Finite Group Databasebeta

← C3×#10596040 #13179320

Subgroups — C3×#10596040 #13179320

Order 1728.

1480 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
3C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
4C3×#64099C213yesmaximal
5C3×#64097C213yesmaximal
6C3×#64079C213yesmaximal
7C3×C4×#634C213yesmaximal
8#10596040C311yesmaximal, characteristic, direct product factor
9C3×#5569—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2213yes—
12C33×C4⋊C4C2213yes—
13C33×C42C2213yes—
14C3×C4⋊(C3⋊Dic3)C2213yes—
15C3×C4⋊(C3⋊Dic3)C2213yes—
16C3×C4⋊(C3⋊Dic3)C2213yes—
17C3×#3750C2211yescharacteristic
18C3×#3750C2213yes—
19C3×#3750C2216yes—
20C3×C4×C3⋊Dic3C2213yes—
21C3×C4×C3⋊Dic3C2213yes—
22C32×C4×Q8—24no—
23C32×C4×Q8D314yes—
24C32×C4×Q8C2×C311yescharacteristic
25Q8×C3⋊Dic3C2×C311yescharacteristic
26Q8×C3⋊Dic3C2×C311yescharacteristic
27#64099C2×C313yes—
28#64097C2×C313yes—
29#64079C2×C313yes—
30C4×#634C2×C313yes—
31C3×Q8×Dic3—34no—
32C3×Q8×Dic3—34no—
33C3×#890—312no—
34C3×#888—312no—
35C3×#870—312no—
36C3×C4×Dic6—312no—
37C33×Q8Q814yescomplemented
38C2×C33×C4C2311yescharacteristic
39C2×C33×C4C2313yes—
40C2×C33×C4C2313yes—
41C2×C3×C3⋊Dic3C2311yescharacteristic
42C2×C3×C3⋊Dic3C2313yes—
43C2×C3×C3⋊Dic3C2313yes—
44C2×C3×C3⋊Dic3C2311yescharacteristic
45C3×#634—26no—
46C3×#557—91no—
47#5569—34no—
48C2×C32×Q8—24no—
49C2×C32×Q8C2×D314yes—
50C2×C32×Q8C22×C311yescharacteristic
51C2×#634C22×C313yes—
52C2×C3×Dic6—312no—
53C32×C4⋊C4—212no—
54C32×C4⋊C4C2×D3112yes—
55C32×C4⋊C4C22×C313yes—
56C32×C42—212no—
57C32×C42C2×D3112yes—
58C32×C42C22×C313yes—
59C4⋊(C3⋊Dic3)C22×C313yes—
60C4⋊(C3⋊Dic3)C22×C313yes—
61C4⋊(C3⋊Dic3)C22×C313yes—
62#3750C22×C311yescharacteristic
63#3750C22×C313yes—
64#3750C22×C316yes—
65C4×C3⋊Dic3C22×C313yes—
66C4×C3⋊Dic3C22×C313yes—
67C3×C4⋊Dic3—312no—
68C3×C4⋊Dic3—312no—
69C3×C4⋊Dic3—312no—
70C3×Dic3⋊C4—324no—
71C3×Dic3⋊C4—34no—
72C3×Dic3⋊C4—312no—
73C3×C4×Dic3—312no—
74C3×C4×Dic3—312no—
75C22×C33C2411yescharacteristic
76C33×C4—23no—
77C33×C4—21no—
78C33×C4C2×Q816yes—
79C3×C3⋊Dic3—23no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—23no—
82C3×C3⋊Dic3#4112yes—
83C3×C4⋊Q8—93no—
84C3×#125—93no—
85C3×#125—93no—
86C3×C4×Q8—24no—
87C3×C4×Q8—91no—
88C3×C4×Q8—91no—
89C3×C4×Q8—93no—
90C3×C4×Q8C3×D314yes—
91C3×C4×Q8C3⋊D311yescharacteristic
92Q8×Dic3—34no—
93Q8×Dic3—34no—
94#890—312no—
95#888—312no—
96#870—312no—
97C4×Dic6—312no—
98C32×Q8—216no—
99C32×Q8Dic6116yescomplemented
100C32×Q8C3×Q814yescomplemented
101C2×C32×C4—24no—
102C2×C32×C4—212no—
103C2×C32×C4—212no—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D3112yes—
106C2×C32×C4C22×D3112yes—
107C2×C32×C4C23×C311yescharacteristic
108C2×C32×C4C23×C313yes—
109C2×C32×C4C23×C313yes—
110C2×C3⋊Dic3C23×C311yescharacteristic
111C2×C3⋊Dic3C23×C313yes—
112C2×C3⋊Dic3C23×C313yes—
113C2×C3⋊Dic3C23×C311yescharacteristic
114#634—26no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—312no—
118C2×C3×Dic3—312no—
119C3×Dic6—624no—
120#557—91noSylow
121C2×C33C22×Q811yescharacteristic
122C2×C33C2×#4111yescharacteristic
123C2×C332−1+411yescharacteristic
124C2×C3×Q8—24no—
125C2×C3×Q8—93no—
126C2×C3×Q8C2×C3×D314yes—
127C2×C3×Q8C2×C3⋊D311yescharacteristic
128C2×Dic6—312no—
129C3×C4⋊C4—212no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—93no—
132C3×C4⋊C4—96no—
133C3×C4⋊C4—93no—
134C3×C4⋊C4—93no—
135C3×C4⋊C4—93no—
136C3×C4⋊C4C2×C3×D3112yes—
137C3×C4⋊C4C2×C3⋊D313yes—
138C3×C42—212no—
139C3×C42—93no—
140C3×C42—93no—
141C3×C42C2×C3×D3112yes—
142C3×C42C2×C3⋊D313yes—
143C4⋊Dic3—312no—
144C4⋊Dic3—312no—
145C4⋊Dic3—312no—
146Dic3⋊C4—34no—
147Dic3⋊C4—312no—
148Dic3⋊C4—324no—
149C4×Dic3—312no—
150C4×Dic3—312no—
151C22×C32—24no—
152C22×C32C23×D314yes—
153C22×C32C24×C311yescharacteristic
154C32×C4—28no—
155C32×C4—24no—
156C32×C4—23no—
157C32×C4—21no—
158C32×C4—224no—
159C32×C4—212no—
160C32×C4—412no—
161C32×C4C2×Dic6124yes—
162C32×C4C2×C3×Q816yes—
163C3⋊Dic3—23no—
164C3⋊Dic3—21no—
165C3⋊Dic3—23no—
166C3⋊Dic3C3×#4112yes—
167C3×Dic3—38no—
168C3×Dic3—64no—
169C3×Dic3—612no—
170C3×Dic3—612no—
171C4⋊Q8—93no—
172#125—93no—
173#125—93no—
174C4×Q8—91no—
175C4×Q8—91no—
176C4×Q8—93no—
177C4×Q8C3×C3⋊D311yescharacteristic
178C33#55711yescharacteristic, complemented, Sylow
179C3×Q8—216no—
180C3×Q8—186no—
181C3×Q8C3×Dic6116yescomplemented
182C3×Q8#63414yescomplemented
183C2×C3×C4—24no—
184C2×C3×C4—212no—
185C2×C3×C4—212no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—93no—
189C2×C3×C4—93no—
190C2×C3×C4C22×C3×D314yes—
191C2×C3×C4C22×C3×D3112yes—
192C2×C3×C4C22×C3×D3112yes—
193C2×C3×C4C22×C3⋊D311yescharacteristic
194C2×C3×C4C22×C3⋊D313yes—
195C2×C3×C4C22×C3⋊D313yes—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—312no—
199C2×Dic3—312no—
200Dic6—624no—
201C2×C32—24no—
202C2×C32—24no—
203C2×C32—24no—
204C2×C32C22×Dic614yes—
205C2×C32#100714yes—
206C2×C32D3×#4114yes—
207C2×C32C22×C3×Q811yescharacteristic
208C2×C32C2×C3×#4111yescharacteristic
209C2×C32C3×2−1+411yescharacteristic
210C2×Q8—93no—
211C2×Q8C2×C3×C3⋊D311yescharacteristic
212C4⋊C4—91no—
213C4⋊C4—93no—
214C4⋊C4—93no—
215C4⋊C4—93no—
216C4⋊C4—93no—
217C4⋊C4—96no—
218C4⋊C4C2×C3×C3⋊D313yes—
219C42—93no—
220C42—93no—
221C42C2×C3×C3⋊D313yes—
222C22×C3—24no—
223C22×C3C23×C3×D314yes—
224C22×C3C23×C3⋊D311yescharacteristic
225C3×C4—28no—
226C3×C4—24no—
227C3×C4—212no—
228C3×C4—21no—
229C3×C4—23no—
230C3×C4—224no—
231C3×C4—412no—
232C3×C4—92no—
233C3×C4—181no—
234C3×C4—183no—
235C3×C4—183no—
236C3×C4C2×C3×Dic6124yes—
237C3×C4C2×#63416yes—
238Dic3—38no—
239Dic3—64no—
240Dic3—612no—
241Dic3—612no—
242C32—24no—
243C32#556914yescomplemented
244C32C3×#55711yescharacteristic, complemented
245Q8—186no—
246Q8C3×#63414yescomplemented
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—93no—
250C2×C4—93no—
251C2×C4C22×C3×C3⋊D311yescharacteristic
252C2×C4C22×C3×C3⋊D313yes—
253C2×C4C22×C3×C3⋊D313yes—
254C2×C3—24no—
255C2×C3—24no—
256C2×C3—24no—
257C2×C3C22×C3×Dic614yes—
258C2×C3C3×#100714yes—
259C2×C3C3×D3×#4114yes—
260C2×C3C22×#63411yescharacteristic
261C2×C3#6436411yescharacteristic
262C2×C3#41×C3⋊D311yescharacteristic
263C22C23×C3×C3⋊D311yescharacteristic
264C4—21no—
265C4—23no—
266C4—92no—
267C4—181no—
268C4—183no—
269C4—183no—
270C4C2×C3×#63416yes—
271C3—24no—
272C3C3×#556914yescomplemented
273C3#1059604011yescharacteristic, direct product factor
274C2C22×C3×#63411yescharacteristic
275C2C3×#6436411yescharacteristic
276C2C3×#41×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×Q8C2
socle, prime radicalC22×C33C24
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#557
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
8. #10596040C3direct (N × H)
37. C33×Q8Q8semidirect
99. C32×Q8Dic6semidirect
100. C32×Q8C3×Q8semidirect
178. C33#557semidirect
181. C3×Q8C3×Dic6semidirect
182. C3×Q8#634semidirect
243. C32#5569semidirect
244. C32C3×#557semidirect
246. Q8C3×#634semidirect
272. C3C3×#5569semidirect
273. C3#10596040direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×Q82
C3×Q8×C3⋊Dic32
C3×#640992
C3×#640972
C3×#640792
C3×C4×#6342
#105960403
C3×#55693