Finite Group Databasebeta

← C3×#10596045 #13179325

Subgroups — C3×#10596045 #13179325

Order 1728.

2044 conjugacy classes of subgroups.

The subgroup lattice has 314 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×C4⋊C4×C3⋊D3C213yesmaximal, complemented
6C3×C4×C4⋊(C3⋊D3)C213yesmaximal, complemented
7C3×#64081C213yesmaximal, complemented
8#10596045C311yesmaximal, characteristic, direct product factor
9C3×#5574—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832C418yescomplemented
12C2×C3×C4⋊(C3⋊D3)C2213yes—
13C2×C3×C4×C3⋊D3C2216yes—
14C2×C3×C4×C3⋊D3C2213yes—
15C33×C4⋊C4C2213yes—
16C33×C42C2213yes—
17C3×#3752C2213yes—
18C3×#3752C2213yes—
19C3×C4⋊(C3⋊Dic3)C2213yes—
20C3×#3750C2211yescharacteristic
21C3×#3750C2213yes—
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
30C4⋊C4×C3⋊D3C2×C313yescomplemented
31C4×C4⋊(C3⋊D3)C2×C313yescomplemented
32#64081C2×C313yescomplemented
33C3×Q8×Dic3—34no—
34C3×#893—312no—
35C3×D3×C4⋊C4—312no—
36C3×C4×D12—312no—
37C3×#872—312no—
38C22×C3×C3⋊D3C2313yes—
39C33×Q8C2×C414yescomplemented
40C2×C33×C4C2311yescharacteristic
41C2×C33×C4C2313yes—
42C2×C33×C4C2313yes—
43C2×C3×C3⋊Dic3C2311yescharacteristic
44C2×C3×C3⋊Dic3C2313yes—
45C2×C3×C3⋊Dic3C2311yescharacteristic
46C3×C4⋊(C3⋊D3)C2×C4112yescomplemented
47C3×C4×C3⋊D3—46no—
48C3×C4×C3⋊D3C2×C4112yescomplemented
49C3×#520—91no—
50#5574—34no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C22×C311yescharacteristic
54#3832C3×C418yescomplemented
55C2×C4⋊(C3⋊D3)C22×C313yes—
56C2×C4×C3⋊D3C22×C316yes—
57C2×C4×C3⋊D3C22×C313yes—
58C3×#239—332no—
59C2×C3×D12—312no—
60C2×C3×C4×D3—312no—
61C2×C3×C4×D3—324no—
62C32×C4⋊C4—212no—
63C32×C4⋊C4C2×D3112yes—
64C32×C4⋊C4C22×C313yes—
65C32×C42—212no—
66C32×C42C2×D3112yes—
67C32×C42C22×C313yes—
68#3752C22×C313yes—
69#3752C22×C313yes—
70C4⋊(C3⋊Dic3)C22×C313yes—
71#3750C22×C311yescharacteristic
72#3750C22×C313yes—
73C4×C3⋊Dic3C22×C313yes—
74C3×(C2×D3)⋊C4—312no—
75C3×(C2×D3)⋊C4—312no—
76C3×C4⋊Dic3—312no—
77C3×Dic3⋊C4—312no—
78C3×Dic3⋊C4—34no—
79C3×C4×Dic3—312no—
80C22×C33C2411yescharacteristic
81C2×C3×C3⋊D3—23no—
82C2×C3×C3⋊D3—23no—
83C2×C3×C3⋊D3C22×C416yes—
84C33×C4—23no—
85C33×C4—21no—
86C33×C4C22×C416yes—
87C3×C3⋊Dic3—23no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3C22×C412yes—
90C2×C3×#41—91no—
91C2×#239—34no—
92C3×C4×Q8—24no—
93C3×C4×Q8—91no—
94C3×C4×Q8C3×D314yescomplemented
95C3×C4×Q8C3⋊D311yescharacteristic, complemented
96C3×C4×D4—93no—
97C3×C4×D4—93no—
98C3×#117—93no—
99C2×C3×C4⋊C4—93no—
100Q8×Dic3—34no—
101#893—312no—
102D3×C4⋊C4—312no—
103C4×D12—312no—
104#872—312no—
105C22×C3⋊D3C23×C313yes—
106C22×C3×D3—312no—
107C32×Q8—216no—
108C32×Q8C4×D3116yescomplemented
109C32×Q8C2×C3×C414yescomplemented
110C2×C32×C4—24no—
111C2×C32×C4—212no—
112C2×C32×C4—212no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D3112yes—
115C2×C32×C4C22×D3112yes—
116C2×C32×C4C23×C311yescharacteristic
117C2×C32×C4C23×C313yes—
118C2×C32×C4C23×C313yes—
119C2×C3⋊Dic3C23×C311yescharacteristic
120C2×C3⋊Dic3C23×C313yes—
121C2×C3⋊Dic3C23×C311yescharacteristic
122C4⋊(C3⋊D3)C2×C3×C4112yescomplemented
123C4×C3⋊D3—46no—
124C4×C3⋊D3C2×C3×C4112yescomplemented
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—312no—
128C3×D12—348no—
129C3×C4×D3—348no—
130C3×C4×D3—1224no—
131#520—91noSylow
132C2×C33C23×C411yescharacteristic
133C2×C33H(5,𝔽2)11yescharacteristic
134C2×C332−1+411yescharacteristic
135C3×C3⋊D3—26no—
136C3×#41—98no—
137C2×C3×Q8—24no—
138C2×C3×Q8C2×C3×D314yes—
139C2×C3×Q8C2×C3⋊D311yescharacteristic
140C2×C3×D4—93no—
141C22×C3×C4—93no—
142C22×C3×C4—96no—
143#239—332no—
144C2×D12—312no—
145C2×C4×D3—312no—
146C2×C4×D3—324no—
147C3×C4⋊C4—212no—
148C3×C4⋊C4—93no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—93no—
151C3×C4⋊C4C2×C3×D3112yes—
152C3×C4⋊C4C2×C3⋊D313yes—
153C3×C22⋊C4—93no—
154C3×C22⋊C4—93no—
155C3×C42—212no—
156C3×C42—93no—
157C3×C42C2×C3×D3112yes—
158C3×C42C2×C3⋊D313yes—
159(C2×D3)⋊C4—312no—
160(C2×D3)⋊C4—312no—
161C4⋊Dic3—312no—
162Dic3⋊C4—34no—
163Dic3⋊C4—312no—
164C4×Dic3—312no—
165C22×C32—24no—
166C22×C32C23×D314yes—
167C22×C32C24×C311yescharacteristic
168C2×C3⋊D3—23no—
169C2×C3⋊D3—23no—
170C2×C3⋊D3C22×C3×C416yes—
171C2×C3×D3—324no—
172C2×C3×D3—612no—
173C2×C3×D3—612no—
174C32×C4—28no—
175C32×C4—24no—
176C32×C4—23no—
177C32×C4—21no—
178C32×C4—224no—
179C32×C4—212no—
180C32×C4—412no—
181C32×C4C2×C4×D3124yes—
182C32×C4C22×C3×C416yes—
183C3⋊Dic3—23no—
184C3⋊Dic3—21no—
185C3⋊Dic3C22×C3×C412yes—
186C3×Dic3—38no—
187C3×Dic3—64no—
188C3×Dic3—612no—
189C2×#41—91no—
190C4×Q8—91no—
191C4×Q8C3×C3⋊D311yescharacteristic, complemented
192C4×D4—93no—
193C4×D4—93no—
194#117—93no—
195C2×C4⋊C4—93no—
196C33#52011yescharacteristic, complemented, Sylow
197C23×C3—93no—
198C22×D3—312no—
199C3×Q8—216no—
200C3×Q8C3×C4×D3116yescomplemented
201C3×Q8C4×C3⋊D314yescomplemented
202C3×D4—912no—
203C2×C3×C4—24no—
204C2×C3×C4—212no—
205C2×C3×C4—212no—
206C2×C3×C4—91no—
207C2×C3×C4—912no—
208C2×C3×C4—91no—
209C2×C3×C4—93no—
210C2×C3×C4—366no—
211C2×C3×C4C22×C3×D314yes—
212C2×C3×C4C22×C3×D3112yes—
213C2×C3×C4C22×C3×D3112yes—
214C2×C3×C4C22×C3⋊D311yescharacteristic
215C2×C3×C4C22×C3⋊D313yes—
216C2×C3×C4C22×C3⋊D313yes—
217C2×Dic3—34no—
218C2×Dic3—34no—
219C2×Dic3—312no—
220D12—348no—
221C4×D3—348no—
222C4×D3—1224no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32C22×C4×D314yes—
227C2×C32#100714yes—
228C2×C32#100914yes—
229C2×C32C23×C3×C411yescharacteristic
230C2×C32C3×H(5,𝔽2)11yescharacteristic
231C2×C32C3×2−1+411yescharacteristic
232C3⋊D3—26no—
233C3×D3—624no—
234#41—98no—
235C2×Q8C2×C3×C3⋊D311yescharacteristic
236C2×D4—93no—
237C22×C4—93no—
238C22×C4—96no—
239C4⋊C4—93no—
240C4⋊C4—93no—
241C4⋊C4—91no—
242C4⋊C4C2×C3×C3⋊D313yes—
243C22⋊C4—93no—
244C22⋊C4—93no—
245C42—93no—
246C42C2×C3×C3⋊D313yes—
247C22×C3—24no—
248C22×C3—96no—
249C22×C3—183no—
250C22×C3—183no—
251C22×C3C23×C3×D314yes—
252C22×C3C23×C3⋊D311yescharacteristic
253C2×D3—324no—
254C2×D3—612no—
255C2×D3—612no—
256C3×C4—28no—
257C3×C4—24no—
258C3×C4—212no—
259C3×C4—21no—
260C3×C4—23no—
261C3×C4—224no—
262C3×C4—412no—
263C3×C4—92no—
264C3×C4—181no—
265C3×C4—183no—
266C3×C4C2×C3×C4×D3124yes—
267C3×C4C2×C4×C3⋊D316yes—
268Dic3—38no—
269Dic3—64no—
270Dic3—612no—
271C32—24no—
272C32#557414yescomplemented
273C32C3×#52011yescharacteristic, complemented
274C23—93no—
275Q8C3×C4×C3⋊D314yescomplemented
276D4—912no—
277C2×C4—91no—
278C2×C4—912no—
279C2×C4—91no—
280C2×C4—93no—
281C2×C4—366no—
282C2×C4C22×C3×C3⋊D311yescharacteristic
283C2×C4C22×C3×C3⋊D313yes—
284C2×C4C22×C3×C3⋊D313yes—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—186no—
289C2×C3C22×C3×C4×D314yes—
290C2×C3C3×#100714yes—
291C2×C3C3×#100914yes—
292C2×C3C22×C4×C3⋊D311yescharacteristic
293C2×C3#6436411yescharacteristic
294C2×C3#6436611yescharacteristic
295D3—624no—
296C22—96no—
297C22—183no—
298C22—183no—
299C22C23×C3×C3⋊D311yescharacteristic
300C4—21no—
301C4—23no—
302C4—92no—
303C4—181no—
304C4—183no—
305C4C2×C3×C4×C3⋊D316yes—
306C3—24no—
307C3C3×#557414yescomplemented
308C3#1059604511yescharacteristic, direct product factor
309C2—186no—
310C2C22×C3×C4×C3⋊D311yescharacteristic
311C2C3×#6436411yescharacteristic
312C2C3×#6436611yescharacteristic
subgroupG/H
centerC22×C3C23×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×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ 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×C4⋊C4×C3⋊D3C2semidirect
6. C3×C4×C4⋊(C3⋊D3)C2semidirect
7. C3×#64081C2semidirect
8. #10596045C3direct (N × H)
11. C3×#3832C4semidirect
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
29. #64102C2×C3semidirect
30. C4⋊C4×C3⋊D3C2×C3semidirect
31. C4×C4⋊(C3⋊D3)C2×C3semidirect
32. #64081C2×C3semidirect
39. C33×Q8C2×C4semidirect
46. C3×C4⋊(C3⋊D3)C2×C4semidirect
48. C3×C4×C3⋊D3C2×C4semidirect
54. #3832C3×C4semidirect
94. C3×C4×Q8C3×D3semidirect
95. C3×C4×Q8C3⋊D3semidirect
108. C32×Q8C4×D3semidirect
109. C32×Q8C2×C3×C4semidirect
122. C4⋊(C3⋊D3)C2×C3×C4semidirect
124. C4×C3⋊D3C2×C3×C4semidirect
191. C4×Q8C3×C3⋊D3semidirect
196. C33#520semidirect
200. C3×Q8C3×C4×D3semidirect
201. C3×Q8C4×C3⋊D3semidirect
272. C32#5574semidirect
273. C32C3×#520semidirect
275. Q8C3×C4×C3⋊D3semidirect
307. C3C3×#5574semidirect
308. C3#10596045direct (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×C4⋊C4×C3⋊D32
C3×C4×C4⋊(C3⋊D3)2
C3×#640812
#105960453
C3×#55743