Finite Group Databasebeta

← A4×#3714 #13166129

Subgroups — A4×#3714 #13166129

Order 1728.

498 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 54 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
1A4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×A4×D12C211yesmaximal, characteristic, complemented
3C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
4C22×#3714C311yesmaximal, characteristic, complemented
5A4×#213—31nomaximal
6A4×D24—31nomaximal
7C2×A4×C3⋊D3—21no—
8C2×C3×D3×A4—21no—
9C32×C4×A4C2211yescharacteristic
10C3×#3714—41nomaximal
11C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
12C22×C3×D12C2×C311yescharacteristic, complemented
13C3×D4×A4—31no—
14A4×D12—21no—
15A4×D12—31no—
16A4×D12—31no—
17A4×D12—61no—
18A4×D12D311yescharacteristic, complemented
19C22×C3×C3⋊C8C2×C311yescharacteristic, complemented
20C3×C8×A4—31no—
21C2×#3714—31no—
22C2×#3714—31no—
23C2×#3714—31no—
24C2×#3714—31no—
25A4×C3⋊C8—21no—
26A4×C3⋊C8D311yescharacteristic, complemented
27C2×C32×A4D411yescharacteristic
28A4×C3⋊D3—41no—
29C3×D3×A4—41no—
30C3×C4⋊(C3⋊D3)—41no—
31C32×D12—41no—
32C32×C3⋊C8—41no—
33C22×#213—31no—
34C22×D24—31no—
35A4×D8—91no—
36C23×C3⋊D3—21no—
37C23×C3×D3—21no—
38C22×C3×A4—61no—
39C2×D3×A4—22no—
40C2×D3×A4—21no—
41C2×D3×A4—62no—
42C2×D3×A4—61no—
43C2×D3×A4—61no—
44C22×C32×C4C22×C311yescharacteristic
45C2×C4⋊(C3⋊D3)—31no—
46C2×C4⋊(C3⋊D3)—31no—
47C2×C4⋊(C3⋊D3)—61no—
48C2×C3×D12—31no—
49C2×C3×D12—31no—
50C2×C3×D12—61no—
51C3×C4×A4—21no—
52C3×C4×A4—21no—
53C3×C4×A4—21no—
54C3×C4×A4—41no—
55C3×C4×A4C2×D311yescharacteristic
56C3×C4×A4C2×D311yescharacteristic
57C3×#213—121no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8—31no—
60C3×D24—121no—
61#3714—32no—
62#3714—31no—
63#3714—31no—
64#3714—31no—
65#3714A411yescharacteristic, direct product factor
66C2×C3×C3⋊D3—81no—
67C2×C32×D3—81no—
68C32×A4D811yescharacteristic, complemented
69C33×C4—41no—
70C22×C3×D4—31no—
71C22×D12—31no—
72C22×D12—31no—
73C22×D12—61no—
74C22×D12C3×D311yescharacteristic, complemented
75D4×A4—31no—
76D4×A4—61no—
77D4×A4—91no—
78C22×C3×C8—31no—
79C2×#213—91no—
80C2×#213—91no—
81C2×#213—91no—
82C2×#213—91no—
83C22×C3⋊C8C3×D311yescharacteristic, complemented
84C2×D24—91no—
85C2×D24—91no—
86C2×D24—91no—
87C2×D24—91no—
88C8×A4—31no—
89C8×A4—61no—
90C23×C32C3×D411yescharacteristic
91C22×C3⋊D3—41no—
92C22×C3⋊D3—61no—
93C22×C3⋊D3—61no—
94C22×C3⋊D3—121no—
95C22×C3×D3—41no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—121no—
99C2×C3×A4—21no—
100C2×C3×A4—21no—
101C2×C3×A4—21no—
102C2×C3×A4—41no—
103C2×C3×A4—121no—
104C2×C3×A4D1211yescharacteristic
105C2×C3×A4#6711yescharacteristic
106D3×A4—42no—
107D3×A4—41no—
108D3×A4—122no—
109D3×A4—121no—
110D3×A4—121no—
111C32×D4—121no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C4⋊(C3⋊D3)—31no—
115C4⋊(C3⋊D3)—61no—
116C4⋊(C3⋊D3)—61no—
117C4⋊(C3⋊D3)C2×A411yescharacteristic, complemented
118C3×D12—31no—
119C3×D12—41no—
120C3×D12—61no—
121C3×D12—61no—
122C3×D12—81no—
123C3×D12—121no—
124C3×D12—121no—
125C3×D12—241no—
126C3×D12C2×A411yescharacteristic, complemented
127C32×C8—121no—
128C3×C3⋊C8—31no—
129C3×C3⋊C8—41no—
130C3×C3⋊C8—81no—
131C3×C3⋊C8C2×A411yescharacteristic, complemented
132C22×D8—91noSylow
133C2×C33—41no—
134C3×C3⋊D3—161no—
135C32×D3—161no—
136C24×C3—61no—
137C23×D3—21no—
138C23×D3—62no—
139C23×D3—61no—
140C23×D3—61no—
141C22×A4—62no—
142C22×A4—61no—
143C22×A4—181no—
144C2×C3×D4—91no—
145C2×C3×D4—91no—
146C2×C3×D4—181no—
147C22×C3×C4—21no—
148C22×C3×C4C2×C3×D311yescharacteristic
149C22×C3×C4C2×C3×D311yescharacteristic
150C2×D12—31no—
151C2×D12—31no—
152C2×D12—61no—
153C2×D12—91no—
154C2×D12—91no—
155C2×D12—91no—
156C2×D12—91no—
157C2×D12—181no—
158C2×D12—181no—
159C2×D12—182no—
160C2×D12—181no—
161C2×D12—181no—
162C4×A4—21no—
163C4×A4—21no—
164C4×A4—41no—
165C4×A4D3211yescharacteristic
166C3×D8—361no—
167C2×C3×C8—91no—
168C2×C3×C8—91no—
169#213—31no—
170#213—91no—
171#213—91no—
172#213—92no—
173#213—91no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176D24—31no—
177D24—91no—
178D24—91no—
179D24—92no—
180D24—91no—
181C22×C32—31no—
182C22×C32—31no—
183C22×C32—241no—
184C22×C32C3×D811yescharacteristic, complemented
185C2×C3⋊D3—21no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—121no—
191C2×C3×D3—21no—
192C2×C3×D3—61no—
193C2×C3×D3—81no—
194C2×C3×D3—82no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—241no—
200C2×C3×D3—241no—
201C2×C3×D3—242no—
202C3×A4—21no—
203C3×A4—21no—
204C3×A4—21no—
205C3×A4—41no—
206C3×A4D2411yescharacteristic, complemented
207C3×A4#21311yescharacteristic, complemented
208C32×C4—31no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4—81no—
212C32×C4—81no—
213C32×C4—81no—
214C32×C4—161no—
215C32×C4C22×A411yescharacteristic
216C22×D4—31no—
217C22×D4—91no—
218C2×D8—271no—
219C2×D8—271no—
220C2×D8—271no—
221C2×D8—271no—
222C22×C8—31no—
223C33—41noSylow
224C23×C3—21no—
225C23×C3—121no—
226C23×C3—181no—
227C23×C3—181no—
228C23×C3—361no—
229C23×C3C3×D1211yescharacteristic
230C23×C3C3×#6711yescharacteristic
231C22×D3—41no—
232C22×D3—61no—
233C22×D3—61no—
234C22×D3—121no—
235C22×D3—122no—
236C22×D3—121no—
237C22×D3—121no—
238C22×D3—181no—
239C22×D3—181no—
240C22×D3—182no—
241C22×D3—181no—
242C22×D3—181no—
243C22×D3—182no—
244C22×D3—361no—
245C22×D3—361no—
246C22×D3—362no—
247C2×A4—21no—
248C2×A4—21no—
249C2×A4—41no—
250C2×A4—122no—
251C2×A4—121no—
252C2×A4—361no—
253C2×A4#62611yescharacteristic
254C3×D4—31no—
255C3×D4—91no—
256C3×D4—121no—
257C3×D4—181no—
258C3×D4—181no—
259C3×D4—241no—
260C3×D4—361no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267D12—31no—
268D12—31no—
269D12—31no—
270D12—61no—
271D12—61no—
272D12—61no—
273D12—91no—
274D12—91no—
275D12—181no—
276D12—181no—
277D12—181no—
278D12—181no—
279D12—181no—
280D12—182no—
281D12—182no—
282D12D3×A411yescharacteristic, complemented
283C3×C8—31no—
284C3×C8—91no—
285C3×C8—121no—
286C3×C8—241no—
287C3⋊C8—31no—
288C3⋊C8D3×A411yescharacteristic, complemented
289C2×C32—31no—
290C2×C32—31no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—81no—
294C2×C32—81no—
295C2×C32—81no—
296C2×C32—161no—
297C2×C32—481no—
298C2×C32D4×A411yescharacteristic
299C3⋊D3—41no—
300C3⋊D3—121no—
301C3×D3—41no—
302C3×D3—121no—
303C3×D3—161no—
304C3×D3—162no—
305C3×D3—481no—
306C3×D3—481no—
307C3×D3—482no—
308C24—61no—
309C24—181no—
310C2×D4—91no—
311C2×D4—91no—
312C2×D4—181no—
313C2×D4—271no—
314C2×D4—271no—
315C2×D4—541no—
316C22×C4C3×D3211yescharacteristic
317D8—91no—
318D8—271no—
319D8—271no—
320D8—271no—
321D8—272no—
322C2×C8—91no—
323C2×C8—91no—
324C22×C3—21no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—61no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—181no—
333C22×C3—241no—
334C22×C3—242no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—721no—
340C22×C3C3×D2411yescharacteristic, complemented
341C22×C3C3×#21311yescharacteristic, complemented
342C2×D3—21no—
343C2×D3—62no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—182no—
352C2×D3—181no—
353C2×D3—181no—
354C2×D3—361no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—362no—
359C2×D3—362no—
360C2×D3—361no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—362no—
365C2×D3—362no—
366A4—21no—
367A4—21no—
368A4—41no—
369A4#371411yescharacteristic, direct product factor
370C3×C4—21no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—41no—
374C3×C4—61no—
375C3×C4—81no—
376C3×C4—81no—
377C3×C4—161no—
378C3×C4C2×D3×A411yescharacteristic
379C3×C4C2×D3×A411yescharacteristic
380C32—41no—
381C32—41no—
382C32—81no—
383C32—81no—
384C32—81no—
385C32—161no—
386C32A4×D811yescharacteristic, complemented
387C23—121no—
388C23—181no—
389C23—181no—
390C23—361no—
391C23—361no—
392C23—541no—
393C23—541no—
394C23—1081no—
395C23C3×#62611yescharacteristic
396D4—31no—
397D4—91no—
398D4—91no—
399D4—181no—
400D4—181no—
401D4—271no—
402D4—541no—
403D4—541no—
404C2×C4—31no—
405C2×C4—31no—
406C8—31no—
407C8—91no—
408C2×C3—21no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—41no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—81no—
417C2×C3—81no—
418C2×C3—121no—
419C2×C3—161no—
420C2×C3—361no—
421C2×C3—481no—
422C2×C3—482no—
423C2×C3—1441no—
424C2×C3A4×D1211yescharacteristic
425C2×C3A4×#6711yescharacteristic
426D3—41no—
427D3—122no—
428D3—121no—
429D3—121no—
430D3—121no—
431D3—362no—
432D3—361no—
433D3—361no—
434C22—31no—
435C22—31no—
436C22—61no—
437C22—181no—
438C22—181no—
439C22—361no—
440C22—361no—
441C22—361no—
442C22—361no—
443C22—541no—
444C22—1081no—
445C22—1081no—
446C22—1081no—
447C22—1081no—
448C22C3×#371411yescharacteristic, complemented
449C4—31no—
450C4D32×A411yescharacteristic
451C3—21no—
452C3—41no—
453C3—81no—
454C3—81no—
455C3—161no—
456C3A4×D2411yescharacteristic, complemented
457C3A4×#21311yescharacteristic, complemented
458C2—31no—
459C2—31no—
460C2—121no—
461C2—361no—
462C2—361no—
463C2—1081no—
464C2A4×#62611yescharacteristic
subgroupG/H
centerC2A4×#626
commutator, Fitting, cosocleC22×C32×C4C22×C3
Frattini, hypercenterC4D32×A4
socleC23×C32C3×D4
hypocenterC22×C32C3×D8
LCSG ▷ C22×C32×C4 ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. A4×C4⋊(C3⋊D3)C2semidirect
2. C3×A4×D12C2semidirect
3. C3×A4×C3⋊C8C2semidirect
4. C22×#3714C3semidirect
11. C22×C4⋊(C3⋊D3)C2×C3semidirect
12. C22×C3×D12C2×C3semidirect
18. A4×D12D3semidirect
19. C22×C3×C3⋊C8C2×C3semidirect
26. A4×C3⋊C8D3semidirect
65. #3714A4direct (N × H)
68. C32×A4D8semidirect
74. C22×D12C3×D3semidirect
83. C22×C3⋊C8C3×D3semidirect
117. C4⋊(C3⋊D3)C2×A4semidirect
126. C3×D12C2×A4semidirect
131. C3×C3⋊C8C2×A4semidirect
184. C22×C32C3×D8semidirect
206. C3×A4D24semidirect
207. C3×A4#213semidirect
282. D12D3×A4semidirect
288. C3⋊C8D3×A4semidirect
340. C22×C3C3×D24semidirect
341. C22×C3C3×#213semidirect
369. A4#3714direct (N × H)
386. C32A4×D8semidirect
448. C22C3×#3714semidirect
456. C3A4×D24semidirect
457. C3A4×#213semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊(C3⋊D3)2
C3×A4×D122
C3×A4×C3⋊C82
C22×#37143
A4×#2133
A4×D243
C3×#37144