Finite Group Databasebeta

← S4×C4⋊(C3⋊D3) #13180745

Subgroups — S4×C4⋊(C3⋊D3) #13180745

Order 1728.

1532 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 101 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
2A4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
4C32×C4×S4C211yesmaximal, characteristic, complemented
5#11724190C212yesmaximal, complemented
6D12×S4—34nomaximal
7D4×C4⋊(C3⋊D3)—31nomaximal
8C2×A4×C3⋊D3C2212yescomplemented
9C2×C3⋊(C3⋊S4)C2212yescomplemented
10C2×C32×S4C2211yescharacteristic
11C3⋊D3×S4—24no—
12C32×C4×A4C2211yescharacteristic, complemented
13D3×C4⋊(C3⋊D3)—41nomaximal
14C32×SL(2,ℤ/4)C2211yescharacteristic, complemented
15C2×D3×S4—38no—
16C2×D4×C3⋊D3—32no—
17C22×C4⋊(C3⋊D3)—31no—
18C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
19A4×D12—34no—
20C4⋊(C3⋊S4)—34no—
21C4⋊(C3⋊S4)—64no—
22C3×C4×S4D314yescomplemented
23#64210—38no—
24C32×C4×D4—31no—
25#64139—31no—
26#64139—31no—
27#64104—32no—
28#64091—32no—
29#64091—32no—
30#64083—31no—
31C4×C4⋊(C3⋊D3)—31no—
32C2×C32×A4C2311yescharacteristic
33C2×D3×C3⋊D3—42no—
34A4×C3⋊D3—22no—
35C3⋊(C3⋊S4)—22no—
36C32×S4D412yescomplemented
37C4⋊(C3⋊(C3⋊D3))—41no—
38C3×C4⋊(C3⋊D3)—41no—
39C32×C4×D3—41no—
40#6292—42no—
41D4×S4—91no—
42D4×D12—94no—
43C23×C3⋊D3—32no—
44C23×C3⋊D3C2×D312yescomplemented
45C2×D3×A4—38no—
46C2×C3⋊S4—38no—
47C2×C3⋊S4—68no—
48C2×C3×S4C2×D314yes—
49D3×S4—616no—
50C2×C32×D4—31no—
51C22×C32×C4—31no—
52C22×C32×C4C2×D311yescharacteristic, complemented
53C2×#638—32no—
54C2×#638—32no—
55D4×C3⋊D3—64no—
56D4×C3⋊D3—64no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—32no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—32no—
62C2×C4⋊(C3⋊D3)—32no—
63C2×C4⋊(C3⋊D3)—32no—
64C2×C4⋊(C3⋊D3)—62no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4×C3⋊D3—32no—
68C3×C4×A4—24no—
69C3×C4×A4C2×D314yescomplemented
70D3×D12—124no—
71C3×SL(2,ℤ/4)C2×D314yescomplemented
72C32×C4⋊C4—31no—
73C32×C22⋊C4—31no—
74C32×C22⋊C4—31no—
75C32×C42—31no—
76#3752—32no—
77#3752—32no—
78#3752—32no—
79C4⋊(C3⋊Dic3)—31no—
80C2×C3⋊(C3⋊D3)—42no—
81C2×C3×C3⋊D3—42no—
82C2×C32×D3—41no—
83C32×A4C2×D411yescharacteristic, complemented
84D3×C3⋊D3—84no—
85C33×C4—41no—
86C32×Dic3—41no—
87C22×S4—92no—
88C2×D3×D4—98no—
89C22×D12—34no—
90C22×D12—94no—
91D4×A4—91no—
92GL(2,ℤ/4)—92no—
93C4⋊S4—91no—
94C4⋊S4—184no—
95C4×S4C3⋊D311yescharacteristic, complemented
96C3×C4×D4—34no—
97#930—94no—
98#930—94no—
99#895—98no—
100#882—98no—
101#882—98no—
102#874—94no—
103C4×D12—94no—
104C23×C32—31no—
105C23×C32C22×D311yescharacteristic
106C22×C3⋊D3—22no—
107C22×C3⋊D3—32no—
108C22×C3⋊D3—32no—
109C22×C3⋊D3—32no—
110C22×C3⋊D3—62no—
111C22×C3⋊D3—62no—
112C22×C3⋊D3—64no—
113C22×C3⋊D3—62no—
114C22×C3⋊D3—122no—
115C2×C3×A4—24no—
116C2×C3×A4C22×D314yes—
117C2×D32—128no—
118D3×A4—68no—
119C3⋊S4—68no—
120C3⋊S4—128no—
121C3×S4D1218yescomplemented
122C32×D4—32no—
123C32×D4—32no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—31no—
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—61no—
130C2×C32×C4—61no—
131#638—64no—
132#638—62no—
133#638—62no—
134C2×C3⋊Dic3—32no—
135C4⋊(C3⋊D3)—32no—
136C4⋊(C3⋊D3)—31no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—62no—
141C4⋊(C3⋊D3)—62no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—124no—
144C4⋊(C3⋊D3)—122no—
145C4⋊(C3⋊D3)—244no—
146C4⋊(C3⋊D3)S411yescharacteristic, direct product factor
147C4×C3⋊D3—64no—
148C3×D12—124no—
149C3×C4×D3—44no—
150#626—128no—
151D42—271noSylow
152C2×C33—41no—
153C3⋊(C3⋊D3)—82no—
154C3×C3⋊D3—82no—
155C32×D3—42no—
156C23×D3—38no—
157C23×D3—98no—
158C22×A4—92no—
159C2×S4—92no—
160C2×S4—188no—
161C2×S4—184no—
162C2×S4C2×C3⋊D311yescharacteristic
163C2×C3×D4—34no—
164C22×C3×C4—34no—
165C22×C3×C4D3214yescomplemented
166C2×#67—98no—
167C2×#67—98no—
168D3×D4—1816no—
169D3×D4—1816no—
170D3×D4—361no—
171C2×D12—94no—
172C2×D12—94no—
173C2×D12—98no—
174C2×D12—98no—
175C2×D12—94no—
176C2×D12—98no—
177C2×D12—98no—
178C2×D12—184no—
179C2×D12—188no—
180C2×D12—184no—
181C2×C4×D3—98no—
182C4×A4—24no—
183C4×A4C2×C3⋊D311yescharacteristic, complemented
184SL(2,ℤ/4)C2×C3⋊D311yescharacteristic, complemented
185C3×C4⋊C4—34no—
186C3×C22⋊C4—34no—
187C3×C22⋊C4—34no—
188C3×C42—34no—
189(C2×D3)⋊C4—98no—
190(C2×D3)⋊C4—98no—
191(C2×D3)⋊C4—98no—
192C4⋊Dic3—94no—
193C22×C32—31no—
194C22×C32—31no—
195C22×C32—32no—
196C22×C32—61no—
197C22×C32—61no—
198C22×C32D3×D411yescharacteristic, complemented
199C2×C3⋊D3—32no—
200C2×C3⋊D3—62no—
201C2×C3⋊D3—62no—
202C2×C3⋊D3—62no—
203C2×C3⋊D3—62no—
204C2×C3⋊D3—62no—
205C2×C3⋊D3—62no—
206C2×C3⋊D3—62no—
207C2×C3⋊D3—128no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—124no—
210C2×C3⋊D3—248no—
211C2×C3⋊D3C2×S412yescomplemented
212C2×C3×D3—44no—
213C2×C3×D3—128no—
214C3×A4—24no—
215C3×A4C2×D1214yescomplemented
216D32—2416no—
217C32×C4—32no—
218C32×C4—31no—
219C32×C4—44no—
220C32×C4—61no—
221C32×C4—61no—
222C32×C4—84no—
223C32×C4C2×S411yescharacteristic, complemented
224C3⋊Dic3—62no—
225C3×Dic3—44no—
226C22×D4—91no—
227C22×D4—271no—
228C22×D4—272no—
229#127—271no—
230#121—272no—
231#121—271no—
232#121—271no—
233#120—272no—
234#120—272no—
235C4×D4—31no—
236C4×D4—271no—
237C33—41noSylow
238C23×C3—34no—
239C23×C3C2×D3214yes—
240C22×D3—68no—
241C22×D3—98no—
242C22×D3—98no—
243C22×D3—98no—
244C22×D3—188no—
245C22×D3—188no—
246C22×D3—188no—
247C22×D3—1816no—
248C22×D3—362no—
249C22×D3—368no—
250C2×A4—24no—
251C2×A4—182no—
252C2×A4C22×C3⋊D311yescharacteristic
253S4—182no—
254S4—368no—
255S4C4⋊(C3⋊D3)12yesdirect product factor
256C3×D4—38no—
257C3×D4—38no—
258C3×D4—361no—
259C2×C3×C4—34no—
260C2×C3×C4—34no—
261C2×C3×C4—34no—
262C2×C3×C4—34no—
263C2×C3×C4—34no—
264C2×C3×C4—64no—
265C2×C3×C4—64no—
266#67—1816no—
267#67—188no—
268#67—188no—
269#67—362no—
270C2×Dic3—98no—
271D12—34no—
272D12—98no—
273D12—94no—
274D12—184no—
275D12—184no—
276D12—1816no—
277D12—188no—
278D12—188no—
279D12—184no—
280D12—361no—
281D12—368no—
282D12—724no—
283C4×D3—41no—
284C4×D3—1816no—
285C2×C32—31no—
286C2×C32—31no—
287C2×C32—44no—
288C2×C32—62no—
289C2×C32—84no—
290C2×C32C22×S411yescharacteristic
291C3⋊D3—22no—
292C3⋊D3—62no—
293C3⋊D3—122no—
294C3⋊D3—248no—
295C3⋊D3—488no—
296C3×D3—48no—
297C3×D3—248no—
298C24—92no—
299C24—272no—
300C2×D4—31no—
301C2×D4—271no—
302C2×D4—272no—
303C2×D4—271no—
304C2×D4—272no—
305C2×D4—272no—
306C2×D4—271no—
307C2×D4—272no—
308C2×D4—272no—
309C2×D4—272no—
310C2×D4—541no—
311C2×D4—542no—
312C2×D4—541no—
313C2×D4—544no—
314C2×D4—544no—
315C22×C4—31no—
316C22×C4—272no—
317C22×C4D3×C3⋊D311yescharacteristic, complemented
318C4⋊C4—31no—
319C4⋊C4—271no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—272no—
323C22⋊C4—272no—
324C22⋊C4—272no—
325C42—31no—
326C22×C3—34no—
327C22×C3—34no—
328C22×C3—38no—
329C22×C3—64no—
330C22×C3—64no—
331C22×C3—362no—
332C22×C3D3×D1214yescomplemented
333C2×D3—38no—
334C2×D3—41no—
335C2×D3—98no—
336C2×D3—188no—
337C2×D3—188no—
338C2×D3—188no—
339C2×D3—188no—
340C2×D3—188no—
341C2×D3—188no—
342C2×D3—188no—
343C2×D3—362no—
344C2×D3—3616no—
345C2×D3—3616no—
346C2×D3—728no—
347C2×D3—724no—
348A4—24no—
349A4C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
350C3×C4—38no—
351C3×C4—34no—
352C3×C4—41no—
353C3×C4—64no—
354C3×C4—64no—
355C3×C4—84no—
356C3×C4D3×S414yescomplemented
357Dic3—41no—
358Dic3—188no—
359C32—44no—
360C32—84no—
361C32D4×S411yescharacteristic, complemented
362C23—31no—
363C23—182no—
364C23—272no—
365C23—272no—
366C23—272no—
367C23—542no—
368C23—544no—
369C23—542no—
370C23—542no—
371C23—1082no—
372C23C2×D3×C3⋊D311yescharacteristic
373D4—32no—
374D4—32no—
375D4—91no—
376D4—272no—
377D4—271no—
378D4—541no—
379D4—544no—
380D4—541no—
381D4—542no—
382D4—542no—
383D4—542no—
384D4—544no—
385D4—542no—
386D4—541no—
387D4—1082no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—272no—
396C2×C4—544no—
397C2×C3—34no—
398C2×C3—34no—
399C2×C3—41no—
400C2×C3—68no—
401C2×C3—84no—
402C2×C3—722no—
403C2×C3C2×D3×S414yes—
404D3—42no—
405D3—68no—
406D3—188no—
407D3—368no—
408D3—722no—
409D3—1448no—
410C22—31no—
411C22—31no—
412C22—32no—
413C22—61no—
414C22—61no—
415C22—92no—
416C22—272no—
417C22—542no—
418C22—542no—
419C22—542no—
420C22—542no—
421C22—542no—
422C22—542no—
423C22—542no—
424C22—1084no—
425C22—1084no—
426C22D3×C4⋊(C3⋊D3)11yescharacteristic, complemented
427C4—32no—
428C4—31no—
429C4—61no—
430C4—61no—
431C4—542no—
432C4C3⋊D3×S411yescharacteristic, complemented
433C3—41no—
434C3—84no—
435C3D12×S414yescomplemented
436C2—31no—
437C2—31no—
438C2—62no—
439C2—182no—
440C2—542no—
441C2—1082no—
442C2C2×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. A4×C4⋊(C3⋊D3)C2semidirect
3. C4⋊(C3⋊(C3⋊S4))C2semidirect
4. C32×C4×S4C2semidirect
5. #11724190C2semidirect
8. C2×A4×C3⋊D3C22semidirect
9. C2×C3⋊(C3⋊S4)C22semidirect
12. C32×C4×A4C22semidirect
14. C32×SL(2,ℤ/4)C22semidirect
18. C22×C4⋊(C3⋊D3)D3semidirect
22. C3×C4×S4D3semidirect
36. C32×S4D4semidirect
44. C23×C3⋊D3C2×D3semidirect
52. C22×C32×C4C2×D3semidirect
69. C3×C4×A4C2×D3semidirect
71. C3×SL(2,ℤ/4)C2×D3semidirect
83. C32×A4C2×D4semidirect
95. C4×S4C3⋊D3semidirect
121. C3×S4D12semidirect
146. C4⋊(C3⋊D3)S4direct (N × H)
165. C22×C3×C4D32semidirect
183. C4×A4C2×C3⋊D3semidirect
184. SL(2,ℤ/4)C2×C3⋊D3semidirect
198. C22×C32D3×D4semidirect
211. C2×C3⋊D3C2×S4semidirect
215. C3×A4C2×D12semidirect
223. C32×C4C2×S4semidirect
255. S4C4⋊(C3⋊D3)direct (N × H)
317. C22×C4D3×C3⋊D3semidirect
332. C22×C3D3×D12semidirect
349. A4C2×C4⋊(C3⋊D3)semidirect
356. C3×C4D3×S4semidirect
361. C32D4×S4semidirect
426. C22D3×C4⋊(C3⋊D3)semidirect
432. C4C3⋊D3×S4semidirect
435. C3D12×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
A4×C4⋊(C3⋊D3)2
C4⋊(C3⋊(C3⋊S4))2
C32×C4×S42
#117241902
D12×S43
D4×C4⋊(C3⋊D3)3
D3×C4⋊(C3⋊D3)4