Finite Group Databasebeta

← C2×C3×A4×M16 #11919776

Subgroups — C2×C3×A4×M16 #11919776

Order 1152.

936 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×A4C211yesmaximal, characteristic
2C3×A4×M16C214yesmaximal, direct product factor
3C2×C3×C8×A4C212yesmaximal, complemented
4C23×C3×M16C311yesmaximal, characteristic, complemented
5C2×A4×M16C313yesmaximal, direct product factor
6C23×C3×A4C411yescharacteristic
7C2×C3×C4×A4C412yes—
8C2×C3×C4×A4C411yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×C4×A4C2212yes—
11C2×C32×M16—41nomaximal
12C3×C8×A4C2214yescomplemented
13C24×C3×C4C2×C311yescharacteristic
14C22×C4×A4C2×C313yes—
15C22×C3×M16—31no—
16C22×C3×M16—32no—
17C22×C3×M16—31no—
18C22×C3×M16—34no—
19C22×C3×M16C2×C314yescomplemented
20C23×C3×C8C2×C312yescomplemented
21A4×M16C2×C3112yesdirect product factor
22C2×C8×A4C2×C316yescomplemented
23C22×C3×A4—22no—
24C22×C3×A4C2×C412yes—
25C22×C3×A4C2×C411yescharacteristic
26C22×C32×C4—41no—
27C3×C4×A4C2×C411yescharacteristic
28C3×C4×A4C2×C412yes—
29C3×C4×A4C2311yescharacteristic
30C32×M16—44no—
31C2×C32×C8—42no—
32C23×M16C3211yescharacteristic, complemented, Sylow
33C25×C3C3×C411yescharacteristic
34C23×A4C3×C413yes—
35C23×C3×C4—31no—
36C23×C3×C4—32no—
37C23×C3×C4—31no—
38C23×C3×C4—31no—
39C23×C3×C4—31no—
40C23×C3×C4—32no—
41C23×C3×C4C3×C412yes—
42C23×C3×C4C3×C411yescharacteristic
43C23×C3×C4C22×C311yescharacteristic
44C23×C3×C4C22×C312yes—
45C2×C4×A4C3×C416yes—
46C2×C4×A4C3×C413yes—
47C2×C4×A4C22×C313yes—
48C2×C4×A4C22×C316yes—
49C2×C3×M16—34no—
50C2×C3×M16—38no—
51C2×C3×M16—32no—
52C2×C3×M16—31no—
53C2×C3×M16—34no—
54C2×C3×M16—34no—
55C2×C3×M16—38no—
56C2×C3×M16—34no—
57C2×C3×M16—32no—
58C2×C3×M16—43no—
59C2×C3×M16A411yescharacteristic, direct product factor
60C22×C3×C8—32no—
61C22×C3×C8—32no—
62C22×C3×C8—34no—
63C22×C3×C8C22×C314yescomplemented
64C8×A4C22×C3112yescomplemented
65C23×C32—41no—
66C2×C3×A4—22no—
67C2×C3×A4M1612yesdirect product factor
68C2×C3×A4C22×C411yescharacteristic
69C2×C32×C4—41no—
70C2×C32×C4—42no—
71C2×C32×C4—42no—
72C2×C32×C4—41no—
73C32×C8—44no—
74C24×C4C2×C3211yescharacteristic
75C22×M16—31no—
76C22×M16—32no—
77C22×M16—31no—
78C22×M16—34no—
79C22×M16C2×C3214yescomplemented
80C23×C8C2×C3212yescomplemented
81C24×C3—22no—
82C24×C3—32no—
83C24×C3—31no—
84C24×C3—31no—
85C24×C3—62no—
86C24×C3C2×C3×C412yes—
87C24×C3C2×C3×C411yescharacteristic
88C22×A4—26no—
89C22×A4C2×C3×C416yes—
90C22×A4C2×C3×C413yes—
91C22×C3×C4—32no—
92C22×C3×C4—34no—
93C22×C3×C4—32no—
94C22×C3×C4—32no—
95C22×C3×C4—34no—
96C22×C3×C4—31no—
97C22×C3×C4—32no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—32no—
101C22×C3×C4—32no—
102C22×C3×C4—32no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—32no—
107C22×C3×C4—31no—
108C22×C3×C4—32no—
109C22×C3×C4—32no—
110C22×C3×C4—32no—
111C22×C3×C4—32no—
112C22×C3×C4—31no—
113C22×C3×C4—32no—
114C22×C3×C4—31no—
115C22×C3×C4—32no—
116C22×C3×C4—43no—
117C22×C3×C4C2×C3×C411yescharacteristic
118C22×C3×C4C2×C3×C412yes—
119C22×C3×C4C2×A411yescharacteristic
120C22×C3×C4C23×C311yescharacteristic
121C4×A4C2×C3×C413yes—
122C4×A4C2×C3×C416yes—
123C4×A4C23×C313yes—
124C3×M16—38no—
125C3×M16—34no—
126C3×M16—38no—
127C3×M16—412no—
128C3×M16C2×A414yesdirect product factor
129C2×C3×C8—34no—
130C2×C3×C8—32no—
131C2×C3×C8—34no—
132C2×C3×C8—34no—
133C2×C3×C8—34no—
134C2×C3×C8—46no—
135C2×C3×C8C2×A412yescomplemented
136C22×C32—41no—
137C22×C32—42no—
138C22×C32—82no—
139C3×A4C2×M1611yescharacteristic, direct product factor
140C32×C4—42no—
141C32×C4—41no—
142C32×C4—41no—
143C25C32×C411yescharacteristic
144C23×C4—31no—
145C23×C4—31no—
146C23×C4—32no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—32no—
150C23×C4C32×C412yes—
151C23×C4C32×C411yescharacteristic
152C23×C4C22×C3211yescharacteristic
153C23×C4C22×C3212yes—
154C2×M16—34no—
155C2×M16—38no—
156C2×M16—32no—
157C2×M16—31no—
158C2×M16—34no—
159C2×M16—34no—
160C2×M16—38no—
161C2×M16—34no—
162C2×M16—32no—
163C2×M16C3×A411yescharacteristic, direct product factor
164C22×C8—32no—
165C22×C8—32no—
166C22×C8—34no—
167C22×C8C22×C3214yescomplemented
168C23×C3—22no—
169C23×C3—31no—
170C23×C3—32no—
171C23×C3—31no—
172C23×C3—32no—
173C23×C3—32no—
174C23×C3—32no—
175C23×C3—31no—
176C23×C3—32no—
177C23×C3—43no—
178C23×C3—62no—
179C23×C3—62no—
180C23×C3—62no—
181C23×C3—64no—
182C23×C3—62no—
183C23×C3—62no—
184C23×C3—64no—
185C23×C3C3×M1612yescomplemented
186C23×C3C4×A411yescharacteristic
187C23×C3C22×C3×C411yescharacteristic
188C2×A4—26no—
189C2×A4C3×M1616yesdirect product factor
190C2×A4C22×C3×C413yes—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—34no—
194C2×C3×C4—34no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—31no—
203C2×C3×C4—32no—
204C2×C3×C4—32no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—32no—
209C2×C3×C4—31no—
210C2×C3×C4—32no—
211C2×C3×C4—31no—
212C2×C3×C4—43no—
213C2×C3×C4—46no—
214C2×C3×C4—43no—
215C2×C3×C4—46no—
216C2×C3×C4C4×A411yescharacteristic
217C2×C3×C4C4×A412yes—
218C2×C3×C4C22×A412yes—
219C2×C3×C4C22×A411yescharacteristic
220C3×C8—34no—
221C3×C8—412no—
222C3×C8C22×A414yescomplemented
223C2×C32—41no—
224C2×C32—42no—
225C2×C32—82no—
226C24—22no—
227C24—32no—
228C24—31no—
229C24—31no—
230C24—62no—
231C24C2×C32×C412yes—
232C24C2×C32×C411yescharacteristic
233C22×C4—32no—
234C22×C4—34no—
235C22×C4—32no—
236C22×C4—32no—
237C22×C4—34no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—32no—
241C22×C4—32no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—32no—
245C22×C4—32no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—32no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—32no—
252C22×C4—32no—
253C22×C4—32no—
254C22×C4—31no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—32no—
258C22×C4C2×C32×C411yescharacteristic
259C22×C4C2×C32×C412yes—
260C22×C4C2×C3×A411yescharacteristic
261C22×C4C23×C3211yescharacteristic
262M16—38no—
263M16—34no—
264M16—38no—
265M16C2×C3×A414yesdirect product factor
266C2×C8—34no—
267C2×C8—32no—
268C2×C8—34no—
269C2×C8—34no—
270C2×C8—34no—
271C2×C8C2×C3×A412yescomplemented
272C22×C3—22no—
273C22×C3—32no—
274C22×C3—31no—
275C22×C3—32no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—43no—
282C22×C3—46no—
283C22×C3—62no—
284C22×C3—64no—
285C22×C3—62no—
286C22×C3—62no—
287C22×C3—62no—
288C22×C3—64no—
289C22×C3—62no—
290C22×C3—86no—
291C22×C3C2×C3×M1611yescharacteristic, complemented
292C22×C3C2×C4×A412yes—
293C22×C3C2×C4×A411yescharacteristic
294A4C2×C3×M1613yesdirect product factor
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—32no—
298C3×C4—43no—
299C3×C4—46no—
300C3×C4—43no—
301C3×C4C2×C4×A412yes—
302C3×C4C2×C4×A411yescharacteristic
303C3×C4C23×A411yescharacteristic
304C32—41noSylow
305C23—22no—
306C23—31no—
307C23—32no—
308C23—31no—
309C23—32no—
310C23—32no—
311C23—32no—
312C23—32no—
313C23—31no—
314C23—62no—
315C23—62no—
316C23—62no—
317C23—64no—
318C23—62no—
319C23—62no—
320C23—64no—
321C23C32×M1612yescomplemented
322C23C3×C4×A411yescharacteristic
323C23C22×C32×C411yescharacteristic
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—34no—
327C2×C4—34no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—31no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—32no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—31no—
345C2×C4C3×C4×A412yes—
346C2×C4C3×C4×A411yescharacteristic
347C2×C4C22×C3×A411yescharacteristic
348C2×C4C22×C3×A412yes—
349C8—34no—
350C8C22×C3×A414yescomplemented
351C2×C3—22no—
352C2×C3—32no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—43no—
356C2×C3—46no—
357C2×C3—62no—
358C2×C3—86no—
359C2×C3A4×M1612yesdirect product factor
360C2×C3C22×C4×A411yescharacteristic
361C22—22no—
362C22—32no—
363C22—31no—
364C22—32no—
365C22—31no—
366C22—31no—
367C22—32no—
368C22—32no—
369C22—32no—
370C22—62no—
371C22—62no—
372C22—64no—
373C22—62no—
374C22—62no—
375C22—64no—
376C22—62no—
377C22C2×C32×M1611yescharacteristic, complemented
378C22C2×C3×C4×A412yes—
379C22C2×C3×C4×A411yescharacteristic
380C4—31no—
381C4—32no—
382C4—31no—
383C4C2×C3×C4×A411yescharacteristic
384C4C2×C3×C4×A412yes—
385C4C23×C3×A411yescharacteristic
386C3—43no—
387C3C2×A4×M1611yescharacteristic, direct product factor
388C2—22no—
389C2—31no—
390C2—32no—
391C2—31no—
392C2—62no—
393C2C3×A4×M1612yesdirect product factor
394C2C22×C3×C4×A411yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutatorC23C22×C32×C4
FrattiniC4C23×C3×A4
FittingC23×C3×M16C3
socleC24×C3C2×C3×C4
prime radicalC23×C3×A4C4
cosocleC22×C4C23×C32
hypercenterC2×C3×M16A4
hypocenterC22C2×C32×M16
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
2. C3×A4×M16C2direct (N × H)
3. C2×C3×C8×A4C2semidirect
4. C23×C3×M16C3semidirect
5. C2×A4×M16C3direct (N × H)
12. C3×C8×A4C22semidirect
19. C22×C3×M16C2×C3semidirect
20. C23×C3×C8C2×C3semidirect
21. A4×M16C2×C3direct (N × H)
22. C2×C8×A4C2×C3semidirect
32. C23×M16C32semidirect
59. C2×C3×M16A4direct (N × H)
63. C22×C3×C8C22×C3semidirect
64. C8×A4C22×C3semidirect
67. C2×C3×A4M16direct (N × H)
79. C22×M16C2×C32semidirect
80. C23×C8C2×C32semidirect
128. C3×M16C2×A4direct (N × H)
135. C2×C3×C8C2×A4semidirect
139. C3×A4C2×M16direct (N × H)
163. C2×M16C3×A4direct (N × H)
167. C22×C8C22×C32semidirect
185. C23×C3C3×M16semidirect
189. C2×A4C3×M16direct (N × H)
222. C3×C8C22×A4semidirect
265. M16C2×C3×A4direct (N × H)
271. C2×C8C2×C3×A4semidirect
291. C22×C3C2×C3×M16semidirect
294. A4C2×C3×M16direct (N × H)
321. C23C32×M16semidirect
350. C8C22×C3×A4semidirect
359. C2×C3A4×M16direct (N × H)
377. C22C2×C32×M16semidirect
387. C3C2×A4×M16direct (N × H)
393. C2C3×A4×M16direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C3×A4×M162
C2×C3×C8×A42
C23×C3×M163
C2×A4×M163
C2×C32×M164