Finite Group Databasebeta

← C2×C32×#1007 #13181719

Subgroups — C2×C32×#1007 #13181719

Order 1728.

5388 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×Q8C211yesmaximal, characteristic, complemented
2C32×#1007C2116yesmaximal, direct product factor
3C2×C32×#239C214yesmaximal, complemented
4C2×C32×D3×Q8C214yesmaximal, complemented
5C2×C32×#235C216yesmaximal, complemented
6C2×C32×2−1+4—31nomaximal
7C2×C3×#1007C314yesmaximal, direct product factor
8C2×C33×Q8C2214yes—
9C2×C33×Q8C2218yescomplemented
10C22×C33×C4C2213yes—
11C2×C32×#67C2214yes—
12C32×#239C22132yescomplemented
13C32×D3×Q8C22132yescomplemented
14C32×#235C22148yescomplemented
15C2×C32×D12C2216yes—
16C2×C32×C4×D3C22112yes—
17C2×C32×Dic6C2216yes—
18C32×2−1+4—316no—
19C2×C32×#41—34no—
20C2×C32×#41—36no—
21C22×C32×Q8—24no—
22C22×C32×Q8—34no—
23C22×C32×Q8D311yescharacteristic, complemented
24C22×C32×Q8C2×C314yescomplemented
25C3×#1007C2×C3164yesdirect product factor
26C2×C3×#239C2×C3116yescomplemented
27C2×C3×D3×Q8C2×C3116yescomplemented
28C2×C3×#235C2×C3124yescomplemented
29C23×C33C2311yescharacteristic
30C22×C32×D3C2314yes—
31C33×Q8C23116yes—
32C2×C33×C4C2316yes—
33C2×C33×C4C23112yes—
34C32×#67C23116yes—
35C2×C32×Dic3C2314yes—
36C32×D12C23124yes—
37C32×C4×D3C23148yes—
38C32×Dic6C23124yes—
39C2×C3×2−1+4—34no—
40C2×#1007C3211yescharacteristic, direct product factor
41C32×#41—332no—
42C32×#41—348no—
43C2×C32×Q8—216no—
44C2×C32×Q8—232no—
45C2×C32×Q8—332no—
46C2×C32×Q8—36no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C2×D318yescomplemented
49C2×C32×Q8C22×C3116yes—
50C2×C32×Q8C22×C3132yescomplemented
51C2×C32×D4—34no—
52C2×C32×D4—36no—
53C22×C32×C4—212no—
54C22×C32×C4—312no—
55C22×C32×C4C2×D313yes—
56C22×C32×C4C22×C3112yes—
57C2×C3×#67C22×C3116yes—
58C3×#239C22×C31128yescomplemented
59C3×D3×Q8C22×C31128yescomplemented
60C3×#235C22×C31192yescomplemented
61C2×C3×D12C22×C3124yes—
62C2×C3×C4×D3C22×C3148yes—
63C2×C3×Dic6C22×C3124yes—
64C22×C33—22no—
65C22×C33C2412yes—
66C22×C33C2411yescharacteristic
67C2×C32×D3—28no—
68C2×C32×D3C2418yes—
69C33×C4C24112yes—
70C32×Dic3C2418yes—
71C3×2−1+4—364no—
72C2×C3×#41—316no—
73C2×C3×#41—324no—
74C22×C3×Q8—24no—
75C22×C3×Q8—316no—
76C22×C3×Q8C3×D314yescomplemented
77C22×C3×Q8C2×C3211yescharacteristic, complemented
78#1007C2×C32116yesdirect product factor
79C2×#239C2×C3214yescomplemented
80C2×D3×Q8C2×C3214yescomplemented
81C2×#235C2×C3216yescomplemented
82C23×C32—24no—
83C23×C32—34no—
84C23×C32C22×D311yescharacteristic
85C23×C32C23×C314yes—
86C22×C3×D3C23×C3116yes—
87C32×Q8—264no—
88C32×Q8—324no—
89C32×Q8C22×D3116yes—
90C32×Q8C23×C3164yes—
91C32×D4—316no—
92C32×D4—324no—
93C2×C32×C4—224no—
94C2×C32×C4—248no—
95C2×C32×C4—34no—
96C2×C32×C4—348no—
97C2×C32×C4C22×D3112yes—
98C2×C32×C4C22×D316yes—
99C2×C32×C4C23×C3124yes—
100C2×C32×C4C23×C3148yes—
101C3×#67C23×C3164yes—
102C2×C3×Dic3C23×C3116yes—
103C3×D12C23×C3196yes—
104C3×C4×D3C23×C31192yes—
105C3×Dic6C23×C3196yes—
106C2×2−1+4—31noSylow
107C2×C33—22no—
108C2×C332−1+412yescomplemented
109C2×C33C2511yescharacteristic
110C32×D3—28no—
111C3×#41—3128no—
112C3×#41—3192no—
113C2×C3×Q8—216no—
114C2×C3×Q8—232no—
115C2×C3×Q8—3128no—
116C2×C3×Q8—324no—
117C2×C3×Q8C2×C3×D3116yes—
118C2×C3×Q8C2×C3×D3132yescomplemented
119C2×C3×Q8C22×C3214yes—
120C2×C3×Q8C22×C3218yescomplemented
121C2×C3×D4—316no—
122C2×C3×D4—324no—
123C22×C3×C4—212no—
124C22×C3×C4—348no—
125C22×C3×C4C2×C3×D3112yes—
126C22×C3×C4C22×C3213yes—
127C2×#67C22×C3214yes—
128#239C22×C32132yescomplemented
129D3×Q8C22×C32132yescomplemented
130#235C22×C32148yescomplemented
131C2×D12C22×C3216yes—
132C2×C4×D3C22×C32112yes—
133C2×Dic6C22×C3216yes—
134C22×C32—216no—
135C22×C32—22no—
136C22×C32—28no—
137C22×C32—28no—
138C22×C32—24no—
139C22×C32—38no—
140C22×C32—68no—
141C22×C32C23×D312yes—
142C22×C32C23×D311yescharacteristic
143C22×C32C24×C318yes—
144C22×C32C24×C314yes—
145C2×C3×D3—232no—
146C2×C3×D3C24×C3132yes—
147C32×C4—248no—
148C32×C4—38no—
149C32×C4C23×D3112yes—
150C32×C4C24×C3148yes—
151C3×Dic3C24×C3132yes—
1522−1+4—316no—
153C2×#41—34no—
154C2×#41—36no—
155C22×Q8—34no—
156C22×Q8C32×D311yescharacteristic, complemented
157C33C2×2−1+411yescharacteristic, complemented, Sylow
158C23×C3—24no—
159C23×C3—316no—
160C23×C3C22×C3×D314yes—
161C23×C3C23×C3211yescharacteristic
162C22×D3C23×C3214yes—
163C3×Q8—264no—
164C3×Q8—396no—
165C3×Q8C22×C3×D3164yes—
166C3×Q8C23×C32116yes—
167C3×D4—364no—
168C3×D4—396no—
169C2×C3×C4—224no—
170C2×C3×C4—248no—
171C2×C3×C4—316no—
172C2×C3×C4—3192no—
173C2×C3×C4C22×C3×D3124yes—
174C2×C3×C4C22×C3×D3148yes—
175C2×C3×C4C23×C3216yes—
176C2×C3×C4C23×C32112yes—
177#67C23×C32116yes—
178C2×Dic3C23×C3214yes—
179D12C23×C32124yes—
180C4×D3C23×C32148yes—
181Dic6C23×C32124yes—
182C2×C32—216no—
183C2×C32—22no—
184C2×C32—28no—
185C2×C32—28no—
186C2×C32—24no—
187C2×C32—68no—
188C2×C32#100712yesdirect product factor
189C2×C32C3×2−1+418yescomplemented
190C2×C32C24×D311yescharacteristic
191C2×C32C25×C314yes—
192C3×D3—232no—
193#41—332no—
194#41—348no—
195C2×Q8—332no—
196C2×Q8—36no—
197C2×Q8C2×C32×D314yes—
198C2×Q8C2×C32×D318yescomplemented
199C2×D4—34no—
200C2×D4—36no—
201C22×C4—312no—
202C22×C4C2×C32×D313yes—
203C22×C3—216no—
204C22×C3—28no—
205C22×C3—22no—
206C22×C3—28no—
207C22×C3—24no—
208C22×C3—332no—
209C22×C3—632no—
210C22×C3C23×C3×D318yes—
211C22×C3C23×C3×D314yes—
212C22×C3C24×C3212yes—
213C22×C3C24×C3211yescharacteristic
214C2×D3—28no—
215C2×D3C24×C3218yes—
216C3×C4—248no—
217C3×C4—332no—
218C3×C4C23×C3×D3148yes—
219C3×C4C24×C32112yes—
220Dic3C24×C3218yes—
221C32—24no—
222C32C2×#100711yescharacteristic, direct product factor
223C32C2×C3×2−1+414yescomplemented
224C23—34no—
225C23C22×C32×D311yescharacteristic
226Q8—324no—
227Q8C22×C32×D3116yes—
228D4—316no—
229D4—324no—
230C2×C4—34no—
231C2×C4—348no—
232C2×C4C22×C32×D316yes—
233C2×C4C22×C32×D3112yes—
234C2×C3—216no—
235C2×C3—28no—
236C2×C3—22no—
237C2×C3—28no—
238C2×C3—24no—
239C2×C3—632no—
240C2×C3C3×#100718yesdirect product factor
241C2×C3C32×2−1+412yescomplemented
242C2×C3C24×C3×D314yes—
243C2×C3C25×C3211yescharacteristic
244D3—28no—
245C22—22no—
246C22—38no—
247C22—68no—
248C22C23×C32×D312yes—
249C22C23×C32×D311yescharacteristic
250C4—38no—
251C4C23×C32×D3112yes—
252C3—24no—
253C3C2×C3×#100714yesdirect product factor
254C3C2×C32×2−1+411yescharacteristic, complemented
255C2—22no—
256C2—68no—
257C2C32×#100712yesdirect product factor
258C2C24×C32×D311yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC2×C3C25×C32
FrattiniC2C24×C32×D3
FittingC22×C33×Q8C2
socleC22×C33C24
hypercenterC22×C32×Q8D3
hypocenterC3C2×C32×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C22×C32×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C33×Q8C2semidirect
2. C32×#1007C2direct (N × H)
3. C2×C32×#239C2semidirect
4. C2×C32×D3×Q8C2semidirect
5. C2×C32×#235C2semidirect
7. C2×C3×#1007C3direct (N × H)
9. C2×C33×Q8C22semidirect
12. C32×#239C22semidirect
13. C32×D3×Q8C22semidirect
14. C32×#235C22semidirect
23. C22×C32×Q8D3semidirect
24. C22×C32×Q8C2×C3semidirect
25. C3×#1007C2×C3direct (N × H)
26. C2×C3×#239C2×C3semidirect
27. C2×C3×D3×Q8C2×C3semidirect
28. C2×C3×#235C2×C3semidirect
40. C2×#1007C32direct (N × H)
48. C2×C32×Q8C2×D3semidirect
50. C2×C32×Q8C22×C3semidirect
58. C3×#239C22×C3semidirect
59. C3×D3×Q8C22×C3semidirect
60. C3×#235C22×C3semidirect
76. C22×C3×Q8C3×D3semidirect
77. C22×C3×Q8C2×C32semidirect
78. #1007C2×C32direct (N × H)
79. C2×#239C2×C32semidirect
80. C2×D3×Q8C2×C32semidirect
81. C2×#235C2×C32semidirect
108. C2×C332−1+4semidirect
118. C2×C3×Q8C2×C3×D3semidirect
120. C2×C3×Q8C22×C32semidirect
128. #239C22×C32semidirect
129. D3×Q8C22×C32semidirect
130. #235C22×C32semidirect
156. C22×Q8C32×D3semidirect
157. C33C2×2−1+4semidirect
188. C2×C32#1007direct (N × H)
189. C2×C32C3×2−1+4semidirect
198. C2×Q8C2×C32×D3semidirect
222. C32C2×#1007direct (N × H)
223. C32C2×C3×2−1+4semidirect
240. C2×C3C3×#1007direct (N × H)
241. C2×C3C32×2−1+4semidirect
253. C3C2×C3×#1007direct (N × H)
254. C3C2×C32×2−1+4semidirect
257. C2C32×#1007direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×Q82
C32×#10072
C2×C32×#2392
C2×C32×D3×Q82
C2×C32×#2352
C2×C32×2−1+43
C2×C3×#10073