Finite Group Databasebeta

← C2×#11724468 #13181401

Subgroups — C2×#11724468 #13181401

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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#11724468C214yesmaximal, direct product factor
3C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
4C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
5C2×#64264—312nomaximal
6C2×#64151—31nomaximal
7C2×#6312—41nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×SL(2,𝔽3)C2212yescomplemented
10C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
11C3⋊(C3⋊#226)C2212yescomplemented
12C22×C32×Q8D311yescharacteristic, complemented
13C2×#3828—31no—
14C22×C3×SL(2,𝔽3)D3112yescomplemented
15#64264—348no—
16C2×C3⋊GL(2,𝔽3)—312no—
17C2×C3⋊#226—312no—
18C2×#3756—31no—
19C2×#3756—31no—
20#64151—34no—
21#64151—34no—
22C2×#3755—31no—
23C2×#3755—31no—
24C2×#3748—31no—
25C23×C33—41no—
26C22×C3⋊(C3⋊D3)—41no—
27#6312—44no—
28C2×C3⋊(C3⋊Dic3)—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30C2×#983—99no—
31C2×#942—94no—
32C2×C32×Q8—31no—
33C2×C32×Q8—32no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C32×Q8C2×D312yescomplemented
36C22×C32×C4—31no—
37C2×#638—61no—
38C2×#638—1212no—
39#3828—34no—
40#3828—62no—
41C2×C4⋊(C3⋊D3)—31no—
42C2×C4×C3⋊D3—61no—
43C2×#634—31no—
44C2×C3×SL(2,𝔽3)C2×D3112yes—
45C2×C3×SL(2,𝔽3)C2×D3124yescomplemented
46C3⋊GL(2,𝔽3)—324no—
47C3⋊#226—324no—
48#3756—32no—
49#3756—34no—
50#3756—32no—
51#3755—32no—
52#3755—34no—
53#3755—32no—
54#3748—34no—
55C2×C3⋊(C3⋊C8)—31no—
56C2×C3⋊(C3⋊C8)—31no—
57C22×C33—41no—
58C22×C33—42no—
59C22×C33—82no—
60C2×C3⋊(C3⋊D3)—42no—
61C2×C3⋊(C3⋊D3)—82no—
62C3⋊(C3⋊Dic3)—42no—
63C22×C3×Q8C3⋊D314yescomplemented
64C2×#235—94no—
65C22×SL(2,𝔽3)C3⋊D319yescomplemented
66#983—936no—
67C2×GL(2,𝔽3)—99no—
68C2×#226—99no—
69C2×#216—94no—
70C2×#216—94no—
71#942—916no—
72#942—916no—
73C2×#215—94no—
74C2×#215—94no—
75C2×#208—94no—
76C23×C32—412no—
77C23×C32S411yescharacteristic
78C22×C3⋊D3—61no—
79C22×C3⋊D3—1212no—
80C32×Q8—31no—
81C32×Q8—32no—
82C32×Q8—61no—
83C32×Q8C22×D311yescharacteristic
84C2×C32×C4—31no—
85C2×C32×C4—32no—
86C2×C32×C4—31no—
87C2×C32×C4—32no—
88#638—64no—
89#638—1248no—
90C2×C3⋊Dic3—61no—
91C2×C3⋊Dic3—1212no—
92C4⋊(C3⋊D3)—32no—
93C4⋊(C3⋊D3)—61no—
94C4×C3⋊D3—64no—
95#634—32no—
96#634—61no—
97C3×SL(2,𝔽3)C22×D3112yes—
98C3⋊(C3⋊C8)—32no—
99C3⋊(C3⋊C8)—32no—
100C2×#137—271noSylow
101C2×C33—42no—
102C2×C33—41no—
103C2×C33—82no—
104C3⋊(C3⋊D3)—82no—
105C2×C3×Q8—34no—
106C2×C3×Q8—38no—
107C2×C3×Q8C2×C3⋊D314yes—
108C2×C3×Q8C2×C3⋊D318yescomplemented
109C22×C3×C4—34no—
110C2×#67—184no—
111C2×#67—369no—
112#235—916no—
113#235—188no—
114C2×D12—94no—
115C2×C4×D3—184no—
116C2×Dic6—94no—
117C2×SL(2,𝔽3)C2×C3⋊D319yes—
118C2×SL(2,𝔽3)C2×C3⋊D3118yescomplemented
119GL(2,𝔽3)—918no—
120#226—918no—
121#216—98no—
122#216—916no—
123#216—98no—
124#215—98no—
125#215—916no—
126#215—98no—
127#208—916no—
128C2×C3⋊C8—94no—
129C2×C3⋊C8—94no—
130C22×C32—22no—
131C22×C32—412no—
132C22×C32—424no—
133C22×C32—824no—
134C22×C32C2×S412yes—
135C22×C32C2×S411yescharacteristic
136C2×C3⋊D3—62no—
137C2×C3⋊D3—1224no—
138C2×C3⋊D3—122no—
139C2×C3⋊D3—2424no—
140C32×C4—31no—
141C32×C4—31no—
142C32×C4—32no—
143C3⋊Dic3—62no—
144C3⋊Dic3—1224no—
145C2×#41—271no—
146C22×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
147#137—274no—
148#137—274no—
149C2×Q16—271no—
150C2×Q16—271no—
151C2×SD16—271no—
152C2×SD16—271no—
153C2×M16—271no—
154C33—41noSylow
155C23×C3—49no—
156C23×C3C3⋊S414yes—
157C22×D3—184no—
158C22×D3—369no—
159C3×Q8—34no—
160C3×Q8—38no—
161C3×Q8—64no—
162C3×Q8C22×C3⋊D314yes—
163C2×C3×C4—34no—
164C2×C3×C4—38no—
165C2×C3×C4—34no—
166C2×C3×C4—38no—
167#67—1816no—
168#67—3636no—
169C2×Dic3—184no—
170C2×Dic3—369no—
171D12—98no—
172D12—184no—
173C4×D3—1816no—
174Dic6—98no—
175Dic6—184no—
176SL(2,𝔽3)C22×C3⋊D319yes—
177C3⋊C8—98no—
178C3⋊C8—98no—
179C2×C32—22no—
180C2×C32—424no—
181C2×C32—412no—
182C2×C32—824no—
183C2×C32#98312yescomplemented
184C2×C32C22×S411yescharacteristic
185C3⋊D3—122no—
186C3⋊D3—2424no—
187#41—274no—
188#41—542no—
189C2×Q8—31no—
190C2×Q8—32no—
191C2×Q8—271no—
192C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
193C2×Q8C2×C3⋊(C3⋊D3)12yescomplemented
194C2×D4—271no—
195C2×D4—541no—
196C22×C4—31no—
197C22×C4—541no—
198Q16—272no—
199Q16—274no—
200Q16—272no—
201SD16—272no—
202SD16—274no—
203SD16—272no—
204M16—274no—
205C2×C8—271no—
206C2×C8—271no—
207C22×C3—28no—
208C22×C3—418no—
209C22×C3—49no—
210C22×C3—818no—
211C22×C3C2×C3⋊S418yes—
212C22×C3C2×C3⋊S414yes—
213C2×D3—188no—
214C2×D3—3618no—
215C2×D3—368no—
216C2×D3—7218no—
217C3×C4—34no—
218C3×C4—34no—
219C3×C4—38no—
220Dic3—188no—
221Dic3—3618no—
222C32—412no—
223C32C2×#98311yescharacteristic, complemented
224C23—541no—
225C23C3⋊(C3⋊S4)11yescharacteristic
226Q8—31no—
227Q8—32no—
228Q8—61no—
229Q8—272no—
230Q8—541no—
231Q8C22×C3⋊(C3⋊D3)11yescharacteristic
232D4—272no—
233D4—541no—
234D4—544no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—544no—
240C2×C4—541no—
241C8—272no—
242C8—272no—
243C2×C3—28no—
244C2×C3—418no—
245C2×C3—49no—
246C2×C3—818no—
247C2×C3#6426418yescomplemented
248C2×C3C22×C3⋊S414yes—
249D3—368no—
250D3—7218no—
251C22—22no—
252C22—542no—
253C22—1082no—
254C22C2×C3⋊(C3⋊S4)12yes—
255C22C2×C3⋊(C3⋊S4)11yescharacteristic
256C4—31no—
257C4—32no—
258C4—31no—
259C4—542no—
260C3—49no—
261C3C2×#6426414yescomplemented
262C2—22no—
263C2—1082no—
264C2#1172446812yesdirect product factor
265C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC22×C32×Q8D3
socleC22×C32C2×S4
hypercenterC23C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. #11724468C2direct (N × H)
3. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
4. C2×C3⋊(C3⋊#226)C2semidirect
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
11. C3⋊(C3⋊#226)C22semidirect
12. C22×C32×Q8D3semidirect
14. C22×C3×SL(2,𝔽3)D3semidirect
35. C2×C32×Q8C2×D3semidirect
45. C2×C3×SL(2,𝔽3)C2×D3semidirect
63. C22×C3×Q8C3⋊D3semidirect
65. C22×SL(2,𝔽3)C3⋊D3semidirect
108. C2×C3×Q8C2×C3⋊D3semidirect
118. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
146. C22×Q8C3⋊(C3⋊D3)semidirect
183. C2×C32#983semidirect
193. C2×Q8C2×C3⋊(C3⋊D3)semidirect
223. C32C2×#983semidirect
247. C2×C3#64264semidirect
261. C3C2×#64264semidirect
264. C2#11724468direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
#117244682
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C3⋊(C3⋊#226)2
C2×#642643
C2×#641513
C2×#63124