Finite Group Databasebeta

← C4×C3⋊D3×S4 #13180743

Subgroups — C4×C3⋊D3×S4 #13180743

Order 1728.

1526 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 113 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×S4C211yesmaximal, characteristic
2C4×A4×C3⋊D3C211yesmaximal, characteristic, complemented
3C4×C3⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
4C32×C4×S4C211yesmaximal, characteristic, complemented
5C3⋊D3×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6#11724186C211yesmaximal, characteristic, complemented
7S4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8C4×D3×S4—34nomaximal
9C4×D4×C3⋊D3—31nomaximal
10C2×A4×C3⋊D3C2211yescharacteristic
11C2×C3⋊(C3⋊S4)C2211yescharacteristic
12C2×C32×S4C2211yescharacteristic
13C3⋊D3×S4C414yesdirect product factor
14C32×C4×A4C2211yescharacteristic, complemented
15C4×D3×C3⋊D3—41nomaximal
16A4×C3⋊Dic3C2211yescharacteristic, complemented
17C3⋊(C3⋊SL(2,ℤ/4))C2211yescharacteristic, complemented
18C32×SL(2,ℤ/4)C2211yescharacteristic, complemented
19C2×D3×S4—34no—
20C2×D4×C3⋊D3—31no—
21C22×C4×C3⋊D3—31no—
22C22×C4×C3⋊D3D311yescharacteristic, complemented
23C4×D3×A4—34no—
24C4×C3⋊S4—34no—
25C4×C3⋊S4—64no—
26C3×C4×S4D314yescomplemented
27D3×SL(2,ℤ/4)—34no—
28(C3⋊S4)⋊C4—34no—
29Dic3×S4—34no—
30C32×C4×D4—31no—
31D4×C3⋊Dic3—31no—
32C4×#638—31no—
33C4×#638—31no—
34#64102—31no—
35C4⋊C4×C3⋊D3—31no—
36#64090—31no—
37#64090—31no—
38C22⋊C4×C3⋊D3—31no—
39C22⋊C4×C3⋊D3—31no—
40C4×C4⋊(C3⋊D3)—31no—
41C42×C3⋊D3—31no—
42C2×C32×A4C2311yescharacteristic
43C2×D3×C3⋊D3—41no—
44A4×C3⋊D3C2×C412yescomplemented
45C3⋊(C3⋊S4)C2×C412yescomplemented
46C32×S4C2×C412yescomplemented
47C4×C3⋊(C3⋊D3)—41no—
48C3×C4×C3⋊D3—41no—
49C32×C4×D3—41no—
50#6289—41no—
51Dic3×C3⋊D3—41no—
52D3×C3⋊Dic3—41no—
53C2×C4×S4—91no—
54C4×D3×D4—94no—
55C23×C3⋊D3—31no—
56C23×C3⋊D3C2×D311yescharacteristic
57C2×D3×A4—34no—
58C2×C3⋊S4—34no—
59C2×C3⋊S4—64no—
60C2×C3×S4C2×D314yes—
61D3×S4—316no—
62C2×C32×D4—31no—
63C22×C32×C4—31no—
64C22×C32×C4C2×D311yescharacteristic, complemented
65C2×#638—31no—
66C2×#638—31no—
67C22×C3⋊Dic3—31no—
68C22×C3⋊Dic3C2×D311yescharacteristic, complemented
69D4×C3⋊D3—34no—
70D4×C3⋊D3—34no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4×C3⋊D3—31no—
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—31no—
75C2×C4×C3⋊D3—31no—
76C2×C4×C3⋊D3—31no—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—31no—
79C2×C4×C3⋊D3—31no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3—61no—
82C2×C4×C3⋊D3—61no—
83C2×C4×C3⋊D3—61no—
84C2×C4×C3⋊D3—61no—
85C3×C4×A4—24no—
86C3×C4×A4C2×D314yescomplemented
87C4×D32—124no—
88Dic3×A4—34no—
89C3⋊SL(2,ℤ/4)—34no—
90C3⋊SL(2,ℤ/4)—64no—
91C3×SL(2,ℤ/4)C2×D314yescomplemented
92C32×C4⋊C4—31no—
93C32×C22⋊C4—31no—
94C32×C22⋊C4—31no—
95C32×C42—31no—
96#3757—31no—
97#3757—31no—
98#3752—31no—
99#3752—31no—
100#3752—31no—
101#3752—31no—
102C4⋊(C3⋊Dic3)—31no—
103#3750—31no—
104#3750—31no—
105C4×C3⋊Dic3—31no—
106C4×C3⋊Dic3—31no—
107C4×C3⋊Dic3—31no—
108C2×C3⋊(C3⋊D3)—41no—
109C2×C3×C3⋊D3—41no—
110C2×C32×D3—41no—
111C32×A4C22×C411yescharacteristic, complemented
112D3×C3⋊D3—44no—
113C33×C4—41no—
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—41no—
117C22×S4—91no—
118C2×D3×D4—94no—
119C22×C4×D3—34no—
120C22×C4×D3—94no—
121C2×C4×A4—91no—
122C2×SL(2,ℤ/4)—91no—
123C4×S4—91no—
124C4×S4—91no—
125C4×S4—91no—
126C4×S4—184no—
127C4×S4C3⋊D311yescharacteristic, direct product factor
128C3×C4×D4—34no—
129D4×Dic3—94no—
130C4×#67—94no—
131C4×#67—94no—
132#893—94no—
133D3×C4⋊C4—94no—
134#67⋊C4—94no—
135#67⋊C4—94no—
136D3×C22⋊C4—94no—
137D3×C22⋊C4—94no—
138C4×D12—94no—
139C42×D3—94no—
140C23×C32—31no—
141C23×C32C22×D311yescharacteristic
142C22×C3⋊D3—31no—
143C22×C3⋊D3—31no—
144C22×C3⋊D3—32no—
145C22×C3⋊D3—31no—
146C22×C3⋊D3—34no—
147C22×C3⋊D3—62no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—61no—
150C22×C3⋊D3C4×D312yescomplemented
151C2×C3×A4—24no—
152C2×C3×A4C22×D314yes—
153C2×D32—124no—
154D3×A4—38no—
155C3⋊S4—38no—
156C3⋊S4—68no—
157C3×S4C4×D318yescomplemented
158C32×D4—32no—
159C32×D4—32no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—31no—
164C2×C32×C4—31no—
165C2×C32×C4—61no—
166C2×C32×C4—61no—
167#638—34no—
168#638—32no—
169#638—32no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C4⋊(C3⋊D3)—34no—
178C4×C3⋊D3—34no—
179C4×C3⋊D3—31no—
180C4×C3⋊D3—31no—
181C4×C3⋊D3—31no—
182C4×C3⋊D3—61no—
183C4×C3⋊D3—61no—
184C4×C3⋊D3—61no—
185C4×C3⋊D3—62no—
186C4×C3⋊D3—61no—
187C4×C3⋊D3—61no—
188C4×C3⋊D3—61no—
189C4×C3⋊D3—61no—
190C4×C3⋊D3—124no—
191C4×C3⋊D3—244no—
192C4×C3⋊D3S411yescharacteristic, direct product factor
193C3×C4×D3—44no—
194C3×C4×D3—124no—
195#624—124no—
196D3×Dic3—124no—
197D3×Dic3—124no—
198C2×C4×D4—271noSylow
199C2×C33—41no—
200C3⋊(C3⋊D3)—42no—
201C3×C3⋊D3—42no—
202C32×D3—42no—
203C23×D3—34no—
204C23×D3—94no—
205C22×A4—91no—
206C2×S4—91no—
207C2×S4—94no—
208C2×S4—184no—
209C2×S4C2×C3⋊D311yescharacteristic
210C2×C3×D4—34no—
211C22×C3×C4—34no—
212C22×C3×C4D3214yescomplemented
213C2×#67—94no—
214C2×#67—94no—
215C22×Dic3—34no—
216C22×Dic3—94no—
217D3×D4—916no—
218D3×D4—916no—
219C2×D12—94no—
220C2×C4×D3—94no—
221C2×C4×D3—94no—
222C2×C4×D3—94no—
223C2×C4×D3—94no—
224C2×C4×D3—94no—
225C2×C4×D3—94no—
226C2×C4×D3—94no—
227C2×C4×D3—94no—
228C2×C4×D3—94no—
229C2×C4×D3—184no—
230C2×C4×D3—184no—
231C2×C4×D3—184no—
232C2×C4×D3—184no—
233C2×C4×D3—361no—
234C4×A4—24no—
235C4×A4—91no—
236C4×A4C2×C3⋊D311yescharacteristic, complemented
237SL(2,ℤ/4)—91no—
238SL(2,ℤ/4)—184no—
239SL(2,ℤ/4)C2×C3⋊D311yescharacteristic, complemented
240C3×C4⋊C4—34no—
241C3×C22⋊C4—34no—
242C3×C22⋊C4—34no—
243C3×C42—34no—
244#217—94no—
245#217—94no—
246(C2×D3)⋊C4—94no—
247(C2×D3)⋊C4—94no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—94no—
250C4⋊Dic3—94no—
251Dic3⋊C4—94no—
252Dic3⋊C4—94no—
253C4×Dic3—94no—
254C4×Dic3—94no—
255C4×Dic3—94no—
256C22×C32—31no—
257C22×C32—31no—
258C22×C32—32no—
259C22×C32—61no—
260C22×C32—61no—
261C22×C32C2×C4×D311yescharacteristic, complemented
262C2×C3⋊D3—31no—
263C2×C3⋊D3—32no—
264C2×C3⋊D3—32no—
265C2×C3⋊D3—32no—
266C2×C3⋊D3—32no—
267C2×C3⋊D3—61no—
268C2×C3⋊D3—64no—
269C2×C3⋊D3—61no—
270C2×C3⋊D3—61no—
271C2×C3⋊D3—64no—
272C2×C3⋊D3—124no—
273C2×C3⋊D3—244no—
274C2×C3⋊D3C2×S411yescharacteristic
275C2×C3×D3—44no—
276C2×C3×D3—124no—
277C3×A4—24no—
278C3×A4C2×C4×D314yescomplemented
279D32—1216no—
280C32×C4—32no—
281C32×C4—31no—
282C32×C4—44no—
283C32×C4—61no—
284C32×C4—61no—
285C32×C4—84no—
286C32×C4C2×S411yescharacteristic, complemented
287C3⋊Dic3—32no—
288C3⋊Dic3—31no—
289C3⋊Dic3—61no—
290C3⋊Dic3—61no—
291C3⋊Dic3—124no—
292C3⋊Dic3—244no—
293C3⋊Dic3C2×S411yescharacteristic, complemented
294C3×Dic3—44no—
295C3×Dic3—124no—
296C22×D4—271no—
297C23×C4—91no—
298C23×C4—271no—
299C4×D4—31no—
300C4×D4—271no—
301C4×D4—271no—
302C4×D4—271no—
303C4×D4—271no—
304C4×D4—271no—
305C4×D4—271no—
306C4×D4—271no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—271no—
309C2×C22⋊C4—271no—
310C2×C42—271no—
311C33—41noSylow
312C23×C3—34no—
313C23×C3C2×D3214yes—
314C22×D3—38no—
315C22×D3—94no—
316C22×D3—94no—
317C22×D3—98no—
318C22×D3—94no—
319C22×D3—916no—
320C22×D3—184no—
321C22×D3—184no—
322C22×D3—188no—
323C22×D3—361no—
324C2×A4—24no—
325C2×A4—92no—
326C2×A4C22×C3⋊D311yescharacteristic
327S4—92no—
328S4—188no—
329S4C4×C3⋊D312yesdirect product factor
330C3×D4—38no—
331C3×D4—38no—
332C2×C3×C4—34no—
333C2×C3×C4—34no—
334C2×C3×C4—34no—
335C2×C3×C4—34no—
336C2×C3×C4—34no—
337C2×C3×C4—64no—
338C2×C3×C4—64no—
339C2×C3×C4—361no—
340#67—916no—
341#67—98no—
342#67—98no—
343C2×Dic3—94no—
344C2×Dic3—94no—
345C2×Dic3—94no—
346C2×Dic3—94no—
347C2×Dic3—94no—
348C2×Dic3—184no—
349C2×Dic3—184no—
350C2×Dic3—361no—
351D12—916no—
352C4×D3—34no—
353C4×D3—41no—
354C4×D3—94no—
355C4×D3—94no—
356C4×D3—916no—
357C4×D3—94no—
358C4×D3—184no—
359C4×D3—184no—
360C4×D3—184no—
361C4×D3—184no—
362C4×D3—184no—
363C4×D3—188no—
364C4×D3—184no—
365C4×D3—184no—
366C4×D3—361no—
367C4×D3—361no—
368C4×D3—361no—
369C4×D3—724no—
370C2×C32—31no—
371C2×C32—31no—
372C2×C32—44no—
373C2×C32—62no—
374C2×C32—84no—
375C2×C32C22×S411yescharacteristic
376C3⋊D3—32no—
377C3⋊D3—62no—
378C3⋊D3—128no—
379C3⋊D3—248no—
380C3⋊D3C4×S412yesdirect product factor
381C3×D3—48no—
382C3×D3—128no—
383C24—91no—
384C24—271no—
385C2×D4—31no—
386C2×D4—271no—
387C2×D4—274no—
388C2×D4—274no—
389C2×D4—271no—
390C2×D4—271no—
391C22×C4—31no—
392C22×C4—91no—
393C22×C4—271no—
394C22×C4—271no—
395C22×C4—271no—
396C22×C4—271no—
397C22×C4—271no—
398C22×C4—271no—
399C22×C4—271no—
400C22×C4—271no—
401C22×C4—271no—
402C22×C4—271no—
403C22×C4—541no—
404C22×C4—541no—
405C22×C4—541no—
406C22×C4—541no—
407C22×C4D3×C3⋊D311yescharacteristic, complemented
408C4⋊C4—31no—
409C4⋊C4—271no—
410C4⋊C4—271no—
411C4⋊C4—271no—
412C22⋊C4—31no—
413C22⋊C4—31no—
414C22⋊C4—271no—
415C22⋊C4—271no—
416C22⋊C4—271no—
417C22⋊C4—271no—
418C22⋊C4—271no—
419C22⋊C4—271no—
420C42—31no—
421C42—271no—
422C42—271no—
423C42—271no—
424C22×C3—34no—
425C22×C3—34no—
426C22×C3—38no—
427C22×C3—64no—
428C22×C3—64no—
429C22×C3—361no—
430C22×C3C4×D3214yescomplemented
431C2×D3—34no—
432C2×D3—41no—
433C2×D3—98no—
434C2×D3—98no—
435C2×D3—94no—
436C2×D3—98no—
437C2×D3—98no—
438C2×D3—1816no—
439C2×D3—184no—
440C2×D3—1816no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—364no—
444C2×D3—361no—
445C2×D3—724no—
446A4—24no—
447A4C2×C4×C3⋊D311yescharacteristic, complemented
448C3×C4—38no—
449C3×C4—34no—
450C3×C4—41no—
451C3×C4—64no—
452C3×C4—64no—
453C3×C4—84no—
454C3×C4—361no—
455C3×C4D3×S414yescomplemented
456Dic3—34no—
457Dic3—41no—
458Dic3—98no—
459Dic3—94no—
460Dic3—184no—
461Dic3—184no—
462Dic3—361no—
463Dic3—724no—
464C32—44no—
465C32—84no—
466C32C2×C4×S411yescharacteristic, complemented
467C23—31no—
468C23—92no—
469C23—271no—
470C23—271no—
471C23—272no—
472C23—271no—
473C23—274no—
474C23—541no—
475C23—541no—
476C23—542no—
477C23C2×D3×C3⋊D311yescharacteristic
478D4—32no—
479D4—32no—
480D4—274no—
481D4—272no—
482D4—274no—
483D4—272no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—31no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—91no—
492C2×C4—271no—
493C2×C4—271no—
494C2×C4—274no—
495C2×C4—271no—
496C2×C4—271no—
497C2×C4—271no—
498C2×C4—271no—
499C2×C4—271no—
500C2×C4—271no—
501C2×C4—541no—
502C2×C4—541no—
503C2×C4—541no—
504C2×C4—541no—
505C2×C4—541no—
506C2×C4—541no—
507C2×C4—542no—
508C2×C4—541no—
509C2×C4—541no—
510C2×C4—541no—
511C2×C3—34no—
512C2×C3—34no—
513C2×C3—41no—
514C2×C3—68no—
515C2×C3—84no—
516C2×C3—362no—
517C2×C3C2×D3×S414yes—
518D3—38no—
519D3—42no—
520D3—98no—
521D3—188no—
522D3—362no—
523D3—728no—
524C22—31no—
525C22—31no—
526C22—32no—
527C22—61no—
528C22—61no—
529C22—91no—
530C22—272no—
531C22—272no—
532C22—272no—
533C22—272no—
534C22—271no—
535C22—541no—
536C22—541no—
537C22—544no—
538C22—544no—
539C22—541no—
540C22C4×D3×C3⋊D311yescharacteristic, complemented
541C4—32no—
542C4—31no—
543C4—61no—
544C4—61no—
545C4—91no—
546C4—272no—
547C4—271no—
548C4—541no—
549C4—541no—
550C4C3⋊D3×S411yescharacteristic, direct product factor
551C3—41no—
552C3—84no—
553C3C4×D3×S414yescomplemented
554C2—31no—
555C2—31no—
556C2—62no—
557C2—92no—
558C2—272no—
559C2—542no—
560C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊D3×S4
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×C3⋊D3×S4
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
prime radicalC2×C3⋊D3×S4C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4×A4×C3⋊D3C2semidirect
3. C4×C3⋊(C3⋊S4)C2semidirect
4. C32×C4×S4C2semidirect
5. C3⋊D3×SL(2,ℤ/4)C2semidirect
6. #11724186C2semidirect
7. S4×C3⋊Dic3C2semidirect
13. C3⋊D3×S4C4direct (N × H)
14. C32×C4×A4C22semidirect
16. A4×C3⋊Dic3C22semidirect
17. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
18. C32×SL(2,ℤ/4)C22semidirect
22. C22×C4×C3⋊D3D3semidirect
26. C3×C4×S4D3semidirect
44. A4×C3⋊D3C2×C4semidirect
45. C3⋊(C3⋊S4)C2×C4semidirect
46. C32×S4C2×C4semidirect
64. C22×C32×C4C2×D3semidirect
68. C22×C3⋊Dic3C2×D3semidirect
86. C3×C4×A4C2×D3semidirect
91. C3×SL(2,ℤ/4)C2×D3semidirect
111. C32×A4C22×C4semidirect
127. C4×S4C3⋊D3direct (N × H)
150. C22×C3⋊D3C4×D3semidirect
157. C3×S4C4×D3semidirect
192. C4×C3⋊D3S4direct (N × H)
212. C22×C3×C4D32semidirect
236. C4×A4C2×C3⋊D3semidirect
239. SL(2,ℤ/4)C2×C3⋊D3semidirect
261. C22×C32C2×C4×D3semidirect
278. C3×A4C2×C4×D3semidirect
286. C32×C4C2×S4semidirect
293. C3⋊Dic3C2×S4semidirect
329. S4C4×C3⋊D3direct (N × H)
380. C3⋊D3C4×S4direct (N × H)
407. C22×C4D3×C3⋊D3semidirect
430. C22×C3C4×D32semidirect
447. A4C2×C4×C3⋊D3semidirect
455. C3×C4D3×S4semidirect
466. C32C2×C4×S4semidirect
540. C22C4×D3×C3⋊D3semidirect
550. C4C3⋊D3×S4direct (N × H)
553. C3C4×D3×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
C3⋊D3×SL(2,ℤ/4)2
#117241862
S4×C3⋊Dic32
C4×D3×S43
C4×D4×C3⋊D33
C4×D3×C3⋊D34