Finite Group Databasebeta

← C3×C4⋊C4×#157 #13173228

Subgroups — C3×C4⋊C4×#157 #13173228

Order 1728.

624 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊#157C211yesmaximal, characteristic
2C2×C3×C4×#157C212yesmaximal
3C2×C3×C4×#157C211yesmaximal, characteristic
4C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
5C3×#63781C212yesmaximal
6C4⋊C4×#157C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2211yescharacteristic
8C22×C3×#157C2212yes—
9C22×C3×#157C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C3×C4⋊#157C414yescomplemented
13C3×C4×#157—24no—
14C3×C4×#157C414yescomplemented
15C33×C4⋊C4C411yescharacteristic, complemented
16C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
17C3×#3750C412yescomplemented
18C2×C4⋊#157C2×C311yescharacteristic
19C2×C4×#157C2×C312yes—
20C2×C4×#157C2×C311yescharacteristic
21C4⋊C4×C3⋊D3C2×C311yescharacteristic
22C3×D3×C4⋊C4—62no—
23#63781C2×C312yes—
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3×#157—22no—
26C2×C3×#157—21no—
27C2×C3×#157—21no—
28C2×C3×#157—22no—
29C2×C3×#157C2×C412yes—
30C2×C3×#157C2×C414yes—
31C2×C3×#157C2×C412yes—
32C2×C3×#157D412yes—
33C2×C3×#157Q812yes—
34C2×C33×C4C2×C412yes—
35C2×C33×C4C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3C2×C412yes—
42C3×C4×C4⋊C4—91nomaximal
43C22×#157C22×C312yes—
44C22×#157C22×C311yescharacteristic
45C22×#157C22×C311yescharacteristic
46C2×C4×C3⋊D3C22×C312yes—
47C2×C4×C3⋊D3C22×C311yescharacteristic
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—64no—
50C4⋊#157C3×C414yescomplemented
51C4×#157—24no—
52C4×#157C3×C414yescomplemented
53C32×C4⋊C4—22no—
54C32×C4⋊C4—42no—
55C32×C4⋊C4C3×C411yescharacteristic, complemented
56C4⋊(C3⋊Dic3)C3×C411yescharacteristic, complemented
57#3750C3×C412yescomplemented
58C3×C4⋊Dic3—62no—
59C3×Dic3⋊C4—64no—
60C22×C33C22×C411yescharacteristic
61C2×C3×C3⋊D3C22×C411yescharacteristic
62C2×C3×C3⋊D3C22×C411yescharacteristic
63C2×C3×C3⋊D3C2×D411yescharacteristic
64C2×C3×C3⋊D3C2×Q811yescharacteristic
65C2×C3×C3⋊D3#4111yescharacteristic
66C2×C3×C3⋊D3#4111yescharacteristic
67C3×#157—22no—
68C3×#157—24no—
69C3×#157C4⋊C414yesdirect product factor
70C33×C4—22no—
71C33×C4C4212yescomplemented
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3C4212yescomplemented
74C2×C3×C4⋊C4—91no—
75C2×C3×C4⋊C4—91no—
76C2×C3×C42—91no—
77C2×C3×C42—92no—
78D3×C4⋊C4—62no—
79C3×#95—92no—
80C22×C3⋊D3C23×C311yescharacteristic
81C22×C3×D3—62no—
82C2×#157—22no—
83C2×#157—21no—
84C2×#157—21no—
85C2×#157—22no—
86C2×#157C2×C3×C412yes—
87C2×#157C2×C3×C414yes—
88C2×#157C2×C3×C412yes—
89C2×#157C3×D412yes—
90C2×#157C3×Q812yes—
91C2×C32×C4—22no—
92C2×C32×C4—24no—
93C2×C32×C4—42no—
94C2×C32×C4—44no—
95C2×C32×C4C2×C3×C412yes—
96C2×C32×C4C2×C3×C411yescharacteristic
97C2×C3⋊Dic3C2×C3×C412yes—
98C2×C3⋊Dic3C2×C3×C411yescharacteristic
99C4×C3⋊D3—22no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3C2×C3×C412yes—
102C4×C3⋊D3C2×C3×C412yes—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—64no—
105C3×C4×D3—64no—
106C3×C4×D3—64no—
107C3×C4×D3—124no—
108C3×C4×D3—124no—
109C4×C4⋊C4—91noSylow
110C2×C33C2×C4211yescharacteristic
111C2×C33C4×D411yescharacteristic
112C2×C33C4×Q811yescharacteristic
113C3×C3⋊D3C2×C4⋊C411yescharacteristic
114C3×C3⋊D3#11711yescharacteristic
115C3×C3⋊D3C4×D411yescharacteristic
116C3×C3⋊D3C4×Q811yescharacteristic
117C22×C3×C4—91no—
118C22×C3×C4—91no—
119C22×C3×C4—91no—
120C22×C3×C4—92no—
121C22×C3×C4—92no—
122C2×C4×D3—62no—
123C2×C4×D3—64no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—94no—
127C3×C4⋊C4—92no—
128C3×C4⋊C4—91no—
129C3×C4⋊C4#15711yescharacteristic, direct product factor
130C3×C42—94no—
131C3×C42—184no—
132C4⋊Dic3—62no—
133Dic3⋊C4—64no—
134C22×C32—22no—
135C22×C32—42no—
136C22×C32C22×C3×C411yescharacteristic
137C2×C3⋊D3C22×C3×C411yescharacteristic
138C2×C3⋊D3C22×C3×C411yescharacteristic
139C2×C3⋊D3C2×C3×D411yescharacteristic
140C2×C3⋊D3C2×C3×Q811yescharacteristic
141C2×C3⋊D3C3×#4111yescharacteristic
142C2×C3⋊D3C3×#4111yescharacteristic
143C2×C3×D3—62no—
144C2×C3×D3—62no—
145C2×C3×D3—62no—
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149#157—22no—
150#157—24no—
151#157C3×C4⋊C414yesdirect product factor
152C32×C4—24no—
153C32×C4—22no—
154C32×C4—44no—
155C32×C4—44no—
156C32×C4—84no—
157C32×C4C3×C4212yescomplemented
158C3⋊Dic3—22no—
159C3⋊Dic3C3×C4212yescomplemented
160C3×Dic3—64no—
161C3×Dic3—124no—
162C2×C4⋊C4—91no—
163C2×C4⋊C4—91no—
164C2×C42—91no—
165C2×C42—92no—
166#95—92no—
167C33C4×C4⋊C411yescharacteristic, complemented, Sylow
168C23×C3—91no—
169C22×D3—62no—
170C2×C3×C4—22no—
171C2×C3×C4—24no—
172C2×C3×C4—42no—
173C2×C3×C4—44no—
174C2×C3×C4—92no—
175C2×C3×C4—92no—
176C2×C3×C4—92no—
177C2×C3×C4—91no—
178C2×C3×C4—92no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—94no—
182C2×C3×C4—92no—
183C2×C3×C4—182no—
184C2×C3×C4—182no—
185C2×C3×C4—182no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4—182no—
189C2×C3×C4C2×#15712yes—
190C2×C3×C4C2×#15711yescharacteristic
191C2×Dic3—62no—
192C2×Dic3—64no—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—124no—
196C4×D3—124no—
197C2×C32—22no—
198C2×C32—22no—
199C2×C32—22no—
200C2×C32—42no—
201C2×C32—42no—
202C2×C32—42no—
203C2×C32C2×C3×C4211yescharacteristic
204C2×C32C3×C4×D411yescharacteristic
205C2×C32C3×C4×Q811yescharacteristic
206C3⋊D3C2×C3×C4⋊C411yescharacteristic
207C3⋊D3C3×#11711yescharacteristic
208C3⋊D3C3×C4×D411yescharacteristic
209C3⋊D3C3×C4×Q811yescharacteristic
210C3×D3—62no—
211C3×D3—62no—
212C3×D3—62no—
213C3×D3—62no—
214C22×C4—91no—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4—92no—
218C22×C4—92no—
219C4⋊C4—94no—
220C4⋊C4—92no—
221C4⋊C4—91no—
222C4⋊C4C3×#15711yescharacteristic, direct product factor
223C42—94no—
224C42—184no—
225C22×C3—22no—
226C22×C3—42no—
227C22×C3—91no—
228C22×C3—91no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—91no—
232C22×C3—91no—
233C22×C3C22×#15711yescharacteristic
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—62no—
238C2×D3—62no—
239C2×D3—62no—
240C3×C4—22no—
241C3×C4—24no—
242C3×C4—44no—
243C3×C4—44no—
244C3×C4—84no—
245C3×C4—94no—
246C3×C4—92no—
247C3×C4—184no—
248C3×C4—182no—
249C3×C4—182no—
250C3×C4C4×#15712yescomplemented
251Dic3—64no—
252Dic3—124no—
253C32—22no—
254C32—42no—
255C32C3×C4×C4⋊C411yescharacteristic, complemented
256C23—91no—
257C2×C4—92no—
258C2×C4—92no—
259C2×C4—92no—
260C2×C4—91no—
261C2×C4—92no—
262C2×C4—92no—
263C2×C4—92no—
264C2×C4—94no—
265C2×C4—92no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—182no—
272C2×C4C2×C3×#15712yes—
273C2×C4C2×C3×#15711yescharacteristic
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—22no—
277C2×C3—42no—
278C2×C3—42no—
279C2×C3—42no—
280C2×C3—91no—
281C2×C3—91no—
282C2×C3—91no—
283C2×C3—91no—
284C2×C3C2×C4×#15711yescharacteristic
285C2×C3D4×#15711yescharacteristic
286C2×C3Q8×#15711yescharacteristic
287D3—62no—
288D3—62no—
289D3—62no—
290D3—62no—
291C22—91no—
292C22—91no—
293C22—91no—
294C22—91no—
295C22—91no—
296C22—91no—
297C22C22×C3×#15711yescharacteristic
298C4—22no—
299C4—94no—
300C4—92no—
301C4—184no—
302C4—182no—
303C4—182no—
304C4C3×C4×#15712yescomplemented
305C3—22no—
306C3—42no—
307C3C4⋊C4×#15711yescharacteristic, direct product factor
308C2—91no—
309C2—91no—
310C2—91no—
311C2—91no—
312C2C2×C3×C4×#15711yescharacteristic
313C2C3×D4×#15711yescharacteristic
314C2C3×Q8×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC2×C32C2×C3×C42
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C4⋊C4×#157C3direct (N × H)
12. C3×C4⋊#157C4semidirect
14. C3×C4×#157C4semidirect
15. C33×C4⋊C4C4semidirect
16. C3×C4⋊(C3⋊Dic3)C4semidirect
17. C3×#3750C4semidirect
50. C4⋊#157C3×C4semidirect
52. C4×#157C3×C4semidirect
55. C32×C4⋊C4C3×C4semidirect
56. C4⋊(C3⋊Dic3)C3×C4semidirect
57. #3750C3×C4semidirect
69. C3×#157C4⋊C4direct (N × H)
71. C33×C4C42semidirect
73. C3×C3⋊Dic3C42semidirect
129. C3×C4⋊C4#157direct (N × H)
151. #157C3×C4⋊C4direct (N × H)
157. C32×C4C3×C42semidirect
159. C3⋊Dic3C3×C42semidirect
167. C33C4×C4⋊C4semidirect
222. C4⋊C4C3×#157direct (N × H)
250. C3×C4C4×#157semidirect
255. C32C3×C4×C4⋊C4semidirect
304. C4C3×C4×#157semidirect
307. 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×C4⋊C4×C3⋊D32
C3×#637812
C4⋊C4×#1573
C3×C4×C4⋊C49