Finite Group Databasebeta

← C2×#11724177 #13180748

Subgroups — C2×#11724177 #13180748

Order 1728.

1022 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1022 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⋊#226)C211yesmaximal, characteristic, complemented
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11724177C214yesmaximal, direct product factor
5C2×#64197—34nomaximal
6C2×#64125—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×D3×C3⋊Dic3—41nomaximal
9C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
10C3⋊(C3⋊#226)C2212yescomplemented
11C32×GL(2,𝔽3)C2212yescomplemented
12C2×#3832D311yescharacteristic, complemented
13C2×#3830—31no—
14C2×C3⋊(#41⋊C3)—34no—
15C2×C3⋊#226—34no—
16C2×C3⋊#226—64no—
17C2×C3×GL(2,𝔽3)D314yescomplemented
18#64197—316no—
19C2×C32×SD16—31no—
20C2×#3756—31no—
21C2×#3753—31no—
22#64125—34no—
23#64125—34no—
24C2×#3744—31no—
25C2×C8×C3⋊D3—31no—
26C22×C32×D3—41no—
27C2×C3⋊(C3⋊Dic3)—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30D3×C3⋊Dic3—44no—
31C2×#985—91no—
32C2×#916—94no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C32×D4—31no—
35C2×#638—61no—
36C22×C3⋊Dic3—61no—
37#3832—32no—
38#3832C2×D312yescomplemented
39#3830—34no—
40#3830—62no—
41C2×C4⋊(C3⋊D3)—31no—
42C2×C4×C3⋊D3—31no—
43C2×#634—31no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yes—
46C2×D3×Dic3—124no—
47C3⋊(#41⋊C3)—38no—
48C3⋊#226—38no—
49C3⋊#226—68no—
50C3×GL(2,𝔽3)C2×D318yescomplemented
51C32×SD16—32no—
52C32×SD16—32no—
53C2×C32×C8—31no—
54#3756—32no—
55#3756—32no—
56#3753—34no—
57C2×C3⋊(C3⋊C8)—31no—
58#3744—34no—
59C8×C3⋊D3—34no—
60C22×C33—41no—
61C2×C32×D3—42no—
62C2×C32×D3—44no—
63C3⋊(C3⋊Dic3)—42no—
64C3×C3⋊Dic3—42no—
65C2×#239—34no—
66C2×#237—94no—
67C2×#41⋊C3—91no—
68#985—94no—
69C2×GL(2,𝔽3)C3⋊D311yescharacteristic, complemented
70C2×#226—91no—
71C2×#226—184no—
72C2×C3×SD16—34no—
73C2×#216—94no—
74C2×#213—94no—
75#916—916no—
76#916—916no—
77C2×#204—94no—
78C2×D3×C8—94no—
79C23×C32—61no—
80C22×C3⋊D3—31no—
81C22×C3×D3—44no—
82C32×Q8—31no—
83C32×Q8C22×D311yescharacteristic
84C32×D4—32no—
85C32×D4—61no—
86C2×C32×C4—31no—
87#638—64no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—64no—
90C2×C3⋊Dic3—124no—
91C2×C3⋊Dic3—244no—
92C2×C3⋊Dic3S411yescharacteristic
93C4⋊(C3⋊D3)—32no—
94C4⋊(C3⋊D3)—61no—
95C4×C3⋊D3—32no—
96C4×C3⋊D3—32no—
97#634—32no—
98#634—61no—
99C2×C3×Dic3—124no—
100C3×SL(2,𝔽3)—24no—
101C3×SL(2,𝔽3)C22×D314yes—
102D3×Dic3—1216no—
103C32×C8—32no—
104C3⋊(C3⋊C8)—32no—
105C2×#135—271noSylow
106C2×C33—42no—
107C2×C33—41no—
108C32×D3—44no—
109C2×C3×Q8D3214yes—
110C2×C3×D4—34no—
111C2×#67—184no—
112C22×Dic3—184no—
113#239—38no—
114#239—98no—
115#237—916no—
116#237—188no—
117C2×D12—94no—
118C2×C4×D3—94no—
119C2×C4×D3—361no—
120C2×Dic6—94no—
121#41⋊C3—92no—
122C2×SL(2,𝔽3)—24no—
123C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
124GL(2,𝔽3)C2×C3⋊D312yescomplemented
125#226—92no—
126#226—188no—
127C3×SD16—38no—
128C3×SD16—38no—
129C2×C3×C8—34no—
130#216—98no—
131#216—98no—
132#213—916no—
133C2×C3⋊C8—94no—
134#204—916no—
135D3×C8—916no—
136C22×C32—44no—
137C22×C32—62no—
138C22×C32—84no—
139C22×C32—122no—
140C22×C32C2×S411yescharacteristic
141C2×C3⋊D3—32no—
142C2×C3⋊D3—62no—
143C2×C3×D3—48no—
144C2×C3×D3—416no—
145C32×C4—31no—
146C32×C4—31no—
147C3⋊Dic3—62no—
148C3⋊Dic3—128no—
149C3⋊Dic3—248no—
150C3⋊Dic3C2×S412yes—
151C3×Dic3—128no—
152C2×#41—91no—
153C2×#41—271no—
154#135—274no—
155#135—274no—
156C2×Q16—271no—
157C2×SD16—31no—
158C2×SD16—271no—
159C2×D8—271no—
160C22×C8—271no—
161C33—41noSylow
162C23×C3—64no—
163C22×D3—41no—
164C22×D3—94no—
165C3×Q8—34no—
166C3×Q8C2×D3214yes—
167C3×D4—38no—
168C3×D4—64no—
169C2×C3×C4—34no—
170C2×C3×C4—361no—
171#67—1816no—
172C2×Dic3—34no—
173C2×Dic3—1816no—
174C2×Dic3—184no—
175C2×Dic3—361no—
176C2×Dic3—724no—
177D12—98no—
178D12—184no—
179C4×D3—98no—
180C4×D3—98no—
181C4×D3—364no—
182Dic6—98no—
183Dic6—184no—
184SL(2,𝔽3)—24no—
185SL(2,𝔽3)C22×C3⋊D311yescharacteristic
186C3×C8—38no—
187C3⋊C8—98no—
188C2×C32—44no—
189C2×C32—48no—
190C2×C32—84no—
191C2×C32—88no—
192C2×C32—122no—
193C2×C32#98512yescomplemented
194C2×C32C22×S411yescharacteristic
195C3⋊D3—62no—
196C3×D3—416no—
197#41—92no—
198#41—272no—
199#41—274no—
200#41—542no—
201C2×Q8—271no—
202C2×Q8D3×C3⋊D311yescharacteristic
203C2×D4—31no—
204C2×D4—271no—
205C2×D4—541no—
206C22×C4—271no—
207C22×C4—541no—
208Q16—272no—
209Q16—272no—
210SD16—32no—
211SD16—32no—
212SD16—274no—
213D8—274no—
214C2×C8—31no—
215C2×C8—274no—
216C2×C8—271no—
217C22×C3—41no—
218C22×C3—68no—
219C22×C3—84no—
220C22×C3—128no—
221C22×C3D3×S414yes—
222C2×D3—42no—
223C2×D3—44no—
224C2×D3—98no—
225C2×D3—188no—
226C3×C4—34no—
227C3×C4—34no—
228C3×C4—362no—
229Dic3—38no—
230Dic3—188no—
231Dic3—362no—
232Dic3—728no—
233C32—44no—
234C32—84no—
235C32C2×#98511yescharacteristic, complemented
236C23—61no—
237C23—271no—
238Q8—31no—
239Q8—272no—
240Q8—541no—
241Q8C2×D3×C3⋊D311yescharacteristic
242D4—32no—
243D4—61no—
244D4—272no—
245D4—541no—
246D4—544no—
247C2×C4—31no—
248C2×C4—91no—
249C2×C4—272no—
250C2×C4—272no—
251C2×C4—544no—
252C2×C4—541no—
253C8—32no—
254C8—272no—
255C2×C3—42no—
256C2×C3—41no—
257C2×C3—88no—
258C2×C3—84no—
259C2×C3—128no—
260C2×C3#6419718yescomplemented
261C2×C3C2×D3×S414yes—
262D3—44no—
263D3—188no—
264C22—62no—
265C22—122no—
266C22—272no—
267C22—542no—
268C22C3⋊D3×S411yescharacteristic
269C4—31no—
270C4—31no—
271C4—92no—
272C4—542no—
273C3—41no—
274C3—84no—
275C3C2×#6419714yescomplemented
276C2—122no—
277C2—542no—
278C2#1172417712yesdirect product factor
279C2C2×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⋊#226)C2semidirect
3. C2×C32×GL(2,𝔽3)C2semidirect
4. #11724177C2direct (N × H)
9. C3⋊(C3⋊(#41⋊C3))C22semidirect
10. C3⋊(C3⋊#226)C22semidirect
11. C32×GL(2,𝔽3)C22semidirect
12. C2×#3832D3semidirect
17. C2×C3×GL(2,𝔽3)D3semidirect
38. #3832C2×D3semidirect
50. C3×GL(2,𝔽3)C2×D3semidirect
69. C2×GL(2,𝔽3)C3⋊D3semidirect
124. GL(2,𝔽3)C2×C3⋊D3semidirect
193. C2×C32#985semidirect
235. C32C2×#985semidirect
260. C2×C3#64197semidirect
275. C3C2×#64197semidirect
278. C2#11724177direct (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⋊#226)2
C2×C32×GL(2,𝔽3)2
#117241772
C2×#641973
C2×#641253
C2×D3×C3⋊Dic34