Finite Group Databasebeta

← C3×A4×C3⋊S4 #72220520

Subgroups — C3×A4×C3⋊S4 #72220520

Order 2592.

953 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C32×A42C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊S4C311yesmaximal, characteristic, complemented
3A4×C3⋊S4C313yesmaximal, direct product factor
4C3×A4×S4—33nomaximal
5C3×A4×#67—31nomaximal
6C3×A4×C3⋊D3—41nomaximal
7C32×C3⋊S4—41nomaximal
8C32×C22⋊A4—21no—
9C22×C32×A4D311yescharacteristic, complemented
10C22×C32×A4C2×C311yescharacteristic, complemented
11C2×C3×D3×A4—31no—
12C2×C3×C3⋊S4—31no—
13C2×C3×C3⋊S4—31no—
14C3×A42—23no—
15C3×A42—23no—
16C3×A42—29no—
17C3×A42D313yescomplemented
18C3×A42C2×C313yescomplemented
19C3×Dic3×A4—31no—
20C33×A4—41no—
21C33×A4—41no—
22C22×C3⋊S4C3211yescharacteristic, complemented
23C22×C3×S4—33no—
24A4×S4—39no—
25C22×C3×#67—31no—
26C3×D4×A4—91no—
27A4×#67—33no—
28C22×C3×C3⋊D3—41no—
29C2×C32×A4—31no—
30C2×C32×A4—31no—
31A4×C3⋊D3—43no—
32C3×D3×A4—61no—
33C3×D3×A4—123no—
34C3×C3⋊S4—31no—
35C3×C3⋊S4—43no—
36C3×C3⋊S4A411yescharacteristic, direct product factor
37C32×S4—123no—
38C32×#67—121no—
39C32×C3⋊D3—161no—
40C24×C32C3×D311yescharacteristic, complemented
41C23×C3×D3—31no—
42C3×C22⋊A4—26no—
43C3×C22⋊A4—23no—
44C3×C22⋊A4—23no—
45C22×C3×A4—21no—
46C22×C3×A4—21no—
47C22×C3×A4—23no—
48C22×C3×A4—23no—
49C22×C3×A4—91no—
50C22×C3×A4C3×D313yescomplemented
51C22×C3×A4C3×D313yescomplemented
52C22×C3×A4C3⋊D311yescharacteristic, complemented
53C22×C3×A4C2×C3211yescharacteristic, complemented
54C2×D3×A4—33no—
55C2×C3⋊S4—31no—
56C2×C3⋊S4—31no—
57C2×C3×S4—93no—
58C2×C3×S4—93no—
59A42—218no—
60A42—29no—
61A42—29no—
62A42C3×D319yescomplemented
63C2×C3×#67—91no—
64C2×C3×#67—91no—
65C2×C3×#67—91no—
66C2×C3×#67—91no—
67C22×C3×Dic3—31no—
68C3×C4×A4—91no—
69Dic3×A4—33no—
70C22×C33—41no—
71C22×C33—41no—
72C2×C3×C3⋊D3—121no—
73C2×C3×C3⋊D3—121no—
74C2×C32×D3—121no—
75C32×A4—43no—
76C32×A4—43no—
77C32×A4—43no—
78C32×A4—43no—
79C32×A4—89no—
80C32×A4—89no—
81C32×A4—83no—
82C32×A4—83no—
83C32×A4—81no—
84C32×A4—81no—
85C32×A4—83no—
86C32×A4—83no—
87C32×A4—241no—
88C32×A4S411yescharacteristic, complemented
89C32×A4C2×A411yescharacteristic, complemented
90C32×Dic3—121no—
91C22×S4—33no—
92C22×C3×D4—91no—
93C22×#67—31no—
94D4×A4—93no—
95C34—161noSylow
96C23×C32—31no—
97C23×C32—31no—
98C23×C32—91no—
99C22×C3⋊D3—41no—
100C22×C3×D3—61no—
101C22×C3×D3—91no—
102C22×C3×D3—91no—
103C22×C3×D3—123no—
104C22×C3×D3—181no—
105C2×C3×A4—31no—
106C2×C3×A4—31no—
107C2×C3×A4—33no—
108C2×C3×A4—33no—
109C2×C3×A4—63no—
110C2×C3×A4—63no—
111C2×C3×A4—61no—
112C2×C3×A4—61no—
113C2×C3×A4—181no—
114D3×A4—63no—
115D3×A4—129no—
116C3⋊S4—31no—
117C3⋊S4C3×A411yescharacteristic, direct product factor
118C3×S4—33no—
119C3×S4—93no—
120C3×S4—129no—
121C32×D4—361no—
122C3×#67—31no—
123C3×#67—91no—
124C3×#67—91no—
125C3×#67—92no—
126C3×#67—91no—
127C3×#67—123no—
128C2×C3×Dic3—91no—
129C2×C3×Dic3—91no—
130C2×C33—121no—
131C2×C33—121no—
132C3×C3⋊D3—41no—
133C3×C3⋊D3—121no—
134C3×C3⋊D3—163no—
135C32×D3—241no—
136C32×D3—483no—
137C24×C3—21no—
138C24×C3—91no—
139C24×C3C32×D311yescharacteristic, complemented
140C24×C3C3×C3⋊D311yescharacteristic, complemented
141C23×D3—31no—
142C22⋊A4—29no—
143C22×A4—23no—
144C22×A4—23no—
145C22×A4—93no—
146C22×A4C32×D313yescomplemented
147C22×A4C3×C3⋊D313yescomplemented
148C2×S4—93no—
149C2×S4—93no—
150C2×C3×D4—271no—
151C2×C3×D4—271no—
152C2×C3×D4—271no—
153C2×C3×D4—271no—
154C22×C3×C4—91no—
155C2×#67—91no—
156C2×#67—91no—
157C2×#67—91no—
158C2×#67—91no—
159C22×Dic3—31no—
160C4×A4—93no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32—43no—
164C22×C32—43no—
165C22×C32—61no—
166C22×C32—83no—
167C22×C32—83no—
168C22×C32—81no—
169C22×C32—81no—
170C22×C32—91no—
171C22×C32—91no—
172C22×C32—91no—
173C22×C32—361no—
174C22×C32C3×S411yescharacteristic, complemented
175C22×C32D3×A411yescharacteristic, complemented
176C2×C3⋊D3—121no—
177C2×C3⋊D3—121no—
178C2×C3×D3—31no—
179C2×C3×D3—91no—
180C2×C3×D3—123no—
181C2×C3×D3—181no—
182C2×C3×D3—181no—
183C2×C3×D3—181no—
184C2×C3×D3—181no—
185C2×C3×D3—363no—
186C2×C3×D3—363no—
187C3×A4—21no—
188C3×A4—21no—
189C3×A4—23no—
190C3×A4—23no—
191C3×A4—49no—
192C3×A4—49no—
193C3×A4—89no—
194C3×A4—89no—
195C3×A4—818no—
196C3×A4—818no—
197C3×A4—86no—
198C3×A4—86no—
199C3×A4—83no—
200C3×A4—83no—
201C3×A4—83no—
202C3×A4—83no—
203C3×A4—89no—
204C3×A4—89no—
205C3×A4—246no—
206C3×A4—243no—
207C3×A4—243no—
208C3×A4C3×S413yescomplemented
209C3×A4C3⋊S411yescharacteristic, direct product factor
210C3×A4D3×A413yescomplemented
211C3×A4C2×C3×A411yescharacteristic, complemented
212C32×C4—361no—
213C3×Dic3—31no—
214C3×Dic3—91no—
215C3×Dic3—123no—
216C22×D4—91noSylow
217C33—41no—
218C33—41no—
219C33—163no—
220C33—163no—
221C33—323no—
222C33—323no—
223C33—329no—
224C33—321no—
225C23×C3—31no—
226C23×C3—31no—
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3—61no—
231C23×C3—91no—
232C23×C3—91no—
233C23×C3—181no—
234C23×C3—181no—
235C23×C3—271no—
236C23×C3—271no—
237C23×C3—541no—
238C22×D3—61no—
239C22×D3—91no—
240C22×D3—91no—
241C22×D3—123no—
242C22×D3—181no—
243C2×A4—33no—
244C2×A4—33no—
245C2×A4—63no—
246C2×A4—63no—
247C2×A4—183no—
248S4—33no—
249S4—93no—
250C3×D4—91no—
251C3×D4—271no—
252C3×D4—271no—
253C3×D4—272no—
254C3×D4—271no—
255C3×D4—363no—
256C2×C3×C4—271no—
257C2×C3×C4—271no—
258#67—31no—
259#67—91no—
260#67—91no—
261#67—92no—
262#67—91no—
263C2×Dic3—91no—
264C2×Dic3—91no—
265C2×C32—31no—
266C2×C32—31no—
267C2×C32—91no—
268C2×C32—121no—
269C2×C32—121no—
270C2×C32—123no—
271C2×C32—123no—
272C2×C32—241no—
273C2×C32—241no—
274C2×C32—243no—
275C2×C32—243no—
276C2×C32—721no—
277C3⋊D3—41no—
278C3⋊D3—121no—
279C3×D3—61no—
280C3×D3—123no—
281C3×D3—181no—
282C3×D3—243no—
283C3×D3—363no—
284C3×D3—489no—
285C24—91no—
286C24C32×C3⋊D311yescharacteristic, complemented
287C2×D4—271no—
288C2×D4—271no—
289C2×D4—271no—
290C2×D4—271no—
291C22×C4—91no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—43no—
295C22×C3—43no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—62no—
299C22×C3—83no—
300C22×C3—83no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—91no—
306C22×C3—91no—
307C22×C3—91no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3—271no—
312C22×C3—363no—
313C22×C3—541no—
314C22×C3—541no—
315C22×C3—541no—
316C22×C3—541no—
317C22×C3C32×S411yescharacteristic, complemented
318C22×C3C3×C3⋊S411yescharacteristic, complemented
319C22×C3C3×D3×A411yescharacteristic, complemented
320C22×C3A4×C3⋊D311yescharacteristic, complemented
321C2×D3—31no—
322C2×D3—91no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—181no—
326C2×D3—181no—
327C2×D3—363no—
328C2×D3—363no—
329A4—23no—
330A4—23no—
331A4—89no—
332A4—89no—
333A4—249no—
334A4C3×C3⋊S413yesdirect product factor
335A4C3×D3×A413yescomplemented
336C3×C4—91no—
337C3×C4—271no—
338C3×C4—363no—
339Dic3—31no—
340Dic3—91no—
341C32—41no—
342C32—41no—
343C32—43no—
344C32—43no—
345C32—83no—
346C32—83no—
347C32—81no—
348C32—81no—
349C32—169no—
350C32—326no—
351C32—329no—
352C32—329no—
353C32—3218no—
354C32—323no—
355C32—323no—
356C32A4×S411yescharacteristic, complemented
357C23—31no—
358C23—31no—
359C23—91no—
360C23—181no—
361C23—271no—
362C23—271no—
363C23—541no—
364D4—91no—
365D4—271no—
366D4—271no—
367D4—272no—
368D4—271no—
369C2×C4—271no—
370C2×C4—271no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—61no—
376C2×C3—61no—
377C2×C3—91no—
378C2×C3—91no—
379C2×C3—123no—
380C2×C3—123no—
381C2×C3—181no—
382C2×C3—181no—
383C2×C3—243no—
384C2×C3—243no—
385C2×C3—541no—
386C2×C3—723no—
387D3—61no—
388D3—123no—
389D3—181no—
390D3—363no—
391C22—61no—
392C22—91no—
393C22—91no—
394C22—91no—
395C22—91no—
396C22—271no—
397C22—541no—
398C22—541no—
399C22—541no—
400C22—541no—
401C22C32×C3⋊S411yescharacteristic, complemented
402C22C3×A4×C3⋊D311yescharacteristic, complemented
403C4—91no—
404C4—271no—
405C3—21no—
406C3—43no—
407C3—43no—
408C3—83no—
409C3—83no—
410C3—329no—
411C3C3×A4×S411yescharacteristic, complemented
412C3A4×C3⋊S411yescharacteristic, direct product factor
413C2—31no—
414C2—31no—
415C2—91no—
416C2—181no—
417C2—541no—
subgroupG/H
center, hypercenterC3A4×C3⋊S4
commutator, cosocle, hypocenterC22×C3×A4C2×C32
FrattiniC1G
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. C32×A42C2semidirect
2. C22×C3×C3⋊S4C3semidirect
3. A4×C3⋊S4C3direct (N × H)
9. C22×C32×A4D3semidirect
10. C22×C32×A4C2×C3semidirect
17. C3×A42D3semidirect
18. C3×A42C2×C3semidirect
22. C22×C3⋊S4C32semidirect
36. C3×C3⋊S4A4direct (N × H)
40. C24×C32C3×D3semidirect
50. C22×C3×A4C3×D3semidirect
51. C22×C3×A4C3×D3semidirect
52. C22×C3×A4C3⋊D3semidirect
53. C22×C3×A4C2×C32semidirect
62. A42C3×D3semidirect
88. C32×A4S4semidirect
89. C32×A4C2×A4semidirect
117. C3⋊S4C3×A4direct (N × H)
139. C24×C3C32×D3semidirect
140. C24×C3C3×C3⋊D3semidirect
146. C22×A4C32×D3semidirect
147. C22×A4C3×C3⋊D3semidirect
174. C22×C32C3×S4semidirect
175. C22×C32D3×A4semidirect
208. C3×A4C3×S4semidirect
209. C3×A4C3⋊S4direct (N × H)
210. C3×A4D3×A4semidirect
211. C3×A4C2×C3×A4semidirect
286. C24C32×C3⋊D3semidirect
317. C22×C3C32×S4semidirect
318. C22×C3C3×C3⋊S4semidirect
319. C22×C3C3×D3×A4semidirect
320. C22×C3A4×C3⋊D3semidirect
334. A4C3×C3⋊S4direct (N × H)
335. A4C3×D3×A4semidirect
356. C32A4×S4semidirect
401. C22C32×C3⋊S4semidirect
402. C22C3×A4×C3⋊D3semidirect
411. C3C3×A4×S4semidirect
412. C3A4×C3⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C32×A422
C22×C3×C3⋊S43
A4×C3⋊S43
C3×A4×S43
C3×A4×#673
C3×A4×C3⋊D34
C32×C3⋊S44