Finite Group Databasebeta

← C2×C3×A4×#41 #11919870

Subgroups — C2×C3×A4×#41 #11919870

Order 1152.

3048 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×A4×#41C218yesmaximal, direct product factor
2C2×C3×Q8×A4C211yesmaximal, characteristic, complemented
3C2×C3×D4×A4C213yesmaximal, complemented
4C22×C3×C4×A4C213yesmaximal, complemented
5C23×C3×#41C311yesmaximal, characteristic, complemented
6C2×A4×#41C313yesmaximal, direct product factor
7C23×C3×A4C2213yes—
8C2×C32×#41—41nomaximal
9C3×Q8×A4C2214yescomplemented
10C3×D4×A4C22112yescomplemented
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2213yes—
13C2×C3×C4×A4C22112yescomplemented
14C22×C3×#41—33no—
15C22×C3×#41—33no—
16C22×C3×#41—38no—
17C22×C3×#41—31no—
18C22×C3×#41—31no—
19C22×C3×#41C2×C318yescomplemented
20C23×C3×Q8C2×C311yescharacteristic, complemented
21C23×C3×D4C2×C313yescomplemented
22C24×C3×C4C2×C313yescomplemented
23A4×#41C2×C3124yesdirect product factor
24C2×Q8×A4C2×C313yescomplemented
25C2×D4×A4C2×C319yescomplemented
26C22×C4×A4C2×C319yescomplemented
27C22×C3×A4—26no—
28C22×C3×A4C2316yes—
29C22×C3×A4C2311yescharacteristic
30C32×#41—48no—
31C2×C32×Q8—41no—
32C2×C32×D4—43no—
33C22×C32×C4—43no—
34C3×C4×A4C2312yes—
35C3×C4×A4C2316yes—
36C23×#41C3211yescharacteristic, complemented, Sylow
37C25×C3C22×C313yes—
38C23×A4C22×C319yes—
39C2×C3×#41—324no—
40C2×C3×#41—324no—
41C2×C3×#41—38no—
42C2×C3×#41—38no—
43C2×C3×#41—324no—
44C2×C3×#41—36no—
45C2×C3×#41—36no—
46C2×C3×#41—324no—
47C2×C3×#41—32no—
48C2×C3×#41—38no—
49C2×C3×#41—38no—
50C2×C3×#41—31no—
51C2×C3×#41—33no—
52C2×C3×#41—33no—
53C2×C3×#41—43no—
54C2×C3×#41A411yescharacteristic, direct product factor
55C22×C3×Q8—33no—
56C22×C3×Q8—34no—
57C22×C3×Q8—31no—
58C22×C3×Q8C22×C314yescomplemented
59C22×C3×D4—33no—
60C22×C3×D4—33no—
61C22×C3×D4—36no—
62C22×C3×D4—312no—
63C22×C3×D4C22×C3112yescomplemented
64C23×C3×C4—33no—
65C23×C3×C4—33no—
66C23×C3×C4—33no—
67C23×C3×C4—33no—
68C23×C3×C4—312no—
69C23×C3×C4C22×C311yescharacteristic
70C23×C3×C4C22×C313yes—
71C23×C3×C4C22×C3112yescomplemented
72Q8×A4C22×C3112yescomplemented
73D4×A4C22×C3136yescomplemented
74C2×C4×A4C22×C313yes—
75C2×C4×A4C22×C319yes—
76C2×C4×A4C22×C3136yescomplemented
77C23×C32—43no—
78C2×C3×A4—26no—
79C2×C3×A4#4112yesdirect product factor
80C2×C3×A4C2411yescharacteristic
81C32×Q8—44no—
82C32×D4—412no—
83C2×C32×C4—43no—
84C2×C32×C4—41no—
85C2×C32×C4—412no—
86C22×#41—33no—
87C22×#41—33no—
88C22×#41—38no—
89C22×#41—31no—
90C22×#41—31no—
91C22×#41C2×C3218yescomplemented
92C23×Q8C2×C3211yescharacteristic, complemented
93C23×D4C2×C3213yescomplemented
94C24×C4C2×C3213yescomplemented
95C24×C3—26no—
96C24×C3—36no—
97C24×C3—33no—
98C24×C3—33no—
99C24×C3—66no—
100C24×C3C23×C316yes—
101C24×C3C23×C311yescharacteristic
102C22×A4—218no—
103C22×A4C23×C3118yes—
104C22×A4C23×C313yes—
105C3×#41—348no—
106C3×#41—348no—
107C3×#41—316no—
108C3×#41—38no—
109C3×#41—324no—
110C3×#41—324no—
111C3×#41—424no—
112C3×#41C2×A418yesdirect product factor
113C2×C3×Q8—312no—
114C2×C3×Q8—312no—
115C2×C3×Q8—32no—
116C2×C3×Q8—34no—
117C2×C3×Q8—34no—
118C2×C3×Q8—33no—
119C2×C3×Q8—43no—
120C2×C3×Q8C2×A411yescharacteristic, complemented
121C2×C3×D4—324no—
122C2×C3×D4—36no—
123C2×C3×D4—312no—
124C2×C3×D4—36no—
125C2×C3×D4—312no—
126C2×C3×D4—312no—
127C2×C3×D4—312no—
128C2×C3×D4—324no—
129C2×C3×D4—33no—
130C2×C3×D4—49no—
131C2×C3×D4C2×A413yescomplemented
132C22×C3×C4—312no—
133C22×C3×C4—312no—
134C22×C3×C4—312no—
135C22×C3×C4—36no—
136C22×C3×C4—33no—
137C22×C3×C4—33no—
138C22×C3×C4—33no—
139C22×C3×C4—31no—
140C22×C3×C4—33no—
141C22×C3×C4—312no—
142C22×C3×C4—312no—
143C22×C3×C4—31no—
144C22×C3×C4—32no—
145C22×C3×C4—33no—
146C22×C3×C4—36no—
147C22×C3×C4—312no—
148C22×C3×C4—312no—
149C22×C3×C4—312no—
150C22×C3×C4—49no—
151C22×C3×C4C2×A413yescomplemented
152C22×C3×C4C23×C312yes—
153C22×C3×C4C23×C316yes—
154C4×A4C23×C316yes—
155C4×A4C23×C3118yes—
156C22×C32—41no—
157C22×C32—46no—
158C22×C32—86no—
159C3×A4C2×#4111yescharacteristic, direct product factor
160C32×C4—42no—
161C32×C4—46no—
162C25C22×C3213yes—
163C2×#41—324no—
164C2×#41—324no—
165C2×#41—38no—
166C2×#41—38no—
167C2×#41—324no—
168C2×#41—36no—
169C2×#41—36no—
170C2×#41—324no—
171C2×#41—32no—
172C2×#41—38no—
173C2×#41—38no—
174C2×#41—31no—
175C2×#41—33no—
176C2×#41—33no—
177C2×#41C3×A411yescharacteristic, direct product factor
178C22×Q8—33no—
179C22×Q8—34no—
180C22×Q8—31no—
181C22×Q8C22×C3214yescomplemented
182C22×D4—33no—
183C22×D4—33no—
184C22×D4—36no—
185C22×D4—312no—
186C22×D4C22×C32112yescomplemented
187C23×C4—33no—
188C23×C4—33no—
189C23×C4—33no—
190C23×C4—33no—
191C23×C4—312no—
192C23×C4C22×C3211yescharacteristic
193C23×C4C22×C3213yes—
194C23×C4C22×C32112yescomplemented
195C23×C3—26no—
196C23×C3—31no—
197C23×C3—32no—
198C23×C3—31no—
199C23×C3—36no—
200C23×C3—36no—
201C23×C3—36no—
202C23×C3—33no—
203C23×C3—36no—
204C23×C3—49no—
205C23×C3—66no—
206C23×C3—66no—
207C23×C3—66no—
208C23×C3—612no—
209C23×C3—66no—
210C23×C3—66no—
211C23×C3—612no—
212C23×C3C3×#4112yescomplemented
213C23×C3C22×A413yes—
214C23×C3C24×C311yescharacteristic
215C2×A4—218no—
216C2×A4C3×#4116yesdirect product factor
217C2×A4C24×C313yes—
218C3×Q8—38no—
219C3×Q8—312no—
220C3×Q8—412no—
221C3×Q8C22×A414yescomplemented
222C3×D4—324no—
223C3×D4—324no—
224C3×D4—312no—
225C3×D4—436no—
226C3×D4C22×A4112yescomplemented
227C2×C3×C4—32no—
228C2×C3×C4—36no—
229C2×C3×C4—324no—
230C2×C3×C4—31no—
231C2×C3×C4—33no—
232C2×C3×C4—312no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—36no—
236C2×C3×C4—36no—
237C2×C3×C4—312no—
238C2×C3×C4—312no—
239C2×C3×C4—32no—
240C2×C3×C4—36no—
241C2×C3×C4—43no—
242C2×C3×C4—49no—
243C2×C3×C4—436no—
244C2×C3×C4C22×A413yes—
245C2×C3×C4C22×A411yescharacteristic
246C2×C3×C4C22×A4112yescomplemented
247C2×C32—42no—
248C2×C32—41no—
249C2×C32—86no—
250C24—26no—
251C24—36no—
252C24—33no—
253C24—33no—
254C24—66no—
255C24C23×C3216yes—
256C24C23×C3211yescharacteristic
257#41—348no—
258#41—348no—
259#41—316no—
260#41—38no—
261#41—324no—
262#41—324no—
263#41C2×C3×A418yesdirect product factor
264C2×Q8—312no—
265C2×Q8—312no—
266C2×Q8—32no—
267C2×Q8—34no—
268C2×Q8—34no—
269C2×Q8—33no—
270C2×Q8C2×C3×A411yescharacteristic, complemented
271C2×D4—324no—
272C2×D4—36no—
273C2×D4—312no—
274C2×D4—36no—
275C2×D4—312no—
276C2×D4—312no—
277C2×D4—312no—
278C2×D4—324no—
279C2×D4—33no—
280C2×D4C2×C3×A413yescomplemented
281C22×C4—312no—
282C22×C4—312no—
283C22×C4—312no—
284C22×C4—36no—
285C22×C4—33no—
286C22×C4—31no—
287C22×C4—33no—
288C22×C4—33no—
289C22×C4—33no—
290C22×C4—312no—
291C22×C4—312no—
292C22×C4—31no—
293C22×C4—32no—
294C22×C4—33no—
295C22×C4—36no—
296C22×C4—312no—
297C22×C4—312no—
298C22×C4—312no—
299C22×C4C2×C3×A413yescomplemented
300C22×C4C23×C3212yes—
301C22×C4C23×C3216yes—
302C22×C3—26no—
303C22×C3—32no—
304C22×C3—31no—
305C22×C3—32no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—36no—
311C22×C3—43no—
312C22×C3—418no—
313C22×C3—66no—
314C22×C3—612no—
315C22×C3—66no—
316C22×C3—66no—
317C22×C3—66no—
318C22×C3—612no—
319C22×C3—66no—
320C22×C3—818no—
321C22×C3C2×C3×#4111yescharacteristic, complemented
322C22×C3C23×A416yes—
323C22×C3C23×A411yescharacteristic
324A4C2×C3×#4113yesdirect product factor
325C3×C4—32no—
326C3×C4—36no—
327C3×C4—46no—
328C3×C4—418no—
329C3×C4C23×A412yes—
330C3×C4C23×A416yes—
331C32—41noSylow
332C23—26no—
333C23—31no—
334C23—32no—
335C23—31no—
336C23—36no—
337C23—36no—
338C23—36no—
339C23—36no—
340C23—33no—
341C23—66no—
342C23—66no—
343C23—66no—
344C23—612no—
345C23—66no—
346C23—66no—
347C23—612no—
348C23C32×#4112yescomplemented
349C23C22×C3×A413yes—
350C23C24×C3211yescharacteristic
351Q8—38no—
352Q8—312no—
353Q8C22×C3×A414yescomplemented
354D4—324no—
355D4—324no—
356D4—312no—
357D4C22×C3×A4112yescomplemented
358C2×C4—32no—
359C2×C4—36no—
360C2×C4—324no—
361C2×C4—31no—
362C2×C4—33no—
363C2×C4—312no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—36no—
367C2×C4—36no—
368C2×C4—312no—
369C2×C4—312no—
370C2×C4—32no—
371C2×C4—36no—
372C2×C4C22×C3×A411yescharacteristic
373C2×C4C22×C3×A413yes—
374C2×C4C22×C3×A4112yescomplemented
375C2×C3—26no—
376C2×C3—32no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—46no—
380C2×C3—43no—
381C2×C3—66no—
382C2×C3—818no—
383C2×C3A4×#4112yesdirect product factor
384C2×C3C24×A411yescharacteristic
385C22—26no—
386C22—31no—
387C22—32no—
388C22—32no—
389C22—32no—
390C22—32no—
391C22—31no—
392C22—31no—
393C22—36no—
394C22—66no—
395C22—66no—
396C22—612no—
397C22—66no—
398C22—66no—
399C22—612no—
400C22—66no—
401C22C2×C32×#4111yescharacteristic, complemented
402C22C23×C3×A416yes—
403C22C23×C3×A411yescharacteristic
404C4—32no—
405C4—36no—
406C4C23×C3×A412yes—
407C4C23×C3×A416yes—
408C3—43no—
409C3C2×A4×#4111yescharacteristic, direct product factor
410C2—26no—
411C2—31no—
412C2—31no—
413C2—32no—
414C2—66no—
415C2C3×A4×#4112yesdirect product factor
416C2C24×C3×A411yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutator, cosocleC23C24×C32
FrattiniC2C24×C3×A4
FittingC23×C3×#41C3
socleC24×C3C23×C3
hypercenterC2×C3×#41A4
hypocenterC22C2×C32×#41
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C3×A4×#41C2direct (N × H)
2. C2×C3×Q8×A4C2semidirect
3. C2×C3×D4×A4C2semidirect
4. C22×C3×C4×A4C2semidirect
5. C23×C3×#41C3semidirect
6. C2×A4×#41C3direct (N × H)
9. C3×Q8×A4C22semidirect
10. C3×D4×A4C22semidirect
13. C2×C3×C4×A4C22semidirect
19. C22×C3×#41C2×C3semidirect
20. C23×C3×Q8C2×C3semidirect
21. C23×C3×D4C2×C3semidirect
22. C24×C3×C4C2×C3semidirect
23. A4×#41C2×C3direct (N × H)
24. C2×Q8×A4C2×C3semidirect
25. C2×D4×A4C2×C3semidirect
26. C22×C4×A4C2×C3semidirect
36. C23×#41C32semidirect
54. C2×C3×#41A4direct (N × H)
58. C22×C3×Q8C22×C3semidirect
63. C22×C3×D4C22×C3semidirect
71. C23×C3×C4C22×C3semidirect
72. Q8×A4C22×C3semidirect
73. D4×A4C22×C3semidirect
76. C2×C4×A4C22×C3semidirect
79. C2×C3×A4#41direct (N × H)
91. C22×#41C2×C32semidirect
92. C23×Q8C2×C32semidirect
93. C23×D4C2×C32semidirect
94. C24×C4C2×C32semidirect
112. C3×#41C2×A4direct (N × H)
120. C2×C3×Q8C2×A4semidirect
131. C2×C3×D4C2×A4semidirect
151. C22×C3×C4C2×A4semidirect
159. C3×A4C2×#41direct (N × H)
177. C2×#41C3×A4direct (N × H)
181. C22×Q8C22×C32semidirect
186. C22×D4C22×C32semidirect
194. C23×C4C22×C32semidirect
212. C23×C3C3×#41semidirect
216. C2×A4C3×#41direct (N × H)
221. C3×Q8C22×A4semidirect
226. C3×D4C22×A4semidirect
246. C2×C3×C4C22×A4semidirect
263. #41C2×C3×A4direct (N × H)
270. C2×Q8C2×C3×A4semidirect
280. C2×D4C2×C3×A4semidirect
299. C22×C4C2×C3×A4semidirect
321. C22×C3C2×C3×#41semidirect
324. A4C2×C3×#41direct (N × H)
348. C23C32×#41semidirect
353. Q8C22×C3×A4semidirect
357. D4C22×C3×A4semidirect
374. C2×C4C22×C3×A4semidirect
383. C2×C3A4×#41direct (N × H)
401. C22C2×C32×#41semidirect
409. C3C2×A4×#41direct (N × H)
415. C2C3×A4×#41direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×#412
C2×C3×Q8×A42
C2×C3×D4×A42
C22×C3×C4×A42
C23×C3×#413
C2×A4×#413
C2×C32×#414