Finite Group Databasebeta

← C3×C4×C4⋊#157 #13173205

Subgroups — C3×C4×C4⋊#157 #13173205

Order 1728.

664 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊#157C212yesmaximal
2C2×C3×C4×#157C212yesmaximal
3C3×C42×C3⋊D3C211yesmaximal, characteristic
4C3×#63781C212yesmaximal
5C4×C4⋊#157C311yesmaximal, characteristic, direct product factor
6C22×C3×#157C2214yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C3×C4⋊#157C418yesdirect product factor
11C3×C4×#157—24no—
12C33×C42C411yescharacteristic, complemented
13C3×C4×C3⋊Dic3C411yescharacteristic, complemented
14C3×C4×C3⋊Dic3C411yescharacteristic, complemented
15C3×C4×C3⋊Dic3C411yescharacteristic, complemented
16C2×C4⋊#157C2×C312yes—
17C2×C4×#157C2×C312yes—
18C42×C3⋊D3C2×C311yescharacteristic
19C3×C42×D3—62no—
20#63781C2×C312yes—
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3×#157—24no—
23C2×C3×#157—24no—
24C2×C3×#157C2×C418yes—
25C2×C33×C4C2×C411yescharacteristic
26C2×C33×C4C2×C411yescharacteristic
27C2×C33×C4C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2×C411yescharacteristic
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3—22no—
32C3×C4×C3⋊D3C2×C414yes—
33C3×C4×C3⋊D3D412yes—
34C3×C4×C3⋊D3Q812yes—
35C3×C4×C4⋊C4—91nomaximal
36C22×#157C22×C314yes—
37C2×C4×C3⋊D3C22×C311yescharacteristic
38C2×C4×C3⋊D3C22×C311yescharacteristic
39C2×C4×C3⋊D3C22×C311yescharacteristic
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—62no—
43C4⋊#157C3×C418yesdirect product factor
44C4×#157—24no—
45C32×C42—22no—
46C32×C42—42no—
47C32×C42C3×C411yescharacteristic, complemented
48C4×C3⋊Dic3C3×C411yescharacteristic, complemented
49C4×C3⋊Dic3C3×C411yescharacteristic, complemented
50C4×C3⋊Dic3C3×C411yescharacteristic, complemented
51C3×C4×Dic3—62no—
52C3×C4×Dic3—62no—
53C3×C4×Dic3—62no—
54C22×C33C22×C411yescharacteristic
55C2×C3×C3⋊D3C22×C412yes—
56C2×C3×C3⋊D3C2×D411yescharacteristic
57C2×C3×C3⋊D3C2×Q811yescharacteristic
58C2×C3×C3⋊D3#4112yes—
59C3×#157—28no—
60C33×C4—21no—
61C33×C4C4212yescomplemented
62C33×C4C4⋊C412yescomplemented
63C3×C3⋊Dic3—21no—
64C3×C3⋊Dic3C4212yescomplemented
65C3×C3⋊Dic3C4⋊C412yescomplemented
66C2×C3×C4⋊C4—92no—
67C2×C3×C42—91no—
68C2×C3×C42—92no—
69C42×D3—62no—
70C3×#95—92no—
71C22×C3⋊D3C23×C311yescharacteristic
72C22×C3×D3—62no—
73C2×#157—24no—
74C2×#157—24no—
75C2×#157C2×C3×C418yes—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—42no—
80C2×C32×C4—42no—
81C2×C32×C4—42no—
82C2×C32×C4C2×C3×C411yescharacteristic
83C2×C32×C4C2×C3×C411yescharacteristic
84C2×C32×C4C2×C3×C411yescharacteristic
85C2×C3⋊Dic3C2×C3×C411yescharacteristic
86C2×C3⋊Dic3C2×C3×C411yescharacteristic
87C2×C3⋊Dic3C2×C3×C411yescharacteristic
88C4×C3⋊D3—22no—
89C4×C3⋊D3C2×C3×C414yes—
90C4×C3⋊D3C3×D412yes—
91C4×C3⋊D3C3×Q812yes—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×C4×D3—68no—
96C3×C4×D3—64no—
97C3×C4×D3—64no—
98C3×C4×D3—68no—
99C4×C4⋊C4—91noSylow
100C2×C33C2×C4211yescharacteristic
101C2×C33C2×C4⋊C411yescharacteristic
102C2×C33#11711yescharacteristic
103C3×C3⋊D3C4×D412yes—
104C3×C3⋊D3C4×Q812yes—
105C22×C3×C4—91no—
106C22×C3×C4—91no—
107C22×C3×C4—94no—
108C22×C3×C4—91no—
109C2×C4×D3—62no—
110C2×C4×D3—62no—
111C2×C4×D3—62no—
112C3×C4⋊C4—98no—
113C3×C42—22no—
114C3×C42—42no—
115C3×C42—91no—
116C3×C42—91no—
117C3×C42—91no—
118C3×C42—184no—
119C3×C42#15711yescharacteristic, complemented
120C4×Dic3—62no—
121C4×Dic3—62no—
122C4×Dic3—62no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C22×C3×C411yescharacteristic
126C2×C3⋊D3C22×C3×C412yes—
127C2×C3⋊D3C2×C3×D411yescharacteristic
128C2×C3⋊D3C2×C3×Q811yescharacteristic
129C2×C3⋊D3C3×#4112yes—
130C2×C3×D3—64no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—64no—
134#157—28no—
135C32×C4—21no—
136C32×C4—24no—
137C32×C4—24no—
138C32×C4—24no—
139C32×C4—44no—
140C32×C4—44no—
141C32×C4—44no—
142C32×C4C3×C4212yescomplemented
143C32×C4C3×C4⋊C412yescomplemented
144C3⋊Dic3—21no—
145C3⋊Dic3C3×C4212yescomplemented
146C3⋊Dic3C3×C4⋊C412yescomplemented
147C3×Dic3—64no—
148C3×Dic3—64no—
149C3×Dic3—64no—
150C2×C4⋊C4—92no—
151C2×C42—91no—
152C2×C42—92no—
153#95—92no—
154C33C4×C4⋊C411yescharacteristic, complemented, Sylow
155C23×C3—91no—
156C22×D3—62no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—94no—
166C2×C3×C4—92no—
167C2×C3×C4—92no—
168C2×C3×C4—98no—
169C2×C3×C4—91no—
170C2×C3×C4—182no—
171C2×C3×C4—184no—
172C2×C3×C4—184no—
173C2×C3×C4C2×#15711yescharacteristic
174C2×C3×C4C2×#15711yescharacteristic
175C2×C3×C4C2×#15711yescharacteristic
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—62no—
179C4×D3—68no—
180C4×D3—64no—
181C4×D3—64no—
182C4×D3—68no—
183C2×C32—22no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32C2×C3×C4211yescharacteristic
190C2×C32C2×C3×C4⋊C411yescharacteristic
191C2×C32C3×#11711yescharacteristic
192C3⋊D3C3×C4×D412yes—
193C3⋊D3C3×C4×Q812yes—
194C3×D3—64no—
195C3×D3—64no—
196C22×C4—91no—
197C22×C4—91no—
198C22×C4—94no—
199C22×C4—91no—
200C4⋊C4—98no—
201C42—91no—
202C42—91no—
203C42—91no—
204C42—184no—
205C42C3×#15711yescharacteristic, complemented
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—92no—
209C22×C3—91no—
210C22×C3—91no—
211C22×C3—92no—
212C22×C3C22×#15711yescharacteristic
213C2×D3—64no—
214C2×D3—62no—
215C2×D3—62no—
216C2×D3—64no—
217C3×C4—24no—
218C3×C4—21no—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—44no—
222C3×C4—44no—
223C3×C4—44no—
224C3×C4—92no—
225C3×C4—92no—
226C3×C4—188no—
227C3×C4—181no—
228C3×C4C4×#15712yescomplemented
229C3×C4C4⋊#15712yesdirect product factor
230Dic3—64no—
231Dic3—64no—
232Dic3—64no—
233C32—22no—
234C32—42no—
235C32C3×C4×C4⋊C411yescharacteristic, complemented
236C23—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—94no—
240C2×C4—92no—
241C2×C4—92no—
242C2×C4—98no—
243C2×C4—91no—
244C2×C4—182no—
245C2×C4—184no—
246C2×C4—184no—
247C2×C4C2×C3×#15711yescharacteristic
248C2×C4C2×C3×#15711yescharacteristic
249C2×C4C2×C3×#15711yescharacteristic
250C2×C3—22no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—42no—
254C2×C3—42no—
255C2×C3—42no—
256C2×C3—92no—
257C2×C3—92no—
258C2×C3C2×C4×#15711yescharacteristic
259C2×C3C2×C4⋊#15711yescharacteristic
260C2×C3#6428611yescharacteristic
261D3—64no—
262D3—64no—
263C22—92no—
264C22—91no—
265C22—91no—
266C22—92no—
267C22C22×C3×#15711yescharacteristic
268C4—21no—
269C4—92no—
270C4—92no—
271C4—188no—
272C4—181no—
273C4C3×C4×#15712yescomplemented
274C4C3×C4⋊#15712yesdirect product factor
275C3—22no—
276C3—42no—
277C3C4×C4⋊#15711yescharacteristic, direct product factor
278C2—92no—
279C2—92no—
280C2C2×C3×C4×#15711yescharacteristic
281C2C2×C3×C4⋊#15711yescharacteristic
282C2C3×#6428611yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×C32C2×C3×C42
FrattiniC22C22×C3×#157
FittingC33×C42C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC3×C42#157
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
5. C4×C4⋊#157C3direct (N × H)
10. C3×C4⋊#157C4direct (N × H)
12. C33×C42C4semidirect
13. C3×C4×C3⋊Dic3C4semidirect
14. C3×C4×C3⋊Dic3C4semidirect
15. C3×C4×C3⋊Dic3C4semidirect
43. C4⋊#157C3×C4direct (N × H)
47. C32×C42C3×C4semidirect
48. C4×C3⋊Dic3C3×C4semidirect
49. C4×C3⋊Dic3C3×C4semidirect
50. C4×C3⋊Dic3C3×C4semidirect
61. C33×C4C42semidirect
62. C33×C4C4⋊C4semidirect
64. C3×C3⋊Dic3C42semidirect
65. C3×C3⋊Dic3C4⋊C4semidirect
119. C3×C42#157semidirect
142. C32×C4C3×C42semidirect
143. C32×C4C3×C4⋊C4semidirect
145. C3⋊Dic3C3×C42semidirect
146. C3⋊Dic3C3×C4⋊C4semidirect
154. C33C4×C4⋊C4semidirect
205. C42C3×#157semidirect
228. C3×C4C4×#157semidirect
229. C3×C4C4⋊#157direct (N × H)
235. C32C3×C4×C4⋊C4semidirect
273. C4C3×C4×#157semidirect
274. C4C3×C4⋊#157direct (N × H)
277. C3C4×C4⋊#157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C3×C42×C3⋊D32
C3×#637812
C4×C4⋊#1573
C3×C4×C4⋊C49