Finite Group Databasebeta

← A4×(C3⋊S4)⋊C3 #72220525

Subgroups — A4×(C3⋊S4)⋊C3 #72220525

Order 2592.

476 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 38 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2A4×C3⋊S4C311yesmaximal, characteristic, complemented
3C3×A4×#67—31nomaximal
4C22×(C3⋊S4)⋊C3C311yesmaximal, characteristic, complemented
5(C3⋊S4)⋊A4C312yesmaximal, complemented
6A4×(C3⋊D3)⋊C3—41nomaximal
7C3×(C3⋊S4)⋊C3—41nomaximal
8C22×C32×A4D311yescharacteristic, complemented
9C2×C3×D3×A4—31no—
10C3×A42—21no—
11C3×A42C2×C311yescharacteristic, complemented
12C3×Dic3×A4—31no—
13C32⋊(C22⋊A4)—21no—
14C22×C22⋊H(3,𝔽3)C2×C311yescharacteristic, complemented
15C2×(C3⋊S4)⋊C3—31no—
16C2×(C3⋊S4)⋊C3—31no—
17(C3×A4)⋊A4—22no—
18(C3×A4)⋊A4C2×C312yescomplemented
19C3×C22⋊H(3,𝔽3)—41no—
20A4×H(3,𝔽3)—41no—
21C22×C3⋊S4C3211yescharacteristic, complemented
22A4×S4—91no—
23C22×C3×#67—31no—
24C3×D4×A4—91no—
25A4×#67—31no—
26A4×#67—32no—
27C2×C32×A4—31no—
28A4×C3⋊D3—41no—
29C3×D3×A4—61no—
30C3×C3⋊S4—41no—
31C32×#67—121no—
32C22×(C3⋊D3)⋊C3—41no—
33C2×C22⋊H(3,𝔽3)—31no—
34(C3⋊D3)⋊A4—42no—
35(C3⋊S4)⋊C3—31no—
36(C3⋊S4)⋊C3—42no—
37(C3⋊S4)⋊C3A411yescharacteristic, direct product factor
38C3×(C3⋊D3)⋊C3—161no—
39C24×C32C3×D311yescharacteristic, complemented
40C23×C3×D3—31no—
41C3×C22⋊A4—22no—
42C3×C22⋊A4—21no—
43C22×C3×A4—21no—
44C22×C3×A4—31no—
45C22×C3×A4—61no—
46C22×C3×A4—91no—
47C22×C3×A4C3×D311yescharacteristic
48C22×C3×A4C3×D312yescomplemented
49C22×C3×A4C2×C3211yescharacteristic, complemented
50C2×D3×A4—31no—
51C2×D3×A4—32no—
52C2×C3⋊S4—31no—
53C2×C3⋊S4—31no—
54A42—31no—
55A42—61no—
56A42—62no—
57A42—61no—
58C2×C3×#67—91no—
59C2×C3×#67—91no—
60C2×C3×#67—91no—
61C2×C3×#67—91no—
62C22×C3×Dic3—31no—
63C3×C4×A4—91no—
64Dic3×A4—31no—
65Dic3×A4—32no—
66C22×C33—41no—
67C2×C32×D3—121no—
68C32×A4—41no—
69C32×A4—41no—
70C32×A4—81no—
71C32×A4—81no—
72C32×A4S411yescharacteristic, complemented
73C32×Dic3—121no—
74C22×H(3,𝔽3)—41no—
75C2×(C3⋊D3)⋊C3—121no—
76C2×(C3⋊D3)⋊C3—121no—
77C22⋊H(3,𝔽3)—42no—
78C22⋊H(3,𝔽3)—42no—
79C22⋊H(3,𝔽3)—82no—
80C22⋊H(3,𝔽3)—81no—
81C22⋊H(3,𝔽3)—81no—
82C22⋊H(3,𝔽3)—82no—
83C22⋊H(3,𝔽3)—241no—
84C22⋊H(3,𝔽3)C2×A411yescharacteristic, complemented
85C22×S4—91no—
86C22×C3×D4—91no—
87C22×#67—31no—
88D4×A4—92no—
89D4×A4—91no—
90C3×H(3,𝔽3)—161noSylow
91C23×C32—31no—
92C23×C32—31no—
93C23×C32—91no—
94C22×C3⋊D3—41no—
95C22×C3×D3—61no—
96C22×C3×D3—91no—
97C22×C3×D3—91no—
98C22×C3×D3—181no—
99C2×C3×A4—31no—
100C2×C3×A4—31no—
101C2×C3×A4—32no—
102C2×C3×A4—31no—
103C2×C3×A4—61no—
104C2×C3×A4—62no—
105C2×C3×A4—61no—
106C2×C3×A4—61no—
107C2×C3×A4—181no—
108D3×A4—61no—
109D3×A4—62no—
110D3×A4—361no—
111C3⋊S4—31no—
112C3⋊S4C3×A411yescharacteristic, complemented
113C3×S4—361no—
114C32×D4—361no—
115C3×#67—31no—
116C3×#67—91no—
117C3×#67—91no—
118C3×#67—92no—
119C3×#67—91no—
120C3×#67—122no—
121C3×#67—121no—
122C2×C3×Dic3—91no—
123C2×C3×Dic3—91no—
124C2×C33—121no—
125C3×C3⋊D3—161no—
126C32×D3—241no—
127C2×H(3,𝔽3)—121no—
128(C3⋊D3)⋊C3—41no—
129(C3⋊D3)⋊C3—121no—
130(C3⋊D3)⋊C3—162no—
131C24×C3—31no—
132C24×C3—91no—
133C24×C3C32×D311yescharacteristic
134C23×D3—31no—
135C22⋊A4—61no—
136C22⋊A4—62no—
137C22×A4—21no—
138C22×A4—32no—
139C22×A4—31no—
140C22×A4—61no—
141C22×A4—92no—
142C22×A4—91no—
143C22×A4(C3⋊D3)⋊C311yescharacteristic, complemented
144C2×S4—271no—
145C2×S4—271no—
146C2×C3×D4—271no—
147C2×C3×D4—271no—
148C2×C3×D4—271no—
149C2×C3×D4—271no—
150C22×C3×C4—91no—
151C2×#67—91no—
152C2×#67—91no—
153C2×#67—91no—
154C2×#67—91no—
155C22×Dic3—31no—
156C4×A4—92no—
157C4×A4—91no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—42no—
161C22×C32—61no—
162C22×C32—81no—
163C22×C32—91no—
164C22×C32—91no—
165C22×C32—91no—
166C22×C32—121no—
167C22×C32—241no—
168C22×C32—361no—
169C22×C32C3×S411yescharacteristic, complemented
170C22×C32D3×A411yescharacteristic, complemented
171C2×C3⋊D3—121no—
172C2×C3⋊D3—121no—
173C2×C3×D3—31no—
174C2×C3×D3—91no—
175C2×C3×D3—122no—
176C2×C3×D3—121no—
177C2×C3×D3—181no—
178C2×C3×D3—181no—
179C2×C3×D3—181no—
180C2×C3×D3—181no—
181C3×A4—21no—
182C3×A4—31no—
183C3×A4—61no—
184C3×A4—82no—
185C3×A4—81no—
186C3×A4—82no—
187C3×A4—81no—
188C3×A4—121no—
189C3×A4—121no—
190C3×A4—242no—
191C3×A4—241no—
192C3×A4—242no—
193C3×A4—241no—
194C3×A4—241no—
195C3×A4—241no—
196C3×A4—242no—
197C3×A4—241no—
198C3×A4C3×S411yescharacteristic
199C3×A4C3×S412yescomplemented
200C3×A4C2×C3×A411yescharacteristic, complemented
201C32×C4—361no—
202C3×Dic3—31no—
203C3×Dic3—91no—
204C3×Dic3—122no—
205C3×Dic3—121no—
206C22×D4—91noSylow
207C33—41no—
208C33—161no—
209C33—321no—
210H(3,𝔽3)—41no—
211H(3,𝔽3)—162no—
212H(3,𝔽3)—322no—
213H(3,𝔽3)—321no—
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—61no—
218C23×C3—91no—
219C23×C3—91no—
220C23×C3—91no—
221C23×C3—181no—
222C23×C3—181no—
223C23×C3—271no—
224C23×C3—271no—
225C23×C3—541no—
226C22×D3—61no—
227C22×D3—91no—
228C22×D3—91no—
229C22×D3—181no—
230C22×D3—361no—
231C2×A4—32no—
232C2×A4—31no—
233C2×A4—61no—
234C2×A4—62no—
235C2×A4—91no—
236C2×A4—181no—
237C2×A4—182no—
238C2×A4—181no—
239S4—91no—
240S4—271no—
241C3×D4—91no—
242C3×D4—271no—
243C3×D4—271no—
244C3×D4—271no—
245C3×D4—272no—
246C3×D4—361no—
247C3×D4—362no—
248C2×C3×C4—271no—
249C2×C3×C4—271no—
250#67—31no—
251#67—91no—
252#67—91no—
253#67—92no—
254#67—91no—
255C2×Dic3—91no—
256C2×Dic3—91no—
257C2×C32—31no—
258C2×C32—31no—
259C2×C32—91no—
260C2×C32—121no—
261C2×C32—121no—
262C2×C32—122no—
263C2×C32—121no—
264C2×C32—241no—
265C2×C32—241no—
266C2×C32—241no—
267C2×C32—242no—
268C2×C32—721no—
269C3⋊D3—41no—
270C3⋊D3—121no—
271C3×D3—61no—
272C3×D3—181no—
273C3×D3—242no—
274C3×D3—241no—
275C3×D3—1441no—
276C24—91no—
277C24C3×(C3⋊D3)⋊C311yescharacteristic, complemented
278C2×D4—271no—
279C2×D4—271no—
280C2×D4—271no—
281C2×D4—271no—
282C22×C4—91no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—41no—
286C22×C3—61no—
287C22×C3—81no—
288C22×C3—91no—
289C22×C3—91no—
290C22×C3—91no—
291C22×C3—91no—
292C22×C3—91no—
293C22×C3—91no—
294C22×C3—91no—
295C22×C3—121no—
296C22×C3—122no—
297C22×C3—181no—
298C22×C3—181no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—241no—
302C22×C3—271no—
303C22×C3—361no—
304C22×C3—362no—
305C22×C3—541no—
306C22×C3—541no—
307C22×C3—541no—
308C22×C3—541no—
309C22×C3C32×S411yescharacteristic
310C22×C3C3×D3×A411yescharacteristic
311C2×D3—31no—
312C2×D3—91no—
313C2×D3—181no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—181no—
317C2×D3—1081no—
318C2×D3—1081no—
319A4—21no—
320A4—32no—
321A4—31no—
322A4—61no—
323A4—241no—
324A4—242no—
325A4—241no—
326A4—242no—
327A4—721no—
328A4—722no—
329A4(C3⋊S4)⋊C311yescharacteristic, direct product factor
330C3×C4—91no—
331C3×C4—271no—
332C3×C4—361no—
333C3×C4—362no—
334Dic3—31no—
335Dic3—91no—
336C32—41no—
337C32—42no—
338C32—41no—
339C32—81no—
340C32—121no—
341C32—241no—
342C32—322no—
343C32—321no—
344C32—481no—
345C32—961no—
346C32—961no—
347C32—962no—
348C32A4×S411yescharacteristic, complemented
349C23—31no—
350C23—31no—
351C23—91no—
352C23—181no—
353C23—271no—
354C23—271no—
355C23—541no—
356D4—91no—
357D4—271no—
358D4—271no—
359D4—272no—
360D4—271no—
361C2×C4—271no—
362C2×C4—271no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—61no—
367C2×C3—91no—
368C2×C3—91no—
369C2×C3—91no—
370C2×C3—121no—
371C2×C3—122no—
372C2×C3—181no—
373C2×C3—181no—
374C2×C3—241no—
375C2×C3—242no—
376C2×C3—361no—
377C2×C3—541no—
378C2×C3—721no—
379C2×C3—721no—
380C2×C3—722no—
381D3—61no—
382D3—181no—
383D3—361no—
384D3—1081no—
385C22—61no—
386C22—91no—
387C22—91no—
388C22—91no—
389C22—91no—
390C22—271no—
391C22—541no—
392C22—541no—
393C22—541no—
394C22—541no—
395C22C3×(C3⋊S4)⋊C311yescharacteristic, complemented
396C22A4×(C3⋊D3)⋊C311yescharacteristic, complemented
397C4—91no—
398C4—271no—
399C3—31no—
400C3—41no—
401C3—81no—
402C3—121no—
403C3—122no—
404C3—241no—
405C3—961no—
406C3—962no—
407C3C3×A4×S411yescharacteristic
408C2—31no—
409C2—31no—
410C2—91no—
411C2—181no—
412C2—541no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×C3×A4C2×C32
FrattiniC3C3×A4×S4
FittingC24×C32C3×D3
socleC24×C3C32×D3
LCSG ▷ C22×C3×A4
UCSC1

Complements

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

NH ≅ G/Ntype
1. A4×C22⋊H(3,𝔽3)C2semidirect
2. A4×C3⋊S4C3semidirect
4. C22×(C3⋊S4)⋊C3C3semidirect
5. (C3⋊S4)⋊A4C3semidirect
8. C22×C32×A4D3semidirect
11. C3×A42C2×C3semidirect
14. C22×C22⋊H(3,𝔽3)C2×C3semidirect
18. (C3×A4)⋊A4C2×C3semidirect
21. C22×C3⋊S4C32semidirect
37. (C3⋊S4)⋊C3A4direct (N × H)
39. C24×C32C3×D3semidirect
48. C22×C3×A4C3×D3semidirect
49. C22×C3×A4C2×C32semidirect
72. C32×A4S4semidirect
84. C22⋊H(3,𝔽3)C2×A4semidirect
112. C3⋊S4C3×A4semidirect
143. C22×A4(C3⋊D3)⋊C3semidirect
169. C22×C32C3×S4semidirect
170. C22×C32D3×A4semidirect
199. C3×A4C3×S4semidirect
200. C3×A4C2×C3×A4semidirect
277. C24C3×(C3⋊D3)⋊C3semidirect
329. A4(C3⋊S4)⋊C3direct (N × H)
348. C32A4×S4semidirect
395. C22C3×(C3⋊S4)⋊C3semidirect
396. C22A4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
A4×C22⋊H(3,𝔽3)2
A4×C3⋊S43
C3×A4×#673
C22×(C3⋊S4)⋊C33
(C3⋊S4)⋊A43
A4×(C3⋊D3)⋊C34
C3×(C3⋊S4)⋊C34