Finite Group Databasebeta

← C2×#11724178 #13180750

Subgroups — C2×#11724178 #13180750

Order 1728.

1086 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 89 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⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4#11724178C214yesmaximal, direct product factor
5C2×#64198—34nomaximal
6C2×#64128—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6289—41nomaximal
9C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
10C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
11C32×#226C2212yescomplemented
12C2×#3832—31no—
13C2×#3832D311yescharacteristic, complemented
14C2×C3⋊(#41⋊C3)—34no—
15C2×C3⋊GL(2,𝔽3)—34no—
16C2×C3⋊GL(2,𝔽3)—64no—
17C2×C3×#226D314yescomplemented
18#64198—316no—
19C2×C32×Q16—31no—
20C2×#3755—31no—
21C2×#3755—31no—
22#64128—34no—
23#64128—34no—
24C2×#3745—31no—
25C2×C8×C3⋊D3—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30#6289—44no—
31C2×#985—91no—
32C2×#919—94no—
33C2×C32×Q8—31no—
34C2×C32×Q8C2×D311yescharacteristic
35#3832—32no—
36#3832—34no—
37#3832—62no—
38#3832C2×D312yescomplemented
39C2×C4⋊(C3⋊D3)—31no—
40C2×C4⋊(C3⋊D3)—31no—
41C2×C4⋊(C3⋊D3)—61no—
42C2×C4×C3⋊D3—31no—
43C2×C4×C3⋊D3—61no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yes—
46C2×#624—124no—
47C3⋊(#41⋊C3)—38no—
48C3⋊GL(2,𝔽3)—38no—
49C3⋊GL(2,𝔽3)—68no—
50C3×#226C2×D318yescomplemented
51C32×Q16—32no—
52C32×Q16—32no—
53C2×C32×C8—31no—
54#3755—34no—
55#3755—32no—
56#3755—32no—
57C2×C3⋊(C3⋊C8)—31no—
58#3745—34no—
59C8×C3⋊D3—34no—
60C22×C33—41no—
61C2×C3⋊(C3⋊D3)—42no—
62C2×C3⋊(C3⋊D3)—44no—
63C3×C3⋊Dic3—42no—
64C32×Dic3—42no—
65C2×#239—34no—
66C2×#239—94no—
67C2×#41⋊C3—91no—
68#985—94no—
69C2×GL(2,𝔽3)—91no—
70C2×GL(2,𝔽3)—184no—
71C2×#226C3⋊D311yescharacteristic, complemented
72C2×C3×Q16—34no—
73C2×#215—94no—
74C2×#215—94no—
75#919—916no—
76#919—916no—
77C2×D24—94no—
78C2×D3×C8—94no—
79C22×C3⋊D3—31no—
80C22×C3⋊D3—61no—
81C22×C3⋊D3—124no—
82C22×C3⋊D3—244no—
83C32×Q8—31no—
84C32×Q8—32no—
85C32×Q8—61no—
86C32×Q8C22×D311yescharacteristic
87C2×C32×C4—31no—
88C2×C32×C4—61no—
89C2×C3⋊Dic3S411yescharacteristic
90C4⋊(C3⋊D3)—32no—
91C4⋊(C3⋊D3)—32no—
92C4⋊(C3⋊D3)—61no—
93C4⋊(C3⋊D3)—61no—
94C4⋊(C3⋊D3)—64no—
95C4×C3⋊D3—32no—
96C4×C3⋊D3—32no—
97C4×C3⋊D3—64no—
98C2×C3×Dic3—44no—
99C2×C3×Dic3—124no—
100C3×SL(2,𝔽3)—24no—
101C3×SL(2,𝔽3)C22×D314yes—
102#624—1216no—
103C32×C8—32no—
104C3⋊(C3⋊C8)—32no—
105C2×#135—271noSylow
106C2×C33—42no—
107C2×C33—41no—
108C3⋊(C3⋊D3)—44no—
109C2×C3×Q8—34no—
110C2×C3×Q8D3214yes—
111#239—38no—
112#239—98no—
113#239—916no—
114#239—188no—
115C2×D12—94no—
116C2×D12—94no—
117C2×D12—184no—
118C2×C4×D3—94no—
119C2×C4×D3—184no—
120C2×C4×D3—361no—
121#41⋊C3—92no—
122C2×SL(2,𝔽3)—24no—
123C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
124GL(2,𝔽3)—92no—
125GL(2,𝔽3)—188no—
126#226C2×C3⋊D312yescomplemented
127C3×Q16—38no—
128C3×Q16—38no—
129C2×C3×C8—34no—
130#215—916no—
131#215—98no—
132#215—98no—
133C2×C3⋊C8—94no—
134D24—916no—
135D3×C8—916no—
136C22×C32—44no—
137C22×C32—84no—
138C22×C32C2×S411yescharacteristic
139C2×C3⋊D3—32no—
140C2×C3⋊D3—62no—
141C2×C3⋊D3—62no—
142C2×C3⋊D3—128no—
143C2×C3⋊D3—122no—
144C2×C3⋊D3—1216no—
145C2×C3⋊D3—248no—
146C2×C3⋊D3—2416no—
147C32×C4—31no—
148C32×C4—31no—
149C32×C4—62no—
150C3⋊Dic3C2×S412yes—
151C3×Dic3—48no—
152C3×Dic3—128no—
153C2×#41—91no—
154C2×#41—271no—
155#135—274no—
156#135—274no—
157C2×Q16—31no—
158C2×SD16—271no—
159C2×SD16—271no—
160C2×D8—271no—
161C22×C8—271no—
162C33—41noSylow
163C22×D3—94no—
164C22×D3—184no—
165C22×D3—361no—
166C22×D3—724no—
167C3×Q8—34no—
168C3×Q8—38no—
169C3×Q8—64no—
170C3×Q8C2×D3214yes—
171C2×C3×C4—34no—
172C2×C3×C4—64no—
173C2×C3×C4—361no—
174C2×Dic3—34no—
175C2×Dic3—41no—
176D12—98no—
177D12—98no—
178D12—184no—
179D12—184no—
180D12—1816no—
181C4×D3—98no—
182C4×D3—98no—
183C4×D3—1816no—
184C4×D3—364no—
185SL(2,𝔽3)—24no—
186SL(2,𝔽3)C22×C3⋊D311yescharacteristic
187C3×C8—38no—
188C3⋊C8—98no—
189C2×C32—44no—
190C2×C32—48no—
191C2×C32—84no—
192C2×C32—88no—
193C2×C32#98512yescomplemented
194C2×C32C22×S411yescharacteristic
195C3⋊D3—62no—
196C3⋊D3—122no—
197C3⋊D3—1216no—
198C3⋊D3—2416no—
199#41—92no—
200#41—272no—
201#41—274no—
202#41—542no—
203C2×Q8—31no—
204C2×Q8D3×C3⋊D311yescharacteristic
205C2×D4—271no—
206C2×D4—271no—
207C2×D4—541no—
208C22×C4—271no—
209C22×C4—541no—
210Q16—32no—
211Q16—32no—
212SD16—274no—
213SD16—272no—
214SD16—272no—
215D8—274no—
216C2×C8—31no—
217C2×C8—274no—
218C2×C8—271no—
219C22×C3—41no—
220C22×C3—84no—
221C22×C3D3×S414yes—
222C2×D3—98no—
223C2×D3—188no—
224C2×D3—188no—
225C2×D3—362no—
226C2×D3—364no—
227C2×D3—368no—
228C2×D3—728no—
229C2×D3—7216no—
230C3×C4—34no—
231C3×C4—34no—
232C3×C4—68no—
233C3×C4—362no—
234Dic3—38no—
235Dic3—42no—
236C32—44no—
237C32—84no—
238C32C2×#98511yescharacteristic, complemented
239C23—271no—
240C23—541no—
241Q8—31no—
242Q8—32no—
243Q8—61no—
244Q8C2×D3×C3⋊D311yescharacteristic
245D4—272no—
246D4—272no—
247D4—541no—
248D4—541no—
249D4—544no—
250C2×C4—31no—
251C2×C4—61no—
252C2×C4—91no—
253C2×C4—272no—
254C2×C4—272no—
255C2×C4—544no—
256C8—32no—
257C8—272no—
258C2×C3—42no—
259C2×C3—41no—
260C2×C3—88no—
261C2×C3—84no—
262C2×C3#6419818yescomplemented
263C2×C3C2×D3×S414yes—
264D3—188no—
265D3—364no—
266D3—368no—
267D3—7216no—
268C22—272no—
269C22—542no—
270C22—542no—
271C22—1082no—
272C22C3⋊D3×S411yescharacteristic
273C4—31no—
274C4—31no—
275C4—62no—
276C4—92no—
277C3—41no—
278C3—84no—
279C3C2×#6419814yescomplemented
280C2—542no—
281C2—1082no—
282C2#1172417812yesdirect product factor
283C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×C3⋊D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C2×C32×#226C2semidirect
4. #11724178C2direct (N × H)
9. C3⋊(C3⋊(#41⋊C3))C22semidirect
10. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
11. C32×#226C22semidirect
13. C2×#3832D3semidirect
17. C2×C3×#226D3semidirect
38. #3832C2×D3semidirect
50. C3×#226C2×D3semidirect
71. C2×#226C3⋊D3semidirect
126. #226C2×C3⋊D3semidirect
193. C2×C32#985semidirect
238. C32C2×#985semidirect
262. C2×C3#64198semidirect
279. C3C2×#64198semidirect
282. C2#11724178direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊(#41⋊C3))2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C32×#2262
#117241782
C2×#641983
C2×#641283
C2×#62894