Finite Group Databasebeta

← C3×#10596047 #13179327

Subgroups — C3×#10596047 #13179327

Order 1728.

2020 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C33×C4×Q8C211yesmaximal, characteristic, complemented
4C3×#64106C213yesmaximal, complemented
5C3×#64104C213yesmaximal, complemented
6C3×#64084C213yesmaximal, complemented
7C3×C4×C4⋊(C3⋊D3)C213yesmaximal, complemented
8#10596047C311yesmaximal, characteristic, direct product factor
9C3×#5576—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—24no—
12C3×Q8×C3⋊D3—24no—
13C2×C3×C4⋊(C3⋊D3)C2213yes—
14C2×C3×C4⋊(C3⋊D3)C2213yes—
15C2×C3×C4×C3⋊D3C2213yes—
16C2×C3×C4×C3⋊D3C2213yes—
17C2×C3×#634C2213yes—
18C33×C4⋊C4C2213yes—
19C33×C42C2213yes—
20C3×#3752C2211yescharacteristic
21C3×#3752C2213yes—
22C3×#3752C2216yes—
23C3×C4⋊(C3⋊Dic3)C2213yes—
24C2×#3832C2×C311yescharacteristic, complemented
25C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×#239—34no—
27C2×C3×D3×Q8—34no—
28C32×C4×Q8—24no—
29C32×C4×Q8D314yescomplemented
30C32×C4×Q8C2×C311yescharacteristic, complemented
31#64106C2×C313yescomplemented
32#64104C2×C313yescomplemented
33#64084C2×C313yescomplemented
34C4×C4⋊(C3⋊D3)C2×C313yescomplemented
35C3×#897—312no—
36C3×#895—312no—
37C3×#875—312no—
38C3×C4×D12—312no—
39C22×C3×C3⋊D3C2313yes—
40C22×C3×C3⋊D3C2311yescharacteristic
41C33×Q8D414yescomplemented
42C2×C33×C4C2313yes—
43C2×C33×C4C2311yescharacteristic
44C2×C33×C4C2313yes—
45C2×C3×C3⋊Dic3C2311yescharacteristic
46C2×C3×C3⋊Dic3C2313yes—
47C3×C4⋊(C3⋊D3)—26no—
48C3×C4⋊(C3⋊D3)—26no—
49C3×C4×C3⋊D3—26no—
50C3×C4×C3⋊D3—26no—
51C3×#634—26no—
52C3×#548—91no—
53#5576—34no—
54C2×C32×Q8—24no—
55C2×C32×Q8C2×D314yes—
56C2×C32×Q8C22×C311yescharacteristic
57#3832—24no—
58Q8×C3⋊D3—24no—
59C2×C4⋊(C3⋊D3)C22×C313yes—
60C2×C4⋊(C3⋊D3)C22×C313yes—
61C2×C4×C3⋊D3C22×C313yes—
62C2×C4×C3⋊D3C22×C313yes—
63C2×#634C22×C313yes—
64C3×#239—616no—
65C3×D3×Q8—616no—
66C2×C3×D12—312no—
67C2×C3×D12—312no—
68C2×C3×C4×D3—312no—
69C2×C3×C4×D3—312no—
70C2×C3×Dic6—312no—
71C32×C4⋊C4—212no—
72C32×C4⋊C4C2×D3112yes—
73C32×C4⋊C4C22×C313yes—
74C32×C42—212no—
75C32×C42C2×D3112yes—
76C32×C42C22×C313yes—
77#3752C22×C311yescharacteristic
78#3752C22×C313yes—
79#3752C22×C316yes—
80C4⋊(C3⋊Dic3)C22×C313yes—
81C3×(C2×D3)⋊C4—324no—
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—312no—
84C3×C4⋊Dic3—312no—
85C22×C33C2411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—23no—
89C2×C3×C3⋊D3—23no—
90C2×C3×C3⋊D3—23no—
91C2×C3×C3⋊D3#4112yescomplemented
92C33×C4—23no—
93C33×C4—21no—
94C33×C4C2×D416yes—
95C3×C3⋊Dic3—23no—
96C3×C3⋊Dic3—21no—
97C2×C3×#41—91no—
98C22×C3×Q8—91no—
99C2×#239—34no—
100C2×D3×Q8—34no—
101C3×#124—93no—
102C3×C22⋊Q8—93no—
103C3×#121—93no—
104C3×C4×Q8—24no—
105C3×C4×Q8C3×D314yescomplemented
106C3×C4×Q8C3⋊D311yescharacteristic, complemented
107C3×C4×D4—93no—
108#897—312no—
109#895—312no—
110#875—312no—
111C4×D12—312no—
112C22×C3⋊D3C23×C313yes—
113C22×C3⋊D3C23×C311yescharacteristic
114C22×C3×D3—34no—
115C22×C3×D3—312no—
116C32×Q8—216no—
117C32×Q8D12116yescomplemented
118C32×Q8C3×D414yescomplemented
119C2×C32×C4—24no—
120C2×C32×C4—212no—
121C2×C32×C4—212no—
122C2×C32×C4C22×D3112yes—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D3112yes—
125C2×C32×C4C23×C313yes—
126C2×C32×C4C23×C311yescharacteristic
127C2×C32×C4C23×C313yes—
128C2×C3⋊Dic3C23×C311yescharacteristic
129C2×C3⋊Dic3C23×C313yes—
130C4⋊(C3⋊D3)—26no—
131C4⋊(C3⋊D3)—26no—
132C4×C3⋊D3—26no—
133C4×C3⋊D3—26no—
134#634—26no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—312no—
137C3×D12—624no—
138C3×D12—624no—
139C3×C4×D3—624no—
140C3×C4×D3—624no—
141C3×Dic6—624no—
142#548—91noSylow
143C2×C33C22×D411yescharacteristic
144C2×C33C2×#4111yescharacteristic
145C2×C332−1+411yescharacteristic
146C3×C3⋊D3—22no—
147C3×C3⋊D3—43no—
148C3×#41—184no—
149C2×C3×Q8—24no—
150C2×C3×Q8—93no—
151C2×C3×Q8—184no—
152C2×C3×Q8C2×C3×D314yes—
153C2×C3×Q8C2×C3⋊D311yescharacteristic
154C2×C3×D4—93no—
155C2×C3×D4—93no—
156C22×C3×C4—93no—
157C22×C3×C4—93no—
158#239—616no—
159D3×Q8—616no—
160C2×D12—312no—
161C2×D12—312no—
162C2×C4×D3—312no—
163C2×C4×D3—312no—
164C2×Dic6—312no—
165C3×C4⋊C4—212no—
166C3×C4⋊C4—93no—
167C3×C4⋊C4C2×C3×D3112yes—
168C3×C4⋊C4C2×C3⋊D313yes—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—93no—
171C3×C22⋊C4—96no—
172C3×C42—212no—
173C3×C42C2×C3×D3112yes—
174C3×C42C2×C3⋊D313yes—
175(C2×D3)⋊C4—34no—
176(C2×D3)⋊C4—312no—
177(C2×D3)⋊C4—324no—
178C4⋊Dic3—312no—
179C22×C32—24no—
180C22×C32C23×D314yes—
181C22×C32C24×C311yescharacteristic
182C2×C3⋊D3—21no—
183C2×C3⋊D3—21no—
184C2×C3⋊D3—23no—
185C2×C3⋊D3—23no—
186C2×C3⋊D3—23no—
187C2×C3⋊D3C3×#4112yescomplemented
188C2×C3×D3—38no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—612no—
192C2×C3×D3—612no—
193C2×C3×D3—612no—
194C32×C4—28no—
195C32×C4—24no—
196C32×C4—23no—
197C32×C4—21no—
198C32×C4—224no—
199C32×C4—212no—
200C32×C4—412no—
201C32×C4C2×D12124yes—
202C32×C4C2×C3×D416yes—
203C3⋊Dic3—23no—
204C3⋊Dic3—21no—
205C3×Dic3—612no—
206C3×Dic3—64no—
207C2×#41—91no—
208C22×Q8—91no—
209#124—93no—
210C22⋊Q8—93no—
211#121—93no—
212C4×Q8C3×C3⋊D311yescharacteristic, complemented
213C4×D4—93no—
214C33#54811yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C23×C3—93no—
217C22×D3—34no—
218C22×D3—312no—
219C3×Q8—216no—
220C3×Q8—186no—
221C3×Q8C3×D12116yescomplemented
222C3×Q8C4⋊(C3⋊D3)14yescomplemented
223C3×D4—186no—
224C3×D4—186no—
225C2×C3×C4—24no—
226C2×C3×C4—212no—
227C2×C3×C4—212no—
228C2×C3×C4—93no—
229C2×C3×C4—91no—
230C2×C3×C4—186no—
231C2×C3×C4—186no—
232C2×C3×C4C22×C3×D3112yes—
233C2×C3×C4C22×C3×D314yes—
234C2×C3×C4C22×C3×D3112yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×C3×C4C22×C3⋊D313yes—
237C2×C3×C4C22×C3⋊D313yes—
238C2×Dic3—34no—
239C2×Dic3—312no—
240D12—624no—
241D12—624no—
242C4×D3—624no—
243C4×D3—624no—
244Dic6—624no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32C22×D1214yes—
249C2×C32#100714yes—
250C2×C32D3×#4114yes—
251C2×C32C22×C3×D411yescharacteristic
252C2×C32C2×C3×#4111yescharacteristic
253C2×C32C3×2−1+411yescharacteristic
254C3⋊D3—22no—
255C3⋊D3—43no—
256C3×D3—68no—
257C3×D3—1212no—
258#41—184no—
259C2×Q8—93no—
260C2×Q8—184no—
261C2×Q8C2×C3×C3⋊D311yescharacteristic
262C2×D4—93no—
263C2×D4—93no—
264C22×C4—93no—
265C22×C4—93no—
266C4⋊C4—93no—
267C4⋊C4C2×C3×C3⋊D313yes—
268C22⋊C4—91no—
269C22⋊C4—93no—
270C22⋊C4—96no—
271C42C2×C3×C3⋊D313yes—
272C22×C3—24no—
273C22×C3—92no—
274C22×C3—181no—
275C22×C3—181no—
276C22×C3—183no—
277C22×C3—183no—
278C22×C3—183no—
279C22×C3C23×C3×D314yes—
280C22×C3C23×C3⋊D311yescharacteristic
281C2×D3—38no—
282C2×D3—64no—
283C2×D3—64no—
284C2×D3—612no—
285C2×D3—612no—
286C2×D3—612no—
287C3×C4—28no—
288C3×C4—24no—
289C3×C4—212no—
290C3×C4—21no—
291C3×C4—23no—
292C3×C4—224no—
293C3×C4—412no—
294C3×C4—181no—
295C3×C4—183no—
296C3×C4C2×C3×D12124yes—
297C3×C4C2×C4⋊(C3⋊D3)16yes—
298Dic3—612no—
299Dic3—64no—
300C32—24no—
301C32#557614yescomplemented
302C32C3×#54811yescharacteristic, complemented
303C23—91no—
304C23—93no—
305Q8—186no—
306Q8C3×C4⋊(C3⋊D3)14yescomplemented
307D4—186no—
308D4—186no—
309C2×C4—93no—
310C2×C4—91no—
311C2×C4—186no—
312C2×C4—186no—
313C2×C4C22×C3×C3⋊D311yescharacteristic
314C2×C4C22×C3×C3⋊D313yes—
315C2×C4C22×C3×C3⋊D313yes—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—182no—
320C2×C3—363no—
321C2×C3C22×C3×D1214yes—
322C2×C3C3×#100714yes—
323C2×C3C3×D3×#4114yes—
324C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
325C2×C3#6436411yescharacteristic
326C2×C3#41×C3⋊D311yescharacteristic
327D3—68no—
328D3—1212no—
329C22—92no—
330C22—181no—
331C22—181no—
332C22—183no—
333C22—183no—
334C22—183no—
335C22C23×C3×C3⋊D311yescharacteristic
336C4—21no—
337C4—23no—
338C4—181no—
339C4—183no—
340C4C2×C3×C4⋊(C3⋊D3)16yes—
341C3—24no—
342C3C3×#557614yescomplemented
343C3#1059604711yescharacteristic, direct product factor
344C2—182no—
345C2—363no—
346C2C22×C3×C4⋊(C3⋊D3)11yescharacteristic
347C2C3×#6436411yescharacteristic
348C2C3×#41×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C24
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C33×C4×Q8C2semidirect
4. C3×#64106C2semidirect
5. C3×#64104C2semidirect
6. C3×#64084C2semidirect
7. C3×C4×C4⋊(C3⋊D3)C2semidirect
8. #10596047C3direct (N × H)
24. C2×#3832C2×C3semidirect
25. C2×Q8×C3⋊D3C2×C3semidirect
29. C32×C4×Q8D3semidirect
30. C32×C4×Q8C2×C3semidirect
31. #64106C2×C3semidirect
32. #64104C2×C3semidirect
33. #64084C2×C3semidirect
34. C4×C4⋊(C3⋊D3)C2×C3semidirect
41. C33×Q8D4semidirect
91. C2×C3×C3⋊D3#41semidirect
105. C3×C4×Q8C3×D3semidirect
106. C3×C4×Q8C3⋊D3semidirect
117. C32×Q8D12semidirect
118. C32×Q8C3×D4semidirect
187. C2×C3⋊D3C3×#41semidirect
212. C4×Q8C3×C3⋊D3semidirect
214. C33#548semidirect
221. C3×Q8C3×D12semidirect
222. C3×Q8C4⋊(C3⋊D3)semidirect
301. C32#5576semidirect
302. C32C3×#548semidirect
306. Q8C3×C4⋊(C3⋊D3)semidirect
342. C3C3×#5576semidirect
343. C3#10596047direct (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
C2×C3×Q8×C3⋊D32
C33×C4×Q82
C3×#641062
C3×#641042
C3×#640842
C3×C4×C4⋊(C3⋊D3)2
#105960473
C3×#55763