Finite Group Databasebeta

← C2×C3×Dic3×S4 #13180721

Subgroups — C2×C3×Dic3×S4 #13180721

Order 1728.

1147 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1147 conjugacy classes of subgroups — too many to draw. Only its 138 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×C32×S4C211yesmaximal, characteristic
2C2×C3×Dic3×A4C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C3×Dic3×S4C214yesmaximal, direct product factor
5C2×C3×C4×S4—31nomaximal
6C2×Dic3×S4C311yesmaximal, characteristic, direct product factor
7C2×C3×D4×Dic3—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C32×S4C414yescomplemented
10C2×C32×S4C2212yes—
11C2×C3×D3×Dic3—41nomaximal
12C3×Dic3×A4C2212yescomplemented
13C3×C3⋊SL(2,ℤ/4)C2212yescomplemented
14C22×C3×S4—21no—
15C22×C3×S4D311yescharacteristic
16C22×C3×S4C2×C311yescharacteristic
17C22×C32×D4—31no—
18C23×C3×Dic3—31no—
19C23×C3×Dic3D311yescharacteristic, complemented
20C2×C3×C4×A4—31no—
21C2×Dic3×A4—21no—
22C2×Dic3×A4C2×C311yescharacteristic, complemented
23C2×C3⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
24C2×C3×SL(2,ℤ/4)—31no—
25C2×C3×SL(2,ℤ/4)—61no—
26C3×C4×S4—34no—
27Dic3×S4C2×C314yesdirect product factor
28C2×C3×#217—31no—
29C2×C3×#217—31no—
30C3×D4×Dic3—34no—
31C3×D4×Dic3—34no—
32C2×C3×C4⋊Dic3—31no—
33C2×C3×C4×Dic3—31no—
34C22×C32×D3—41no—
35C2×C32×A4C2×C412yescomplemented
36C2×C32×A4C2311yescharacteristic
37C32×S4C2×C414yescomplemented
38C2×C3×C3⋊Dic3—41no—
39C2×C32×Dic3—41no—
40C3×D3×Dic3—44no—
41C2×C4×S4—31no—
42C2×C3×C4×D4—91no—
43C2×D4×Dic3—31no—
44C24×C32—31no—
45C24×C32C2×D311yescharacteristic
46C22×C3×A4—21no—
47C22×C3×A4—21no—
48C22×C3×A4—21no—
49C22×C3×A4—41no—
50C22×C3×A4C2×D311yescharacteristic
51C22×C3×A4C22×C311yescharacteristic
52C2×C3×S4—24no—
53C2×C3×S4—22no—
54C2×C3×S4Dic314yesdirect product factor
55C2×C3×S4C3×C414yescomplemented
56C2×C3×S4C2×D312yes—
57C2×C3×S4C22×C312yes—
58C2×C32×D4—32no—
59C2×C32×D4—34no—
60C2×C32×D4—34no—
61C2×C32×D4—32no—
62C22×C32×C4—31no—
63C22×C3×Dic3—31no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—34no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3—62no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3C2×D312yescomplemented
73C2×C3×C4×D3—121no—
74C3×C4×A4—32no—
75C2×D3×Dic3—41no—
76Dic3×A4—22no—
77Dic3×A4C22×C312yescomplemented
78C3⋊SL(2,ℤ/4)C22×C312yescomplemented
79C3×SL(2,ℤ/4)—32no—
80C3×SL(2,ℤ/4)—62no—
81C3×#217—34no—
82C3×#217—32no—
83C3×#217—32no—
84C3×C4⋊Dic3—34no—
85C3×C4×Dic3—34no—
86C22×C33—41no—
87C2×C32×D3—44no—
88C2×C32×D3—42no—
89C32×A4C22×C411yescharacteristic, complemented
90C3×C3⋊Dic3—42no—
91C32×Dic3—42no—
92C22×S4C3×D311yescharacteristic
93C22×C3×D4—31no—
94C22×C3×D4—31no—
95C22×C3×D4—61no—
96C23×C3×C4—31no—
97C23×C3×C4—91no—
98C23×Dic3—31no—
99C23×Dic3C3×D311yescharacteristic, complemented
100C2×C4×A4—31no—
101C2×C4×A4—61no—
102C2×SL(2,ℤ/4)—31no—
103C2×SL(2,ℤ/4)—61no—
104C4×S4—34no—
105C3×C4×D4—94no—
106C3×C4×D4—94no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C22⋊C4—91no—
109C2×C3×C22⋊C4—91no—
110C2×C3×C42—91no—
111C2×#217—31no—
112C2×#217—31no—
113D4×Dic3—34no—
114D4×Dic3—34no—
115C2×C4⋊Dic3—31no—
116C2×C4×Dic3—31no—
117C23×C32—31no—
118C23×C32—31no—
119C23×C32—32no—
120C23×C32—32no—
121C23×C32—34no—
122C23×C32—62no—
123C23×C32—61no—
124C23×C32—61no—
125C23×C32C4×D312yescomplemented
126C23×C32C22×D311yescharacteristic
127C22×C3×D3—41no—
128C22×C3×D3—41no—
129C22×C3×D3—81no—
130C2×C3×A4—22no—
131C2×C3×A4—22no—
132C2×C3×A4—21no—
133C2×C3×A4—21no—
134C2×C3×A4—22no—
135C2×C3×A4—21no—
136C2×C3×A4—42no—
137C2×C3×A4—41no—
138C2×C3×A4C2×Dic312yescomplemented
139C2×C3×A4C2×C3×C412yescomplemented
140C2×C3×A4C22×D311yescharacteristic
141C2×C3×A4C23×C311yescharacteristic
142C3×S4—24no—
143C3×S4C2×Dic314yesdirect product factor
144C3×S4C2×C3×C414yescomplemented
145C32×D4—38no—
146C32×D4—34no—
147C32×D4—34no—
148C2×C32×C4—32no—
149C2×C32×C4—34no—
150C2×C32×C4—121no—
151C2×C3⋊Dic3—41no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—41no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—81no—
166C2×C3×Dic3—121no—
167C2×C3×Dic3—241no—
168C2×C3×Dic3S411yescharacteristic, direct product factor
169C3×C4×D3—124no—
170D3×Dic3—44no—
171C2×C4×D4—91noSylow
172C2×C33—42no—
173C2×C33—41no—
174C32×D3—44no—
175C24×C3—21no—
176C24×C3—31no—
177C24×C3—31no—
178C24×C3—61no—
179C24×C3D3211yescharacteristic
180C24×C3C2×C3×D311yescharacteristic
181C22×A4—21no—
182C22×A4—21no—
183C22×A4—41no—
184C22×A4C2×C3×D311yescharacteristic
185C2×S4C3×Dic314yesdirect product factor
186C2×S4C2×C3×D312yes—
187C2×C3×D4—32no—
188C2×C3×D4—32no—
189C2×C3×D4—34no—
190C2×C3×D4—34no—
191C2×C3×D4—34no—
192C2×C3×D4—34no—
193C2×C3×D4—32no—
194C2×C3×D4—32no—
195C2×C3×D4—64no—
196C2×C3×D4—62no—
197C2×C3×D4—62no—
198C2×C3×D4—64no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C22×C3×C4—61no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—92no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4—94no—
209C22×C3×C4—181no—
210C22×C3×C4—182no—
211C22×C3×C4—181no—
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—32no—
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—34no—
218C22×Dic3—61no—
219C22×Dic3—62no—
220C22×Dic3—61no—
221C22×Dic3C2×C3×D312yescomplemented
222C2×C4×D3—121no—
223C4×A4—32no—
224C4×A4—62no—
225SL(2,ℤ/4)—32no—
226SL(2,ℤ/4)—62no—
227C3×C4⋊C4—94no—
228C3×C22⋊C4—94no—
229C3×C22⋊C4—92no—
230C3×C22⋊C4—92no—
231C3×C42—94no—
232#217—34no—
233#217—32no—
234#217—32no—
235C4⋊Dic3—34no—
236C4×Dic3—34no—
237C22×C32—32no—
238C22×C32—31no—
239C22×C32—32no—
240C22×C32—32no—
241C22×C32—31no—
242C22×C32—31no—
243C22×C32—34no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32—61no—
247C22×C32—62no—
248C22×C32—64no—
249C22×C32—64no—
250C22×C32—62no—
251C22×C32—81no—
252C22×C32—81no—
253C22×C32—81no—
254C22×C32—161no—
255C22×C32C2×C4×D311yescharacteristic, complemented
256C22×C32C2×S411yescharacteristic
257C2×C3×D3—44no—
258C2×C3×D3—44no—
259C2×C3×D3—42no—
260C2×C3×D3—42no—
261C2×C3×D3—82no—
262C2×C3×D3—84no—
263C3×A4—21no—
264C3×A4—21no—
265C3×A4—21no—
266C3×A4—41no—
267C3×A4C22×Dic311yescharacteristic, complemented
268C3×A4C22×C3×C411yescharacteristic, complemented
269C32×C4—34no—
270C32×C4—122no—
271C3⋊Dic3—42no—
272C3×Dic3—32no—
273C3×Dic3—42no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C3×Dic3—82no—
277C3×Dic3—122no—
278C3×Dic3—242no—
279C3×Dic3C2×S412yesdirect product factor
280C22×D4—31no—
281C23×C4—31no—
282C23×C4—91no—
283C4×D4—94no—
284C4×D4—94no—
285C2×C4⋊C4—91no—
286C2×C22⋊C4—91no—
287C2×C22⋊C4—91no—
288C2×C42—91no—
289C33—41noSylow
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—31no—
293C23×C3—31no—
294C23×C3—31no—
295C23×C3—31no—
296C23×C3—32no—
297C23×C3—32no—
298C23×C3—32no—
299C23×C3—32no—
300C23×C3—34no—
301C23×C3—34no—
302C23×C3—61no—
303C23×C3—61no—
304C23×C3—62no—
305C23×C3—62no—
306C23×C3—64no—
307C23×C3—61no—
308C23×C3—61no—
309C23×C3—62no—
310C23×C3—62no—
311C23×C3—61no—
312C23×C3—61no—
313C23×C3—121no—
314C23×C3—121no—
315C23×C3—122no—
316C23×C3D3×Dic312yescomplemented
317C23×C3C3×C4×D312yescomplemented
318C23×C3C2×D3211yescharacteristic
319C23×C3C22×C3×D311yescharacteristic
320C22×D3—41no—
321C2×A4—22no—
322C2×A4—22no—
323C2×A4—21no—
324C2×A4—21no—
325C2×A4—42no—
326C2×A4—41no—
327C2×A4C2×C3×Dic312yescomplemented
328C2×A4C22×C3×D311yescharacteristic
329S4C2×C3×Dic314yesdirect product factor
330C3×D4—38no—
331C3×D4—38no—
332C3×D4—34no—
333C3×D4—34no—
334C3×D4—34no—
335C3×D4—34no—
336C3×D4—64no—
337C3×D4—68no—
338C3×D4—64no—
339C2×C3×C4—32no—
340C2×C3×C4—32no—
341C2×C3×C4—34no—
342C2×C3×C4—34no—
343C2×C3×C4—31no—
344C2×C3×C4—62no—
345C2×C3×C4—64no—
346C2×C3×C4—92no—
347C2×C3×C4—92no—
348C2×C3×C4—91no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4—121no—
353C2×C3×C4—184no—
354C2×C3×C4—181no—
355C2×C3×C4—184no—
356C2×C3×C4—182no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4—241no—
360C2×Dic3—31no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—61no—
367C2×Dic3—64no—
368C2×Dic3—62no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—64no—
372C2×Dic3—121no—
373C2×Dic3—241no—
374C2×Dic3C3×S411yescharacteristic, direct product factor
375C4×D3—124no—
376C2×C32—31no—
377C2×C32—31no—
378C2×C32—32no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—42no—
382C2×C32—42no—
383C2×C32—64no—
384C2×C32—82no—
385C2×C32—81no—
386C2×C32—81no—
387C2×C32—81no—
388C2×C32—82no—
389C2×C32—82no—
390C2×C32—162no—
391C2×C32—161no—
392C2×C32C4×S412yescomplemented
393C2×C32C22×S411yescharacteristic
394C3×D3—44no—
395C3×D3—44no—
396C3×D3—84no—
397C24—31no—
398C24C3×D3211yescharacteristic
399C2×D4—32no—
400C2×D4—34no—
401C2×D4—34no—
402C2×D4—32no—
403C22×C4—31no—
404C22×C4—32no—
405C22×C4—91no—
406C22×C4—91no—
407C22×C4—92no—
408C22×C4—91no—
409C22×C4—91no—
410C22×C4—94no—
411C22×C4—181no—
412C22×C4—182no—
413C22×C4—181no—
414C4⋊C4—94no—
415C22⋊C4—94no—
416C22⋊C4—92no—
417C22⋊C4—92no—
418C42—94no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—32no—
422C22×C3—32no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—32no—
430C22×C3—32no—
431C22×C3—32no—
432C22×C3—32no—
433C22×C3—34no—
434C22×C3—34no—
435C22×C3—41no—
436C22×C3—62no—
437C22×C3—61no—
438C22×C3—62no—
439C22×C3—61no—
440C22×C3—62no—
441C22×C3—61no—
442C22×C3—64no—
443C22×C3—61no—
444C22×C3—61no—
445C22×C3—62no—
446C22×C3—64no—
447C22×C3—62no—
448C22×C3—64no—
449C22×C3—62no—
450C22×C3—62no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—124no—
456C22×C3—122no—
457C22×C3—121no—
458C22×C3—122no—
459C22×C3—124no—
460C22×C3—161no—
461C22×C3C2×D3×Dic311yescharacteristic, complemented
462C22×C3C2×C3×C4×D311yescharacteristic, complemented
463C22×C3D3×S411yescharacteristic
464C22×C3C2×C3×S411yescharacteristic
465C2×D3—42no—
466C2×D3—44no—
467A4—21no—
468A4—21no—
469A4—41no—
470A4C22×C3×Dic311yescharacteristic, complemented
471C3×C4—34no—
472C3×C4—34no—
473C3×C4—32no—
474C3×C4—64no—
475C3×C4—92no—
476C3×C4—122no—
477C3×C4—182no—
478C3×C4—182no—
479C3×C4—242no—
480Dic3—32no—
481Dic3—62no—
482Dic3—62no—
483Dic3—122no—
484Dic3—242no—
485Dic3C2×C3×S412yesdirect product factor
486C32—41no—
487C32—41no—
488C32—81no—
489C32—81no—
490C32—81no—
491C32—161no—
492C32C2×C4×S411yescharacteristic, complemented
493C23—31no—
494C23—31no—
495C23—32no—
496C23—32no—
497C23—34no—
498C23—61no—
499C23—62no—
500C23—61no—
501C23C3×D3×Dic312yescomplemented
502C23C2×C3×D3211yescharacteristic
503D4—38no—
504D4—34no—
505D4—34no—
506C2×C4—32no—
507C2×C4—34no—
508C2×C4—31no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—91no—
512C2×C4—92no—
513C2×C4—92no—
514C2×C4—92no—
515C2×C4—184no—
516C2×C4—181no—
517C2×C4—184no—
518C2×C4—181no—
519C2×C4—182no—
520C2×C4—181no—
521C2×C3—22no—
522C2×C3—21no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—32no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—32no—
529C2×C3—42no—
530C2×C3—41no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—62no—
534C2×C3—64no—
535C2×C3—64no—
536C2×C3—82no—
537C2×C3—82no—
538C2×C3—81no—
539C2×C3—81no—
540C2×C3—124no—
541C2×C3—162no—
542C2×C3—161no—
543C2×C3Dic3×S412yesdirect product factor
544C2×C3C3×C4×S412yescomplemented
545C2×C3C2×D3×S411yescharacteristic
546C2×C3C22×C3×S411yescharacteristic
547D3—44no—
548C22—32no—
549C22—31no—
550C22—32no—
551C22—32no—
552C22—31no—
553C22—31no—
554C22—34no—
555C22—61no—
556C22—62no—
557C22—64no—
558C22—64no—
559C22—62no—
560C22C2×C3×D3×Dic311yescharacteristic, complemented
561C22C3×D3×S411yescharacteristic
562C4—34no—
563C4—32no—
564C4—92no—
565C4—182no—
566C4—182no—
567C3—21no—
568C3—41no—
569C3—81no—
570C3—81no—
571C3—161no—
572C3C2×Dic3×S411yescharacteristic, direct product factor
573C3C2×C3×C4×S411yescharacteristic, complemented
574C2—31no—
575C2—31no—
576C2—32no—
577C2—64no—
578C2C3×Dic3×S412yesdirect product factor
579C2C2×C3×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, hypocenterC3×A4C22×C3×C4
FrattiniC2C2×C3×D3×S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C32×S4C2
cosocleC2×C3×A4C23×C3
LCSG ▷ C3×A4
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3×A4C2semidirect
3. C2×C3×C3⋊SL(2,ℤ/4)C2semidirect
4. C3×Dic3×S4C2direct (N × H)
6. C2×Dic3×S4C3direct (N × H)
9. C2×C32×S4C4semidirect
12. C3×Dic3×A4C22semidirect
13. C3×C3⋊SL(2,ℤ/4)C22semidirect
19. C23×C3×Dic3D3semidirect
22. C2×Dic3×A4C2×C3semidirect
23. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
27. Dic3×S4C2×C3direct (N × H)
35. C2×C32×A4C2×C4semidirect
37. C32×S4C2×C4semidirect
54. C2×C3×S4Dic3direct (N × H)
55. C2×C3×S4C3×C4semidirect
72. C22×C3×Dic3C2×D3semidirect
77. Dic3×A4C22×C3semidirect
78. C3⋊SL(2,ℤ/4)C22×C3semidirect
89. C32×A4C22×C4semidirect
99. C23×Dic3C3×D3semidirect
125. C23×C32C4×D3semidirect
138. C2×C3×A4C2×Dic3semidirect
139. C2×C3×A4C2×C3×C4semidirect
143. C3×S4C2×Dic3direct (N × H)
144. C3×S4C2×C3×C4semidirect
168. C2×C3×Dic3S4direct (N × H)
185. C2×S4C3×Dic3direct (N × H)
221. C22×Dic3C2×C3×D3semidirect
255. C22×C32C2×C4×D3semidirect
267. C3×A4C22×Dic3semidirect
268. C3×A4C22×C3×C4semidirect
279. C3×Dic3C2×S4direct (N × H)
316. C23×C3D3×Dic3semidirect
317. C23×C3C3×C4×D3semidirect
327. C2×A4C2×C3×Dic3semidirect
329. S4C2×C3×Dic3direct (N × H)
374. C2×Dic3C3×S4direct (N × H)
392. C2×C32C4×S4semidirect
461. C22×C3C2×D3×Dic3semidirect
462. C22×C3C2×C3×C4×D3semidirect
470. A4C22×C3×Dic3semidirect
485. Dic3C2×C3×S4direct (N × H)
492. C32C2×C4×S4semidirect
501. C23C3×D3×Dic3semidirect
543. C2×C3Dic3×S4direct (N × H)
544. C2×C3C3×C4×S4semidirect
560. C22C2×C3×D3×Dic3semidirect
572. C3C2×Dic3×S4direct (N × H)
573. C3C2×C3×C4×S4semidirect
578. C2C3×Dic3×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×S42
C2×C3×Dic3×A42
C2×C3×C3⋊SL(2,ℤ/4)2
C3×Dic3×S42
C2×C3×C4×S43
C2×Dic3×S43
C2×C3×D4×Dic33
C2×C3×D3×Dic34