Finite Group Databasebeta

← C2×C32×#985 #13181363

Subgroups — C2×C32×#985 #13181363

Order 1728.

1022 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1022 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C32×#985C214yesmaximal, direct product factor
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C32×#226C211yesmaximal, characteristic, complemented
5C2×C3×#985C314yesmaximal, direct product factor
6C2×C32×#135—31nomaximal
7C2×C32×C4×D3—41nomaximal
8C32×#41⋊C3C2212yescomplemented
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C32×GL(2,𝔽3)C2212yescomplemented
11C32×#226C2212yescomplemented
12C2×C32×#41—31no—
13C2×C32×#41D311yescharacteristic, complemented
14C2×C3×#41⋊C3—24no—
15C2×C3×#41⋊C3C2×C314yescomplemented
16C3×#985C2×C3116yesdirect product factor
17C2×C3×GL(2,𝔽3)C2×C314yescomplemented
18C2×C3×#226C2×C314yescomplemented
19C32×#135—34no—
20C32×#135—34no—
21C2×C32×Q16—31no—
22C2×C32×SD16—31no—
23C2×C32×SD16—31no—
24C2×C32×D8—31no—
25C22×C32×C8—31no—
26C22×C32×D3—41no—
27C2×C33×C4—41no—
28C2×C32×Dic3—41no—
29C32×C4×D3—44no—
30C32×SL(2,𝔽3)C2311yescharacteristic
31C2×#985C3211yescharacteristic, direct product factor
32C2×C3×#135—34no—
33C32×#41—32no—
34C32×#41—34no—
35C32×#41—62no—
36C32×#41C2×D312yescomplemented
37C2×C32×Q8—31no—
38C2×C32×Q8C2×D311yescharacteristic
39C2×C32×D4—31no—
40C2×C32×D4—31no—
41C2×C32×D4—61no—
42C22×C32×C4—31no—
43C22×C32×C4—61no—
44C2×C3×C4×D3—44no—
45C3×#41⋊C3—28no—
46C3×#41⋊C3C22×C318yescomplemented
47C2×C3×SL(2,𝔽3)—24no—
48C2×C3×SL(2,𝔽3)C22×C314yes—
49C3×GL(2,𝔽3)C22×C318yescomplemented
50C3×#226C22×C318yescomplemented
51C32×Q16—32no—
52C32×Q16—32no—
53C32×SD16—34no—
54C32×SD16—32no—
55C32×SD16—32no—
56C32×D8—34no—
57C2×C32×C8—31no—
58C2×C32×C8—31no—
59C2×C32×C8—34no—
60C22×C33—41no—
61C2×C32×D3—42no—
62C2×C32×D3—44no—
63C33×C4—42no—
64C32×Dic3—42no—
65C2×C3×#41—34no—
66C2×C3×#41C3×D314yescomplemented
67C2×#41⋊C3—24no—
68C2×#41⋊C3C2×C3211yescharacteristic, complemented
69#985C2×C3214yesdirect product factor
70C2×GL(2,𝔽3)C2×C3211yescharacteristic, complemented
71C2×#226C2×C3211yescharacteristic, complemented
72C3×#135—316no—
73C3×#135—316no—
74C2×C3×Q16—34no—
75C2×C3×SD16—34no—
76C2×C3×SD16—34no—
77C2×C3×D8—34no—
78C22×C3×C8—34no—
79C23×C32—31no—
80C23×C32—61no—
81C22×C3×D3—44no—
82C32×Q8—31no—
83C32×Q8—32no—
84C32×Q8—61no—
85C32×Q8C22×D311yescharacteristic
86C32×D4—32no—
87C32×D4—32no—
88C32×D4—61no—
89C32×D4—61no—
90C32×D4—64no—
91C2×C32×C4—31no—
92C2×C32×C4—32no—
93C2×C32×C4—32no—
94C2×C32×C4—44no—
95C2×C32×C4—61no—
96C2×C32×C4—64no—
97C2×C32×C4—84no—
98C2×C32×C4S411yescharacteristic
99C2×C3×Dic3—44no—
100C3×C4×D3—416no—
101C3×SL(2,𝔽3)—24no—
102C3×SL(2,𝔽3)C23×C314yes—
103C32×C8—32no—
104C32×C8—32no—
105C2×#135—31noSylow
106C2×C33—42no—
107C2×C33—41no—
108C32×D3—44no—
109C3×#41—38no—
110C3×#41—316no—
111C3×#41—68no—
112C3×#41C2×C3×D318yescomplemented
113C2×C3×Q8—34no—
114C2×C3×Q8C2×C3×D314yes—
115C2×C3×D4—34no—
116C2×C3×D4—34no—
117C2×C3×D4—64no—
118C22×C3×C4—34no—
119C22×C3×C4—64no—
120C2×C4×D3—41no—
121#41⋊C3—28no—
122#41⋊C3C22×C3212yescomplemented
123C2×SL(2,𝔽3)—24no—
124C2×SL(2,𝔽3)C22×C3211yescharacteristic
125GL(2,𝔽3)C22×C3212yescomplemented
126#226C22×C3212yescomplemented
127C3×Q16—38no—
128C3×Q16—38no—
129C3×SD16—316no—
130C3×SD16—38no—
131C3×SD16—38no—
132C3×D8—316no—
133C2×C3×C8—34no—
134C2×C3×C8—34no—
135C2×C3×C8—316no—
136C22×C32—32no—
137C22×C32—44no—
138C22×C32—62no—
139C22×C32—62no—
140C22×C32—84no—
141C22×C32—122no—
142C22×C32C2×S411yescharacteristic
143C2×C3×D3—48no—
144C2×C3×D3—416no—
145C32×C4—31no—
146C32×C4—31no—
147C32×C4—48no—
148C32×C4—62no—
149C32×C4—88no—
150C32×C4C2×S412yes—
151C3×Dic3—48no—
152C2×#41—31no—
153C2×#41C32×D311yescharacteristic, complemented
154#135—34no—
155#135—34no—
156C2×Q16—31no—
157C2×SD16—31no—
158C2×SD16—31no—
159C2×D8—31no—
160C22×C8—31no—
161C33—41noSylow
162C23×C3—34no—
163C23×C3—64no—
164C22×D3—41no—
165C3×Q8—34no—
166C3×Q8—38no—
167C3×Q8—64no—
168C3×Q8C22×C3×D314yes—
169C3×D4—38no—
170C3×D4—38no—
171C3×D4—64no—
172C3×D4—64no—
173C3×D4—616no—
174C2×C3×C4—34no—
175C2×C3×C4—38no—
176C2×C3×C4—38no—
177C2×C3×C4—41no—
178C2×C3×C4—64no—
179C2×C3×C4—616no—
180C2×C3×C4—84no—
181C2×C3×C4C3×S414yes—
182C2×Dic3—41no—
183C4×D3—44no—
184SL(2,𝔽3)—24no—
185SL(2,𝔽3)C23×C3211yescharacteristic
186C3×C8—38no—
187C3×C8—38no—
188C2×C32—44no—
189C2×C32—48no—
190C2×C32—62no—
191C2×C32—84no—
192C2×C32—88no—
193C2×C32—122no—
194C2×C32#98512yesdirect product factor
195C2×C32C22×S411yescharacteristic
196C3×D3—416no—
197#41—32no—
198#41—34no—
199#41—62no—
200#41C2×C32×D312yescomplemented
201C2×Q8—31no—
202C2×Q8C2×C32×D311yescharacteristic
203C2×D4—31no—
204C2×D4—31no—
205C2×D4—61no—
206C22×C4—31no—
207C22×C4—61no—
208Q16—32no—
209Q16—32no—
210SD16—34no—
211SD16—32no—
212SD16—32no—
213D8—34no—
214C2×C8—31no—
215C2×C8—31no—
216C2×C8—34no—
217C22×C3—38no—
218C22×C3—41no—
219C22×C3—68no—
220C22×C3—68no—
221C22×C3—84no—
222C22×C3—128no—
223C22×C3C2×C3×S414yes—
224C2×D3—42no—
225C2×D3—44no—
226C3×C4—34no—
227C3×C4—34no—
228C3×C4—42no—
229C3×C4—68no—
230C3×C4—88no—
231C3×C4C2×C3×S418yes—
232Dic3—42no—
233C32—44no—
234C32—84no—
235C32C2×#98511yescharacteristic, direct product factor
236C23—31no—
237C23—61no—
238Q8—31no—
239Q8—32no—
240Q8—61no—
241Q8C22×C32×D311yescharacteristic
242D4—32no—
243D4—32no—
244D4—61no—
245D4—61no—
246D4—64no—
247C2×C4—31no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—61no—
251C2×C4—64no—
252C2×C4C32×S411yescharacteristic
253C8—32no—
254C8—32no—
255C2×C3—42no—
256C2×C3—41no—
257C2×C3—68no—
258C2×C3—88no—
259C2×C3—84no—
260C2×C3—128no—
261C2×C3C3×#98518yesdirect product factor
262C2×C3C22×C3×S414yes—
263D3—44no—
264C22—32no—
265C22—62no—
266C22—62no—
267C22—122no—
268C22C2×C32×S411yescharacteristic
269C4—31no—
270C4—31no—
271C4—62no—
272C4C2×C32×S412yes—
273C3—41no—
274C3—84no—
275C3C2×C3×#98514yesdirect product factor
276C2—62no—
277C2—122no—
278C2C32×#98512yesdirect product factor
279C2C22×C32×S411yescharacteristic
subgroupG/H
center, hypercenterC2×C32×C4S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C32×#985C2direct (N × H)
3. C2×C32×GL(2,𝔽3)C2semidirect
4. C2×C32×#226C2semidirect
5. C2×C3×#985C3direct (N × H)
8. C32×#41⋊C3C22semidirect
10. C32×GL(2,𝔽3)C22semidirect
11. C32×#226C22semidirect
13. C2×C32×#41D3semidirect
15. C2×C3×#41⋊C3C2×C3semidirect
16. C3×#985C2×C3direct (N × H)
17. C2×C3×GL(2,𝔽3)C2×C3semidirect
18. C2×C3×#226C2×C3semidirect
31. C2×#985C32direct (N × H)
36. C32×#41C2×D3semidirect
46. C3×#41⋊C3C22×C3semidirect
49. C3×GL(2,𝔽3)C22×C3semidirect
50. C3×#226C22×C3semidirect
66. C2×C3×#41C3×D3semidirect
68. C2×#41⋊C3C2×C32semidirect
69. #985C2×C32direct (N × H)
70. C2×GL(2,𝔽3)C2×C32semidirect
71. C2×#226C2×C32semidirect
112. C3×#41C2×C3×D3semidirect
122. #41⋊C3C22×C32semidirect
125. GL(2,𝔽3)C22×C32semidirect
126. #226C22×C32semidirect
153. C2×#41C32×D3semidirect
194. C2×C32#985direct (N × H)
200. #41C2×C32×D3semidirect
235. C32C2×#985direct (N × H)
261. C2×C3C3×#985direct (N × H)
275. C3C2×C3×#985direct (N × H)
278. C2C32×#985direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C32×#9852
C2×C32×GL(2,𝔽3)2
C2×C32×#2262
C2×C3×#9853
C2×C32×#1353
C2×C32×C4×D34