Finite Group Databasebeta

← C2×C32×#983 #13181359

Subgroups — C2×C32×#983 #13181359

Order 1728.

1086 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 174 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
2C32×#983C214yesmaximal, direct product factor
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C32×#226C211yesmaximal, characteristic, complemented
5C2×C3×#983C314yesmaximal, direct product factor
6C2×C32×#137—31nomaximal
7C2×C32×#67—41nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×SL(2,𝔽3)C2212yescomplemented
10C32×GL(2,𝔽3)C2212yescomplemented
11C32×#226C2212yescomplemented
12C2×C32×#41—31no—
13C22×C32×Q8D311yescharacteristic, complemented
14C22×C3×SL(2,𝔽3)—24no—
15C22×C3×SL(2,𝔽3)C2×C314yescomplemented
16C3×#983C2×C3116yesdirect product factor
17C2×C3×GL(2,𝔽3)C2×C314yescomplemented
18C2×C3×#226C2×C314yescomplemented
19C32×#137—34no—
20C32×#137—34no—
21C2×C32×Q16—31no—
22C2×C32×Q16—31no—
23C2×C32×SD16—31no—
24C2×C32×SD16—31no—
25C2×C32×M16—31no—
26C23×C33—41no—
27C22×C32×D3—41no—
28C32×#67—44no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)C2311yescharacteristic
31C2×#983C3211yescharacteristic, direct product factor
32C2×C3×#137—34no—
33C32×#41—34no—
34C32×#41—62no—
35C2×C32×Q8—31no—
36C2×C32×Q8—32no—
37C2×C32×Q8—31no—
38C2×C32×Q8C2×D311yescharacteristic
39C2×C32×Q8C2×D312yescomplemented
40C2×C32×D4—31no—
41C2×C32×D4—61no—
42C22×C32×C4—31no—
43C22×C32×C4—61no—
44C2×C3×#67—44no—
45C2×C3×SL(2,𝔽3)—24no—
46C2×C3×SL(2,𝔽3)—28no—
47C2×C3×SL(2,𝔽3)C22×C314yes—
48C2×C3×SL(2,𝔽3)C22×C318yescomplemented
49C3×GL(2,𝔽3)C22×C318yescomplemented
50C3×#226C22×C318yescomplemented
51C32×Q16—32no—
52C32×Q16—34no—
53C32×Q16—32no—
54C32×SD16—32no—
55C32×SD16—34no—
56C32×SD16—32no—
57C32×M16—34no—
58C2×C32×C8—31no—
59C2×C32×C8—31no—
60C22×C33—41no—
61C22×C33—42no—
62C22×C33—82no—
63C2×C32×D3—42no—
64C2×C32×D3—82no—
65C32×Dic3—42no—
66C2×C3×#41—34no—
67C22×C3×Q8C3×D314yescomplemented
68C22×SL(2,𝔽3)—24no—
69C22×SL(2,𝔽3)C2×C3211yescharacteristic, complemented
70#983C2×C3214yesdirect product factor
71C2×GL(2,𝔽3)C2×C3211yescharacteristic, complemented
72C2×#226C2×C3211yescharacteristic, complemented
73C3×#137—316no—
74C3×#137—316no—
75C2×C3×Q16—34no—
76C2×C3×Q16—34no—
77C2×C3×SD16—34no—
78C2×C3×SD16—34no—
79C2×C3×M16—34no—
80C23×C32—44no—
81C23×C32—61no—
82C23×C32—84no—
83C23×C32S411yescharacteristic
84C22×C3×D3—44no—
85C32×Q8—31no—
86C32×Q8—32no—
87C32×Q8—32no—
88C32×Q8—61no—
89C32×Q8—61no—
90C32×Q8C22×D311yescharacteristic
91C32×D4—32no—
92C32×D4—61no—
93C32×D4—64no—
94C2×C32×C4—32no—
95C2×C32×C4—32no—
96C2×C32×C4—31no—
97C2×C32×C4—31no—
98C2×C32×C4—64no—
99C2×C32×C4—61no—
100C3×#67—416no—
101C2×C3×Dic3—44no—
102C3×SL(2,𝔽3)—24no—
103C3×SL(2,𝔽3)C23×C314yes—
104C32×C8—32no—
105C32×C8—32no—
106C2×#137—31noSylow
107C2×C33—42no—
108C2×C33—41no—
109C2×C33—82no—
110C32×D3—82no—
111C3×#41—316no—
112C3×#41—68no—
113C2×C3×Q8—34no—
114C2×C3×Q8—38no—
115C2×C3×Q8—34no—
116C2×C3×Q8C2×C3×D314yes—
117C2×C3×Q8C2×C3×D318yescomplemented
118C2×C3×D4—34no—
119C2×C3×D4—64no—
120C22×C3×C4—34no—
121C22×C3×C4—64no—
122C2×#67—41no—
123C2×SL(2,𝔽3)—24no—
124C2×SL(2,𝔽3)—28no—
125C2×SL(2,𝔽3)C22×C3211yescharacteristic
126C2×SL(2,𝔽3)C22×C3212yescomplemented
127GL(2,𝔽3)C22×C3212yescomplemented
128#226C22×C3212yescomplemented
129C3×Q16—38no—
130C3×Q16—316no—
131C3×Q16—38no—
132C3×SD16—38no—
133C3×SD16—316no—
134C3×SD16—38no—
135C3×M16—316no—
136C2×C3×C8—34no—
137C2×C3×C8—34no—
138C22×C32—22no—
139C22×C32—44no—
140C22×C32—48no—
141C22×C32—62no—
142C22×C32—88no—
143C22×C32—816no—
144C22×C32—88no—
145C22×C32—84no—
146C22×C32—122no—
147C22×C32C2×S412yes—
148C22×C32C2×S411yescharacteristic
149C2×C3×D3—48no—
150C2×C3×D3—88no—
151C32×C4—31no—
152C32×C4—32no—
153C32×C4—31no—
154C32×C4—62no—
155C3×Dic3—48no—
156C2×#41—31no—
157C22×Q8C32×D311yescharacteristic, complemented
158#137—34no—
159#137—34no—
160C2×Q16—31no—
161C2×Q16—31no—
162C2×SD16—31no—
163C2×SD16—31no—
164C2×M16—31no—
165C33—41noSylow
166C23×C3—41no—
167C23×C3—64no—
168C23×C3—84no—
169C23×C3C3×S414yes—
170C22×D3—41no—
171C3×Q8—34no—
172C3×Q8—38no—
173C3×Q8—38no—
174C3×Q8—64no—
175C3×Q8—64no—
176C3×Q8C22×C3×D314yes—
177C3×D4—38no—
178C3×D4—64no—
179C3×D4—616no—
180C2×C3×C4—34no—
181C2×C3×C4—38no—
182C2×C3×C4—34no—
183C2×C3×C4—38no—
184C2×C3×C4—616no—
185C2×C3×C4—64no—
186#67—44no—
187C2×Dic3—41no—
188SL(2,𝔽3)—24no—
189SL(2,𝔽3)C23×C3211yescharacteristic
190C3×C8—38no—
191C3×C8—38no—
192C2×C32—22no—
193C2×C32—44no—
194C2×C32—48no—
195C2×C32—88no—
196C2×C32—816no—
197C2×C32—84no—
198C2×C32—88no—
199C2×C32—122no—
200C2×C32#98312yesdirect product factor
201C2×C32C22×S411yescharacteristic
202C3×D3—88no—
203#41—34no—
204#41—62no—
205C2×Q8—31no—
206C2×Q8—32no—
207C2×Q8—31no—
208C2×Q8C2×C32×D311yescharacteristic
209C2×Q8C2×C32×D312yescomplemented
210C2×D4—31no—
211C2×D4—61no—
212C22×C4—31no—
213C22×C4—61no—
214Q16—32no—
215Q16—34no—
216Q16—32no—
217SD16—32no—
218SD16—34no—
219SD16—32no—
220M16—34no—
221C2×C8—31no—
222C2×C8—31no—
223C22×C3—28no—
224C22×C3—41no—
225C22×C3—42no—
226C22×C3—68no—
227C22×C3—82no—
228C22×C3—816no—
229C22×C3—88no—
230C22×C3—84no—
231C22×C3—128no—
232C22×C3C2×C3×S418yes—
233C22×C3C2×C3×S414yes—
234C2×D3—42no—
235C2×D3—82no—
236C3×C4—34no—
237C3×C4—38no—
238C3×C4—34no—
239C3×C4—68no—
240Dic3—42no—
241C32—44no—
242C32—84no—
243C32C2×#98311yescharacteristic, direct product factor
244C23—61no—
245C23C32×S411yescharacteristic
246Q8—31no—
247Q8—32no—
248Q8—32no—
249Q8—61no—
250Q8—61no—
251Q8C22×C32×D311yescharacteristic
252D4—32no—
253D4—61no—
254D4—64no—
255C2×C4—31no—
256C2×C4—32no—
257C2×C4—31no—
258C2×C4—32no—
259C2×C4—64no—
260C2×C4—61no—
261C8—32no—
262C8—32no—
263C2×C3—28no—
264C2×C3—42no—
265C2×C3—41no—
266C2×C3—82no—
267C2×C3—816no—
268C2×C3—88no—
269C2×C3—84no—
270C2×C3—128no—
271C2×C3C3×#98318yesdirect product factor
272C2×C3C22×C3×S414yes—
273D3—82no—
274C22—22no—
275C22—62no—
276C22—122no—
277C22C2×C32×S412yes—
278C22C2×C32×S411yescharacteristic
279C4—31no—
280C4—32no—
281C4—31no—
282C4—62no—
283C3—41no—
284C3—84no—
285C3C2×C3×#98314yesdirect product factor
286C2—22no—
287C2—122no—
288C2C32×#98312yesdirect product factor
289C2C22×C32×S411yescharacteristic
subgroupG/H
center, socleC22×C32C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC22×C32×Q8D3
hypercenterC23×C32S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C32 ◁ C23×C32

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C32×#983C2direct (N × H)
3. C2×C32×GL(2,𝔽3)C2semidirect
4. C2×C32×#226C2semidirect
5. C2×C3×#983C3direct (N × H)
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C32×GL(2,𝔽3)C22semidirect
11. C32×#226C22semidirect
13. C22×C32×Q8D3semidirect
15. C22×C3×SL(2,𝔽3)C2×C3semidirect
16. C3×#983C2×C3direct (N × H)
17. C2×C3×GL(2,𝔽3)C2×C3semidirect
18. C2×C3×#226C2×C3semidirect
31. C2×#983C32direct (N × H)
39. C2×C32×Q8C2×D3semidirect
48. C2×C3×SL(2,𝔽3)C22×C3semidirect
49. C3×GL(2,𝔽3)C22×C3semidirect
50. C3×#226C22×C3semidirect
67. C22×C3×Q8C3×D3semidirect
69. C22×SL(2,𝔽3)C2×C32semidirect
70. #983C2×C32direct (N × H)
71. C2×GL(2,𝔽3)C2×C32semidirect
72. C2×#226C2×C32semidirect
117. C2×C3×Q8C2×C3×D3semidirect
126. C2×SL(2,𝔽3)C22×C32semidirect
127. GL(2,𝔽3)C22×C32semidirect
128. #226C22×C32semidirect
157. C22×Q8C32×D3semidirect
200. C2×C32#983direct (N × H)
209. C2×Q8C2×C32×D3semidirect
243. C32C2×#983direct (N × H)
271. C2×C3C3×#983direct (N × H)
285. C3C2×C3×#983direct (N × H)
288. C2C32×#983direct (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
C32×#9832
C2×C32×GL(2,𝔽3)2
C2×C32×#2262
C2×C3×#9833
C2×C32×#1373
C2×C32×#674