
166C22×C3—49no—
167C22×C3—68no—
168C22×C3C2×C3⋊S414yes—
169C2×D3—1816no—
170C2×D3—3636no—
171C2×D3—3632no—
172C2×D3—3616no—
173C2×D3—7236no—
174C3×C4—34no—
175C3×C4—34no—
176C3×C4—418no—
177C3×C4C2×C3⋊S418yes—
178C32—412no—
179C32C2×#98611yescharacteristic, complemented
180C23—31no—
181C23—544no—
182C23—542no—
183C23—1084no—
184Q8—31no—
185Q8C22×C3⋊(C3⋊D3)11yescharacteristic
186D4—32no—
187D4—61no—
188D4—274no—
189D4—542no—
190D4—544no—
191C2×C4—31no—
192C2×C4—32no—
193C2×C4—32no—
194C2×C4C3⋊(C3⋊S4)11yescharacteristic
195C8—274no—
196C2×C3—418no—
197C2×C3—49no—
198C2×C3—68no—
199C2×C3#6426718yescomplemented
200C2×C3C22×C3⋊S414yes—
201D3—3616no—
202D3—7236no—
203C22—32no—
204C22—62no—
205C22—544no—
206C22—1088no—
207C22—1084no—
208C22C2×C3⋊(C3⋊S4)11yescharacteristic
209C4—31no—
210C4—31no—
211C4C2×C3⋊(C3⋊S4)12yes—
212C3—49no—
213C3C2×#6426714yescomplemented
214C2—62no—
215C2—1084no—
216C2#1172447112yesdirect product factor
217C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C4C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. #11724471C2direct (N × H)
3. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
7. C32×#41⋊C3C22semidirect
8. C2×C32×SL(2,𝔽3)C22semidirect
9. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
10. C2×C32×#41D3semidirect
12. C2×C3×#41⋊C3D3semidirect
23. C32×SL(2,𝔽3)C23semidirect
28. C32×#41C2×D3semidirect
37. C3×#41⋊C3C2×D3semidirect
38. C2×C3×SL(2,𝔽3)C2×D3semidirect
49. C2×C3×#41C3⋊D3semidirect
51. C2×#41⋊C3C3⋊D3semidirect
77. C3×SL(2,𝔽3)C22×D3semidirect
85. C3×#41C2×C3⋊D3semidirect
94. #41⋊C3C2×C3⋊D3semidirect
95. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
115. C2×#41C3⋊(C3⋊D3)semidirect
141. SL(2,𝔽3)C22×C3⋊D3semidirect
146. C2×C32#986semidirect
152. #41C2×C3⋊(C3⋊D3)semidirect
179. C32C2×#986semidirect
199. C2×C3#64267semidirect
213. C3C2×#64267semidirect
216. C2#11724471direct (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
#117244712
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×#642673
C2×#641583
C2×C4⋊(C3⋊(C3⋊D3))4