Finite Group Databasebeta

← C4×C4⋊(C3⋊#157) #13173331

Subgroups — C4×C4⋊(C3⋊#157) #13173331

Order 1728.

664 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊#157)C212yesmaximal
2C2×C4×C3⋊#157C212yesmaximal
3C3×C42×C3⋊D3C211yesmaximal, characteristic
4#11723106C212yesmaximal
5C4×C4⋊#157—31nomaximal
6C22×C3⋊#157C2214yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C4⋊(C3⋊#157)C418yesdirect product factor
11C4×C3⋊#157—24no—
12C33×C42C411yescharacteristic, complemented
13C3×C4×C3⋊Dic3C411yescharacteristic, complemented
14C3×C4×C3⋊Dic3C411yescharacteristic, complemented
15C3×C4×C3⋊Dic3C411yescharacteristic, complemented
16C2×C4⋊#157—32no—
17C2×C4×#157—32no—
18C42×C3⋊D3D311yescharacteristic
19C3×C42×D3—62no—
20#63781—32no—
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—
35C4×C4⋊Dic3—91nomaximal
36C22×#157—34no—
37C2×C4×C3⋊D3C2×D311yescharacteristic
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—62no—
43C4⋊#157—38no—
44C4×#157—64no—
45C32×C42—22no—
46C32×C42—42no—
47C32×C42Dic311yescharacteristic, complemented
48C4×C3⋊Dic3Dic311yescharacteristic, complemented
49C4×C3⋊Dic3Dic311yescharacteristic, complemented
50C4×C3⋊Dic3Dic311yescharacteristic, 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×C42—91no—
67C2×C4⋊Dic3—92no—
68C2×C4×Dic3—92no—
69C42×D3—62no—
70C3⋊#95—92no—
71C22×C3⋊D3C22×D311yescharacteristic
72C22×C3×D3—62no—
73C2×#157—38no—
74C2×#157—64no—
75C2×#157—64no—
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×Dic311yescharacteristic
83C2×C32×C4C2×Dic311yescharacteristic
84C2×C32×C4C2×Dic311yescharacteristic
85C2×C3⋊Dic3C2×Dic311yescharacteristic
86C2×C3⋊Dic3C2×Dic311yescharacteristic
87C2×C3⋊Dic3C2×Dic311yescharacteristic
88C4×C3⋊D3—22no—
89C4×C3⋊D3Dic612yes—
90C4×C3⋊D3C4×D314yes—
91C4×C3⋊D3D1212yes—
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—271noSylow
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—91no—
108C22×Dic3—94no—
109C2×C4×D3—62no—
110C2×C4×D3—62no—
111C2×C4×D3—62no—
112C3×C42—22no—
113C3×C42—42no—
114C3×C42—91no—
115C3×C42—91no—
116C3×C42—91no—
117C3×C42#15711yescharacteristic, complemented
118C4⋊Dic3—98no—
119C4×Dic3—62no—
120C4×Dic3—62no—
121C4×Dic3—62no—
122C4×Dic3—184no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C22×Dic311yescharacteristic
126C2×C3⋊D3C2×Dic611yescharacteristic
127C2×C3⋊D3C2×C4×D312yes—
128C2×C3⋊D3C2×D1211yescharacteristic
129C2×C3⋊D3#23512yes—
130C2×C3×D3—64no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—64no—
134#157—68no—
135C32×C4—21no—
136C32×C4—24no—
137C32×C4—24no—
138C32×C4—24no—
139C32×C4—44no—
140C32×C4—44no—
141C32×C4—44no—
142C32×C4C4×Dic312yescomplemented
143C32×C4C4⋊Dic312yescomplemented
144C3⋊Dic3—21no—
145C3⋊Dic3C4×Dic312yescomplemented
146C3⋊Dic3C4⋊Dic312yescomplemented
147C3×Dic3—64no—
148C3×Dic3—64no—
149C3×Dic3—64no—
150C2×C4⋊C4—272no—
151C2×C42—91no—
152C2×C42—272no—
153#95—272no—
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—91no—
169C2×C3×C4—182no—
170C2×C3×C4C2×#15711yescharacteristic
171C2×C3×C4C2×#15711yescharacteristic
172C2×C3×C4C2×#15711yescharacteristic
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—98no—
177C2×Dic3—184no—
178C2×Dic3—184no—
179C4×D3—64no—
180C4×D3—68no—
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×C4×Dic311yescharacteristic
190C2×C32C2×C4⋊Dic311yescharacteristic
191C2×C32#92611yescharacteristic
192C3⋊D3C4×Dic612yes—
193C3⋊D3C4×D1212yes—
194C3×D3—64no—
195C3×D3—64no—
196C22×C4—91no—
197C22×C4—91no—
198C22×C4—91no—
199C22×C4—274no—
200C4⋊C4—278no—
201C42—91no—
202C42—91no—
203C42—91no—
204C42—544no—
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—62no—
214C2×D3—64no—
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—181no—
227C3×C4C4×#15712yescomplemented
228C3×C4C4⋊#15712yescomplemented
229Dic3—64no—
230Dic3—64no—
231Dic3—64no—
232Dic3—188no—
233C32—22no—
234C32—42no—
235C32C4×C4⋊Dic311yescharacteristic, complemented
236C23—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—94no—
240C2×C4—92no—
241C2×C4—92no—
242C2×C4—91no—
243C2×C4—182no—
244C2×C4—278no—
245C2×C4—544no—
246C2×C4—544no—
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—181no—
272C4—548no—
273C4C4×C3⋊#15712yescomplemented
274C4C4⋊(C3⋊#157)12yesdirect product factor
275C3—22no—
276C3—42no—
277C3C4×C4⋊#15711yescharacteristic, complemented
278C2—92no—
279C2—92no—
280C2C2×C4×C3⋊#15711yescharacteristic
281C2C2×C4⋊(C3⋊#157)11yescharacteristic
282C2#1172451411yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊#157
commutatorC2×C33C2×C42
FrattiniC22C22×C3⋊#157
FittingC33×C42C4
socleC22×C33C22×C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC42C3⋊#157
hypocenterC33C4×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
10. C4⋊(C3⋊#157)C4direct (N × H)
12. C33×C42C4semidirect
13. C3×C4×C3⋊Dic3C4semidirect
14. C3×C4×C3⋊Dic3C4semidirect
15. C3×C4×C3⋊Dic3C4semidirect
47. C32×C42Dic3semidirect
48. C4×C3⋊Dic3Dic3semidirect
49. C4×C3⋊Dic3Dic3semidirect
50. C4×C3⋊Dic3Dic3semidirect
61. C33×C4C42semidirect
62. C33×C4C4⋊C4semidirect
64. C3×C3⋊Dic3C42semidirect
65. C3×C3⋊Dic3C4⋊C4semidirect
117. C3×C42#157semidirect
142. C32×C4C4×Dic3semidirect
143. C32×C4C4⋊Dic3semidirect
145. C3⋊Dic3C4×Dic3semidirect
146. C3⋊Dic3C4⋊Dic3semidirect
154. C33C4×C4⋊C4semidirect
205. C42C3⋊#157semidirect
227. C3×C4C4×#157semidirect
228. C3×C4C4⋊#157semidirect
235. C32C4×C4⋊Dic3semidirect
273. C4C4×C3⋊#157semidirect
274. C4C4⋊(C3⋊#157)direct (N × H)
277. C3C4×C4⋊#157semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
C2×C4×C3⋊#1572
C3×C42×C3⋊D32
#117231062
C4×C4⋊#1573
C4×C4⋊Dic39