Finite Group Databasebeta

← C2×A4×C3⋊S4 #13181813

Subgroups — C2×A4×C3⋊S4 #13181813

Order 1728.

926 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 926 conjugacy classes of subgroups — too many to draw. Only its 57 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×A42C211yesmaximal, characteristic, complemented
2A4×C3⋊S4C212yesmaximal, direct product factor
3C23×C3⋊S4C311yesmaximal, characteristic, complemented
4C2×A4×S4—33nomaximal
5C2×A4×#67—31nomaximal
6C2×A4×C3⋊D3—41nomaximal
7C2×C3×C3⋊S4—41nomaximal
8C3×A42C2211yescharacteristic, complemented
9C2×C3×C22⋊A4—21no—
10C23×C3×A4D311yescharacteristic, complemented
11C23×C3×A4C2×C311yescharacteristic, complemented
12C22×D3×A4—31no—
13C22×C3⋊S4—31no—
14C22×C3⋊S4—31no—
15C22×C3⋊S4—32no—
16C22×C3⋊S4C2×C312yescomplemented
17C2×A42—23no—
18C2×A42D313yescomplemented
19A4×S4—36no—
20A4×#67—32no—
21A4×#67—32no—
22C2×Dic3×A4—31no—
23C2×C32×A4—41no—
24C2×C32×A4—41no—
25A4×C3⋊D3—42no—
26C3×C3⋊S4—42no—
27C23×S4—33no—
28C23×#67—31no—
29C2×D4×A4—91no—
30C23×C3⋊D3—41no—
31C3×C22⋊A4—21no—
32C22×C3×A4—31no—
33C22×C3×A4—31no—
34C22×C3×A4—31no—
35C22×C3×A4—31no—
36C22×C3×A4C2×D311yescharacteristic, complemented
37C22×C3×A4C22×C311yescharacteristic, complemented
38C2×D3×A4—32no—
39C2×D3×A4—61no—
40C2×D3×A4—61no—
41C2×D3×A4—123no—
42C2×C3⋊S4—32no—
43C2×C3⋊S4—31no—
44C2×C3⋊S4—32no—
45C2×C3⋊S4—32no—
46C2×C3⋊S4—32no—
47C2×C3⋊S4A411yescharacteristic, direct product factor
48C2×C3×S4—123no—
49A42—23no—
50A42C2×D313yescomplemented
51C2×C3×#67—121no—
52Dic3×A4—32no—
53C2×C3×C3⋊D3—161no—
54C32×A4—41no—
55C32×A4—41no—
56C25×C3C3×D311yescharacteristic, complemented
57C24×D3—31no—
58C2×C22⋊A4—23no—
59C23×A4—21no—
60C23×A4—91no—
61C23×A4C3×D313yescomplemented
62C23×A4C3⋊D311yescharacteristic, complemented
63C22×S4—36no—
64C22×S4—93no—
65C22×S4—96no—
66C22×S4—93no—
67C22×#67—32no—
68C22×#67—32no—
69C22×#67—91no—
70C22×#67—92no—
71C22×#67—91no—
72C22×#67—91no—
73C22×#67—92no—
74C22×#67—91no—
75C23×Dic3—31no—
76D4×A4—92no—
77D4×A4—92no—
78C2×C4×A4—91no—
79C23×C32—41no—
80C23×C32—41no—
81C22×C3⋊D3—42no—
82C22×C3⋊D3—121no—
83C22×C3⋊D3—122no—
84C22×C3⋊D3—121no—
85C22×C3×D3—121no—
86C2×C3×A4—31no—
87C2×C3×A4—31no—
88C2×C3×A4—31no—
89C2×C3×A4—31no—
90C2×C3×A4—43no—
91C2×C3×A4—43no—
92C2×C3×A4—83no—
93C2×C3×A4—81no—
94C2×C3×A4—81no—
95C2×C3×A4—83no—
96C2×C3×A4—241no—
97C2×C3×A4S411yescharacteristic, complemented
98C2×C3×A4C2×A411yescharacteristic, complemented
99D3×A4—62no—
100D3×A4—126no—
101C3⋊S4—32no—
102C3⋊S4C2×A412yesdirect product factor
103C3×S4—126no—
104C3×#67—122no—
105C3×#67—122no—
106C2×C3×Dic3—121no—
107C23×D4—91noSylow
108C2×C33—161no—
109C3×C3⋊D3—162no—
110C24×C3—31no—
111C24×C3—31no—
112C24×C3—31no—
113C24×C3—31no—
114C24×C3—91no—
115C24×C3—91no—
116C24×C3C2×C3×D311yescharacteristic, complemented
117C23×D3—32no—
118C23×D3—61no—
119C23×D3—61no—
120C23×D3—91no—
121C23×D3—92no—
122C23×D3—91no—
123C23×D3—123no—
124C23×D3—181no—
125C23×D3—181no—
126C22⋊A4—23no—
127C22×A4—21no—
128C22×A4—31no—
129C22×A4—33no—
130C22×A4—31no—
131C22×A4—33no—
132C22×A4—61no—
133C22×A4—61no—
134C22×A4—92no—
135C22×A4—181no—
136C22×A4—181no—
137C22×A4C2×C3×D313yescomplemented
138C22×A4C2×C3⋊D311yescharacteristic, complemented
139C2×S4—33no—
140C2×S4—96no—
141C2×S4—96no—
142C2×S4—96no—
143C2×S4—93no—
144C2×S4—96no—
145C2×C3×D4—361no—
146C2×#67—31no—
147C2×#67—91no—
148C2×#67—92no—
149C2×#67—92no—
150C2×#67—92no—
151C2×#67—92no—
152C2×#67—91no—
153C2×#67—92no—
154C2×#67—94no—
155C2×#67—92no—
156C2×#67—94no—
157C2×#67—92no—
158C2×#67—91no—
159C2×#67—92no—
160C2×#67—92no—
161C2×#67—92no—
162C2×#67—92no—
163C2×#67—92no—
164C2×#67—92no—
165C22×Dic3—32no—
166C22×Dic3—91no—
167C22×Dic3—92no—
168C22×Dic3—91no—
169C4×A4—92no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32—121no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—122no—
178C2×C3⋊D3—122no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—122no—
181C2×C3⋊D3—122no—
182C2×C3×D3—122no—
183C2×C3×D3—241no—
184C2×C3×D3—241no—
185C2×C3×D3—483no—
186C3×A4—43no—
187C3×A4—43no—
188C3×A4—83no—
189C3×A4—81no—
190C3×A4—81no—
191C3×A4—83no—
192C3×A4—241no—
193C3×A4C2×S411yescharacteristic, complemented
194C3×A4C22×A411yescharacteristic, complemented
195C3×Dic3—122no—
196C25—91no—
197C25C3×C3⋊D311yescharacteristic, complemented
198C22×D4—92no—
199C22×D4—92no—
200C22×D4—271no—
201C22×D4—272no—
202C22×D4—271no—
203C22×D4—271no—
204C22×D4—272no—
205C22×D4—271no—
206C23×C4—91no—
207C33—161noSylow
208C23×C3—31no—
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—31no—
212C23×C3—43no—
213C23×C3—41no—
214C23×C3—61no—
215C23×C3—81no—
216C23×C3—91no—
217C23×C3—91no—
218C23×C3—91no—
219C23×C3—91no—
220C23×C3—91no—
221C23×C3—91no—
222C23×C3—91no—
223C23×C3—91no—
224C23×C3—92no—
225C23×C3—91no—
226C23×C3—181no—
227C23×C3—181no—
228C23×C3—361no—
229C23×C3C3×S411yescharacteristic, complemented
230C23×C3D3×A411yescharacteristic, complemented
231C22×D3—31no—
232C22×D3—62no—
233C22×D3—92no—
234C22×D3—92no—
235C22×D3—91no—
236C22×D3—92no—
237C22×D3—92no—
238C22×D3—126no—
239C22×D3—181no—
240C22×D3—181no—
241C22×D3—181no—
242C22×D3—181no—
243C22×D3—184no—
244C22×D3—182no—
245C22×D3—181no—
246C22×D3—182no—
247C22×D3—181no—
248C22×D3—181no—
249C22×D3—181no—
250C22×D3—182no—
251C22×D3—363no—
252C22×D3—366no—
253C22×D3—363no—
254C2×A4—21no—
255C2×A4—31no—
256C2×A4—31no—
257C2×A4—33no—
258C2×A4—33no—
259C2×A4—61no—
260C2×A4—61no—
261C2×A4—83no—
262C2×A4—83no—
263C2×A4—182no—
264C2×A4—243no—
265C2×A4C3⋊S411yescharacteristic, direct product factor
266C2×A4D3×A413yescomplemented
267S4—36no—
268S4—96no—
269C3×D4—362no—
270C3×D4—362no—
271C2×C3×C4—361no—
272#67—32no—
273#67—32no—
274#67—92no—
275#67—92no—
276#67—92no—
277#67—92no—
278#67—92no—
279#67—94no—
280#67—94no—
281#67—92no—
282C2×Dic3—31no—
283C2×Dic3—92no—
284C2×Dic3—92no—
285C2×Dic3—91no—
286C2×Dic3—92no—
287C2×Dic3—92no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—121no—
291C2×C32—121no—
292C2×C32—121no—
293C2×C32—121no—
294C2×C32—163no—
295C2×C32—323no—
296C2×C32—321no—
297C3⋊D3—42no—
298C3⋊D3—122no—
299C3×D3—242no—
300C3×D3—486no—
301C24—31no—
302C24—31no—
303C24—31no—
304C24—31no—
305C24—92no—
306C24—91no—
307C24—91no—
308C24—181no—
309C24—181no—
310C24—271no—
311C24—271no—
312C24—272no—
313C24—541no—
314C24—541no—
315C24C2×C3×C3⋊D311yescharacteristic, complemented
316C2×D4—91no—
317C2×D4—271no—
318C2×D4—272no—
319C2×D4—272no—
320C2×D4—272no—
321C2×D4—272no—
322C2×D4—272no—
323C2×D4—271no—
324C2×D4—274no—
325C2×D4—272no—
326C2×D4—272no—
327C2×D4—274no—
328C2×D4—271no—
329C2×D4—272no—
330C2×D4—272no—
331C2×D4—272no—
332C2×D4—272no—
333C2×D4—272no—
334C2×D4—272no—
335C22×C4—92no—
336C22×C4—271no—
337C22×C4—272no—
338C22×C4—271no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—43no—
344C22×C3—41no—
345C22×C3—61no—
346C22×C3—81no—
347C22×C3—91no—
348C22×C3—91no—
349C22×C3—91no—
350C22×C3—91no—
351C22×C3—91no—
352C22×C3—91no—
353C22×C3—91no—
354C22×C3—91no—
355C22×C3—92no—
356C22×C3—91no—
357C22×C3—121no—
358C22×C3—123no—
359C22×C3—121no—
360C22×C3—123no—
361C22×C3—181no—
362C22×C3—181no—
363C22×C3—241no—
364C22×C3—241no—
365C22×C3—362no—
366C22×C3—721no—
367C22×C3—721no—
368C22×C3C2×C3×S411yescharacteristic, complemented
369C22×C3C2×D3×A411yescharacteristic, complemented
370C2×D3—32no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—92no—
374C2×D3—123no—
375C2×D3—182no—
376C2×D3—182no—
377C2×D3—182no—
378C2×D3—182no—
379C2×D3—181no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—181no—
383C2×D3—182no—
384C2×D3—182no—
385C2×D3—366no—
386C2×D3—366no—
387C2×D3—363no—
388C2×D3—366no—
389C2×D3—366no—
390A4—21no—
391A4—83no—
392A4—83no—
393A4—243no—
394A4C2×C3⋊S411yescharacteristic, direct product factor
395A4C2×D3×A413yescomplemented
396C3×C4—362no—
397Dic3—32no—
398Dic3—92no—
399C32—41no—
400C32—41no—
401C32—163no—
402C32—323no—
403C32—321no—
404C23—31no—
405C23—31no—
406C23—31no—
407C23—31no—
408C23—61no—
409C23—91no—
410C23—91no—
411C23—91no—
412C23—91no—
413C23—91no—
414C23—91no—
415C23—91no—
416C23—91no—
417C23—91no—
418C23—92no—
419C23—91no—
420C23—181no—
421C23—181no—
422C23—182no—
423C23—271no—
424C23—272no—
425C23—272no—
426C23—272no—
427C23—272no—
428C23—541no—
429C23—542no—
430C23—541no—
431C23—541no—
432C23—541no—
433C23—541no—
434C23—542no—
435C23—541no—
436C23—542no—
437C23—541no—
438C23—541no—
439C23—544no—
440C23C3×C3⋊S411yescharacteristic, complemented
441C23A4×C3⋊D311yescharacteristic, complemented
442D4—92no—
443D4—92no—
444D4—272no—
445D4—272no—
446D4—272no—
447D4—272no—
448D4—272no—
449D4—274no—
450D4—274no—
451D4—272no—
452C2×C4—91no—
453C2×C4—272no—
454C2×C4—272no—
455C2×C4—271no—
456C2×C4—272no—
457C2×C4—272no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—43no—
463C2×C3—41no—
464C2×C3—81no—
465C2×C3—91no—
466C2×C3—91no—
467C2×C3—121no—
468C2×C3—123no—
469C2×C3—123no—
470C2×C3—121no—
471C2×C3—241no—
472C2×C3—241no—
473C2×C3—323no—
474C2×C3—722no—
475C2×C3A4×S411yescharacteristic, complemented
476D3—62no—
477D3—126no—
478D3—182no—
479D3—366no—
480C22—31no—
481C22—31no—
482C22—31no—
483C22—31no—
484C22—61no—
485C22—91no—
486C22—91no—
487C22—91no—
488C22—91no—
489C22—91no—
490C22—91no—
491C22—91no—
492C22—91no—
493C22—92no—
494C22—92no—
495C22—91no—
496C22—181no—
497C22—181no—
498C22—181no—
499C22—181no—
500C22—272no—
501C22—542no—
502C22—542no—
503C22—542no—
504C22—542no—
505C22—541no—
506C22—542no—
507C22—542no—
508C22—542no—
509C22—542no—
510C22—541no—
511C22C2×C3×C3⋊S411yescharacteristic, complemented
512C22C2×A4×C3⋊D311yescharacteristic, complemented
513C4—92no—
514C4—272no—
515C3—43no—
516C3—41no—
517C3—81no—
518C3—323no—
519C3C2×A4×S411yescharacteristic, complemented
520C2—31no—
521C2—31no—
522C2—31no—
523C2—31no—
524C2—91no—
525C2—91no—
526C2—182no—
527C2—542no—
528C2A4×C3⋊S411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2A4×C3⋊S4
commutator, cosocle, hypocenterC22×C3×A4C22×C3
FrattiniC1G
Fitting, socleC25×C3C3×D3
LCSG ▷ C22×C3×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×A42C2semidirect
2. A4×C3⋊S4C2direct (N × H)
3. C23×C3⋊S4C3semidirect
8. C3×A42C22semidirect
10. C23×C3×A4D3semidirect
11. C23×C3×A4C2×C3semidirect
16. C22×C3⋊S4C2×C3semidirect
18. C2×A42D3semidirect
36. C22×C3×A4C2×D3semidirect
37. C22×C3×A4C22×C3semidirect
47. C2×C3⋊S4A4direct (N × H)
50. A42C2×D3semidirect
56. C25×C3C3×D3semidirect
61. C23×A4C3×D3semidirect
62. C23×A4C3⋊D3semidirect
97. C2×C3×A4S4semidirect
98. C2×C3×A4C2×A4semidirect
102. C3⋊S4C2×A4direct (N × H)
116. C24×C3C2×C3×D3semidirect
137. C22×A4C2×C3×D3semidirect
138. C22×A4C2×C3⋊D3semidirect
193. C3×A4C2×S4semidirect
194. C3×A4C22×A4semidirect
197. C25C3×C3⋊D3semidirect
229. C23×C3C3×S4semidirect
230. C23×C3D3×A4semidirect
265. C2×A4C3⋊S4direct (N × H)
266. C2×A4D3×A4semidirect
315. C24C2×C3×C3⋊D3semidirect
368. C22×C3C2×C3×S4semidirect
369. C22×C3C2×D3×A4semidirect
394. A4C2×C3⋊S4direct (N × H)
395. A4C2×D3×A4semidirect
440. C23C3×C3⋊S4semidirect
441. C23A4×C3⋊D3semidirect
475. C2×C3A4×S4semidirect
511. C22C2×C3×C3⋊S4semidirect
512. C22C2×A4×C3⋊D3semidirect
519. C3C2×A4×S4semidirect
528. C2A4×C3⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×A422
A4×C3⋊S42
C23×C3⋊S43
C2×A4×S43
C2×A4×#673
C2×A4×C3⋊D34
C2×C3×C3⋊S44