Finite Group Databasebeta

← C3×C4×#3832 #13179324

Subgroups — C3×C4×#3832 #13179324

Order 1728.

2140 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C33×C4×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#64102C213yesmaximal, complemented
5C3×#64101C213yesmaximal, complemented
6C3×C4×C4⋊(C3⋊D3)C213yesmaximal, complemented
7C3×C42×C3⋊D3C213yesmaximal, complemented
8C4×#3832C311yesmaximal, characteristic, direct product factor
9C3×C4×#239—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832C418yesdirect product factor
12C2×C3×C4⋊(C3⋊D3)C2213yes—
13C2×C3×C4×C3⋊D3C2213yes—
14C2×C3×C4×C3⋊D3C2213yes—
15C2×C3×C4×C3⋊D3C2213yes—
16C33×C4⋊C4C2213yes—
17C33×C42C2213yes—
18C3×#3752C2216yes—
19C3×C4⋊(C3⋊Dic3)C2213yes—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C3×C4×C3⋊Dic3C2213yes—
22C3×C4×C3⋊Dic3C2213yes—
23C2×#3832C2×C311yescharacteristic
24C2×C3×#239—34no—
25C32×C4×Q8—24no—
26C32×C4×Q8D314yescomplemented
27C32×C4×Q8C2×C311yescharacteristic, complemented
28Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
29#64102C2×C313yescomplemented
30#64101C2×C313yescomplemented
31C4×C4⋊(C3⋊D3)C2×C313yescomplemented
32C42×C3⋊D3C2×C313yescomplemented
33C3×Q8×Dic3—34no—
34C3×#893—312no—
35C3×#892—312no—
36C3×C4×D12—312no—
37C3×C42×D3—312no—
38C22×C3×C3⋊D3C2313yes—
39C33×Q8C2×C414yescomplemented
40C2×C33×C4C2313yes—
41C2×C33×C4C2313yes—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3C2313yes—
44C2×C3×C3⋊Dic3C2311yescharacteristic
45C2×C3×C3⋊Dic3C2311yescharacteristic
46C3×C4⋊(C3⋊D3)C2×C4112yescomplemented
47C3×C4×C3⋊D3—26no—
48C3×C4×C3⋊D3—26no—
49C3×C4×C3⋊D3C2×C4112yescomplemented
50C3×C4×#41—91no—
51C4×#239—34no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C22×C311yescharacteristic
55#3832C3×C418yesdirect product factor
56C2×C4⋊(C3⋊D3)C22×C313yes—
57C2×C4×C3⋊D3C22×C313yes—
58C2×C4×C3⋊D3C22×C313yes—
59C2×C4×C3⋊D3C22×C313yes—
60C3×#239—332no—
61C2×C3×D12—312no—
62C2×C3×C4×D3—312no—
63C2×C3×C4×D3—312no—
64C2×C3×C4×D3—312no—
65C32×C4⋊C4—212no—
66C32×C4⋊C4C2×D3112yes—
67C32×C4⋊C4C22×C313yes—
68C32×C42—212no—
69C32×C42C2×D3112yes—
70C32×C42C22×C313yes—
71#3752C22×C316yes—
72C4⋊(C3⋊Dic3)C22×C313yes—
73C4×C3⋊Dic3C22×C311yescharacteristic
74C4×C3⋊Dic3C22×C313yes—
75C4×C3⋊Dic3C22×C313yes—
76C3×(C2×D3)⋊C4—324no—
77C3×C4⋊Dic3—312no—
78C3×C4×Dic3—312no—
79C3×C4×Dic3—312no—
80C3×C4×Dic3—34no—
81C22×C33C2411yescharacteristic
82C2×C3×C3⋊D3—23no—
83C2×C3×C3⋊D3—23no—
84C2×C3×C3⋊D3C22×C416yes—
85C33×C4—23no—
86C33×C4C22×C416yes—
87C33×C4#4112yescomplemented
88C3×C3⋊Dic3—23no—
89C3×C3⋊Dic3C22×C412yes—
90C3×C3⋊Dic3#4112yescomplemented
91C2×C3×#41—91no—
92C2×#239—34no—
93C3×C4×Q8—24no—
94C3×C4×Q8—91no—
95C3×C4×Q8C3×D314yescomplemented
96C3×C4×Q8C3⋊D311yescharacteristic, complemented
97C3×C4×D4—93no—
98C3×C4×D4—93no—
99C3×#117—93no—
100C2×C3×C42—93no—
101Q8×Dic3—34no—
102#893—312no—
103#892—312no—
104C4×D12—312no—
105C42×D3—312no—
106C22×C3⋊D3C23×C313yes—
107C22×C3×D3—312no—
108C32×Q8—216no—
109C32×Q8C4×D3116yescomplemented
110C32×Q8C2×C3×C414yescomplemented
111C2×C32×C4—24no—
112C2×C32×C4—212no—
113C2×C32×C4—212no—
114C2×C32×C4C22×D3112yes—
115C2×C32×C4C22×D3112yes—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C23×C313yes—
118C2×C32×C4C23×C313yes—
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3C23×C313yes—
121C2×C3⋊Dic3C23×C311yescharacteristic
122C2×C3⋊Dic3C23×C311yescharacteristic
123C4⋊(C3⋊D3)C2×C3×C4112yescomplemented
124C4×C3⋊D3—26no—
125C4×C3⋊D3—26no—
126C4×C3⋊D3C2×C3×C4112yescomplemented
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—312no—
130C3×D12—348no—
131C3×C4×D3—348no—
132C3×C4×D3—624no—
133C3×C4×D3—624no—
134C4×#41—91noSylow
135C2×C33C23×C411yescharacteristic
136C2×C33C2×#4111yescharacteristic
137C2×C33C2×#4111yescharacteristic
138C3×C3⋊D3—26no—
139C3×#41—98no—
140C2×C3×Q8—24no—
141C2×C3×Q8C2×C3×D314yes—
142C2×C3×Q8C2×C3⋊D311yescharacteristic
143C2×C3×D4—93no—
144C22×C3×C4—93no—
145C22×C3×C4—93no—
146C22×C3×C4—93no—
147#239—332no—
148C2×D12—312no—
149C2×C4×D3—312no—
150C2×C4×D3—312no—
151C2×C4×D3—312no—
152C3×C4⋊C4—212no—
153C3×C4⋊C4—93no—
154C3×C4⋊C4C2×C3×D3112yes—
155C3×C4⋊C4C2×C3⋊D313yes—
156C3×C22⋊C4—96no—
157C3×C42—212no—
158C3×C42—93no—
159C3×C42—93no—
160C3×C42—91no—
161C3×C42C2×C3×D3112yes—
162C3×C42C2×C3⋊D313yes—
163(C2×D3)⋊C4—324no—
164C4⋊Dic3—312no—
165C4×Dic3—34no—
166C4×Dic3—312no—
167C4×Dic3—312no—
168C22×C32—24no—
169C22×C32C23×D314yes—
170C22×C32C24×C311yescharacteristic
171C2×C3⋊D3—23no—
172C2×C3⋊D3—23no—
173C2×C3⋊D3C22×C3×C416yes—
174C2×C3×D3—324no—
175C2×C3×D3—612no—
176C2×C3×D3—612no—
177C32×C4—23no—
178C32×C4—224no—
179C32×C4—28no—
180C32×C4—212no—
181C32×C4—412no—
182C32×C4C2×C4×D3124yes—
183C32×C4#23918yescomplemented
184C32×C4C22×C3×C416yes—
185C32×C4C3×#4112yescomplemented
186C3⋊Dic3—23no—
187C3⋊Dic3C22×C3×C412yes—
188C3⋊Dic3C3×#4112yescomplemented
189C3×Dic3—38no—
190C3×Dic3—38no—
191C3×Dic3—612no—
192C2×#41—91no—
193C4×Q8—91no—
194C4×Q8C3×C3⋊D311yescharacteristic, complemented
195C4×D4—93no—
196C4×D4—93no—
197#117—93no—
198C2×C42—93no—
199C33C4×#4111yescharacteristic, complemented, Sylow
200C23×C3—93no—
201C22×D3—312no—
202C3×Q8—216no—
203C3×Q8C3×C4×D3116yescomplemented
204C3×Q8C4×C3⋊D314yescomplemented
205C3×D4—912no—
206C2×C3×C4—24no—
207C2×C3×C4—212no—
208C2×C3×C4—212no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—912no—
212C2×C3×C4—93no—
213C2×C3×C4—186no—
214C2×C3×C4—186no—
215C2×C3×C4C22×C3×D3112yes—
216C2×C3×C4C22×C3×D3112yes—
217C2×C3×C4C22×C3×D314yes—
218C2×C3×C4C22×C3⋊D313yes—
219C2×C3×C4C22×C3⋊D313yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—34no—
222C2×Dic3—34no—
223C2×Dic3—312no—
224D12—348no—
225C4×D3—348no—
226C4×D3—624no—
227C4×D3—624no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32C22×C4×D314yes—
232C2×C32C2×#23914yes—
233C2×C32D3×#4114yes—
234C2×C32C23×C3×C411yescharacteristic
235C2×C32C2×C3×#4111yescharacteristic
236C2×C32C2×C3×#4111yescharacteristic
237C3⋊D3—26no—
238C3×D3—624no—
239#41—98no—
240C2×Q8C2×C3×C3⋊D311yescharacteristic
241C2×D4—93no—
242C22×C4—93no—
243C22×C4—93no—
244C22×C4—93no—
245C4⋊C4—93no—
246C4⋊C4C2×C3×C3⋊D313yes—
247C22⋊C4—96no—
248C42—91no—
249C42—93no—
250C42—93no—
251C42C2×C3×C3⋊D313yes—
252C22×C3—24no—
253C22×C3—96no—
254C22×C3—183no—
255C22×C3—183no—
256C22×C3C23×C3×D314yes—
257C22×C3C23×C3⋊D311yescharacteristic
258C2×D3—324no—
259C2×D3—612no—
260C2×D3—612no—
261C3×C4—212no—
262C3×C4—23no—
263C3×C4—224no—
264C3×C4—28no—
265C3×C4—412no—
266C3×C4—92no—
267C3×C4—92no—
268C3×C4—183no—
269C3×C4C2×C3×C4×D3124yes—
270C3×C4C3×#23918yescomplemented
271C3×C4C2×C4×C3⋊D316yes—
272C3×C4#383212yesdirect product factor
273Dic3—38no—
274Dic3—38no—
275Dic3—612no—
276C32—24no—
277C32C4×#23914yescomplemented
278C32C3×C4×#4111yescharacteristic, complemented
279C23—93no—
280Q8C3×C4×C3⋊D314yescomplemented
281D4—912no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—912no—
285C2×C4—93no—
286C2×C4—186no—
287C2×C4—186no—
288C2×C4C22×C3×C3⋊D313yes—
289C2×C4C22×C3×C3⋊D313yes—
290C2×C4C22×C3×C3⋊D311yescharacteristic
291C2×C3—24no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—186no—
295C2×C3C22×C3×C4×D314yes—
296C2×C3C2×C3×#23914yes—
297C2×C3C3×D3×#4114yes—
298C2×C3C22×C4×C3⋊D311yescharacteristic
299C2×C3C2×#383211yescharacteristic
300C2×C3#41×C3⋊D311yescharacteristic
301D3—624no—
302C22—96no—
303C22—183no—
304C22—183no—
305C22C23×C3×C3⋊D311yescharacteristic
306C4—23no—
307C4—92no—
308C4—92no—
309C4—183no—
310C4C2×C3×C4×C3⋊D316yes—
311C4C3×#383212yesdirect product factor
312C3—24no—
313C3C3×C4×#23914yescomplemented
314C3C4×#383211yescharacteristic, direct product factor
315C2—186no—
316C2C22×C3×C4×C3⋊D311yescharacteristic
317C2C2×C3×#383211yescharacteristic
318C2C3×#41×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC2×C32C23×C3×C4
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C24
prime radicalC2×C3×#3832C2
cosocleC22×C32C24×C3
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×C4×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C33×C4×Q8C2semidirect
3. C3×Q8×C3⋊Dic3C2semidirect
4. C3×#64102C2semidirect
5. C3×#64101C2semidirect
6. C3×C4×C4⋊(C3⋊D3)C2semidirect
7. C3×C42×C3⋊D3C2semidirect
8. C4×#3832C3direct (N × H)
11. C3×#3832C4direct (N × H)
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
29. #64102C2×C3semidirect
30. #64101C2×C3semidirect
31. C4×C4⋊(C3⋊D3)C2×C3semidirect
32. C42×C3⋊D3C2×C3semidirect
39. C33×Q8C2×C4semidirect
46. C3×C4⋊(C3⋊D3)C2×C4semidirect
49. C3×C4×C3⋊D3C2×C4semidirect
55. #3832C3×C4direct (N × H)
87. C33×C4#41semidirect
90. C3×C3⋊Dic3#41semidirect
95. C3×C4×Q8C3×D3semidirect
96. C3×C4×Q8C3⋊D3semidirect
109. C32×Q8C4×D3semidirect
110. C32×Q8C2×C3×C4semidirect
123. C4⋊(C3⋊D3)C2×C3×C4semidirect
126. C4×C3⋊D3C2×C3×C4semidirect
183. C32×C4#239semidirect
185. C32×C4C3×#41semidirect
188. C3⋊Dic3C3×#41semidirect
194. C4×Q8C3×C3⋊D3semidirect
199. C33C4×#41semidirect
203. C3×Q8C3×C4×D3semidirect
204. C3×Q8C4×C3⋊D3semidirect
270. C3×C4C3×#239semidirect
272. C3×C4#3832direct (N × H)
277. C32C4×#239semidirect
278. C32C3×C4×#41semidirect
280. Q8C3×C4×C3⋊D3semidirect
311. C4C3×#3832direct (N × H)
313. C3C3×C4×#239semidirect
314. C3C4×#3832direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C33×C4×Q82
C3×Q8×C3⋊Dic32
C3×#641022
C3×#641012
C3×C4×C4⋊(C3⋊D3)2
C3×C42×C3⋊D32
C4×#38323
C3×C4×#2393