Finite Group Databasebeta

← C2×C32×#1010 #13181722

Subgroups — C2×C32×#1010 #13181722

Order 1728.

5420 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C32×#1010C2116yesmaximal, direct product factor
3C2×C32×D3×Q8C212yesmaximal, complemented
4C2×C32×#237C216yesmaximal, complemented
5C2×C32×#235C213yesmaximal, complemented
6C22×C32×Dic6C213yesmaximal, complemented
7C2×C32×2−1+4—31nomaximal
8C2×C3×#1010C314yesmaximal, direct product factor
9C33×#41C2218yescomplemented
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2213yes—
12C22×C33×C4C2213yes—
13C2×C32×#67C2216yes—
14C22×C32×Dic3C2216yes—
15C32×D3×Q8C22116yescomplemented
16C32×#237C22148yescomplemented
17C32×#235C22124yescomplemented
18C2×C32×D12C2211yescharacteristic
19C2×C32×C4×D3C2216yes—
20C2×C32×Dic6C2213yes—
21C2×C32×Dic6C2216yes—
22C2×C32×Dic6C22124yescomplemented
23C32×2−1+4—316no—
24C2×C32×#41—24no—
25C2×C32×#41—36no—
26C2×C32×#41—33no—
27C2×C32×#41D311yescharacteristic, complemented
28C2×C32×#41C2×C314yescomplemented
29C22×C32×Q8—32no—
30C22×C32×Q8—33no—
31C3×#1010C2×C3164yesdirect product factor
32C2×C3×D3×Q8C2×C318yescomplemented
33C2×C3×#237C2×C3124yescomplemented
34C2×C3×#235C2×C3112yescomplemented
35C22×C3×Dic6C2×C3112yescomplemented
36C23×C33C2313yes—
37C22×C32×D3C2312yes—
38C33×Q8C2314yes—
39C33×D4C23112yes—
40C2×C33×C4C2311yescharacteristic
41C2×C33×C4C2313yes—
42C2×C33×C4C23112yes—
43C32×#67C23124yes—
44C2×C32×Dic3C2316yes—
45C2×C32×Dic3C23124yes—
46C32×D12C2314yes—
47C32×C4×D3C23124yes—
48C32×Dic6C23112yes—
49C32×Dic6C23124yes—
50C2×C3×2−1+4—34no—
51C2×#1010C3211yescharacteristic, direct product factor
52C32×#41—232no—
53C32×#41—348no—
54C32×#41—324no—
55C32×#41C2×D318yescomplemented
56C32×#41C22×C3132yescomplemented
57C2×C32×Q8—24no—
58C2×C32×Q8—316no—
59C2×C32×Q8—36no—
60C2×C32×Q8—33no—
61C2×C32×Q8—324no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×Q8C22×C314yes—
64C2×C32×D4—212no—
65C2×C32×D4—36no—
66C2×C32×D4—31no—
67C2×C32×D4C2×D313yes—
68C2×C32×D4C22×C3112yes—
69C22×C32×C4—212no—
70C22×C32×C4—36no—
71C22×C32×C4—36no—
72C22×C32×C4C2×D313yes—
73C22×C32×C4C22×C3112yes—
74C2×C3×#67C22×C3124yes—
75C22×C3×Dic3C22×C3124yes—
76C3×D3×Q8C22×C3164yescomplemented
77C3×#237C22×C31192yescomplemented
78C3×#235C22×C3196yescomplemented
79C2×C3×D12C22×C314yes—
80C2×C3×C4×D3C22×C3124yes—
81C2×C3×Dic6C22×C3112yes—
82C2×C3×Dic6C22×C3124yes—
83C2×C3×Dic6C22×C3196yescomplemented
84C22×C33—26no—
85C22×C33C2416yes—
86C22×C33C2411yescharacteristic
87C2×C32×D3—24no—
88C2×C32×D3C2414yes—
89C33×C4C2412yes—
90C33×C4C2416yes—
91C32×Dic3C24112yes—
92C3×2−1+4—364no—
93C2×C3×#41—24no—
94C2×C3×#41—324no—
95C2×C3×#41—312no—
96C2×C3×#41C3×D314yescomplemented
97C2×C3×#41C2×C3211yescharacteristic, complemented
98C22×C3×Q8—38no—
99C22×C3×Q8—312no—
100#1010C2×C32116yesdirect product factor
101C2×D3×Q8C2×C3212yescomplemented
102C2×#237C2×C3216yescomplemented
103C2×#235C2×C3213yescomplemented
104C22×Dic6C2×C3213yescomplemented
105C23×C32—212no—
106C23×C32—32no—
107C23×C32C22×D313yes—
108C23×C32C23×C3112yes—
109C22×C3×D3C23×C318yes—
110C32×Q8—216no—
111C32×Q8—324no—
112C32×Q8—312no—
113C32×Q8C22×D314yes—
114C32×Q8C23×C3116yes—
115C32×D4—248no—
116C32×D4—324no—
117C32×D4—34no—
118C32×D4C22×D3112yes—
119C32×D4C23×C3148yes—
120C2×C32×C4—24no—
121C2×C32×C4—212no—
122C2×C32×C4—248no—
123C2×C32×C4—324no—
124C2×C32×C4—36no—
125C2×C32×C4—324no—
126C2×C32×C4C22×D3112yes—
127C2×C32×C4C22×D313yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C23×C314yes—
130C2×C32×C4C23×C3112yes—
131C2×C32×C4C23×C3148yes—
132C3×#67C23×C3196yes—
133C2×C3×Dic3C23×C3124yes—
134C2×C3×Dic3C23×C3196yes—
135C3×D12C23×C3116yes—
136C3×C4×D3C23×C3196yes—
137C3×Dic6C23×C3148yes—
138C3×Dic6C23×C3196yes—
139C2×2−1+4—31noSylow
140C2×C33—26no—
141C2×C332−1+412yescomplemented
142C2×C33C2511yescharacteristic
143C32×D3—24no—
144C3×#41—232no—
145C3×#41—3192no—
146C3×#41—396no—
147C3×#41C2×C3×D3132yescomplemented
148C3×#41C22×C3218yescomplemented
149C2×C3×Q8—24no—
150C2×C3×Q8—364no—
151C2×C3×Q8—324no—
152C2×C3×Q8—312no—
153C2×C3×Q8—396no—
154C2×C3×Q8C2×C3×D314yes—
155C2×C3×Q8C22×C3211yescharacteristic
156C2×C3×D4—212no—
157C2×C3×D4—324no—
158C2×C3×D4—34no—
159C2×C3×D4C2×C3×D3112yes—
160C2×C3×D4C22×C3213yes—
161C22×C3×C4—212no—
162C22×C3×C4—324no—
163C22×C3×C4—324no—
164C22×C3×C4C2×C3×D3112yes—
165C22×C3×C4C22×C3213yes—
166C2×#67C22×C3216yes—
167C22×Dic3C22×C3216yes—
168D3×Q8C22×C32116yescomplemented
169#237C22×C32148yescomplemented
170#235C22×C32124yescomplemented
171C2×D12C22×C3211yescharacteristic
172C2×C4×D3C22×C3216yes—
173C2×Dic6C22×C3213yes—
174C2×Dic6C22×C3216yes—
175C2×Dic6C22×C32124yescomplemented
176C22×C32—26no—
177C22×C32—24no—
178C22×C32—224no—
179C22×C32—224no—
180C22×C32—34no—
181C22×C32—424no—
182C22×C32—64no—
183C22×C32C23×D316yes—
184C22×C32C23×D311yescharacteristic
185C22×C32C24×C3124yes—
186C22×C32C24×C314yes—
187C2×C3×D3—216no—
188C2×C3×D3C24×C3116yes—
189C32×C4—28no—
190C32×C4—224no—
191C32×C4—312no—
192C32×C4C23×D316yes—
193C32×C4C23×D312yes—
194C32×C4C24×C318yes—
195C32×C4C24×C3124yes—
196C3×Dic3C24×C3148yes—
1972−1+4—316no—
198C2×#41—36no—
199C2×#41—33no—
200C2×#41C32×D311yescharacteristic, complemented
201C22×Q8—32no—
202C22×Q8—33no—
203C33C2×2−1+411yescharacteristic, complemented, Sylow
204C23×C3—212no—
205C23×C3—38no—
206C23×C3C22×C3×D3112yes—
207C23×C3C23×C3213yes—
208C22×D3C23×C3212yes—
209C3×Q8—216no—
210C3×Q8—396no—
211C3×Q8—348no—
212C3×Q8C22×C3×D3116yes—
213C3×Q8C23×C3214yes—
214C3×D4—248no—
215C3×D4—396no—
216C3×D4—316no—
217C3×D4C22×C3×D3148yes—
218C3×D4C23×C32112yes—
219C2×C3×C4—24no—
220C2×C3×C4—212no—
221C2×C3×C4—248no—
222C2×C3×C4—324no—
223C2×C3×C4—396no—
224C2×C3×C4—396no—
225C2×C3×C4C22×C3×D3112yes—
226C2×C3×C4C22×C3×D314yes—
227C2×C3×C4C22×C3×D3148yes—
228C2×C3×C4C23×C3211yescharacteristic
229C2×C3×C4C23×C3213yes—
230C2×C3×C4C23×C32112yes—
231#67C23×C32124yes—
232C2×Dic3C23×C3216yes—
233C2×Dic3C23×C32124yes—
234D12C23×C3214yes—
235C4×D3C23×C32124yes—
236Dic6C23×C32112yes—
237Dic6C23×C32124yes—
238C2×C32—26no—
239C2×C32—28no—
240C2×C32—24no—
241C2×C32—224no—
242C2×C32—424no—
243C2×C32—64no—
244C2×C32#101012yesdirect product factor
245C2×C32C3×2−1+418yescomplemented
246C2×C32C24×D311yescharacteristic
247C2×C32C25×C314yes—
248C3×D3—216no—
249#41—348no—
250#41—324no—
251#41C2×C32×D318yescomplemented
252C2×Q8—316no—
253C2×Q8—36no—
254C2×Q8—33no—
255C2×Q8—324no—
256C2×Q8C2×C32×D311yescharacteristic
257C2×D4—36no—
258C2×D4—31no—
259C2×D4C2×C32×D313yes—
260C22×C4—36no—
261C22×C4—36no—
262C22×C4C2×C32×D313yes—
263C22×C3—224no—
264C22×C3—26no—
265C22×C3—24no—
266C22×C3—224no—
267C22×C3—316no—
268C22×C3—424no—
269C22×C3—616no—
270C22×C3C23×C3×D3124yes—
271C22×C3C23×C3×D314yes—
272C22×C3C24×C3216yes—
273C22×C3C24×C3211yescharacteristic
274C2×D3—24no—
275C2×D3C24×C3214yes—
276C3×C4—28no—
277C3×C4—224no—
278C3×C4—348no—
279C3×C4C23×C3×D3124yes—
280C3×C4C23×C3×D318yes—
281C3×C4C24×C3212yes—
282C3×C4C24×C3216yes—
283Dic3C24×C32112yes—
284C32—24no—
285C32C2×#101011yescharacteristic, direct product factor
286C32C2×C3×2−1+414yescomplemented
287C23—32no—
288C23C22×C32×D313yes—
289Q8—324no—
290Q8—312no—
291Q8C22×C32×D314yes—
292D4—324no—
293D4—34no—
294D4C22×C32×D3112yes—
295C2×C4—36no—
296C2×C4—324no—
297C2×C4—324no—
298C2×C4C22×C32×D313yes—
299C2×C4C22×C32×D311yescharacteristic
300C2×C4C22×C32×D3112yes—
301C2×C3—224no—
302C2×C3—26no—
303C2×C3—28no—
304C2×C3—24no—
305C2×C3—424no—
306C2×C3—616no—
307C2×C3C3×#101018yesdirect product factor
308C2×C3C32×2−1+412yescomplemented
309C2×C3C24×C3×D314yes—
310C2×C3C25×C3211yescharacteristic
311D3—24no—
312C22—26no—
313C22—34no—
314C22—64no—
315C22C23×C32×D316yes—
316C22C23×C32×D311yescharacteristic
317C4—312no—
318C4C23×C32×D316yes—
319C4C23×C32×D312yes—
320C3—24no—
321C3C2×C3×#101014yesdirect product factor
322C3C2×C32×2−1+411yescharacteristic, complemented
323C2—26no—
324C2—64no—
325C2C32×#101012yesdirect product factor
326C2C24×C32×D311yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC2×C3C25×C32
FrattiniC2C24×C32×D3
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×C32×#41D3
hypocenterC3C2×C32×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C32×#1010C2direct (N × H)
3. C2×C32×D3×Q8C2semidirect
4. C2×C32×#237C2semidirect
5. C2×C32×#235C2semidirect
6. C22×C32×Dic6C2semidirect
8. C2×C3×#1010C3direct (N × H)
9. C33×#41C22semidirect
15. C32×D3×Q8C22semidirect
16. C32×#237C22semidirect
17. C32×#235C22semidirect
22. C2×C32×Dic6C22semidirect
27. C2×C32×#41D3semidirect
28. C2×C32×#41C2×C3semidirect
31. C3×#1010C2×C3direct (N × H)
32. C2×C3×D3×Q8C2×C3semidirect
33. C2×C3×#237C2×C3semidirect
34. C2×C3×#235C2×C3semidirect
35. C22×C3×Dic6C2×C3semidirect
51. C2×#1010C32direct (N × H)
55. C32×#41C2×D3semidirect
56. C32×#41C22×C3semidirect
76. C3×D3×Q8C22×C3semidirect
77. C3×#237C22×C3semidirect
78. C3×#235C22×C3semidirect
83. C2×C3×Dic6C22×C3semidirect
96. C2×C3×#41C3×D3semidirect
97. C2×C3×#41C2×C32semidirect
100. #1010C2×C32direct (N × H)
101. C2×D3×Q8C2×C32semidirect
102. C2×#237C2×C32semidirect
103. C2×#235C2×C32semidirect
104. C22×Dic6C2×C32semidirect
141. C2×C332−1+4semidirect
147. C3×#41C2×C3×D3semidirect
148. C3×#41C22×C32semidirect
168. D3×Q8C22×C32semidirect
169. #237C22×C32semidirect
170. #235C22×C32semidirect
175. C2×Dic6C22×C32semidirect
200. C2×#41C32×D3semidirect
203. C33C2×2−1+4semidirect
244. C2×C32#1010direct (N × H)
245. C2×C32C3×2−1+4semidirect
251. #41C2×C32×D3semidirect
285. C32C2×#1010direct (N × H)
286. C32C2×C3×2−1+4semidirect
307. C2×C3C3×#1010direct (N × H)
308. C2×C3C32×2−1+4semidirect
321. C3C2×C3×#1010direct (N × H)
322. C3C2×C32×2−1+4semidirect
325. C2C32×#1010direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C32×#10102
C2×C32×D3×Q82
C2×C32×#2372
C2×C32×#2352
C22×C32×Dic62
C2×C32×2−1+43
C2×C3×#10103