Finite Group Databasebeta

← C3⋊D3×C4⋊S4 #13180744

Subgroups — C3⋊D3×C4⋊S4 #13180744

Order 1728.

1526 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 95 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×C3⋊D3×S4C212yesmaximal, complemented
2C4×A4×C3⋊D3C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
4C32×C4⋊S4C211yesmaximal, characteristic, complemented
5#11724187C212yesmaximal, complemented
6D3×C4⋊S4—34nomaximal
7C3⋊D3×#121—31nomaximal
8C2×A4×C3⋊D3C2211yescharacteristic
9C2×C3⋊(C3⋊S4)C2212yescomplemented
10C2×C32×S4C2212yescomplemented
11C3⋊D3×S4—24no—
12C32×C4×A4C2211yescharacteristic, complemented
13C3⋊D3×D12—41nomaximal
14A4×C3⋊Dic3C2211yescharacteristic, complemented
15C2×D3×S4—38no—
16C2×D4×C3⋊D3—31no—
17C2×D4×C3⋊D3—32no—
18C22×C4×C3⋊D3D311yescharacteristic, complemented
19C4×D3×A4—34no—
20C4⋊(C3⋊S4)—34no—
21C4⋊(C3⋊S4)—64no—
22C3×C4⋊S4D314yescomplemented
23#64207—38no—
24C32×#121—31no—
25#64148—32no—
26#64147—31no—
27#64139—31no—
28#64104—31no—
29C4⋊C4×C3⋊D3—31no—
30#64093—32no—
31C22⋊C4×C3⋊D3—32no—
32C2×C32×A4C2311yescharacteristic
33C2×D3×C3⋊D3—42no—
34A4×C3⋊D3D412yescomplemented
35C3⋊(C3⋊S4)—22no—
36C32×S4—22no—
37C4⋊(C3⋊(C3⋊D3))—41no—
38C3×C4×C3⋊D3—41no—
39C32×D12—41no—
40#6291—42no—
41C2×C4⋊S4—91no—
42D3×#121—94no—
43C23×C3⋊D3—32no—
44C23×C3⋊D3C2×D311yescharacteristic
45C2×D3×A4—34no—
46C2×C3⋊S4—38no—
47C2×C3⋊S4—68no—
48C2×C3×S4C2×D318yescomplemented
49D3×S4—616no—
50C2×C32×D4—31no—
51C2×C32×D4—32no—
52C22×C32×C4C2×D311yescharacteristic, complemented
53C2×#638—32no—
54C2×#638—32no—
55C2×#638—32no—
56C22×C3⋊Dic3C2×D311yescharacteristic, complemented
57D4×C3⋊D3—61no—
58D4×C3⋊D3—61no—
59D4×C3⋊D3—61no—
60D4×C3⋊D3—64no—
61D4×C3⋊D3—64no—
62D4×C3⋊D3—61no—
63C2×C4⋊(C3⋊D3)—32no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—31no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—32no—
70C3×C4×A4—24no—
71C3×C4×A4C2×D314yescomplemented
72D3×D12—124no—
73Dic3×A4—34no—
74C32×C4⋊C4—31no—
75C32×C22⋊C4—32no—
76#3757—32no—
77#3752—32no—
78#3752—32no—
79C4⋊(C3⋊Dic3)—31no—
80#3750—32no—
81C2×C3⋊(C3⋊D3)—42no—
82C2×C3×C3⋊D3—41no—
83C2×C32×D3—42no—
84C32×A4C2×D411yescharacteristic, complemented
85D3×C3⋊D3—84no—
86C33×C4—41no—
87C3×C3⋊Dic3—41no—
88C22×S4—92no—
89C2×D3×D4—98no—
90C2×D3×D4—94no—
91C22×C4×D3—34no—
92C2×C4×A4—91no—
93C4⋊S4—92no—
94C4⋊S4—91no—
95C4⋊S4—184no—
96C4⋊S4C3⋊D311yescharacteristic, direct product factor
97C3×#121—34no—
98#939—98no—
99#938—94no—
100#930—94no—
101#895—94no—
102D3×C4⋊C4—94no—
103#884—98no—
104D3×C22⋊C4—98no—
105C23×C32—32no—
106C23×C32C22×D311yescharacteristic
107C22×C3⋊D3—31no—
108C22×C3⋊D3—31no—
109C22×C3⋊D3—32no—
110C22×C3⋊D3—32no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—62no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—62no—
115C22×C3⋊D3D1212yescomplemented
116C2×C3×A4—24no—
117C2×C3×A4C22×D314yes—
118C2×D32—128no—
119D3×A4—38no—
120C3⋊S4—68no—
121C3⋊S4—128no—
122C3×S4—28no—
123C32×D4—61no—
124C32×D4—62no—
125C32×D4—62no—
126C32×D4—61no—
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—32no—
130#638—62no—
131#638—64no—
132#638—62no—
133#638—62no—
134#638—62no—
135C2×C3⋊Dic3—31no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—32no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—124no—
142C4⋊(C3⋊D3)—244no—
143C4×C3⋊D3—31no—
144C4×C3⋊D3—31no—
145C4×C3⋊D3—31no—
146C4×C3⋊D3—61no—
147C4×C3⋊D3—64no—
148C4×C3⋊D3S411yescharacteristic, complemented
149C3×D12—44no—
150C3×C4×D3—124no—
151#626—128no—
152C2×#121—271noSylow
153C2×C33—41no—
154C3⋊(C3⋊D3)—82no—
155C3×C3⋊D3—42no—
156C32×D3—82no—
157C23×D3—34no—
158C23×D3—98no—
159C22×A4—91no—
160C2×S4—92no—
161C2×S4—188no—
162C2×S4—184no—
163C2×S4C2×C3⋊D312yescomplemented
164C2×C3×D4—34no—
165C2×C3×D4—38no—
166C22×C3×C4D3214yescomplemented
167C2×#67—98no—
168C2×#67—98no—
169C2×#67—98no—
170C22×Dic3—34no—
171D3×D4—184no—
172D3×D4—184no—
173D3×D4—184no—
174D3×D4—184no—
175D3×D4—1816no—
176D3×D4—1816no—
177C2×D12—98no—
178C2×D12—94no—
179C2×D12—361no—
180C2×C4×D3—94no—
181C2×C4×D3—94no—
182C2×C4×D3—94no—
183C2×C4×D3—94no—
184C2×C4×D3—98no—
185C4×A4—24no—
186C4×A4—91no—
187C4×A4C2×C3⋊D311yescharacteristic, complemented
188C3×C4⋊C4—34no—
189C3×C22⋊C4—38no—
190#217—98no—
191(C2×D3)⋊C4—98no—
192(C2×D3)⋊C4—98no—
193C4⋊Dic3—94no—
194Dic3⋊C4—98no—
195C22×C32—31no—
196C22×C32—31no—
197C22×C32—62no—
198C22×C32—62no—
199C22×C32—62no—
200C22×C32C2×D1211yescharacteristic, complemented
201C2×C3⋊D3—31no—
202C2×C3⋊D3—32no—
203C2×C3⋊D3—32no—
204C2×C3⋊D3—32no—
205C2×C3⋊D3—61no—
206C2×C3⋊D3—62no—
207C2×C3⋊D3—62no—
208C2×C3⋊D3—62no—
209C2×C3⋊D3—128no—
210C2×C3⋊D3—124no—
211C2×C3⋊D3—124no—
212C2×C3⋊D3—248no—
213C2×C3⋊D3C2×S411yescharacteristic
214C2×C3×D3—48no—
215C2×C3×D3—124no—
216C3×A4—24no—
217C3×A4D3×D414yescomplemented
218D32—2416no—
219C32×C4—31no—
220C32×C4—44no—
221C32×C4—62no—
222C32×C4—84no—
223C32×C4C2×S411yescharacteristic, complemented
224C3⋊Dic3—31no—
225C3⋊Dic3—62no—
226C3⋊Dic3C2×S411yescharacteristic, complemented
227C3×Dic3—124no—
228C22×D4—272no—
229C22×D4—271no—
230C23×C4—91no—
231#121—31no—
232#121—272no—
233#121—271no—
234#121—272no—
235#121—271no—
236#121—271no—
237C2×C4⋊C4—271no—
238C2×C22⋊C4—272no—
239C33—41noSylow
240C23×C3—38no—
241C23×C3C2×D3214yes—
242C22×D3—38no—
243C22×D3—94no—
244C22×D3—94no—
245C22×D3—98no—
246C22×D3—98no—
247C22×D3—188no—
248C22×D3—1816no—
249C22×D3—188no—
250C22×D3—1816no—
251C22×D3—362no—
252C2×A4—24no—
253C2×A4—92no—
254C2×A4C22×C3⋊D311yescharacteristic
255S4—22no—
256S4—182no—
257S4—368no—
258C3×D4—64no—
259C3×D4—68no—
260C3×D4—68no—
261C3×D4—64no—
262C2×C3×C4—34no—
263C2×C3×C4—34no—
264C2×C3×C4—38no—
265C2×C3×C4—361no—
266#67—188no—
267#67—188no—
268#67—1816no—
269#67—188no—
270#67—188no—
271C2×Dic3—94no—
272C2×Dic3—94no—
273C2×Dic3—98no—
274D12—41no—
275D12—184no—
276D12—1816no—
277D12—184no—
278D12—361no—
279D12—362no—
280D12—724no—
281C4×D3—34no—
282C4×D3—94no—
283C4×D3—94no—
284C4×D3—94no—
285C4×D3—184no—
286C4×D3—1816no—
287C2×C32—31no—
288C2×C32—31no—
289C2×C32—44no—
290C2×C32—84no—
291C2×C32—122no—
292C2×C32C22×S411yescharacteristic
293C3⋊D3—32no—
294C3⋊D3—122no—
295C3⋊D3—248no—
296C3⋊D3—488no—
297C3⋊D3C4⋊S412yesdirect product factor
298C3×D3—88no—
299C3×D3—128no—
300C24—91no—
301C24—272no—
302C2×D4—31no—
303C2×D4—32no—
304C2×D4—271no—
305C2×D4—272no—
306C2×D4—272no—
307C2×D4—272no—
308C2×D4—272no—
309C2×D4—541no—
310C2×D4—541no—
311C2×D4—541no—
312C2×D4—541no—
313C2×D4—544no—
314C2×D4—544no—
315C22×C4—91no—
316C22×C4—271no—
317C22×C4—271no—
318C22×C4—271no—
319C22×C4—271no—
320C22×C4—272no—
321C22×C4D3×C3⋊D311yescharacteristic, complemented
322C4⋊C4—31no—
323C4⋊C4—272no—
324C4⋊C4—271no—
325C22⋊C4—32no—
326C22⋊C4—272no—
327C22⋊C4—272no—
328C22⋊C4—272no—
329C22×C3—34no—
330C22×C3—34no—
331C22×C3—68no—
332C22×C3—68no—
333C22×C3—68no—
334C22×C3—361no—
335C22×C3D3×D1214yescomplemented
336C2×D3—34no—
337C2×D3—42no—
338C2×D3—98no—
339C2×D3—98no—
340C2×D3—94no—
341C2×D3—98no—
342C2×D3—184no—
343C2×D3—188no—
344C2×D3—188no—
345C2×D3—188no—
346C2×D3—362no—
347C2×D3—3616no—
348C2×D3—3616no—
349C2×D3—728no—
350C2×D3—724no—
351A4—24no—
352A4D4×C3⋊D311yescharacteristic, complemented
353C3×C4—34no—
354C3×C4—41no—
355C3×C4—68no—
356C3×C4—84no—
357C3×C4—361no—
358C3×C4D3×S414yescomplemented
359Dic3—34no—
360Dic3—94no—
361Dic3—188no—
362C32—44no—
363C32—84no—
364C32C2×C4⋊S411yescharacteristic, complemented
365C23—32no—
366C23—92no—
367C23—271no—
368C23—271no—
369C23—272no—
370C23—272no—
371C23—542no—
372C23—544no—
373C23—542no—
374C23—544no—
375C23C2×D3×C3⋊D311yescharacteristic
376D4—61no—
377D4—62no—
378D4—62no—
379D4—61no—
380D4—541no—
381D4—544no—
382D4—542no—
383D4—542no—
384D4—542no—
385D4—544no—
386D4—542no—
387D4—541no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—32no—
391C2×C4—91no—
392C2×C4—271no—
393C2×C4—271no—
394C2×C4—271no—
395C2×C4—271no—
396C2×C4—271no—
397C2×C4—272no—
398C2×C4—541no—
399C2×C4—544no—
400C2×C3—34no—
401C2×C3—34no—
402C2×C3—41no—
403C2×C3—84no—
404C2×C3—128no—
405C2×C3—362no—
406C2×C3C2×D3×S414yes—
407D3—38no—
408D3—82no—
409D3—98no—
410D3—368no—
411D3—722no—
412D3—1448no—
413C22—31no—
414C22—31no—
415C22—62no—
416C22—62no—
417C22—62no—
418C22—91no—
419C22—272no—
420C22—272no—
421C22—272no—
422C22—271no—
423C22—541no—
424C22—542no—
425C22—542no—
426C22—542no—
427C22—1084no—
428C22—1084no—
429C22C3⋊D3×D1211yescharacteristic, complemented
430C4—31no—
431C4—62no—
432C4—91no—
433C4—271no—
434C4—542no—
435C4C3⋊D3×S411yescharacteristic, complemented
436C3—41no—
437C3—84no—
438C3D3×C4⋊S414yescomplemented
439C2—31no—
440C2—31no—
441C2—92no—
442C2—122no—
443C2—272no—
444C2—1082no—
445C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×S4
commutator, cosocleC2×C32×A4C23
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
hypercenterC4C3⋊D3×S4
hypocenterC32×A4C2×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×S4C2semidirect
2. C4×A4×C3⋊D3C2semidirect
3. C4⋊(C3⋊(C3⋊S4))C2semidirect
4. C32×C4⋊S4C2semidirect
5. #11724187C2semidirect
9. C2×C3⋊(C3⋊S4)C22semidirect
10. C2×C32×S4C22semidirect
12. C32×C4×A4C22semidirect
14. A4×C3⋊Dic3C22semidirect
18. C22×C4×C3⋊D3D3semidirect
22. C3×C4⋊S4D3semidirect
34. A4×C3⋊D3D4semidirect
48. C2×C3×S4C2×D3semidirect
52. C22×C32×C4C2×D3semidirect
56. C22×C3⋊Dic3C2×D3semidirect
71. C3×C4×A4C2×D3semidirect
84. C32×A4C2×D4semidirect
96. C4⋊S4C3⋊D3direct (N × H)
115. C22×C3⋊D3D12semidirect
148. C4×C3⋊D3S4semidirect
163. C2×S4C2×C3⋊D3semidirect
166. C22×C3×C4D32semidirect
187. C4×A4C2×C3⋊D3semidirect
200. C22×C32C2×D12semidirect
217. C3×A4D3×D4semidirect
223. C32×C4C2×S4semidirect
226. C3⋊Dic3C2×S4semidirect
297. C3⋊D3C4⋊S4direct (N × H)
321. C22×C4D3×C3⋊D3semidirect
335. C22×C3D3×D12semidirect
352. A4D4×C3⋊D3semidirect
358. C3×C4D3×S4semidirect
364. C32C2×C4⋊S4semidirect
429. C22C3⋊D3×D12semidirect
435. C4C3⋊D3×S4semidirect
438. C3D3×C4⋊S4semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×S42
C4×A4×C3⋊D32
C4⋊(C3⋊(C3⋊S4))2
C32×C4⋊S42
#117241872
D3×C4⋊S43
C3⋊D3×#1213
C3⋊D3×D124