Finite Group Databasebeta

← C3×#10596043 #13179323

Subgroups — C3×#10596043 #13179323

Order 1728.

1876 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C33×C4×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#64101C213yesmaximal
5C3×#64096C213yesmaximal, complemented
6C3×#64081C213yesmaximal
7C3×C4×#634C213yesmaximal, complemented
8#10596043C311yesmaximal, characteristic, direct product factor
9C3×#5572—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3C418yescomplemented
12C2×C3×C4×C3⋊D3C2213yes—
13C2×C3×#634C2213yes—
14C33×C4⋊C4C2213yes—
15C33×C42C2213yes—
16C3×#3752C2211yescharacteristic
17C3×#3752C2213yes—
18C3×C4⋊(C3⋊Dic3)C2213yes—
19C3×#3750C2213yes—
20C3×#3750C2213yes—
21C3×C4×C3⋊Dic3C2216yes—
22C3×C4×C3⋊Dic3C2213yes—
23C2×Q8×C3⋊D3C2×C311yescharacteristic
24C2×C3×D3×Q8—34no—
25C32×C4×Q8—24no—
26C32×C4×Q8D314yescomplemented
27C32×C4×Q8C2×C311yescharacteristic, complemented
28Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
29#64101C2×C313yes—
30#64096C2×C313yescomplemented
31#64081C2×C313yes—
32C4×#634C2×C313yescomplemented
33C3×Q8×Dic3—34no—
34C3×#892—312no—
35C3×#887—312no—
36C3×#872—312no—
37C3×C4×Dic6—312no—
38C22×C3×C3⋊D3C2311yescharacteristic
39C33×Q8C2×C414yes—
40C2×C33×C4C2313yes—
41C2×C33×C4C2311yescharacteristic
42C2×C33×C4C2313yes—
43C2×C3×C3⋊Dic3C2311yescharacteristic
44C2×C3×C3⋊Dic3C2313yes—
45C2×C3×C3⋊Dic3C2313yes—
46C3×C4×C3⋊D3C2×C4112yes—
47C3×#634C2×C4112yes—
48C3×#519—91no—
49#5572—34no—
50C2×C32×Q8—24no—
51C2×C32×Q8C2×D314yes—
52C2×C32×Q8C22×C311yescharacteristic
53Q8×C3⋊D3C3×C418yescomplemented
54C2×C4×C3⋊D3C22×C313yes—
55C2×#634C22×C313yes—
56C3×D3×Q8—332no—
57C2×C3×C4×D3—312no—
58C2×C3×Dic6—312no—
59C32×C4⋊C4—212no—
60C32×C4⋊C4C2×D3112yes—
61C32×C4⋊C4C22×C313yes—
62C32×C42—212no—
63C32×C42C2×D3112yes—
64C32×C42C22×C313yes—
65#3752C22×C311yescharacteristic
66#3752C22×C313yes—
67C4⋊(C3⋊Dic3)C22×C313yes—
68#3750C22×C313yes—
69#3750C22×C313yes—
70C4×C3⋊Dic3C22×C316yes—
71C4×C3⋊Dic3C22×C313yes—
72C3×(C2×D3)⋊C4—312no—
73C3×(C2×D3)⋊C4—34no—
74C3×C4⋊Dic3—312no—
75C3×Dic3⋊C4—312no—
76C3×Dic3⋊C4—312no—
77C3×C4×Dic3—324no—
78C3×C4×Dic3—312no—
79C22×C33C2411yescharacteristic
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—21no—
82C2×C3×C3⋊D3C22×C412yes—
83C33×C4—21no—
84C33×C4—23no—
85C33×C4C22×C416yes—
86C3×C3⋊Dic3—23no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3C22×C416yes—
89C22×C3×Q8—91no—
90C2×D3×Q8—34no—
91C3×C4×Q8—24no—
92C3×C4×Q8—91no—
93C3×C4×Q8—93no—
94C3×C4×Q8—93no—
95C3×C4×Q8C3×D314yescomplemented
96C3×C4×Q8C3⋊D311yescharacteristic, complemented
97C3×#117—93no—
98C3×#117—93no—
99Q8×Dic3—34no—
100#892—312no—
101#887—312no—
102#872—312no—
103C4×Dic6—312no—
104C22×C3⋊D3C23×C311yescharacteristic
105C22×C3×D3—34no—
106C32×Q8—216no—
107C32×Q8C4×D3116yes—
108C32×Q8C2×C3×C414yes—
109C2×C32×C4—24no—
110C2×C32×C4—212no—
111C2×C32×C4—212no—
112C2×C32×C4C22×D3112yes—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D3112yes—
115C2×C32×C4C23×C313yes—
116C2×C32×C4C23×C311yescharacteristic
117C2×C32×C4C23×C313yes—
118C2×C3⋊Dic3C23×C311yescharacteristic
119C2×C3⋊Dic3C23×C313yes—
120C2×C3⋊Dic3C23×C313yes—
121C4×C3⋊D3C2×C3×C4112yes—
122#634C2×C3×C4112yes—
123C2×C3×Dic3—312no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—312no—
126C3×C4×D3—348no—
127C3×Dic6—348no—
128#519—91noSylow
129C2×C33C23×C411yescharacteristic
130C2×C332−1+411yescharacteristic
131C2×C332−1+411yescharacteristic
132C3×C3⋊D3—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—98no—
135C2×C3×Q8—93no—
136C2×C3×Q8C2×C3×D314yes—
137C2×C3×Q8C2×C3⋊D311yescharacteristic
138C22×C3×C4—93no—
139D3×Q8—332no—
140C2×C4×D3—312no—
141C2×Dic6—312no—
142C3×C4⋊C4—212no—
143C3×C4⋊C4—93no—
144C3×C4⋊C4—93no—
145C3×C4⋊C4—93no—
146C3×C4⋊C4C2×C3×D3112yes—
147C3×C4⋊C4C2×C3⋊D313yes—
148C3×C22⋊C4—93no—
149C3×C22⋊C4—91no—
150C3×C42—212no—
151C3×C42—96no—
152C3×C42—93no—
153C3×C42C2×C3×D3112yes—
154C3×C42C2×C3⋊D313yes—
155(C2×D3)⋊C4—34no—
156(C2×D3)⋊C4—312no—
157C4⋊Dic3—312no—
158Dic3⋊C4—312no—
159Dic3⋊C4—312no—
160C4×Dic3—324no—
161C4×Dic3—312no—
162C22×C32—24no—
163C22×C32C23×D314yes—
164C22×C32C24×C311yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3C22×C3×C412yes—
168C2×C3×D3—38no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C32×C4—28no—
172C32×C4—24no—
173C32×C4—21no—
174C32×C4—23no—
175C32×C4—224no—
176C32×C4—212no—
177C32×C4—412no—
178C32×C4C2×C4×D3124yes—
179C32×C4C22×C3×C416yes—
180C3⋊Dic3—23no—
181C3⋊Dic3—21no—
182C3⋊Dic3C22×C3×C416yes—
183C3×Dic3—324no—
184C3×Dic3—612no—
185C3×Dic3—64no—
186C22×Q8—91no—
187C4×Q8—91no—
188C4×Q8—93no—
189C4×Q8—93no—
190C4×Q8C3×C3⋊D311yescharacteristic, complemented
191#117—93no—
192#117—93no—
193C33#51911yescharacteristic, complemented, Sylow
194C23×C3—91no—
195C22×D3—34no—
196C3×Q8—216no—
197C3×Q8—912no—
198C3×Q8C3×C4×D3116yes—
199C3×Q8C4×C3⋊D314yes—
200C2×C3×C4—24no—
201C2×C3×C4—212no—
202C2×C3×C4—212no—
203C2×C3×C4—93no—
204C2×C3×C4—912no—
205C2×C3×C4—93no—
206C2×C3×C4—91no—
207C2×C3×C4C22×C3×D3112yes—
208C2×C3×C4C22×C3×D314yes—
209C2×C3×C4C22×C3×D3112yes—
210C2×C3×C4C22×C3⋊D313yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×C3×C4C22×C3⋊D313yes—
213C2×Dic3—312no—
214C2×Dic3—34no—
215C2×Dic3—312no—
216C4×D3—348no—
217Dic6—348no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32C22×C4×D314yes—
222C2×C32#100714yes—
223C2×C32#101014yes—
224C2×C32C23×C3×C411yescharacteristic
225C2×C32C3×2−1+411yescharacteristic
226C2×C32C3×2−1+411yescharacteristic
227C3⋊D3—22no—
228C3×D3—68no—
229C2×Q8—98no—
230C2×Q8—93no—
231C2×Q8C2×C3×C3⋊D311yescharacteristic
232C22×C4—93no—
233C4⋊C4—93no—
234C4⋊C4—93no—
235C4⋊C4—93no—
236C4⋊C4C2×C3×C3⋊D313yes—
237C22⋊C4—93no—
238C22⋊C4—91no—
239C42—96no—
240C42—93no—
241C42C2×C3×C3⋊D313yes—
242C22×C3—24no—
243C22×C3—92no—
244C22×C3—181no—
245C22×C3—181no—
246C22×C3C23×C3×D314yes—
247C22×C3C23×C3⋊D311yescharacteristic
248C2×D3—38no—
249C2×D3—64no—
250C2×D3—64no—
251C3×C4—28no—
252C3×C4—24no—
253C3×C4—212no—
254C3×C4—21no—
255C3×C4—23no—
256C3×C4—224no—
257C3×C4—412no—
258C3×C4—96no—
259C3×C4—183no—
260C3×C4—181no—
261C3×C4C2×C3×C4×D3124yes—
262C3×C4C2×C4×C3⋊D316yes—
263Dic3—324no—
264Dic3—612no—
265Dic3—64no—
266C32—24no—
267C32#557214yescomplemented
268C32C3×#51911yescharacteristic, complemented
269C23—91no—
270Q8—912no—
271Q8C3×C4×C3⋊D314yes—
272C2×C4—93no—
273C2×C4—912no—
274C2×C4—93no—
275C2×C4—91no—
276C2×C4C22×C3×C3⋊D313yes—
277C2×C4C22×C3×C3⋊D311yescharacteristic
278C2×C4C22×C3×C3⋊D313yes—
279C2×C3—24no—
280C2×C3—24no—
281C2×C3—24no—
282C2×C3—182no—
283C2×C3C22×C3×C4×D314yes—
284C2×C3C3×#100714yes—
285C2×C3C3×#101014yes—
286C2×C3C22×C4×C3⋊D311yescharacteristic
287C2×C3#6436411yescharacteristic
288C2×C3#6436711yescharacteristic
289D3—68no—
290C22—92no—
291C22—181no—
292C22—181no—
293C22C23×C3×C3⋊D311yescharacteristic
294C4—21no—
295C4—23no—
296C4—96no—
297C4—183no—
298C4—181no—
299C4C2×C3×C4×C3⋊D316yes—
300C3—24no—
301C3C3×#557214yescomplemented
302C3#1059604311yescharacteristic, direct product factor
303C2—182no—
304C2C22×C3×C4×C3⋊D311yescharacteristic
305C2C3×#6436411yescharacteristic
306C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutatorC2×C32C23×C3×C4
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×Q8C2
socleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C32C24×C3
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#519
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
5. C3×#64096C2semidirect
7. C3×C4×#634C2semidirect
8. #10596043C3direct (N × H)
11. C3×Q8×C3⋊D3C4semidirect
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
30. #64096C2×C3semidirect
32. C4×#634C2×C3semidirect
53. Q8×C3⋊D3C3×C4semidirect
95. C3×C4×Q8C3×D3semidirect
96. C3×C4×Q8C3⋊D3semidirect
190. C4×Q8C3×C3⋊D3semidirect
193. C33#519semidirect
267. C32#5572semidirect
268. C32C3×#519semidirect
301. C3C3×#5572semidirect
302. C3#10596043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C33×C4×Q82
C3×Q8×C3⋊Dic32
C3×#641012
C3×#640962
C3×#640812
C3×C4×#6342
#105960433
C3×#55723