Finite Group Databasebeta

← C2×D12×#157 #13180969

Subgroups — C2×D12×#157 #13180969

Order 1728.

1432 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C22×D3×#157C212yesmaximal, complemented
2C2×C3⋊D3×D12C211yesmaximal, characteristic
3C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
4C2×C3×C4×#157C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊#157C212yesmaximal, complemented
6D12×#157C218yesmaximal, direct product factor
7C2×D4×#157—31nomaximal
8C22×D3×C3⋊D3C2212yes—
9C22×C3⋊#157C2212yes—
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×D3×#157—28no—
13C2×D3×#157C2218yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C32×D12C411yescharacteristic, complemented
17C2×#6291C412yescomplemented
18C3⋊D3×D12C414yescomplemented
19C3⋊D3×D12C2214yes—
20C4⋊(C3⋊#157)C2214yescomplemented
21C3×C4×#157C2214yescomplemented
22(C2×D3)⋊#157C2218yescomplemented
23C23×#157—32no—
24C2×D4×C3⋊D3—31no—
25C2×D3×D12—62no—
26C2×C22⋊#157—32no—
27D4×#157—38no—
28C2×C4⋊#157—31no—
29C2×C4×#157D311yescharacteristic, complemented
30C22×C3⋊(C3⋊D3)C2×C412yescomplemented
31C22×C3×C3⋊D3C2311yescharacteristic
32C22×C32×D3C2×C412yescomplemented
33C2×D3×C3⋊D3—24no—
34C2×D3×C3⋊D3—24no—
35C2×D3×C3⋊D3C2×C414yescomplemented
36C2×D3×C3⋊D3C2314yes—
37C2×C3⋊#157—24no—
38C2×C3⋊#157C2314yes—
39C2×C3×#157—22no—
40C2×C3×#157D414yescomplemented
41C2×C3×#157C2312yes—
42C2×C3×#157C2312yes—
43D3×#157—216no—
44C2×C33×C4C2×C411yescharacteristic, complemented
45C4⋊(C3⋊(C3⋊D3))C2×C414yescomplemented
46C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
47C3×C4×C3⋊D3C2×C412yescomplemented
48C3×C4×C3⋊D3C2312yes—
49C32×D12C2×C414yescomplemented
50#6291C2×C418yescomplemented
51C2×C4×D12—91nomaximal
52C23×C3⋊D3—32no—
53C22×D32—64no—
54C22×#157—32no—
55C22×#157—38no—
56C22×#157—68no—
57C22×#157C2×D311yescharacteristic
58C22×#157C2×D311yescharacteristic
59C2×C32×D4—31no—
60C2×#638—32no—
61D4×C3⋊D3—34no—
62D4×C3⋊D3—34no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—62no—
65C2×C4⋊(C3⋊D3)—122no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×D12—22no—
68C2×C3×C4×D3—62no—
69C2×#626—64no—
70D3×D12—68no—
71D3×D12—68no—
72C22⋊#157—38no—
73C4⋊#157—34no—
74C4×#157C2×D314yescomplemented
75C22×C33C22×C411yescharacteristic
76C2×C3⋊(C3⋊D3)—24no—
77C2×C3⋊(C3⋊D3)C22×C414yescomplemented
78C2×C3×C3⋊D3C22×C411yescharacteristic
79C2×C3×C3⋊D3C2×D412yes—
80C2×C3×C3⋊D3#4112yes—
81C2×C3×C3⋊D3C2411yescharacteristic
82C2×C32×D3—24no—
83C2×C32×D3C22×C414yescomplemented
84D3×C3⋊D3—28no—
85D3×C3⋊D3—24no—
86D3×C3⋊D3—24no—
87C3⋊#157—24no—
88C3×#157—22no—
89C3×#157C2×D414yescomplemented
90C33×C4C22×C412yescomplemented
91C3×C3⋊Dic3C22×C412yescomplemented
92C2×D3×D4—182no—
93C22×D12—91no—
94C22×C4×D3—92no—
95C2×C3×C42—91no—
96C2×(C2×D3)⋊C4—92no—
97C2×C4⋊Dic3—91no—
98C4×D12—98no—
99C23×C32—32no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—34no—
102C22×C3⋊D3—34no—
103C22×C3⋊D3—64no—
104C22×C3⋊D3—64no—
105C22×C3⋊D3—64no—
106C22×C3⋊D3—124no—
107C22×C3⋊D3C22×D311yescharacteristic
108C22×C3×D3—24no—
109C22×C3×D3—62no—
110C2×D32—68no—
111C2×D32—68no—
112C2×D32—128no—
113C2×D32—128no—
114C2×#157—22no—
115C2×#157—34no—
116C2×#157—64no—
117C2×#157—616no—
118C2×#157D1214yesdirect product factor
119C2×#157C22×D312yes—
120C2×#157C22×D312yes—
121C32×D4—34no—
122C2×C32×C4—22no—
123C2×C32×C4—42no—
124C2×C32×C4C4×D311yescharacteristic, complemented
125#638—38no—
126C2×C3⋊Dic3C4×D311yescharacteristic, complemented
127C4⋊(C3⋊D3)—34no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—128no—
130C4×C3⋊D3C4×D312yescomplemented
131C4×C3⋊D3C22×D312yes—
132C2×C3×Dic3—62no—
133C3×D12—28no—
134C3×C4×D3—64no—
135C3×C4×D3—64no—
136#626—616no—
137C2×C4×D4—271noSylow
138C2×C33C4×D412yescomplemented
139C2×C33C23×C411yescharacteristic
140C3⋊(C3⋊D3)—24no—
141C3×C3⋊D3C4×D412yescomplemented
142C3×C3⋊D3C22×D411yescharacteristic
143C3×C3⋊D3C2×#4111yescharacteristic
144C32×D3—24no—
145C23×D3—92no—
146C23×D3—184no—
147C2×C3×D4—62no—
148C22×C3×C4—91no—
149C22×C3×C4—91no—
150C22×C3×C4—91no—
151C2×#67—184no—
152C22×Dic3—92no—
153D3×D4—188no—
154D3×D4—188no—
155C2×D12—92no—
156C2×D12—91no—
157C2×D12—94no—
158C2×D12—94no—
159C2×D12—182no—
160C2×D12—362no—
161C2×D12#15711yescharacteristic, direct product factor
162C2×C4×D3—62no—
163C2×C4×D3—98no—
164C2×C4×D3—188no—
165C3×C42—94no—
166(C2×D3)⋊C4—98no—
167C4⋊Dic3—94no—
168C22×C32—22no—
169C22×C32—34no—
170C22×C32—42no—
171C22×C32—64no—
172C22×C32C2×C4×D311yescharacteristic
173C2×C3⋊D3—34no—
174C2×C3⋊D3—68no—
175C2×C3⋊D3—64no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—64no—
179C2×C3⋊D3—128no—
180C2×C3⋊D3—128no—
181C2×C3⋊D3—248no—
182C2×C3⋊D3C2×C4×D311yescharacteristic
183C2×C3⋊D3C2×D1212yes—
184C2×C3⋊D3#23512yes—
185C2×C3⋊D3C23×D311yescharacteristic
186C2×C3×D3—28no—
187C2×C3×D3—48no—
188C2×C3×D3—62no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—62no—
192D32—128no—
193D32—1216no—
194D32—128no—
195#157—22no—
196#157—64no—
197#157C2×D1214yesdirect product factor
198C32×C4—24no—
199C32×C4—44no—
200C32×C4C2×C4×D312yescomplemented
201C3⋊Dic3C2×C4×D312yescomplemented
202C3×Dic3—64no—
203C22×D4—271no—
204C23×C4—272no—
205C4×D4—278no—
206C2×C4⋊C4—271no—
207C2×C22⋊C4—272no—
208C2×C42—91no—
209C33C2×C4×D411yescharacteristic, complemented, Sylow
210C23×C3—64no—
211C23×C3—91no—
212C22×D3—62no—
213C22×D3—92no—
214C22×D3—94no—
215C22×D3—94no—
216C22×D3—188no—
217C22×D3—188no—
218C22×D3—184no—
219C22×D3—184no—
220C22×D3—184no—
221C22×D3—364no—
222C22×D3—368no—
223C22×D3—368no—
224C22×D3C2×#15712yescomplemented
225C3×D4—68no—
226C2×C3×C4—22no—
227C2×C3×C4—42no—
228C2×C3×C4—94no—
229C2×C3×C4—92no—
230C2×C3×C4—91no—
231C2×C3×C4—92no—
232C2×C3×C4—92no—
233C2×C3×C4—92no—
234C2×C3×C4—182no—
235C2×C3×C4C2×#15711yescharacteristic, complemented
236#67—1816no—
237C2×Dic3—62no—
238C2×Dic3—94no—
239C2×Dic3—184no—
240D12—98no—
241D12—94no—
242D12—188no—
243D12—368no—
244D12C2×#15714yesdirect product factor
245C4×D3—64no—
246C4×D3—64no—
247C4×D3—1816no—
248C2×C32—24no—
249C2×C32—22no—
250C2×C32—44no—
251C2×C32—42no—
252C2×C32—64no—
253C2×C32C4×D1212yescomplemented
254C2×C32C22×C4×D311yescharacteristic
255C3⋊D3—64no—
256C3⋊D3—128no—
257C3⋊D3—248no—
258C3⋊D3C4×D1212yescomplemented
259C3⋊D3C22×D1211yescharacteristic
260C3⋊D3C2×#23511yescharacteristic
261C3×D3—48no—
262C3×D3—64no—
263C3×D3—62no—
264C3×D3—62no—
265C24—272no—
266C2×D4—31no—
267C2×D4—274no—
268C2×D4—274no—
269C2×D4—272no—
270C2×D4—271no—
271C22×C4—91no—
272C22×C4—91no—
273C22×C4—91no—
274C22×C4—278no—
275C22×C4—272no—
276C22×C4—548no—
277C4⋊C4—274no—
278C22⋊C4—278no—
279C42—94no—
280C22×C3—22no—
281C22×C3—42no—
282C22×C3—68no—
283C22×C3—92no—
284C22×C3—92no—
285C22×C3—91no—
286C22×C3—91no—
287C22×C3—128no—
288C22×C3C22×#15711yescharacteristic
289C2×D3—24no—
290C2×D3—62no—
291C2×D3—62no—
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—94no—
295C2×D3—188no—
296C2×D3—184no—
297C2×D3—184no—
298C2×D3—188no—
299C2×D3—184no—
300C2×D3—368no—
301C2×D3—368no—
302C2×D3—3616no—
303C2×D3—368no—
304C2×D3—368no—
305C2×D3—728no—
306C2×D3C22×#15714yescomplemented
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—94no—
310C3×C4—92no—
311C3×C4—182no—
312C3×C4C22×#15712yescomplemented
313Dic3—64no—
314Dic3—184no—
315C32—22no—
316C32—42no—
317C32C2×C4×D1211yescharacteristic, complemented
318C23—32no—
319C23—91no—
320C23—274no—
321C23—274no—
322C23—272no—
323C23—544no—
324C23—544no—
325D4—34no—
326D4—278no—
327D4—274no—
328C2×C4—92no—
329C2×C4—94no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—182no—
335C2×C4—274no—
336C2×C4—5416no—
337C2×C4—544no—
338C2×C4D3×#15711yescharacteristic, complemented
339C2×C3—22no—
340C2×C3—24no—
341C2×C3—42no—
342C2×C3—44no—
343C2×C3—91no—
344C2×C3—91no—
345C2×C3—92no—
346C2×C3—128no—
347C2×C3D4×#15712yescomplemented
348C2×C3C23×#15711yescharacteristic
349D3—24no—
350D3—64no—
351D3—62no—
352D3—62no—
353D3—184no—
354D3—368no—
355D3—728no—
356C22—34no—
357C22—64no—
358C22—91no—
359C22—91no—
360C22—92no—
361C22—92no—
362C22—274no—
363C22—544no—
364C22—544no—
365C22—548no—
366C22—544no—
367C22C2×D3×#15711yescharacteristic
368C4—94no—
369C4—92no—
370C4—182no—
371C4—544no—
372C4C2×D3×#15712yescomplemented
373C3—22no—
374C3—42no—
375C3C2×D4×#15711yescharacteristic, complemented
376C2—64no—
377C2—91no—
378C2—91no—
379C2—92no—
380C2—544no—
381C2D12×#15712yesdirect product factor
382C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC2×C3⋊D3×D12C2
cosocleC2×C3×C3⋊D3C24
hypercenterC2×C4D3×#157
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D3×#157C2semidirect
3. C2×C4⋊(C3⋊#157)C2semidirect
4. C2×C3×C4×#157C2semidirect
5. C2×(C2×D3)⋊#157C2semidirect
6. D12×#157C2direct (N × H)
13. C2×D3×#157C22semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
16. C2×C32×D12C4semidirect
17. C2×#6291C4semidirect
18. C3⋊D3×D12C4semidirect
20. C4⋊(C3⋊#157)C22semidirect
21. C3×C4×#157C22semidirect
22. (C2×D3)⋊#157C22semidirect
29. C2×C4×#157D3semidirect
30. C22×C3⋊(C3⋊D3)C2×C4semidirect
32. C22×C32×D3C2×C4semidirect
35. C2×D3×C3⋊D3C2×C4semidirect
40. C2×C3×#157D4semidirect
44. C2×C33×C4C2×C4semidirect
45. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
46. C2×C3×C3⋊Dic3C2×C4semidirect
47. C3×C4×C3⋊D3C2×C4semidirect
49. C32×D12C2×C4semidirect
50. #6291C2×C4semidirect
74. C4×#157C2×D3semidirect
77. C2×C3⋊(C3⋊D3)C22×C4semidirect
83. C2×C32×D3C22×C4semidirect
89. C3×#157C2×D4semidirect
90. C33×C4C22×C4semidirect
91. C3×C3⋊Dic3C22×C4semidirect
118. C2×#157D12direct (N × H)
124. C2×C32×C4C4×D3semidirect
126. C2×C3⋊Dic3C4×D3semidirect
130. C4×C3⋊D3C4×D3semidirect
138. C2×C33C4×D4semidirect
141. C3×C3⋊D3C4×D4semidirect
161. C2×D12#157direct (N × H)
197. #157C2×D12direct (N × H)
200. C32×C4C2×C4×D3semidirect
201. C3⋊Dic3C2×C4×D3semidirect
209. C33C2×C4×D4semidirect
224. C22×D3C2×#157semidirect
235. C2×C3×C4C2×#157semidirect
244. D12C2×#157direct (N × H)
253. C2×C32C4×D12semidirect
258. C3⋊D3C4×D12semidirect
306. C2×D3C22×#157semidirect
312. C3×C4C22×#157semidirect
317. C32C2×C4×D12semidirect
338. C2×C4D3×#157semidirect
347. C2×C3D4×#157semidirect
372. C4C2×D3×#157semidirect
375. C3C2×D4×#157semidirect
381. C2D12×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×C3⋊D3×D122
C2×C4⋊(C3⋊#157)2
C2×C3×C4×#1572
C2×(C2×D3)⋊#1572
D12×#1572
C2×D4×#1573
C2×C4×D129