Finite Group Databasebeta

← C2×C3×D4×#157 #13181495

Subgroups — C2×C3×D4×#157 #13181495

Order 1728.

1432 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#157C212yesmaximal, complemented
2C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
3C2×C3×C22⋊#157C212yesmaximal, complemented
4C3×D4×#157C218yesmaximal, direct product factor
5C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
6C2×C3×C4×#157C211yesmaximal, characteristic, complemented
7C2×D4×#157C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3C2212yes—
9C22×C3×#157—28no—
10C22×C3×#157C2212yes—
11C22×C3×#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2218yescomplemented
14C2×C33×D4C411yescharacteristic, complemented
15C2×C3×#638C412yescomplemented
16C3×D4×C3⋊D3C414yescomplemented
17C3×D4×C3⋊D3C2214yes—
18C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C3×C22⋊#157C2218yescomplemented
21C3×C4⋊#157C2214yescomplemented
22C3×C4×#157C2214yescomplemented
23C23×#157C2×C312yescomplemented
24C2×D4×C3⋊D3C2×C311yescharacteristic
25C2×C3×D3×D4—62no—
26C2×C22⋊#157C2×C312yescomplemented
27D4×#157C2×C318yesdirect product factor
28C2×C4⋊#157C2×C311yescharacteristic, complemented
29C2×C4×#157C2×C311yescharacteristic, complemented
30C23×C33C2×C412yescomplemented
31C22×C3×C3⋊D3—24no—
32C22×C3×C3⋊D3—24no—
33C22×C3×C3⋊D3C2×C414yescomplemented
34C22×C3×C3⋊D3C2×C412yescomplemented
35C22×C3×C3⋊D3C2314yes—
36C22×C3×C3⋊D3C2311yescharacteristic
37C2×C3×#157—24no—
38C2×C3×#157—22no—
39C2×C3×#157—216no—
40C2×C3×#157D414yesdirect product factor
41C2×C3×#157C2314yes—
42C2×C3×#157C2312yes—
43C2×C3×#157C2312yes—
44C33×D4C2×C414yescomplemented
45C2×C33×C4C2×C411yescharacteristic, complemented
46C3×#638C2×C418yescomplemented
47C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
48C3×C4⋊(C3⋊D3)C2×C414yescomplemented
49C3×C4×C3⋊D3C2×C412yescomplemented
50C3×C4×C3⋊D3C2312yes—
51C2×C3×C4×D4—91nomaximal
52C23×C3⋊D3C22×C312yes—
53C23×C3×D3—64no—
54C22×#157—28no—
55C22×#157C22×C311yescharacteristic
56C22×#157C22×C312yes—
57C22×#157C22×C311yescharacteristic
58C22×#157C22×C318yescomplemented
59C2×C32×D4—22no—
60C2×C32×D4—42no—
61C2×C32×D4C3×C411yescharacteristic, complemented
62C2×#638C3×C412yescomplemented
63D4×C3⋊D3C3×C414yescomplemented
64D4×C3⋊D3C22×C314yes—
65C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
66C2×C4×C3⋊D3C22×C311yescharacteristic
67C2×C3×#67—64no—
68C3×D3×D4—68no—
69C3×D3×D4—68no—
70C2×C3×D12—62no—
71C2×C3×C4×D3—62no—
72C22⋊#157C22×C318yescomplemented
73C4⋊#157C22×C314yescomplemented
74C4×#157C22×C314yescomplemented
75C22×C33—24no—
76C22×C33C22×C411yescharacteristic
77C22×C33C22×C414yescomplemented
78C2×C3×C3⋊D3—24no—
79C2×C3×C3⋊D3—28no—
80C2×C3×C3⋊D3—24no—
81C2×C3×C3⋊D3—24no—
82C2×C3×C3⋊D3C22×C411yescharacteristic
83C2×C3×C3⋊D3C22×C414yescomplemented
84C2×C3×C3⋊D3C2×D412yes—
85C2×C3×C3⋊D3#4112yes—
86C2×C3×C3⋊D3C2411yescharacteristic
87C3×#157—24no—
88C3×#157—22no—
89C3×#157C2×D414yesdirect product factor
90C33×C4C22×C412yescomplemented
91C3×C3⋊Dic3C22×C412yescomplemented
92C22×C3×D4—91no—
93C23×C3×C4—92no—
94C2×D3×D4—62no—
95C3×C4×D4—98no—
96C2×C3×C4⋊C4—91no—
97C2×C3×C22⋊C4—92no—
98C2×C3×C42—91no—
99C23×C32—24no—
100C23×C32—44no—
101C23×C32C2×C3×C412yescomplemented
102C22×C3⋊D3—24no—
103C22×C3⋊D3—24no—
104C22×C3⋊D3C2×C3×C414yescomplemented
105C22×C3⋊D3C2×C3×C412yescomplemented
106C22×C3⋊D3C23×C314yes—
107C22×C3⋊D3C23×C311yescharacteristic
108C22×C3×D3—62no—
109C22×C3×D3—64no—
110C22×C3×D3—68no—
111C22×C3×D3—68no—
112C22×C3×D3—128no—
113C22×C3×D3—128no—
114C2×#157—24no—
115C2×#157—22no—
116C2×#157—216no—
117C2×#157C3×D414yesdirect product factor
118C2×#157C23×C314yes—
119C2×#157C23×C312yes—
120C2×#157C23×C312yes—
121C32×D4—28no—
122C32×D4—48no—
123C32×D4C2×C3×C414yescomplemented
124C2×C32×C4—22no—
125C2×C32×C4—42no—
126C2×C32×C4C2×C3×C411yescharacteristic, complemented
127#638C2×C3×C418yescomplemented
128C2×C3⋊Dic3C2×C3×C411yescharacteristic, complemented
129C4⋊(C3⋊D3)C2×C3×C414yescomplemented
130C4×C3⋊D3C2×C3×C412yescomplemented
131C4×C3⋊D3C23×C312yes—
132C3×#67—616no—
133C2×C3×Dic3—62no—
134C3×D12—68no—
135C3×C4×D3—64no—
136C3×C4×D3—64no—
137C2×C4×D4—91noSylow
138C2×C33—24no—
139C2×C33C4×D412yescomplemented
140C2×C33C23×C411yescharacteristic
141C3×C3⋊D3—24no—
142C3×C3⋊D3C4×D412yescomplemented
143C3×C3⋊D3C22×D411yescharacteristic
144C3×C3⋊D3C2×#4111yescharacteristic
145C24×C3—92no—
146C23×D3—64no—
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—92no—
150C2×C3×D4—94no—
151C2×C3×D4—94no—
152C2×C3×D4—91no—
153C2×C3×D4#15711yescharacteristic, direct product factor
154C22×C3×C4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—91no—
157C22×C3×C4—92no—
158C22×C3×C4—98no—
159C22×C3×C4—188no—
160C2×#67—64no—
161D3×D4—68no—
162D3×D4—68no—
163C2×D12—62no—
164C2×C4×D3—62no—
165C3×C4⋊C4—94no—
166C3×C22⋊C4—98no—
167C3×C42—94no—
168C22×C32—22no—
169C22×C32—28no—
170C22×C32—24no—
171C22×C32—42no—
172C22×C32—48no—
173C22×C32—48no—
174C22×C32—88no—
175C22×C32C22×C3×C411yescharacteristic
176C22×C32C22×C3×C414yescomplemented
177C2×C3⋊D3—24no—
178C2×C3⋊D3—28no—
179C2×C3⋊D3—24no—
180C2×C3⋊D3—24no—
181C2×C3⋊D3C22×C3×C411yescharacteristic
182C2×C3⋊D3C22×C3×C414yescomplemented
183C2×C3⋊D3C2×C3×D412yes—
184C2×C3⋊D3C3×#4112yes—
185C2×C3⋊D3C24×C311yescharacteristic
186C2×C3×D3—62no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C2×C3×D3—62no—
190C2×C3×D3—68no—
191C2×C3×D3—128no—
192C2×C3×D3—1216no—
193C2×C3×D3—128no—
194C2×C3×D3—128no—
195#157—24no—
196#157—22no—
197#157C2×C3×D414yesdirect product factor
198C32×C4—24no—
199C32×C4—44no—
200C32×C4C22×C3×C412yescomplemented
201C3⋊Dic3C22×C3×C412yescomplemented
202C3×Dic3—64no—
203C22×D4—91no—
204C23×C4—92no—
205C4×D4—98no—
206C2×C4⋊C4—91no—
207C2×C22⋊C4—92no—
208C2×C42—91no—
209C33C2×C4×D411yescharacteristic, complemented, Sylow
210C23×C3—24no—
211C23×C3—44no—
212C23×C3—91no—
213C23×C3—92no—
214C23×C3—94no—
215C23×C3—94no—
216C23×C3—184no—
217C23×C3—184no—
218C23×C3C2×#15712yescomplemented
219C22×D3—62no—
220C22×D3—64no—
221C22×D3—68no—
222C22×D3—68no—
223C22×D3—128no—
224C22×D3—128no—
225C3×D4—28no—
226C3×D4—48no—
227C3×D4—98no—
228C3×D4—94no—
229C3×D4C2×#15714yesdirect product factor
230C2×C3×C4—22no—
231C2×C3×C4—42no—
232C2×C3×C4—94no—
233C2×C3×C4—92no—
234C2×C3×C4—91no—
235C2×C3×C4—92no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—94no—
239C2×C3×C4—1816no—
240C2×C3×C4—182no—
241C2×C3×C4—184no—
242C2×C3×C4C2×#15711yescharacteristic, complemented
243#67—616no—
244C2×Dic3—62no—
245D12—68no—
246C4×D3—64no—
247C4×D3—64no—
248C2×C32—24no—
249C2×C32—22no—
250C2×C32—24no—
251C2×C32—44no—
252C2×C32—42no—
253C2×C32—48no—
254C2×C32—88no—
255C2×C32C3×C4×D412yescomplemented
256C2×C32C23×C3×C411yescharacteristic
257C3⋊D3—24no—
258C3⋊D3C3×C4×D412yescomplemented
259C3⋊D3C22×C3×D411yescharacteristic
260C3⋊D3C2×C3×#4111yescharacteristic
261C3×D3—64no—
262C3×D3—62no—
263C3×D3—62no—
264C3×D3—128no—
265C24—92no—
266C2×D4—92no—
267C2×D4—94no—
268C2×D4—94no—
269C2×D4—91no—
270C2×D4C3×#15711yescharacteristic, direct product factor
271C22×C4—91no—
272C22×C4—91no—
273C22×C4—91no—
274C22×C4—92no—
275C22×C4—98no—
276C22×C4—188no—
277C4⋊C4—94no—
278C22⋊C4—98no—
279C42—94no—
280C22×C3—24no—
281C22×C3—22no—
282C22×C3—28no—
283C22×C3—48no—
284C22×C3—42no—
285C22×C3—48no—
286C22×C3—88no—
287C22×C3—92no—
288C22×C3—92no—
289C22×C3—91no—
290C22×C3—91no—
291C22×C3—94no—
292C22×C3—184no—
293C22×C3—184no—
294C22×C3—188no—
295C22×C3—184no—
296C22×C3C22×#15711yescharacteristic
297C22×C3C22×#15714yescomplemented
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—64no—
301C2×D3—64no—
302C2×D3—68no—
303C2×D3—128no—
304C2×D3—1216no—
305C2×D3—128no—
306C2×D3—128no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—94no—
310C3×C4—92no—
311C3×C4—184no—
312C3×C4—182no—
313C3×C4C22×#15712yescomplemented
314Dic3—64no—
315C32—22no—
316C32—42no—
317C32C2×C3×C4×D411yescharacteristic, complemented
318C23—91no—
319C23—92no—
320C23—94no—
321C23—94no—
322C23—184no—
323C23—184no—
324C23C2×C3×#15712yescomplemented
325D4—98no—
326D4—94no—
327D4C2×C3×#15714yesdirect product factor
328C2×C4—92no—
329C2×C4—94no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—94no—
335C2×C4—1816no—
336C2×C4—184no—
337C2×C4—182no—
338C2×C4C2×C3×#15711yescharacteristic, complemented
339C2×C3—24no—
340C2×C3—22no—
341C2×C3—24no—
342C2×C3—48no—
343C2×C3—42no—
344C2×C3—44no—
345C2×C3—88no—
346C2×C3—91no—
347C2×C3—91no—
348C2×C3—92no—
349C2×C3—184no—
350C2×C3D4×#15712yesdirect product factor
351C2×C3C23×#15711yescharacteristic
352D3—64no—
353D3—62no—
354D3—62no—
355D3—128no—
356C22—24no—
357C22—91no—
358C22—91no—
359C22—92no—
360C22—92no—
361C22—94no—
362C22—184no—
363C22—184no—
364C22—188no—
365C22—184no—
366C22C22×C3×#15711yescharacteristic
367C22C22×C3×#15714yescomplemented
368C4—94no—
369C4—92no—
370C4—184no—
371C4—182no—
372C4C22×C3×#15712yescomplemented
373C3—22no—
374C3—42no—
375C3C2×D4×#15711yescharacteristic, direct product factor
376C2—24no—
377C2—91no—
378C2—91no—
379C2—92no—
380C2—184no—
381C2C3×D4×#15712yesdirect product factor
382C2C23×C3×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC2×C32C23×C3×C4
FrattiniC2C23×C3×#157
FittingC2×C33×D4C4
socleC22×C33C22×C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C3⋊D3C24×C3
hypercenterC2×C3×D4#157
hypocenterC32C2×C3×C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×#157C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C3×D4×#157C2direct (N × H)
5. C2×C3×C4⋊#157C2semidirect
6. C2×C3×C4×#157C2semidirect
7. C2×D4×#157C3direct (N × H)
13. C22×C3×#157C22semidirect
14. C2×C33×D4C4semidirect
15. C2×C3×#638C4semidirect
16. C3×D4×C3⋊D3C4semidirect
18. C2×C3×C4⋊(C3⋊D3)C4semidirect
20. C3×C22⋊#157C22semidirect
21. C3×C4⋊#157C22semidirect
22. C3×C4×#157C22semidirect
23. C23×#157C2×C3semidirect
26. C2×C22⋊#157C2×C3semidirect
27. D4×#157C2×C3direct (N × H)
28. C2×C4⋊#157C2×C3semidirect
29. C2×C4×#157C2×C3semidirect
30. C23×C33C2×C4semidirect
33. C22×C3×C3⋊D3C2×C4semidirect
34. C22×C3×C3⋊D3C2×C4semidirect
40. C2×C3×#157D4direct (N × H)
44. C33×D4C2×C4semidirect
45. C2×C33×C4C2×C4semidirect
46. C3×#638C2×C4semidirect
47. C2×C3×C3⋊Dic3C2×C4semidirect
48. C3×C4⋊(C3⋊D3)C2×C4semidirect
49. C3×C4×C3⋊D3C2×C4semidirect
58. C22×#157C22×C3semidirect
61. C2×C32×D4C3×C4semidirect
62. C2×#638C3×C4semidirect
63. D4×C3⋊D3C3×C4semidirect
65. C2×C4⋊(C3⋊D3)C3×C4semidirect
72. C22⋊#157C22×C3semidirect
73. C4⋊#157C22×C3semidirect
74. C4×#157C22×C3semidirect
77. C22×C33C22×C4semidirect
83. C2×C3×C3⋊D3C22×C4semidirect
89. C3×#157C2×D4direct (N × H)
90. C33×C4C22×C4semidirect
91. C3×C3⋊Dic3C22×C4semidirect
101. C23×C32C2×C3×C4semidirect
104. C22×C3⋊D3C2×C3×C4semidirect
105. C22×C3⋊D3C2×C3×C4semidirect
117. C2×#157C3×D4direct (N × H)
123. C32×D4C2×C3×C4semidirect
126. C2×C32×C4C2×C3×C4semidirect
127. #638C2×C3×C4semidirect
128. C2×C3⋊Dic3C2×C3×C4semidirect
129. C4⋊(C3⋊D3)C2×C3×C4semidirect
130. C4×C3⋊D3C2×C3×C4semidirect
139. C2×C33C4×D4semidirect
142. C3×C3⋊D3C4×D4semidirect
153. C2×C3×D4#157direct (N × H)
176. C22×C32C22×C3×C4semidirect
182. C2×C3⋊D3C22×C3×C4semidirect
197. #157C2×C3×D4direct (N × H)
200. C32×C4C22×C3×C4semidirect
201. C3⋊Dic3C22×C3×C4semidirect
209. C33C2×C4×D4semidirect
218. C23×C3C2×#157semidirect
229. C3×D4C2×#157direct (N × H)
242. C2×C3×C4C2×#157semidirect
255. C2×C32C3×C4×D4semidirect
258. C3⋊D3C3×C4×D4semidirect
270. C2×D4C3×#157direct (N × H)
297. C22×C3C22×#157semidirect
313. C3×C4C22×#157semidirect
317. C32C2×C3×C4×D4semidirect
324. C23C2×C3×#157semidirect
327. D4C2×C3×#157direct (N × H)
338. C2×C4C2×C3×#157semidirect
350. C2×C3D4×#157direct (N × H)
367. C22C22×C3×#157semidirect
372. C4C22×C3×#157semidirect
375. C3C2×D4×#157direct (N × H)
381. C2C3×D4×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×C3×D4×C3⋊D32
C2×C3×C22⋊#1572
C3×D4×#1572
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C2×D4×#1573
C2×C3×C4×D49