Finite Group Databasebeta

← C2×#11724179 #13180751

Subgroups — C2×#11724179 #13180751

Order 1728.

1182 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1182 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×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11724179C214yesmaximal, direct product factor
5C2×#64199—34nomaximal
6C2×#64123—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
8C2×#6291—41nomaximal
9C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
10C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
11C32×GL(2,𝔽3)C2212yescomplemented
12C2×#3832D311yescharacteristic, complemented
13C2×D4×C3⋊D3—31no—
14C2×C3⋊(#41⋊C3)—34no—
15C2×C3⋊GL(2,𝔽3)—34no—
16C2×C3⋊GL(2,𝔽3)—64no—
17C2×C3×GL(2,𝔽3)D314yescomplemented
18#64199—316no—
19C2×C32×SD16—31no—
20C2×#3755—31no—
21C2×#3753—31no—
22#64123—34no—
23#64123—34no—
24C2×#3745—31no—
25C2×#3743—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C22×C32×D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)C2311yescharacteristic, complemented
30#6291—44no—
31C2×#986—91no—
32C2×#914—94no—
33C23×C3⋊D3—61no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C32×D4—31no—
36C2×#638—61no—
37#3832—32no—
38#3832C2×D312yescomplemented
39D4×C3⋊D3—34no—
40D4×C3⋊D3—62no—
41C2×C4⋊(C3⋊D3)—31no—
42C2×C4⋊(C3⋊D3)—31no—
43C2×C4×C3⋊D3—31no—
44C2×C3×SL(2,𝔽3)—24no—
45C2×C3×SL(2,𝔽3)C2×D314yescomplemented
46C2×#626—124no—
47C3⋊(#41⋊C3)—38no—
48C3⋊GL(2,𝔽3)—38no—
49C3⋊GL(2,𝔽3)—68no—
50C3×GL(2,𝔽3)C2×D318yescomplemented
51C32×SD16—32no—
52C32×SD16—32no—
53C2×C32×C8—31no—
54#3755—32no—
55#3755—32no—
56#3753—34no—
57C2×C3⋊(C3⋊C8)—31no—
58#3745—34no—
59#3743—34no—
60C22×C33—41no—
61C2×C3⋊(C3⋊D3)—42no—
62C2×C3⋊(C3⋊D3)—82no—
63C2×C32×D3—42no—
64C2×C32×D3—82no—
65C3×C3⋊Dic3—42no—
66C2×#239—34no—
67C2×D3×D4—94no—
68C2×#41⋊C3—91no—
69#986—94no—
70C2×GL(2,𝔽3)—91no—
71C2×GL(2,𝔽3)—184no—
72C2×GL(2,𝔽3)C3⋊D311yescharacteristic, complemented
73C2×C3×SD16—34no—
74C2×#215—94no—
75C2×#213—94no—
76#914—916no—
77#914—916no—
78C2×D24—94no—
79C2×ΓL(1,𝔽25)—94no—
80C23×C32—61no—
81C22×C3⋊D3—31no—
82C22×C3⋊D3—64no—
83C22×C3⋊D3—61no—
84C22×C3⋊D3—124no—
85C22×C3⋊D3—124no—
86C22×C3⋊D3—244no—
87C22×C3×D3—44no—
88C32×Q8—31no—
89C32×Q8C22×D311yescharacteristic
90C32×D4—32no—
91C32×D4—61no—
92C2×C32×C4—31no—
93#638—64no—
94C2×C3⋊Dic3S411yescharacteristic
95C4⋊(C3⋊D3)—32no—
96C4⋊(C3⋊D3)—32no—
97C4⋊(C3⋊D3)—61no—
98C4⋊(C3⋊D3)—61no—
99C4×C3⋊D3—32no—
100C4×C3⋊D3—32no—
101C2×C3×Dic3—124no—
102C3×SL(2,𝔽3)—24no—
103C3×SL(2,𝔽3)C22×D314yescomplemented
104#626—1216no—
105C32×C8—32no—
106C3⋊(C3⋊C8)—32no—
107C2×AGL(1,ℤ/8)—271noSylow
108C2×C33—42no—
109C2×C33—41no—
110C3⋊(C3⋊D3)—82no—
111C32×D3—82no—
112C23×D3—184no—
113C2×C3×Q8D3214yes—
114C2×C3×D4—34no—
115C2×#67—184no—
116#239—38no—
117#239—98no—
118D3×D4—916no—
119D3×D4—188no—
120C2×D12—94no—
121C2×D12—94no—
122C2×D12—361no—
123C2×C4×D3—94no—
124#41⋊C3—92no—
125C2×SL(2,𝔽3)—24no—
126C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic, complemented
127GL(2,𝔽3)—92no—
128GL(2,𝔽3)—188no—
129GL(2,𝔽3)C2×C3⋊D312yescomplemented
130C3×SD16—38no—
131C3×SD16—38no—
132C2×C3×C8—34no—
133#215—98no—
134#215—98no—
135#213—916no—
136C2×C3⋊C8—94no—
137D24—916no—
138ΓL(1,𝔽25)—916no—
139C22×C32—44no—
140C22×C32—62no—
141C22×C32—84no—
142C22×C32—122no—
143C22×C32C2×S411yescharacteristic
144C2×C3⋊D3—32no—
145C2×C3⋊D3—62no—
146C2×C3⋊D3—62no—
147C2×C3⋊D3—128no—
148C2×C3⋊D3—128no—
149C2×C3⋊D3—122no—
150C2×C3⋊D3—248no—
151C2×C3⋊D3—248no—
152C2×C3⋊D3—2416no—
153C2×C3×D3—48no—
154C2×C3×D3—88no—
155C32×C4—31no—
156C32×C4—31no—
157C3⋊Dic3C2×S412yes—
158C3×Dic3—128no—
159C2×#41—91no—
160C22×D4—271no—
161AGL(1,ℤ/8)—274no—
162AGL(1,ℤ/8)—274no—
163C2×SD16—31no—
164C2×SD16—271no—
165C2×D8—271no—
166C2×D8—271no—
167C2×M16—271no—
168C33—41noSylow
169C23×C3—64no—
170C22×D3—41no—
171C22×D3—94no—
172C22×D3—1816no—
173C22×D3—184no—
174C22×D3—361no—
175C22×D3—3616no—
176C22×D3—724no—
177C3×Q8—34no—
178C3×Q8C2×D3214yes—
179C3×D4—38no—
180C3×D4—64no—
181C2×C3×C4—34no—
182C2×C3×C4—361no—
183#67—1816no—
184C2×Dic3—34no—
185D12—98no—
186D12—98no—
187D12—184no—
188D12—184no—
189D12—364no—
190C4×D3—98no—
191C4×D3—98no—
192SL(2,𝔽3)—24no—
193SL(2,𝔽3)C22×C3⋊D311yescharacteristic, complemented
194C3×C8—38no—
195C3⋊C8—98no—
196C2×C32—44no—
197C2×C32—48no—
198C2×C32—84no—
199C2×C32—88no—
200C2×C32—122no—
201C2×C32#98612yescomplemented
202C2×C32C22×S411yescharacteristic
203C3⋊D3—62no—
204C3⋊D3—122no—
205C3⋊D3—248no—
206C3⋊D3—2416no—
207C3×D3—88no—
208C24—541no—
209#41—92no—
210#41—272no—
211C2×Q8D3×C3⋊D311yescharacteristic
212C2×D4—31no—
213C2×D4—271no—
214C2×D4—271no—
215C2×D4—274no—
216C2×D4—541no—
217C2×D4—542no—
218C22×C4—271no—
219SD16—32no—
220SD16—32no—
221SD16—272no—
222SD16—272no—
223D8—274no—
224D8—274no—
225M16—274no—
226C2×C8—31no—
227C2×C8—271no—
228C22×C3—41no—
229C22×C3—68no—
230C22×C3—84no—
231C22×C3—128no—
232C22×C3D3×S414yes—
233C2×D3—42no—
234C2×D3—82no—
235C2×D3—98no—
236C2×D3—188no—
237C2×D3—188no—
238C2×D3—362no—
239C2×D3—3632no—
240C2×D3—368no—
241C2×D3—728no—
242C2×D3—722no—
243C2×D3—7216no—
244C3×C4—34no—
245C3×C4—34no—
246C3×C4—362no—
247Dic3—38no—
248C32—44no—
249C32—84no—
250C32C2×#98611yescharacteristic, complemented
251C23—61no—
252C23—271no—
253C23—544no—
254C23—541no—
255C23—1084no—
256Q8—31no—
257Q8C2×D3×C3⋊D311yescharacteristic
258D4—32no—
259D4—61no—
260D4—272no—
261D4—272no—
262D4—541no—
263D4—541no—
264D4—544no—
265C2×C4—31no—
266C2×C4—91no—
267C2×C4—272no—
268C2×C4—272no—
269C8—32no—
270C8—272no—
271C2×C3—42no—
272C2×C3—41no—
273C2×C3—88no—
274C2×C3—84no—
275C2×C3—128no—
276C2×C3#6419918yescomplemented
277C2×C3C2×D3×S414yes—
278D3—82no—
279D3—188no—
280D3—368no—
281D3—722no—
282D3—7216no—
283C22—62no—
284C22—122no—
285C22—272no—
286C22—542no—
287C22—542no—
288C22—1088no—
289C22—1082no—
290C22C3⋊D3×S411yescharacteristic
291C4—31no—
292C4—31no—
293C4—92no—
294C3—41no—
295C3—84no—
296C3C2×#6419914yescomplemented
297C2—122no—
298C2—542no—
299C2—1082no—
300C2#1172417912yesdirect product factor
301C2C2×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×GL(2,𝔽3)C2semidirect
4. #11724179C2direct (N × H)
7. C2×C32×SL(2,𝔽3)C22semidirect
9. C3⋊(C3⋊(#41⋊C3))C22semidirect
10. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
11. C32×GL(2,𝔽3)C22semidirect
12. C2×#3832D3semidirect
17. C2×C3×GL(2,𝔽3)D3semidirect
29. C32×SL(2,𝔽3)C23semidirect
38. #3832C2×D3semidirect
45. C2×C3×SL(2,𝔽3)C2×D3semidirect
50. C3×GL(2,𝔽3)C2×D3semidirect
72. C2×GL(2,𝔽3)C3⋊D3semidirect
103. C3×SL(2,𝔽3)C22×D3semidirect
126. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
129. GL(2,𝔽3)C2×C3⋊D3semidirect
193. SL(2,𝔽3)C22×C3⋊D3semidirect
201. C2×C32#986semidirect
250. C32C2×#986semidirect
276. C2×C3#64199semidirect
296. C3C2×#64199semidirect
300. C2#11724179direct (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×GL(2,𝔽3)2
#117241792
C2×#641993
C2×#641233
C2×#62914