Finite Group Databasebeta

← A4×#3802 #13180888

Subgroups — A4×#3802 #13180888

Order 1728.

1410 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1410 conjugacy classes of subgroups — too many to draw. Only its 102 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×D32×A4C212yesmaximal, complemented
2C4×A4×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×A4×D12C212yesmaximal, complemented
4A4×#625C212yesmaximal, complemented
5C22×#3802C311yesmaximal, characteristic, complemented
6D3×D4×A4—32nomaximal
7C2×A4×C3⋊D3C2211yescharacteristic
8C2×C3×D3×A4C2214yescomplemented
9D32×A4—24no—
10C32×C4×A4C2211yescharacteristic, complemented
11C3×#3802—41nomaximal
12A4×C3⋊Dic3C2211yescharacteristic, complemented
13C23×D32C2×C312yescomplemented
14C22×D3×A4—34no—
15C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
16C22×C3×D12C2×C312yescomplemented
17C3×D4×A4—32no—
18C22×#625C2×C312yescomplemented
19C2×#3802—31no—
20C2×#3802—32no—
21C2×#3802—31no—
22C2×#3802—32no—
23C2×#3802—32no—
24A4×#67—34no—
25A4×D12—22no—
26A4×D12D312yescomplemented
27C4×D3×A4—32no—
28C4×D3×A4—61no—
29C2×C32×A4C2311yescharacteristic
30C2×C3×D32—42no—
31A4×C3⋊D3D412yescomplemented
32C3×D3×A4—24no—
33C3×C4×C3⋊D3—41no—
34C32×D12—42no—
35C3×#625—42no—
36C22×D3×D4—32no—
37C2×D4×A4—91no—
38C23×C3⋊D3C22×C311yescharacteristic
39C23×C3×D3C22×C314yescomplemented
40C22×C3×A4—34no—
41C22×D32—24no—
42C22×D32—32no—
43C22×D32—32no—
44C22×D32—34no—
45C22×D32—64no—
46C2×D3×A4—24no—
47C2×D3×A4—32no—
48C2×D3×A4—68no—
49C2×D3×A4—61no—
50C2×D3×A4C2×D314yescomplemented
51C22×C32×C4C22×C311yescharacteristic, complemented
52C22×C3⋊Dic3C22×C311yescharacteristic, complemented
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—31no—
57C3×D3×D4—122no—
58C2×C3×D12—32no—
59C2×C3×D12—34no—
60C2×C3×D12—32no—
61C3×C4×A4—22no—
62C3×C4×A4—21no—
63C3×C4×A4—41no—
64C3×C4×A4C2×D312yescomplemented
65C2×#625—32no—
66C2×#625—34no—
67C2×#625—32no—
68#3802—32no—
69#3802—32no—
70#3802—38no—
71#3802—32no—
72#3802—32no—
73#3802—31no—
74#3802—32no—
75#3802—32no—
76#3802A411yescharacteristic, direct product factor
77Dic3×A4—32no—
78Dic3×A4—61no—
79C2×C3×C3⋊D3—41no—
80C2×C32×D3—44no—
81C32×A4C2×D411yescharacteristic, complemented
82C3×D32—84no—
83C33×C4—41no—
84C3×C3⋊Dic3—41no—
85C24×D3—34no—
86C23×A4—92no—
87C22×C3×D4—32no—
88C22×#67—34no—
89C2×D3×D4—92no—
90C2×D3×D4—92no—
91C2×D3×D4—92no—
92C2×D3×D4—94no—
93C2×D3×D4—92no—
94C2×D3×D4—94no—
95C22×D12C3×D312yescomplemented
96C22×C4×D3—32no—
97C22×C4×D3—61no—
98D4×A4—32no—
99D4×A4—62no—
100D4×A4—92no—
101C2×C4×A4—91no—
102C23×C32C23×C311yescharacteristic
103C22×C3⋊D3—31no—
104C22×C3⋊D3—31no—
105C22×C3⋊D3—32no—
106C22×C3⋊D3C3×D412yescomplemented
107C22×C3×D3—24no—
108C22×C3×D3—34no—
109C22×C3×D3—34no—
110C22×C3×D3—64no—
111C22×C3×D3—124no—
112C2×C3×A4—22no—
113C2×C3×A4—21no—
114C2×C3×A4—41no—
115C2×C3×A4—64no—
116C2×C3×A4C22×D312yes—
117C2×D32—34no—
118C2×D32—32no—
119C2×D32—34no—
120C2×D32—64no—
121C2×D32—64no—
122C2×D32—68no—
123C2×D32—68no—
124C2×D32—64no—
125C2×D32—64no—
126C2×D32C2×A412yescomplemented
127D3×A4—24no—
128D3×A4—34no—
129D3×A4—44no—
130D3×A4—62no—
131C32×D4—122no—
132C2×C32×C4—31no—
133C2×C32×C4—31no—
134C2×C3⋊Dic3—31no—
135C2×C3⋊Dic3—31no—
136C4×C3⋊D3—32no—
137C4×C3⋊D3—31no—
138C4×C3⋊D3—31no—
139C4×C3⋊D3—31no—
140C4×C3⋊D3C2×A411yescharacteristic, complemented
141C3×#67—124no—
142C3×D12—34no—
143C3×D12—34no—
144C3×D12—32no—
145C3×D12—42no—
146C3×D12—82no—
147C3×D12C2×A412yescomplemented
148C3×C4×D3—122no—
149C3×C4×D3—241no—
150#625—34no—
151#625—34no—
152#625—32no—
153#625C2×A412yescomplemented
154C23×D4—91noSylow
155C2×C33—41no—
156C3×C3⋊D3—42no—
157C32×D3—84no—
158C24×C3—34no—
159C23×D3—32no—
160C23×D3—61no—
161C23×D3—68no—
162C23×D3—94no—
163C23×D3—94no—
164C23×D3—94no—
165C23×D3—94no—
166C23×D3—188no—
167C23×D3C2×C3×D314yescomplemented
168C22×A4—34no—
169C22×A4—64no—
170C22×A4—91no—
171C22×A4—184no—
172C2×C3×D4—92no—
173C2×C3×D4—92no—
174C2×C3×D4—94no—
175C2×C3×D4—361no—
176C22×C3×C4—21no—
177C22×C3×C4C2×C3×D312yescomplemented
178C2×#67—94no—
179C2×#67—94no—
180C2×#67—94no—
181C2×#67—94no—
182C22×Dic3—32no—
183C22×Dic3—61no—
184D3×D4—32no—
185D3×D4—92no—
186D3×D4—94no—
187D3×D4—92no—
188D3×D4—94no—
189D3×D4—94no—
190D3×D4—94no—
191D3×D4—98no—
192D3×D4—98no—
193D3×D4—94no—
194D3×D4—92no—
195C2×D12—32no—
196C2×D12—34no—
197C2×D12—32no—
198C2×C4×D3—92no—
199C2×C4×D3—92no—
200C2×C4×D3—92no—
201C2×C4×D3—92no—
202C2×C4×D3—181no—
203C2×C4×D3—181no—
204C2×C4×D3—181no—
205C2×C4×D3—181no—
206C4×A4—22no—
207C4×A4—41no—
208C4×A4—91no—
209C4×A4D3211yescharacteristic, complemented
210C22×C32—31no—
211C22×C32—31no—
212C22×C32—124no—
213C22×C32C2×C3×D411yescharacteristic, complemented
214C2×C3⋊D3—32no—
215C2×C3⋊D3—31no—
216C2×C3⋊D3—32no—
217C2×C3⋊D3—32no—
218C2×C3⋊D3—32no—
219C2×C3⋊D3C22×A411yescharacteristic
220C2×C3×D3—34no—
221C2×C3×D3—44no—
222C2×C3×D3—64no—
223C2×C3×D3—64no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—84no—
227C2×C3×D3—122no—
228C2×C3×D3—241no—
229C2×C3×D3—248no—
230C2×C3×D3C22×A414yescomplemented
231C3×A4—22no—
232C3×A4—21no—
233C3×A4—41no—
234C3×A4D3×D412yescomplemented
235D32—24no—
236D32—68no—
237D32—68no—
238D32—64no—
239C32×C4—31no—
240C32×C4—42no—
241C32×C4—81no—
242C32×C4—82no—
243C32×C4—161no—
244C32×C4C22×A411yescharacteristic, complemented
245C3⋊Dic3—31no—
246C3⋊Dic3C22×A411yescharacteristic, complemented
247C3×Dic3—122no—
248C3×Dic3—241no—
249C25—92no—
250C22×D4—32no—
251C22×D4—92no—
252C22×D4—271no—
253C22×D4—271no—
254C22×D4—272no—
255C22×D4—272no—
256C22×D4—272no—
257C23×C4—91no—
258C33—41noSylow
259C23×C3—21no—
260C23×C3—64no—
261C23×C3—94no—
262C23×C3—94no—
263C23×C3—184no—
264C23×C3—362no—
265C23×C3C22×C3×D312yes—
266C22×D3—24no—
267C22×D3—34no—
268C22×D3—34no—
269C22×D3—34no—
270C22×D3—34no—
271C22×D3—62no—
272C22×D3—64no—
273C22×D3—94no—
274C22×D3—92no—
275C22×D3—94no—
276C22×D3—98no—
277C22×D3—92no—
278C22×D3—94no—
279C22×D3—94no—
280C22×D3—181no—
281C22×D3—181no—
282C22×D3—182no—
283C22×D3—188no—
284C22×D3—188no—
285C22×D3—188no—
286C22×D3—188no—
287C22×D3—188no—
288C22×D3—188no—
289C22×D3—188no—
290C22×D3—188no—
291C2×A4—22no—
292C2×A4—41no—
293C2×A4—64no—
294C2×A4—92no—
295C2×A4—124no—
296C2×A4C2×D3211yescharacteristic
297C3×D4—32no—
298C3×D4—94no—
299C3×D4—94no—
300C3×D4—92no—
301C3×D4—122no—
302C3×D4—242no—
303C3×D4—362no—
304C2×C3×C4—32no—
305C2×C3×C4—32no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—361no—
309#67—34no—
310#67—94no—
311#67—94no—
312#67—98no—
313#67—94no—
314C2×Dic3—92no—
315C2×Dic3—92no—
316C2×Dic3—181no—
317C2×Dic3—181no—
318D12—34no—
319D12—34no—
320D12—32no—
321D12D3×A412yescomplemented
322C4×D3—32no—
323C4×D3—61no—
324C4×D3—92no—
325C4×D3—92no—
326C4×D3—94no—
327C4×D3—92no—
328C4×D3—181no—
329C4×D3—181no—
330C4×D3—181no—
331C4×D3—182no—
332C2×C32—31no—
333C2×C32—31no—
334C2×C32—42no—
335C2×C32—81no—
336C2×C32—82no—
337C2×C32—161no—
338C2×C32—244no—
339C2×C32C23×A411yescharacteristic
340C3⋊D3—32no—
341C3⋊D3D4×A412yescomplemented
342C3×D3—24no—
343C3×D3—64no—
344C3×D3—84no—
345C3×D3—124no—
346C3×D3—164no—
347C3×D3—242no—
348C24—34no—
349C24—91no—
350C24—184no—
351C24—272no—
352C24—274no—
353C24—272no—
354C24—544no—
355C2×D4—92no—
356C2×D4—92no—
357C2×D4—94no—
358C2×D4—91no—
359C2×D4—271no—
360C2×D4—272no—
361C2×D4—272no—
362C2×D4—272no—
363C2×D4—272no—
364C2×D4—272no—
365C2×D4—278no—
366C2×D4—272no—
367C2×D4—272no—
368C2×D4—272no—
369C2×D4—274no—
370C22×C4—91no—
371C22×C4—271no—
372C22×C4—271no—
373C22×C4—271no—
374C22×C4—271no—
375C22×C4C3×D3211yescharacteristic, complemented
376C22×C3—21no—
377C22×C3—32no—
378C22×C3—32no—
379C22×C3—34no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—94no—
383C22×C3—124no—
384C22×C3—184no—
385C22×C3—184no—
386C22×C3—184no—
387C22×C3—184no—
388C22×C3—244no—
389C22×C3—361no—
390C22×C3—724no—
391C22×C3C3×D3×D412yescomplemented
392C2×D3—32no—
393C2×D3—34no—
394C2×D3—61no—
395C2×D3—68no—
396C2×D3—64no—
397C2×D3—64no—
398C2×D3—64no—
399C2×D3—64no—
400C2×D3—94no—
401C2×D3—94no—
402C2×D3—94no—
403C2×D3—92no—
404C2×D3—94no—
405C2×D3—182no—
406C2×D3—182no—
407C2×D3—181no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—188no—
411C2×D3—188no—
412C2×D3—188no—
413C2×D3—1816no—
414C2×D3C2×D3×A414yescomplemented
415A4—22no—
416A4—41no—
417A4#380211yescharacteristic, direct product factor
418C3×C4—21no—
419C3×C4—32no—
420C3×C4—41no—
421C3×C4—61no—
422C3×C4—82no—
423C3×C4—161no—
424C3×C4—361no—
425C3×C4C2×D3×A412yescomplemented
426Dic3—32no—
427Dic3—61no—
428Dic3—92no—
429Dic3—181no—
430C32—42no—
431C32—81no—
432C32—82no—
433C32—161no—
434C32C2×D4×A411yescharacteristic, complemented
435C23—64no—
436C23—94no—
437C23—92no—
438C23—94no—
439C23—92no—
440C23—184no—
441C23—271no—
442C23—274no—
443C23—271no—
444C23—272no—
445C23—272no—
446C23—274no—
447C23—544no—
448C23—544no—
449C23—548no—
450C23—544no—
451C23—544no—
452C23—548no—
453C23C2×C3×D3211yescharacteristic
454D4—32no—
455D4—94no—
456D4—94no—
457D4—92no—
458D4—92no—
459D4—272no—
460D4—274no—
461D4—274no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—91no—
465C2×C4—271no—
466C2×C4—271no—
467C2×C4—272no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—271no—
471C2×C3—21no—
472C2×C3—32no—
473C2×C3—32no—
474C2×C3—41no—
475C2×C3—61no—
476C2×C3—61no—
477C2×C3—64no—
478C2×C3—82no—
479C2×C3—161no—
480C2×C3—184no—
481C2×C3—244no—
482C2×C3—362no—
483C2×C3—484no—
484C2×C3C22×D3×A412yes—
485D3—24no—
486D3—34no—
487D3—62no—
488D3—64no—
489D3—94no—
490D3—182no—
491C22—31no—
492C22—31no—
493C22—34no—
494C22—91no—
495C22—94no—
496C22—184no—
497C22—184no—
498C22—184no—
499C22—184no—
500C22—184no—
501C22—272no—
502C22—272no—
503C22—272no—
504C22—272no—
505C22—271no—
506C22—548no—
507C22—544no—
508C22—548no—
509C22C3×#380211yescharacteristic, complemented
510C4—31no—
511C4—91no—
512C4—271no—
513C4D32×A411yescharacteristic, complemented
514C3—21no—
515C3—41no—
516C3—82no—
517C3—161no—
518C3D3×D4×A412yescomplemented
519C2—31no—
520C2—31no—
521C2—64no—
522C2—92no—
523C2—184no—
524C2—272no—
525C2C2×D32×A411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×A4
commutator, socle, cosocleC23×C32C23×C3
FittingC22×C32×C4C22×C3
hypercenterC4D32×A4
hypocenterC22×C32C2×C3×D4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D32×A4C2semidirect
2. C4×A4×C3⋊D3C2semidirect
3. C3×A4×D12C2semidirect
4. A4×#625C2semidirect
5. C22×#3802C3semidirect
8. C2×C3×D3×A4C22semidirect
10. C32×C4×A4C22semidirect
12. A4×C3⋊Dic3C22semidirect
13. C23×D32C2×C3semidirect
15. C22×C4×C3⋊D3C2×C3semidirect
16. C22×C3×D12C2×C3semidirect
18. C22×#625C2×C3semidirect
26. A4×D12D3semidirect
31. A4×C3⋊D3D4semidirect
39. C23×C3×D3C22×C3semidirect
50. C2×D3×A4C2×D3semidirect
51. C22×C32×C4C22×C3semidirect
52. C22×C3⋊Dic3C22×C3semidirect
64. C3×C4×A4C2×D3semidirect
76. #3802A4direct (N × H)
81. C32×A4C2×D4semidirect
95. C22×D12C3×D3semidirect
106. C22×C3⋊D3C3×D4semidirect
126. C2×D32C2×A4semidirect
140. C4×C3⋊D3C2×A4semidirect
147. C3×D12C2×A4semidirect
153. #625C2×A4semidirect
167. C23×D3C2×C3×D3semidirect
177. C22×C3×C4C2×C3×D3semidirect
209. C4×A4D32semidirect
213. C22×C32C2×C3×D4semidirect
230. C2×C3×D3C22×A4semidirect
234. C3×A4D3×D4semidirect
244. C32×C4C22×A4semidirect
246. C3⋊Dic3C22×A4semidirect
321. D12D3×A4semidirect
341. C3⋊D3D4×A4semidirect
375. C22×C4C3×D32semidirect
391. C22×C3C3×D3×D4semidirect
414. C2×D3C2×D3×A4semidirect
417. A4#3802direct (N × H)
425. C3×C4C2×D3×A4semidirect
434. C32C2×D4×A4semidirect
509. C22C3×#3802semidirect
513. C4D32×A4semidirect
518. C3D3×D4×A4semidirect

Maximal subgroups

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

subgroupindex
C2×D32×A42
C4×A4×C3⋊D32
C3×A4×D122
A4×#6252
C22×#38023
D3×D4×A43
C3×#38024