Finite Group Databasebeta

← C3×(C3⋊S4)⋊Q8 #13180681

Subgroups — C3×(C3⋊S4)⋊Q8 #13180681

Order 1728.

734 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C3×A4×Dic6C211yesmaximal, characteristic, complemented
2C3×C4×C3⋊S4C211yesmaximal, characteristic
3C32×A4⋊Q8C211yesmaximal, characteristic, complemented
4C3×(C3⋊S4)⋊C4C212yesmaximal
5C3×#64204C212yesmaximal, complemented
6C3×Q8×S4—31nomaximal
7(C3⋊S4)⋊Q8C311yesmaximal, characteristic, direct product factor
8C3×#5632—31nomaximal
9C2×C3×C3⋊S4C2211yescharacteristic
10C32×C4×A4C2211yescharacteristic
11C3×#3797—41nomaximal
12C3×Dic3×A4C2212yes—
13C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic
14C32×SL(2,ℤ/4)C2212yes—
15C2×C3×D3×Q8—31no—
16C22×C3×Dic6D311yescharacteristic, complemented
17C3×Q8×A4—31no—
18A4×Dic6—21no—
19A4×Dic6C2×C311yescharacteristic, complemented
20C4×C3⋊S4C2×C311yescharacteristic
21C3×C4×S4—31no—
22C3×C4×S4—32no—
23C3×C4×S4—61no—
24C3×A4⋊Q8—21no—
25C3×A4⋊Q8—32no—
26C3×A4⋊Q8D311yescharacteristic, complemented
27C3×A4⋊Q8C2×C311yescharacteristic, complemented
28(C3⋊S4)⋊C4C2×C312yes—
29#64204C2×C312yescomplemented
30C32×C22⋊Q8—31no—
31C3×#944—31no—
32C3×C4×#67—31no—
33C3×#897—31no—
34C3×#896—32no—
35C3×#888—32no—
36C3×#887—31no—
37C3×#67⋊C4—32no—
38C3×#878—32no—
39C2×C32×A4C2311yescharacteristic
40C3×C3⋊S4Q812yescomplemented
41C3×C4×C3⋊D3—41no—
42C32×Dic6—41no—
43C32×Dic6—41no—
44C3×#627—42no—
45C3×#624—42no—
46Q8×S4—31no—
47C3×D4×Q8—91no—
48#5632—31no—
49C2×C3⋊S4C22×C311yescharacteristic
50C2×C3×S4—31no—
51C2×C3×S4—61no—
52C2×C32×Q8—31no—
53C22×C32×C4C2×D311yescharacteristic
54C2×C3×#67—31no—
55C22×C3×Dic3C2×D312yes—
56C3×D3×Q8—61no—
57C3×D3×Q8—62no—
58C3×D3×Q8—61no—
59C3×D3×Q8—121no—
60C2×C3×C4×D3—32no—
61C2×C3×C4×D3—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—32no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—32no—
66C3×C4×A4—21no—
67C3×C4×A4—21no—
68C3×C4×A4—21no—
69C3×C4×A4—32no—
70C3×C4×A4—41no—
71C3×C4×A4C2×D311yescharacteristic
72C3×C4×A4C22×C311yescharacteristic
73#3797—41no—
74Dic3×A4—22no—
75Dic3×A4C22×C312yes—
76C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
77C3×SL(2,ℤ/4)—22no—
78C3×SL(2,ℤ/4)—31no—
79C3×SL(2,ℤ/4)—61no—
80C3×SL(2,ℤ/4)C2×D312yes—
81C3×SL(2,ℤ/4)C22×C312yes—
82C32×C4⋊C4—31no—
83C32×C4⋊C4—32no—
84C32×C22⋊C4—32no—
85C3×#217—31no—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—31no—
88C3×C4⋊Dic3—32no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—31no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—31no—
95C2×C3×C3⋊D3—41no—
96C32×A4C2×Q811yescharacteristic, complemented
97C33×C4—41no—
98C3×C3⋊Dic3—41no—
99C32×Dic3—42no—
100C32×Dic3—42no—
101C22×C3×Q8—31no—
102C22×C3×Q8—91no—
103C2×D3×Q8—31no—
104C22×Dic6C3×D311yescharacteristic, complemented
105Q8×A4—31no—
106Q8×A4—61no—
107C4×S4—31no—
108C4×S4—32no—
109C4×S4—61no—
110A4⋊Q8—32no—
111A4⋊Q8C3×D311yescharacteristic, complemented
112C3×C4⋊Q8—92no—
113C3×C4⋊Q8—91no—
114C3×C22⋊Q8—31no—
115C3×C22⋊Q8—31no—
116C3×C22⋊Q8—61no—
117C3×C22⋊Q8—92no—
118C3×C22⋊Q8—92no—
119C3×C22⋊Q8—91no—
120C3×C4×Q8—91no—
121C3×C4×D4—92no—
122C3×C4×D4—91no—
123#944—31no—
124C4×#67—31no—
125#897—31no—
126#896—32no—
127#888—32no—
128#887—31no—
129#67⋊C4—32no—
130#878—32no—
131C23×C32C22×D311yescharacteristic
132C22×C3×D3—31no—
133C2×C3×A4—21no—
134C2×C3×A4—21no—
135C2×C3×A4—21no—
136C2×C3×A4—41no—
137C2×C3×A4C22×D311yescharacteristic
138C2×C3×A4C23×C311yescharacteristic
139C3⋊S4C3×Q812yescomplemented
140C3×S4—32no—
141C3×S4—62no—
142C32×Q8—61no—
143C32×Q8—61no—
144C32×Q8—121no—
145C2×C32×C4—31no—
146C2×C32×C4—31no—
147C2×C32×C4—32no—
148C2×C32×C4—32no—
149C4×C3⋊D3—41no—
150C3×#67—32no—
151C3×#67—32no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C3×C4×D3—62no—
158C3×C4×D3—61no—
159C3×C4×D3—61no—
160C3×C4×D3—62no—
161C3×C4×D3—122no—
162C3×C4×D3—121no—
163C3×C4×D3—241no—
164C3×Dic6—31no—
165C3×Dic6—32no—
166C3×Dic6—41no—
167C3×Dic6—41no—
168C3×Dic6—41no—
169C3×Dic6—61no—
170C3×Dic6—62no—
171C3×Dic6—62no—
172C3×Dic6—81no—
173C3×Dic6—81no—
174C3×Dic6—122no—
175C3×Dic6S411yescharacteristic, complemented
176#627—42no—
177#624—42no—
178D4×Q8—91noSylow
179C2×C33—41no—
180C3×C3⋊D3—42no—
181C2×S4—31no—
182C2×S4—61no—
183C2×C3×Q8—31no—
184C2×C3×Q8—31no—
185C2×C3×Q8—61no—
186C2×C3×Q8—91no—
187C2×C3×Q8—91no—
188C2×C3×Q8—92no—
189C2×C3×Q8—92no—
190C2×C3×Q8—181no—
191C2×C3×Q8—182no—
192C2×C3×Q8—181no—
193C2×C3×D4—91no—
194C22×C3×C4—21no—
195C22×C3×C4—32no—
196C22×C3×C4—91no—
197C22×C3×C4—92no—
198C22×C3×C4D3211yescharacteristic
199C22×C3×C4C2×C3×D311yescharacteristic
200C2×#67—31no—
201C22×Dic3C2×C3×D312yes—
202D3×Q8—61no—
203D3×Q8—62no—
204D3×Q8—61no—
205D3×Q8—121no—
206C2×C4×D3—32no—
207C2×C4×D3—31no—
208C2×Dic6—31no—
209C2×Dic6—32no—
210C2×Dic6—31no—
211C2×Dic6—32no—
212C4×A4—21no—
213C4×A4—21no—
214C4×A4—32no—
215C4×A4—41no—
216C4×A4—62no—
217C4×A4C2×C3×D311yescharacteristic
218SL(2,ℤ/4)—31no—
219SL(2,ℤ/4)—61no—
220SL(2,ℤ/4)C2×C3×D312yes—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—32no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—62no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—92no—
232C3×C22⋊C4—32no—
233C3×C22⋊C4—32no—
234C3×C22⋊C4—62no—
235C3×C22⋊C4—91no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—91no—
238C3×C42—91no—
239C3×C42—92no—
240#217—31no—
241(C2×D3)⋊C4—32no—
242(C2×D3)⋊C4—31no—
243C4⋊Dic3—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—32no—
247Dic3⋊C4—31no—
248C4×Dic3—32no—
249C4×Dic3—31no—
250C22×C32—31no—
251C22×C32—31no—
252C22×C32D3×Q811yescharacteristic, complemented
253C2×C3⋊D3—41no—
254C2×C3×D3—32no—
255C2×C3×D3—61no—
256C2×C3×D3—61no—
257C2×C3×D3—121no—
258C2×C3×D3—241no—
259C3×A4—21no—
260C3×A4—21no—
261C3×A4—21no—
262C3×A4—41no—
263C3×A4D3×Q811yescharacteristic, complemented
264C3×A4C2×C3×Q811yescharacteristic, complemented
265C32×C4—31no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—62no—
269C32×C4—62no—
270C32×C4—81no—
271C32×C4—81no—
272C32×C4—81no—
273C32×C4—122no—
274C32×C4—161no—
275C32×C4C2×S411yescharacteristic
276C3⋊Dic3—41no—
277C3×Dic3—32no—
278C3×Dic3—32no—
279C3×Dic3—42no—
280C3×Dic3—42no—
281C3×Dic3—42no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C3×Dic3—82no—
285C3×Dic3—82no—
286C3×Dic3—121no—
287C3×Dic3—241no—
288C3×Dic3C2×S412yes—
289C22×Q8—31no—
290C22×Q8—91no—
291C4⋊Q8—92no—
292C4⋊Q8—91no—
293C22⋊Q8—31no—
294C22⋊Q8—92no—
295C22⋊Q8—92no—
296C22⋊Q8—91no—
297C4×Q8—91no—
298C4×D4—92no—
299C4×D4—91no—
300C33—41noSylow
301C23×C3—21no—
302C23×C3—91no—
303C23×C3C2×D3211yescharacteristic
304C23×C3C22×C3×D311yescharacteristic
305C22×D3—31no—
306C2×A4—21no—
307C2×A4—21no—
308C2×A4—41no—
309C2×A4C22×C3×D311yescharacteristic
310S4—32no—
311S4—62no—
312C3×Q8—31no—
313C3×Q8—61no—
314C3×Q8—61no—
315C3×Q8—61no—
316C3×Q8—61no—
317C3×Q8—92no—
318C3×Q8—91no—
319C3×Q8—121no—
320C3×Q8—121no—
321C3×Q8—121no—
322C3×Q8—182no—
323C3×Q8—181no—
324C3×Q8—182no—
325C3×Q8—241no—
326C3×D4—92no—
327C3×D4—92no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—32no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—92no—
341C2×C3×C4—91no—
342C2×C3×C4—92no—
343C2×C3×C4—91no—
344C2×C3×C4—91no—
345C2×C3×C4—182no—
346C2×C3×C4—181no—
347C2×C3×C4—182no—
348C2×C3×C4—181no—
349#67—32no—
350#67—32no—
351C2×Dic3—31no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C4×D3—62no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—62no—
360C4×D3—122no—
361C4×D3—121no—
362C4×D3—241no—
363Dic6—31no—
364Dic6—32no—
365Dic6—41no—
366Dic6—61no—
367Dic6—62no—
368Dic6—62no—
369Dic6—122no—
370Dic6C3×S411yescharacteristic, complemented
371C2×C32—31no—
372C2×C32—31no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32—81no—
376C2×C32—81no—
377C2×C32—81no—
378C2×C32—161no—
379C2×C32C22×S411yescharacteristic
380C3⋊D3—42no—
381C3×D3—62no—
382C3×D3—122no—
383C3×D3—242no—
384C2×Q8—31no—
385C2×Q8—91no—
386C2×Q8—91no—
387C2×Q8—92no—
388C2×Q8—92no—
389C2×Q8—181no—
390C2×Q8—182no—
391C2×Q8—181no—
392C2×D4—91no—
393C22×C4—32no—
394C22×C4—91no—
395C22×C4—92no—
396C22×C4C3×D3211yescharacteristic
397C4⋊C4—31no—
398C4⋊C4—32no—
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—92no—
402C4⋊C4—92no—
403C4⋊C4—92no—
404C22⋊C4—32no—
405C22⋊C4—91no—
406C22⋊C4—92no—
407C22⋊C4—91no—
408C42—91no—
409C42—92no—
410C22×C3—21no—
411C22×C3—31no—
412C22×C3—31no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—92no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3#379711yescharacteristic, complemented
421C22×C3C3×D3×Q811yescharacteristic, complemented
422C2×D3—32no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—121no—
426C2×D3—241no—
427A4—21no—
428A4—21no—
429A4—41no—
430A4C3×D3×Q811yescharacteristic, complemented
431C3×C4—21no—
432C3×C4—31no—
433C3×C4—32no—
434C3×C4—31no—
435C3×C4—41no—
436C3×C4—61no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4—62no—
440C3×C4—62no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4—92no—
444C3×C4—92no—
445C3×C4—122no—
446C3×C4—122no—
447C3×C4—122no—
448C3×C4—161no—
449C3×C4—181no—
450C3×C4—181no—
451C3×C4—242no—
452C3×C4D3×S411yescharacteristic
453C3×C4C2×C3×S411yescharacteristic
454Dic3—32no—
455Dic3—32no—
456Dic3—42no—
457Dic3—61no—
458Dic3—61no—
459Dic3—121no—
460Dic3—241no—
461Dic3C2×C3×S412yes—
462C32—41no—
463C32—41no—
464C32—81no—
465C32—81no—
466C32—81no—
467C32—161no—
468C32Q8×S411yescharacteristic, complemented
469C23—91no—
470C23C2×C3×D3211yescharacteristic
471Q8—31no—
472Q8—61no—
473Q8—61no—
474Q8—92no—
475Q8—91no—
476Q8—182no—
477Q8—181no—
478Q8—182no—
479D4—92no—
480D4—92no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—92no—
486C2×C4—91no—
487C2×C4—92no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—182no—
491C2×C4—181no—
492C2×C4—182no—
493C2×C4—181no—
494C2×C3—21no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—41no—
500C2×C3—61no—
501C2×C3—61no—
502C2×C3—81no—
503C2×C3—81no—
504C2×C3—161no—
505C2×C3—182no—
506C2×C3C2×D3×S411yescharacteristic
507C2×C3C22×C3×S411yescharacteristic
508D3—62no—
509D3—122no—
510D3—242no—
511C22—31no—
512C22—31no—
513C22—92no—
514C22—181no—
515C22—181no—
516C22C3×#379711yescharacteristic, complemented
517C4—31no—
518C4—32no—
519C4—62no—
520C4—62no—
521C4—92no—
522C4—92no—
523C4—181no—
524C4—181no—
525C4C3×D3×S411yescharacteristic
526C3—21no—
527C3—41no—
528C3—81no—
529C3—81no—
530C3—161no—
531C3(C3⋊S4)⋊Q811yescharacteristic, direct product factor
532C3C3×Q8×S411yescharacteristic, complemented
533C2—31no—
534C2—31no—
535C2—182no—
536C2C2×C3×D3×S411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C2×C3×D3×S4
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
prime radicalC2×C3×C3⋊S4C22
hypercenterC3×C4D3×S4
hypocenterC3×A4C2×C3×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×A4×Dic6C2semidirect
3. C32×A4⋊Q8C2semidirect
5. C3×#64204C2semidirect
7. (C3⋊S4)⋊Q8C3direct (N × H)
16. C22×C3×Dic6D3semidirect
19. A4×Dic6C2×C3semidirect
26. C3×A4⋊Q8D3semidirect
27. C3×A4⋊Q8C2×C3semidirect
29. #64204C2×C3semidirect
40. C3×C3⋊S4Q8semidirect
96. C32×A4C2×Q8semidirect
104. C22×Dic6C3×D3semidirect
111. A4⋊Q8C3×D3semidirect
139. C3⋊S4C3×Q8semidirect
175. C3×Dic6S4semidirect
252. C22×C32D3×Q8semidirect
263. C3×A4D3×Q8semidirect
264. C3×A4C2×C3×Q8semidirect
370. Dic6C3×S4semidirect
420. C22×C3#3797semidirect
421. C22×C3C3×D3×Q8semidirect
430. A4C3×D3×Q8semidirect
468. C32Q8×S4semidirect
516. C22C3×#3797semidirect
531. C3(C3⋊S4)⋊Q8direct (N × H)
532. C3C3×Q8×S4semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
C3×C4×C3⋊S42
C32×A4⋊Q82
C3×(C3⋊S4)⋊C42
C3×#642042
C3×Q8×S43
(C3⋊S4)⋊Q83
C3×#56323
C3×#37974