Finite Group Databasebeta

← C3×(C3⋊S4)⋊A4 #72220522

Subgroups — C3×(C3⋊S4)⋊A4 #72220522

Order 2592.

545 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C3×(C3×A4)⋊A4C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊S4C311yesmaximal, characteristic, complemented
3C3×A4×#67—31nomaximal
4(C3⋊S4)⋊A4C313yesmaximal, direct product factor
5C3×(C3⋊D3)⋊A4—41nomaximal
6C3×(C3⋊S4)⋊C3—41nomaximal
7C32×C22⋊A4—21no—
8C22×C32×A4D311yescharacteristic, complemented
9C22×C32×A4C2×C311yescharacteristic, complemented
10C2×C3×D3×A4—31no—
11C2×C3×C3⋊S4—31no—
12C2×C3×C3⋊S4—31no—
13C3×Dic3×A4—31no—
14(C3×A4)⋊A4—23no—
15(C3×A4)⋊A4C2×C313yescomplemented
16C3×C22⋊H(3,𝔽3)—41no—
17C3×C22⋊H(3,𝔽3)—41no—
18C22×C3⋊S4C3211yescharacteristic, complemented
19C22×C3×S4—91no—
20C22×C3×#67—31no—
21C3×D4×A4—91no—
22A4×#67—33no—
23C22×C3×C3⋊D3—41no—
24C2×C32×A4—31no—
25C2×C32×A4—31no—
26C3×D3×A4—61no—
27C3×C3⋊S4—31no—
28C3×C3⋊S4A411yescharacteristic, complemented
29C32×#67—121no—
30(C3⋊D3)⋊A4—43no—
31(C3⋊S4)⋊C3—43no—
32C3×(C3⋊D3)⋊C3—161no—
33C24×C32C3×D311yescharacteristic
34C23×C3×D3—31no—
35C3×C22⋊A4—23no—
36C3×C22⋊A4—62no—
37C3×C22⋊A4—61no—
38C22×C3×A4—21no—
39C22×C3×A4—31no—
40C22×C3×A4—31no—
41C22×C3×A4—61no—
42C22×C3×A4—61no—
43C22×C3×A4—91no—
44C22×C3×A4C3×D313yescomplemented
45C22×C3×A4C2×C3211yescharacteristic, complemented
46C2×D3×A4—33no—
47C2×C3⋊S4—31no—
48C2×C3⋊S4—31no—
49C2×C3×S4—93no—
50C2×C3×S4—93no—
51C2×C3×#67—91no—
52C2×C3×#67—91no—
53C2×C3×#67—91no—
54C2×C3×#67—91no—
55C22×C3×Dic3—31no—
56C3×C4×A4—91no—
57Dic3×A4—33no—
58C22×C33—41no—
59C22×C33—41no—
60C2×C3×C3⋊D3—121no—
61C2×C3×C3⋊D3—121no—
62C2×C32×D3—121no—
63C32×A4—81no—
64C32×A4—81no—
65C32×A4—241no—
66C32×A4S411yescharacteristic, complemented
67C32×A4C2×A411yescharacteristic, complemented
68C32×Dic3—121no—
69C22⋊H(3,𝔽3)—43no—
70C22⋊H(3,𝔽3)—43no—
71C22⋊H(3,𝔽3)—83no—
72C22⋊H(3,𝔽3)—83no—
73C22×S4—91no—
74C22×C3×D4—91no—
75C22×#67—31no—
76D4×A4—93no—
77C3×H(3,𝔽3)—161noSylow
78C23×C32—31no—
79C23×C32—31no—
80C23×C32—91no—
81C22×C3⋊D3—41no—
82C22×C3×D3—61no—
83C22×C3×D3—91no—
84C22×C3×D3—91no—
85C22×C3×D3—181no—
86C22×C3×D3—361no—
87C2×C3×A4—31no—
88C2×C3×A4—31no—
89C2×C3×A4—33no—
90C2×C3×A4—33no—
91C2×C3×A4—63no—
92C2×C3×A4—63no—
93C2×C3×A4—61no—
94C2×C3×A4—61no—
95C2×C3×A4—181no—
96D3×A4—63no—
97C3⋊S4—31no—
98C3⋊S4C3×A411yescharacteristic, complemented
99C3×S4—91no—
100C3×S4—93no—
101C32×D4—361no—
102C3×#67—31no—
103C3×#67—91no—
104C3×#67—91no—
105C3×#67—92no—
106C3×#67—91no—
107C3×#67—123no—
108C2×C3×Dic3—91no—
109C2×C3×Dic3—91no—
110C2×C33—121no—
111C2×C33—121no—
112C3×C3⋊D3—41no—
113C3×C3⋊D3—121no—
114C32×D3—241no—
115(C3⋊D3)⋊C3—163no—
116C24×C3—21no—
117C24×C3—91no—
118C24×C3(C3⋊D3)⋊C311yescharacteristic, complemented
119C24×C3C32×D311yescharacteristic
120C23×D3—31no—
121C22⋊A4—63no—
122C22×A4—31no—
123C22×A4—33no—
124C22×A4—61no—
125C22×A4—93no—
126C2×S4—93no—
127C2×S4—93no—
128C2×C3×D4—271no—
129C2×C3×D4—271no—
130C2×C3×D4—271no—
131C2×C3×D4—271no—
132C22×C3×C4—91no—
133C2×#67—91no—
134C2×#67—91no—
135C2×#67—91no—
136C2×#67—91no—
137C22×Dic3—31no—
138C4×A4—93no—
139C22×C32—41no—
140C22×C32—43no—
141C22×C32—61no—
142C22×C32—81no—
143C22×C32—91no—
144C22×C32—91no—
145C22×C32—91no—
146C22×C32—121no—
147C22×C32—121no—
148C22×C32—241no—
149C22×C32—241no—
150C22×C32—361no—
151C22×C32C3×S411yescharacteristic
152C22×C32D3×A411yescharacteristic
153C2×C3⋊D3—121no—
154C2×C3⋊D3—121no—
155C2×C3×D3—31no—
156C2×C3×D3—91no—
157C2×C3×D3—123no—
158C2×C3×D3—181no—
159C2×C3×D3—181no—
160C2×C3×D3—181no—
161C2×C3×D3—181no—
162C2×C3×D3—363no—
163C2×C3×D3—363no—
164C3×A4—21no—
165C3×A4—31no—
166C3×A4—31no—
167C3×A4—61no—
168C3×A4—61no—
169C3×A4—83no—
170C3×A4—83no—
171C3×A4—243no—
172C3×A4—242no—
173C3×A4—242no—
174C3×A4—246no—
175C3×A4—243no—
176C3×A4—241no—
177C3×A4—241no—
178C3×A4C3×S413yescomplemented
179C3×A4C2×C3×A411yescharacteristic, complemented
180C32×C4—361no—
181C3×Dic3—31no—
182C3×Dic3—91no—
183C3×Dic3—123no—
184C22×D4—91noSylow
185C33—41no—
186C33—41no—
187C33—321no—
188H(3,𝔽3)—163no—
189H(3,𝔽3)—323no—
190C23×C3—31no—
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—61no—
195C23×C3—61no—
196C23×C3—91no—
197C23×C3—91no—
198C23×C3—181no—
199C23×C3—181no—
200C23×C3—271no—
201C23×C3—271no—
202C23×C3—541no—
203C22×D3—61no—
204C22×D3—91no—
205C22×D3—91no—
206C22×D3—181no—
207C22×D3—361no—
208C2×A4—33no—
209C2×A4—33no—
210C2×A4—63no—
211C2×A4—63no—
212C2×A4—183no—
213S4—91no—
214S4—93no—
215C3×D4—91no—
216C3×D4—271no—
217C3×D4—271no—
218C3×D4—272no—
219C3×D4—271no—
220C3×D4—363no—
221C2×C3×C4—271no—
222C2×C3×C4—271no—
223#67—31no—
224#67—91no—
225#67—91no—
226#67—92no—
227#67—91no—
228C2×Dic3—91no—
229C2×Dic3—91no—
230C2×C32—31no—
231C2×C32—31no—
232C2×C32—91no—
233C2×C32—121no—
234C2×C32—121no—
235C2×C32—123no—
236C2×C32—123no—
237C2×C32—241no—
238C2×C32—241no—
239C2×C32—243no—
240C2×C32—243no—
241C2×C32—721no—
242C3⋊D3—41no—
243C3⋊D3—121no—
244C3×D3—61no—
245C3×D3—181no—
246C3×D3—243no—
247C3×D3—361no—
248C3×D3—363no—
249C24—91no—
250C24C3×(C3⋊D3)⋊C311yescharacteristic, complemented
251C2×D4—271no—
252C2×D4—271no—
253C2×D4—271no—
254C2×D4—271no—
255C22×C4—91no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—62no—
261C22×C3—91no—
262C22×C3—91no—
263C22×C3—91no—
264C22×C3—91no—
265C22×C3—91no—
266C22×C3—91no—
267C22×C3—91no—
268C22×C3—121no—
269C22×C3—123no—
270C22×C3—181no—
271C22×C3—181no—
272C22×C3—181no—
273C22×C3—241no—
274C22×C3—271no—
275C22×C3—363no—
276C22×C3—541no—
277C22×C3—541no—
278C22×C3—541no—
279C22×C3—541no—
280C22×C3(C3⋊S4)⋊C311yescharacteristic, complemented
281C22×C3(C3⋊D3)⋊A411yescharacteristic, complemented
282C22×C3C32×S411yescharacteristic
283C22×C3C3×D3×A411yescharacteristic
284C2×D3—31no—
285C2×D3—91no—
286C2×D3—181no—
287C2×D3—181no—
288C2×D3—181no—
289C2×D3—181no—
290C2×D3—363no—
291C2×D3—363no—
292A4—33no—
293A4—31no—
294A4—61no—
295A4—249no—
296A4—243no—
297A4—243no—
298C3×C4—91no—
299C3×C4—271no—
300C3×C4—363no—
301Dic3—31no—
302Dic3—91no—
303C32—43no—
304C32—41no—
305C32—81no—
306C32—121no—
307C32—121no—
308C32—241no—
309C32—241no—
310C32—323no—
311C32—962no—
312C32—961no—
313C32A4×S411yescharacteristic
314C23—31no—
315C23—31no—
316C23—91no—
317C23—181no—
318C23—271no—
319C23—271no—
320C23—541no—
321D4—91no—
322D4—271no—
323D4—271no—
324D4—272no—
325D4—271no—
326C2×C4—271no—
327C2×C4—271no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—61no—
333C2×C3—61no—
334C2×C3—91no—
335C2×C3—91no—
336C2×C3—123no—
337C2×C3—123no—
338C2×C3—181no—
339C2×C3—181no—
340C2×C3—243no—
341C2×C3—243no—
342C2×C3—541no—
343C2×C3—723no—
344D3—61no—
345D3—181no—
346D3—361no—
347D3—363no—
348C22—61no—
349C22—91no—
350C22—91no—
351C22—91no—
352C22—91no—
353C22—271no—
354C22—541no—
355C22—541no—
356C22—541no—
357C22—541no—
358C22C3×(C3⋊S4)⋊C311yescharacteristic, complemented
359C22C3×(C3⋊D3)⋊A411yescharacteristic, complemented
360C4—91no—
361C4—271no—
362C3—21no—
363C3—121no—
364C3—123no—
365C3—241no—
366C3—963no—
367C3(C3⋊S4)⋊A411yescharacteristic, direct product factor
368C3C3×A4×S411yescharacteristic
369C2—31no—
370C2—31no—
371C2—91no—
372C2—181no—
373C2—541no—
subgroupG/H
center, hypercenterC3(C3⋊S4)⋊A4
commutator, cosocle, hypocenterC22×C3×A4C2×C32
FrattiniC3C3×A4×S4
Fitting, socleC24×C32C3×D3
LCSG ▷ C22×C3×A4
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×(C3×A4)⋊A4C2semidirect
2. C22×C3×C3⋊S4C3semidirect
4. (C3⋊S4)⋊A4C3direct (N × H)
8. C22×C32×A4D3semidirect
9. C22×C32×A4C2×C3semidirect
15. (C3×A4)⋊A4C2×C3semidirect
18. C22×C3⋊S4C32semidirect
28. C3×C3⋊S4A4semidirect
44. C22×C3×A4C3×D3semidirect
45. C22×C3×A4C2×C32semidirect
66. C32×A4S4semidirect
67. C32×A4C2×A4semidirect
98. C3⋊S4C3×A4semidirect
118. C24×C3(C3⋊D3)⋊C3semidirect
178. C3×A4C3×S4semidirect
179. C3×A4C2×C3×A4semidirect
250. C24C3×(C3⋊D3)⋊C3semidirect
280. C22×C3(C3⋊S4)⋊C3semidirect
281. C22×C3(C3⋊D3)⋊A4semidirect
358. C22C3×(C3⋊S4)⋊C3semidirect
359. C22C3×(C3⋊D3)⋊A4semidirect
367. C3(C3⋊S4)⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C3×(C3×A4)⋊A42
C22×C3×C3⋊S43
C3×A4×#673
(C3⋊S4)⋊A43
C3×(C3⋊D3)⋊A44
C3×(C3⋊S4)⋊C34