Finite Group Databasebeta

← C3×A4×#117 #11919670

Subgroups — C3×A4×#117 #11919670

Order 1152.

1032 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×A4C211yesmaximal, characteristic
2C3×A4×C4⋊C4C212yesmaximal, complemented
3C3×A4×C22⋊C4C212yesmaximal
4C3×C42×A4C212yesmaximal, complemented
5C22×C3×#117C311yesmaximal, characteristic, complemented
6A4×#117C313yesmaximal, direct product factor
7C23×C3×A4C2211yescharacteristic
8C2×C3×C4×A4C414yescomplemented
9C2×C3×C4×A4C2214yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C32×#117—41nomaximal
13C24×C3×C4C2×C311yescharacteristic
14C22×C4×A4C2×C313yes—
15C2×C3×#117—31no—
16C2×C3×#117—32no—
17C2×C3×#117—32no—
18C2×C3×#117—32no—
19C2×C3×#117—31no—
20C22×C3×C4⋊C4C2×C312yescomplemented
21C22×C3×C22⋊C4C2×C312yes—
22C22×C3×C42C2×C312yescomplemented
23A4×C4⋊C4C2×C316yescomplemented
24A4×C22⋊C4C2×C316yes—
25C42×A4C2×C316yescomplemented
26C22×C3×A4—22no—
27C22×C3×A4C2×C412yes—
28C22×C3×A4C2311yescharacteristic
29C22×C32×C4—41no—
30C3×C4×A4—24no—
31C3×C4×A4C2×C412yes—
32C3×C4×A4C2×C412yes—
33C32×C4⋊C4—42no—
34C32×C22⋊C4—42no—
35C32×C42—42no—
36C22×#117C3211yescharacteristic, complemented, Sylow
37C25×C3C22×C311yescharacteristic
38C23×A4C22×C313yes—
39C23×C3×C4—31no—
40C23×C3×C4—34no—
41C23×C3×C4—31no—
42C23×C3×C4—31no—
43C23×C3×C4—31no—
44C23×C3×C4C3×C414yescomplemented
45C23×C3×C4C22×C314yes—
46C23×C3×C4C22×C311yescharacteristic
47C23×C3×C4C22×C311yescharacteristic
48C2×C4×A4C3×C4112yescomplemented
49C2×C4×A4C22×C3112yes—
50C2×C4×A4C22×C313yes—
51C2×C4×A4C22×C313yes—
52C3×#117—32no—
53C3×#117—32no—
54C3×#117—38no—
55C3×#117—32no—
56C3×#117—31no—
57C3×#117—32no—
58C3×#117—32no—
59C3×#117—32no—
60C3×#117—43no—
61C3×#117A411yescharacteristic, direct product factor
62C2×C3×C4⋊C4—32no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C4⋊C4—34no—
65C2×C3×C22⋊C4—32no—
66C2×C3×C22⋊C4—32no—
67C2×C3×C22⋊C4—34no—
68C2×C3×C42—32no—
69C2×C3×C42—34no—
70C2×C3×C42—32no—
71C23×C32—41no—
72C2×C3×A4—22no—
73C2×C3×A4C22×C411yescharacteristic
74C2×C3×A4#4112yes—
75C2×C32×C4—44no—
76C2×C32×C4—41no—
77C2×C32×C4—44no—
78C2×C32×C4—41no—
79C24×C4C2×C3211yescharacteristic
80C2×#117—31no—
81C2×#117—32no—
82C2×#117—32no—
83C2×#117—32no—
84C2×#117—31no—
85C22×C4⋊C4C2×C3212yescomplemented
86C22×C22⋊C4C2×C3212yes—
87C22×C42C2×C3212yescomplemented
88C24×C3—22no—
89C24×C3—31no—
90C24×C3—32no—
91C24×C3—31no—
92C24×C3—62no—
93C24×C3C2×C3×C412yes—
94C24×C3C23×C311yescharacteristic
95C22×A4—26no—
96C22×A4C2×C3×C416yes—
97C22×A4C23×C313yes—
98C22×C3×C4—24no—
99C22×C3×C4—32no—
100C22×C3×C4—34no—
101C22×C3×C4—34no—
102C22×C3×C4—34no—
103C22×C3×C4—31no—
104C22×C3×C4—34no—
105C22×C3×C4—34no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—34no—
113C22×C3×C4—34no—
114C22×C3×C4—34no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—34no—
118C22×C3×C4—34no—
119C22×C3×C4—43no—
120C22×C3×C4—64no—
121C22×C3×C4C2×C3×C412yes—
122C22×C3×C4C2×C3×C412yes—
123C22×C3×C4C2×A411yescharacteristic
124C4×A4—212no—
125C4×A4C2×C3×C416yes—
126C4×A4C2×C3×C416yes—
127C3×C4⋊C4—34no—
128C3×C4⋊C4—34no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—46no—
131C3×C4⋊C4C2×A412yescomplemented
132C3×C22⋊C4—34no—
133C3×C22⋊C4—32no—
134C3×C22⋊C4—34no—
135C3×C22⋊C4—46no—
136C3×C22⋊C4C2×A412yes—
137C3×C42—34no—
138C3×C42—34no—
139C3×C42—32no—
140C3×C42—46no—
141C3×C42C2×A412yescomplemented
142C22×C32—42no—
143C22×C32—41no—
144C22×C32—82no—
145C3×A4#11711yescharacteristic, direct product factor
146C32×C4—42no—
147C32×C4—42no—
148C32×C4—84no—
149C25C22×C3211yescharacteristic
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4—34no—
155C23×C4C32×C414yescomplemented
156C23×C4C22×C3214yes—
157C23×C4C22×C3211yescharacteristic
158C23×C4C22×C3211yescharacteristic
159#117—32no—
160#117—32no—
161#117—38no—
162#117—32no—
163#117—31no—
164#117—32no—
165#117—32no—
166#117—32no—
167#117C3×A411yescharacteristic, direct product factor
168C2×C4⋊C4—32no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—34no—
171C2×C22⋊C4—32no—
172C2×C22⋊C4—32no—
173C2×C22⋊C4—34no—
174C2×C42—32no—
175C2×C42—32no—
176C2×C42—34no—
177C23×C3—22no—
178C23×C3—31no—
179C23×C3—32no—
180C23×C3—31no—
181C23×C3—32no—
182C23×C3—32no—
183C23×C3—32no—
184C23×C3—32no—
185C23×C3—31no—
186C23×C3—43no—
187C23×C3—62no—
188C23×C3—62no—
189C23×C3—62no—
190C23×C3—64no—
191C23×C3—62no—
192C23×C3—62no—
193C23×C3—64no—
194C23×C3C22×C3×C411yescharacteristic
195C23×C3C3×#4112yes—
196C23×C3C22×A411yescharacteristic
197C2×A4—26no—
198C2×A4C22×C3×C413yes—
199C2×A4C3×#4116yes—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—38no—
203C2×C3×C4—34no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—32no—
207C2×C3×C4—34no—
208C2×C3×C4—32no—
209C2×C3×C4—34no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—34no—
215C2×C3×C4—43no—
216C2×C3×C4—43no—
217C2×C3×C4—412no—
218C2×C3×C4—412no—
219C2×C3×C4—64no—
220C2×C3×C4—64no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4C4×A414yescomplemented
224C2×C3×C4C22×A414yes—
225C2×C3×C4C22×A411yescharacteristic
226C2×C3×C4C22×A411yescharacteristic
227C2×C32—41no—
228C2×C32—42no—
229C2×C32—82no—
230C24—22no—
231C24—31no—
232C24—32no—
233C24—31no—
234C24—62no—
235C24C2×C32×C412yes—
236C24C23×C3211yescharacteristic
237C22×C4—24no—
238C22×C4—32no—
239C22×C4—34no—
240C22×C4—34no—
241C22×C4—34no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—34no—
248C22×C4—34no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—34no—
252C22×C4—34no—
253C22×C4—34no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—34no—
257C22×C4—34no—
258C22×C4—64no—
259C22×C4C2×C32×C412yes—
260C22×C4C2×C32×C412yes—
261C22×C4C2×C3×A411yescharacteristic
262C4⋊C4—34no—
263C4⋊C4—34no—
264C4⋊C4—32no—
265C4⋊C4C2×C3×A412yescomplemented
266C22⋊C4—34no—
267C22⋊C4—32no—
268C22⋊C4—34no—
269C22⋊C4C2×C3×A412yes—
270C42—34no—
271C42—34no—
272C42—32no—
273C42C2×C3×A412yescomplemented
274C22×C3—22no—
275C22×C3—31no—
276C22×C3—32no—
277C22×C3—32no—
278C22×C3—31no—
279C22×C3—31no—
280C22×C3—32no—
281C22×C3—32no—
282C22×C3—32no—
283C22×C3—46no—
284C22×C3—43no—
285C22×C3—62no—
286C22×C3—64no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—64no—
291C22×C3—62no—
292C22×C3—86no—
293C22×C3C3×#11711yescharacteristic, complemented
294C22×C3C2×C4×A412yes—
295C22×C3C23×A411yescharacteristic
296A4C3×#11713yesdirect product factor
297C3×C4—24no—
298C3×C4—32no—
299C3×C4—32no—
300C3×C4—46no—
301C3×C4—46no—
302C3×C4—64no—
303C3×C4—812no—
304C3×C4C2×C4×A412yes—
305C3×C4C2×C4×A412yes—
306C32—41noSylow
307C23—22no—
308C23—31no—
309C23—32no—
310C23—31no—
311C23—32no—
312C23—32no—
313C23—32no—
314C23—32no—
315C23—31no—
316C23—62no—
317C23—62no—
318C23—62no—
319C23—64no—
320C23—62no—
321C23—62no—
322C23—64no—
323C23C22×C32×C411yescharacteristic
324C23C32×#4112yes—
325C23C22×C3×A411yescharacteristic
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—38no—
329C2×C4—34no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—34no—
337C2×C4—34no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—34no—
341C2×C4—64no—
342C2×C4—64no—
343C2×C4—64no—
344C2×C4—64no—
345C2×C4C3×C4×A414yescomplemented
346C2×C4C22×C3×A414yes—
347C2×C4C22×C3×A411yescharacteristic
348C2×C4C22×C3×A411yescharacteristic
349C2×C3—22no—
350C2×C3—31no—
351C2×C3—32no—
352C2×C3—31no—
353C2×C3—43no—
354C2×C3—46no—
355C2×C3—62no—
356C2×C3—86no—
357C2×C3C22×C4×A411yescharacteristic
358C2×C3A4×#4112yes—
359C22—22no—
360C22—31no—
361C22—32no—
362C22—32no—
363C22—31no—
364C22—31no—
365C22—32no—
366C22—32no—
367C22—32no—
368C22—62no—
369C22—62no—
370C22—64no—
371C22—62no—
372C22—62no—
373C22—64no—
374C22—62no—
375C22C32×#11711yescharacteristic, complemented
376C22C2×C3×C4×A412yes—
377C22C23×C3×A411yescharacteristic
378C4—24no—
379C4—32no—
380C4—32no—
381C4—64no—
382C4C2×C3×C4×A412yes—
383C4C2×C3×C4×A412yes—
384C3—43no—
385C3A4×#11711yescharacteristic, direct product factor
386C2—22no—
387C2—31no—
388C2—31no—
389C2—32no—
390C2—62no—
391C2C22×C3×C4×A411yescharacteristic
392C2C3×A4×#4112yes—
subgroupG/H
centerC2×C3×C4C22×A4
commutatorC23C22×C32×C4
FrattiniC22C23×C3×A4
FittingC22×C3×#117C3
socleC24×C3C23×C3
prime radicalC23×C3×A4C22
cosocleC24C23×C32
hypercenterC3×#117A4
hypocenterC22C32×#117
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C3×A4×C4⋊C4C2semidirect
4. C3×C42×A4C2semidirect
5. C22×C3×#117C3semidirect
6. A4×#117C3direct (N × H)
8. C2×C3×C4×A4C4semidirect
20. C22×C3×C4⋊C4C2×C3semidirect
22. C22×C3×C42C2×C3semidirect
23. A4×C4⋊C4C2×C3semidirect
25. C42×A4C2×C3semidirect
36. C22×#117C32semidirect
44. C23×C3×C4C3×C4semidirect
48. C2×C4×A4C3×C4semidirect
61. C3×#117A4direct (N × H)
85. C22×C4⋊C4C2×C32semidirect
87. C22×C42C2×C32semidirect
131. C3×C4⋊C4C2×A4semidirect
141. C3×C42C2×A4semidirect
145. C3×A4#117direct (N × H)
155. C23×C4C32×C4semidirect
167. #117C3×A4direct (N × H)
223. C2×C3×C4C4×A4semidirect
265. C4⋊C4C2×C3×A4semidirect
273. C42C2×C3×A4semidirect
293. C22×C3C3×#117semidirect
296. A4C3×#117direct (N × H)
345. C2×C4C3×C4×A4semidirect
375. C22C32×#117semidirect
385. C3A4×#117direct (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×C4⋊C42
C3×A4×C22⋊C42
C3×C42×A42
C22×C3×#1173
A4×#1173
C32×#1174