Finite Group Databasebeta

← A4×#3798 #13180884

Subgroups — A4×#3798 #13180884

Order 1728.

1002 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C4×A4×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×C4×D3×A4C212yesmaximal, complemented
3A4×#627C211yesmaximal, characteristic, complemented
4A4×#626C212yesmaximal, complemented
5A4×#625C211yesmaximal, characteristic, complemented
6C22×#3798C311yesmaximal, characteristic, complemented
7A4×#235—32nomaximal
8C2×A4×C3⋊D3C2211yescharacteristic
9C2×C3×D3×A4C2212yes—
10C32×C4×A4C2211yescharacteristic
11C3×#3798—41nomaximal
12A4×C3⋊Dic3C2211yescharacteristic
13C3×Dic3×A4C2212yes—
14C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
15C22×C3×C4×D3C2×C312yescomplemented
16C2×C3×C4×A4—32no—
17C22×#627C2×C311yescharacteristic, complemented
18C22×#626C2×C312yescomplemented
19C22×#625C2×C311yescharacteristic, complemented
20C2×#3798—31no—
21C2×#3798—31no—
22C2×#3798—32no—
23C2×#3798—31no—
24C2×#3798—32no—
25C2×#3798—31no—
26A4×#67—32no—
27A4×#67—32no—
28A4×D12—32no—
29C4×D3×A4—22no—
30C4×D3×A4—32no—
31C4×D3×A4—61no—
32C4×D3×A4D312yescomplemented
33A4×Dic6—32no—
34C2×C32×A4C2311yescharacteristic
35A4×C3⋊D3—21no—
36C3×D3×A4—22no—
37C3×C4×C3⋊D3—41no—
38C32×C4×D3—42no—
39C3×#627—41no—
40C3×#626—42no—
41C3×#625—41no—
42C22×#235—32no—
43A4×#41—91no—
44C23×C3⋊D3C22×C311yescharacteristic
45C23×C3×D3C22×C312yes—
46C22×C3×A4—32no—
47C2×D3×A4—22no—
48C2×D3×A4—32no—
49C2×D3×A4—61no—
50C2×D3×A4C2×D312yes—
51C22×C32×C4C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C311yescharacteristic
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—31no—
57C22×C3×Dic3C22×C312yes—
58C3×#235—122no—
59C2×C3×C4×D3—32no—
60C2×C3×C4×D3—32no—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—32no—
63C3×C4×A4—22no—
64C3×C4×A4—21no—
65C3×C4×A4—32no—
66C3×C4×A4—41no—
67C3×C4×A4C2×D312yes—
68C2×#627—31no—
69C2×#627—31no—
70C2×#627—32no—
71C2×#626—32no—
72C2×#626—32no—
73C2×#626—32no—
74C2×#626—32no—
75C2×#625—31no—
76C2×#625—32no—
77C2×#625—31no—
78#3798—32no—
79#3798—34no—
80#3798—32no—
81#3798—34no—
82#3798—32no—
83#3798—31no—
84#3798—32no—
85#3798—31no—
86#3798—32no—
87#3798—31no—
88#3798A411yescharacteristic, direct product factor
89Dic3×A4—22no—
90Dic3×A4—32no—
91Dic3×A4—61no—
92Dic3×A4C2×D312yes—
93C2×C3×C3⋊D3—41no—
94C2×C32×D3—42no—
95C32×A4#4111yescharacteristic, complemented
96C33×C4—41no—
97C3×C3⋊Dic3—41no—
98C32×Dic3—42no—
99C23×C3×C4—32no—
100C22×#67—32no—
101C22×#67—32no—
102C2×#235—92no—
103C2×#235—92no—
104C2×#235—92no—
105C2×#235—92no—
106C2×#235—92no—
107C2×#235—92no—
108C2×#235—92no—
109C2×#235—92no—
110C22×D12—32no—
111C22×C4×D3—32no—
112C22×C4×D3—61no—
113C22×C4×D3C3×D312yescomplemented
114C22×Dic6—32no—
115Q8×A4—91no—
116D4×A4—92no—
117D4×A4—91no—
118C2×C4×A4—32no—
119C2×C4×A4—62no—
120C2×C4×A4—91no—
121C23×C32C23×C311yescharacteristic
122C22×C3⋊D3—21no—
123C22×C3⋊D3—31no—
124C22×C3⋊D3—31no—
125C22×C3⋊D3—61no—
126C22×C3×D3—22no—
127C22×C3×D3—32no—
128C22×C3×D3—32no—
129C22×C3×D3—62no—
130C2×C3×A4—22no—
131C2×C3×A4—21no—
132C2×C3×A4—41no—
133C2×C3×A4—62no—
134C2×C3×A4C22×D312yes—
135D3×A4—24no—
136D3×A4—22no—
137D3×A4—62no—
138D3×A4—62no—
139C2×C32×C4—31no—
140C2×C32×C4—31no—
141C2×C32×C4—122no—
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—31no—
144C4×C3⋊D3—32no—
145C4×C3⋊D3—31no—
146C4×C3⋊D3—31no—
147C4×C3⋊D3—31no—
148C4×C3⋊D3C2×A411yescharacteristic, complemented
149C3×#67—122no—
150C3×#67—122no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C3×D12—122no—
154C3×C4×D3—34no—
155C3×C4×D3—32no—
156C3×C4×D3—32no—
157C3×C4×D3—32no—
158C3×C4×D3—42no—
159C3×C4×D3—82no—
160C3×C4×D3—122no—
161C3×C4×D3—241no—
162C3×C4×D3C2×A412yescomplemented
163C3×Dic6—122no—
164#627—32no—
165#627—31no—
166#627—32no—
167#627C2×A411yescharacteristic, complemented
168#626—34no—
169#626—32no—
170#626—32no—
171#626—32no—
172#626C2×A412yescomplemented
173#625—32no—
174#625—32no—
175#625—31no—
176#625C2×A411yescharacteristic, complemented
177C22×#41—91noSylow
178C2×C33—41no—
179C3×C3⋊D3—81no—
180C32×D3—82no—
181C24×C3—32no—
182C23×D3—32no—
183C23×D3—61no—
184C23×D3C2×C3×D312yes—
185C22×A4—32no—
186C22×A4—62no—
187C22×A4—91no—
188C3×#41—361no—
189C22×C3×C4—21no—
190C22×C3×C4—32no—
191C22×C3×C4—92no—
192C22×C3×C4—92no—
193C22×C3×C4—92no—
194C22×C3×C4—92no—
195C22×C3×C4C2×C3×D312yes—
196C2×#67—92no—
197C2×#67—92no—
198C2×#67—92no—
199C2×#67—92no—
200C2×#67—92no—
201C2×#67—92no—
202C2×#67—92no—
203C2×#67—92no—
204C22×Dic3—32no—
205C22×Dic3—61no—
206C22×Dic3C2×C3×D312yes—
207#235—32no—
208#235—92no—
209#235—92no—
210#235—92no—
211#235—92no—
212#235—92no—
213#235—92no—
214#235—94no—
215#235—94no—
216#235—94no—
217#235—94no—
218#235—94no—
219#235—94no—
220#235—94no—
221#235—92no—
222C2×D12—92no—
223C2×D12—92no—
224C2×D12—92no—
225C2×D12—92no—
226C2×C4×D3—32no—
227C2×C4×D3—32no—
228C2×C4×D3—32no—
229C2×C4×D3—32no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—181no—
235C2×C4×D3—181no—
236C2×C4×D3—181no—
237C2×C4×D3—181no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C2×Dic6—92no—
242C4×A4—22no—
243C4×A4—32no—
244C4×A4—41no—
245C4×A4—62no—
246C4×A4—91no—
247C4×A4D3211yescharacteristic
248C22×C32—31no—
249C22×C32—31no—
250C22×C32—122no—
251C22×C32C3×#4111yescharacteristic, complemented
252C2×C3⋊D3—31no—
253C2×C3⋊D3—61no—
254C2×C3⋊D3—61no—
255C2×C3⋊D3—61no—
256C2×C3⋊D3—61no—
257C2×C3⋊D3C22×A411yescharacteristic
258C2×C3×D3—32no—
259C2×C3×D3—42no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—82no—
265C2×C3×D3—122no—
266C2×C3×D3—241no—
267C2×C3×D3C22×A412yes—
268C3×A4—22no—
269C3×A4—21no—
270C3×A4—41no—
271C3×A4#23512yescomplemented
272C32×C4—31no—
273C32×C4—42no—
274C32×C4—81no—
275C32×C4—82no—
276C32×C4—122no—
277C32×C4—161no—
278C32×C4C22×A411yescharacteristic
279C3⋊Dic3—31no—
280C3⋊Dic3C22×A411yescharacteristic
281C3×Dic3—32no—
282C3×Dic3—42no—
283C3×Dic3—82no—
284C3×Dic3—122no—
285C3×Dic3—241no—
286C3×Dic3C22×A412yes—
287C2×#41—271no—
288C2×#41—272no—
289C2×#41—271no—
290C2×#41—272no—
291C2×#41—271no—
292C2×#41—271no—
293C22×Q8—91no—
294C22×D4—92no—
295C22×D4—91no—
296C23×C4—32no—
297C23×C4—91no—
298C33—41noSylow
299C23×C3—21no—
300C23×C3—62no—
301C23×C3—92no—
302C23×C3—92no—
303C23×C3—182no—
304C23×C3C22×C3×D312yes—
305C22×D3—22no—
306C22×D3—32no—
307C22×D3—32no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—92no—
312C22×D3—92no—
313C22×D3—181no—
314C22×D3—181no—
315C22×D3—182no—
316C22×D3—182no—
317C2×A4—22no—
318C2×A4—41no—
319C2×A4—64no—
320C2×A4—62no—
321C2×A4—181no—
322C2×A4C2×D3211yescharacteristic
323C3×Q8—361no—
324C3×D4—362no—
325C3×D4—361no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—94no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—122no—
338C2×C3×C4—242no—
339C2×C3×C4—361no—
340#67—32no—
341#67—32no—
342#67—92no—
343#67—92no—
344#67—92no—
345#67—94no—
346#67—92no—
347#67—94no—
348#67—92no—
349#67—92no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356D12—32no—
357D12—92no—
358D12—92no—
359D12—94no—
360D12—92no—
361C4×D3—34no—
362C4×D3—32no—
363C4×D3—32no—
364C4×D3—32no—
365C4×D3—32no—
366C4×D3—61no—
367C4×D3—92no—
368C4×D3—92no—
369C4×D3—94no—
370C4×D3—92no—
371C4×D3—181no—
372C4×D3—181no—
373C4×D3—181no—
374C4×D3—182no—
375C4×D3D3×A412yescomplemented
376Dic6—32no—
377Dic6—92no—
378Dic6—92no—
379Dic6—94no—
380Dic6—92no—
381C2×C32—31no—
382C2×C32—31no—
383C2×C32—42no—
384C2×C32—81no—
385C2×C32—82no—
386C2×C32—161no—
387C2×C32—242no—
388C2×C32C23×A411yescharacteristic
389C3⋊D3—21no—
390C3⋊D3—61no—
391C3×D3—22no—
392C3×D3—62no—
393C3×D3—82no—
394C3×D3—84no—
395C3×D3—242no—
396C3×D3—242no—
397C24—32no—
398C24—91no—
399#41—91no—
400#41—272no—
401#41—271no—
402#41—272no—
403#41—271no—
404#41—271no—
405#41—274no—
406#41—274no—
407#41—272no—
408#41—272no—
409#41—272no—
410C2×Q8—271no—
411C2×Q8—272no—
412C2×Q8—271no—
413C2×D4—272no—
414C2×D4—272no—
415C2×D4—272no—
416C2×D4—272no—
417C2×D4—271no—
418C2×D4—271no—
419C2×D4—272no—
420C22×C4—32no—
421C22×C4—92no—
422C22×C4—92no—
423C22×C4—92no—
424C22×C4—92no—
425C22×C4—91no—
426C22×C4—271no—
427C22×C4—271no—
428C22×C4—271no—
429C22×C4—271no—
430C22×C4C3×D3211yescharacteristic
431C22×C3—21no—
432C22×C3—32no—
433C22×C3—32no—
434C22×C3—32no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—92no—
438C22×C3—122no—
439C22×C3—182no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3—242no—
444C22×C3—361no—
445C22×C3C3×#23512yescomplemented
446C2×D3—32no—
447C2×D3—32no—
448C2×D3—61no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—92no—
454C2×D3—181no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3C2×D3×A412yes—
464A4—22no—
465A4—41no—
466A4#379811yescharacteristic, direct product factor
467C3×C4—21no—
468C3×C4—32no—
469C3×C4—32no—
470C3×C4—41no—
471C3×C4—61no—
472C3×C4—82no—
473C3×C4—92no—
474C3×C4—122no—
475C3×C4—161no—
476C3×C4—242no—
477C3×C4—361no—
478C3×C4C2×D3×A412yes—
479Dic3—32no—
480Dic3—32no—
481Dic3—61no—
482Dic3—92no—
483Dic3—181no—
484Dic3C2×D3×A412yes—
485C32—42no—
486C32—81no—
487C32—82no—
488C32—161no—
489C32A4×#4111yescharacteristic, complemented
490C23—62no—
491C23—92no—
492C23—92no—
493C23—182no—
494C23—181no—
495C23—271no—
496C23—271no—
497C23—541no—
498C23C2×C3×D3211yescharacteristic
499Q8—91no—
500Q8—272no—
501Q8—271no—
502Q8—272no—
503D4—92no—
504D4—91no—
505D4—272no—
506D4—272no—
507D4—272no—
508D4—271no—
509D4—274no—
510D4—272no—
511D4—272no—
512C2×C4—31no—
513C2×C4—32no—
514C2×C4—31no—
515C2×C4—92no—
516C2×C4—92no—
517C2×C4—92no—
518C2×C4—94no—
519C2×C4—92no—
520C2×C4—91no—
521C2×C4—92no—
522C2×C4—271no—
523C2×C4—271no—
524C2×C4—272no—
525C2×C4—271no—
526C2×C4—271no—
527C2×C4—271no—
528C2×C3—21no—
529C2×C3—32no—
530C2×C3—32no—
531C2×C3—41no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—62no—
535C2×C3—82no—
536C2×C3—161no—
537C2×C3—182no—
538C2×C3—242no—
539C2×C3—244no—
540C2×C3—721no—
541C2×C3C22×D3×A412yes—
542D3—22no—
543D3—62no—
544D3—62no—
545D3—62no—
546D3—182no—
547D3—182no—
548C22—31no—
549C22—31no—
550C22—32no—
551C22—92no—
552C22—91no—
553C22—182no—
554C22—182no—
555C22—182no—
556C22—182no—
557C22—271no—
558C22—541no—
559C22—541no—
560C22—541no—
561C22—541no—
562C22C3×#379811yescharacteristic, complemented
563C4—31no—
564C4—32no—
565C4—92no—
566C4—91no—
567C4—271no—
568C4D32×A411yescharacteristic
569C3—21no—
570C3—41no—
571C3—82no—
572C3—161no—
573C3A4×#23512yescomplemented
574C2—31no—
575C2—31no—
576C2—62no—
577C2—182no—
578C2—181no—
579C2—541no—
580C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC4D32×A4
commutator, socle, cosocleC23×C32C23×C3
FrattiniC2C2×D32×A4
FittingC22×C32×C4C22×C3
hypocenterC22×C32C3×#41
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×A4×C3⋊D3C2semidirect
2. C3×C4×D3×A4C2semidirect
3. A4×#627C2semidirect
4. A4×#626C2semidirect
5. A4×#625C2semidirect
6. C22×#3798C3semidirect
14. C22×C4×C3⋊D3C2×C3semidirect
15. C22×C3×C4×D3C2×C3semidirect
17. C22×#627C2×C3semidirect
18. C22×#626C2×C3semidirect
19. C22×#625C2×C3semidirect
32. C4×D3×A4D3semidirect
88. #3798A4direct (N × H)
95. C32×A4#41semidirect
113. C22×C4×D3C3×D3semidirect
148. C4×C3⋊D3C2×A4semidirect
162. C3×C4×D3C2×A4semidirect
167. #627C2×A4semidirect
172. #626C2×A4semidirect
176. #625C2×A4semidirect
251. C22×C32C3×#41semidirect
271. C3×A4#235semidirect
375. C4×D3D3×A4semidirect
445. C22×C3C3×#235semidirect
466. A4#3798direct (N × H)
489. C32A4×#41semidirect
562. C22C3×#3798semidirect
573. C3A4×#235semidirect

Maximal subgroups

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

subgroupindex
C4×A4×C3⋊D32
C3×C4×D3×A42
A4×#6272
A4×#6262
A4×#6252
C22×#37983
A4×#2353
C3×#37984