Finite Group Databasebeta

← C2×C3×C4×SL(2,ℤ/4) #11922762

Subgroups — C2×C3×C4×SL(2,ℤ/4) #11922762

Order 1152.

1212 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 216 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
1C22×C3×C4×A4C211yesmaximal, characteristic
2C22×C3×SL(2,ℤ/4)C212yesmaximal
3C3×C4×SL(2,ℤ/4)C214yesmaximal, direct product factor
4C2×C4×SL(2,ℤ/4)C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C22⋊C4—31nomaximal
6C23×C3×A4C2211yescharacteristic
7C2×C3×C4×A4C414yescomplemented
8C2×C3×C4×A4C2212yes—
9C2×C3×SL(2,ℤ/4)C418yesdirect product factor
10C2×C3×SL(2,ℤ/4)C2214yes—
11C2×C3×C4×Dic3—41nomaximal
12C24×C3×C4D311yescharacteristic
13C22×C4×A4—21no—
14C22×C4×A4C2×C311yescharacteristic
15C22×SL(2,ℤ/4)C2×C312yes—
16C22×C3×C22⋊C4—32no—
17C22×C3×C42—31no—
18C22×C3×C42—31no—
19C4×SL(2,ℤ/4)C2×C314yesdirect product factor
20C3×C4×C22⋊C4—34no—
21C3×C4×C22⋊C4—34no—
22C2×C3×#95—32no—
23C22×C3×A4C2×C414yes—
24C22×C3×A4C2×C412yes—
25C22×C3×A4C2311yescharacteristic
26C22×C32×C4—41no—
27C22×C3×Dic3—42no—
28C3×C4×A4C2×C414yescomplemented
29C3×SL(2,ℤ/4)C2×C418yesdirect product factor
30C3×C4×Dic3—44no—
31C2×C4×C22⋊C4—31noSylow
32C25×C3C2×D311yescharacteristic
33C23×A4—21no—
34C23×A4C22×C311yescharacteristic
35C23×C3×C4—31no—
36C23×C3×C4—34no—
37C23×C3×C4—31no—
38C23×C3×C4—32no—
39C23×C3×C4—32no—
40C23×C3×C4—32no—
41C23×C3×C4Dic314yescomplemented
42C23×C3×C4C2×D312yes—
43C2×C4×A4—24no—
44C2×C4×A4—22no—
45C2×C4×A4C3×C414yescomplemented
46C2×C4×A4C22×C312yes—
47C2×SL(2,ℤ/4)C3×C418yesdirect product factor
48C2×SL(2,ℤ/4)C22×C314yes—
49C2×C3×C22⋊C4—34no—
50C2×C3×C22⋊C4—38no—
51C2×C3×C22⋊C4—38no—
52C2×C3×C22⋊C4—34no—
53C2×C3×C42—34no—
54C2×C3×C42—34no—
55C2×C3×C42—61no—
56C2×C3×C42—64no—
57C2×C3×C42—62no—
58C2×C3×C42—61no—
59C2×C4×Dic3—41no—
60C3×#95—38no—
61C23×C32—41no—
62C2×C3×A4C4214yescomplemented
63C2×C3×A4C22×C412yes—
64C2×C3×A4C22×C411yescharacteristic
65C2×C32×C4—44no—
66C2×C32×C4—42no—
67C2×C3×Dic3—48no—
68C2×C3×Dic3—44no—
69C24×C4C3×D311yescharacteristic
70C22×C22⋊C4—32no—
71C22×C42—31no—
72C22×C42—31no—
73C4×C22⋊C4—34no—
74C4×C22⋊C4—34no—
75C2×#95—32no—
76C24×C3—31no—
77C24×C3—34no—
78C24×C3—31no—
79C24×C3—32no—
80C24×C3C4×D314yes—
81C24×C3C2×Dic312yes—
82C24×C3C22×D311yescharacteristic
83C22×A4—24no—
84C22×A4—22no—
85C22×A4—21no—
86C22×A4C2×C3×C414yes—
87C22×A4C2×C3×C412yes—
88C22×A4C23×C311yescharacteristic
89C22×C3×C4—31no—
90C22×C3×C4—34no—
91C22×C3×C4—34no—
92C22×C3×C4—32no—
93C22×C3×C4—32no—
94C22×C3×C4—32no—
95C22×C3×C4—34no—
96C22×C3×C4—38no—
97C22×C3×C4—34no—
98C22×C3×C4—34no—
99C22×C3×C4—34no—
100C22×C3×C4—38no—
101C22×C3×C4—38no—
102C22×C3×C4—41no—
103C22×C3×C4—64no—
104C22×C3×C4—62no—
105C22×C3×C4—61no—
106C22×C3×C4—64no—
107C22×C3×C4—62no—
108C22×C3×C4—64no—
109C22×C3×C4—64no—
110C22×C3×C4—62no—
111C22×C3×C4—81no—
112C22×C3×C4C2×Dic314yescomplemented
113C22×C3×C4S411yescharacteristic
114C22×Dic3—42no—
115C4×A4—24no—
116C4×A4C2×C3×C414yescomplemented
117SL(2,ℤ/4)C2×C3×C418yesdirect product factor
118C3×C22⋊C4—316no—
119C3×C22⋊C4—38no—
120C3×C22⋊C4—38no—
121C3×C42—64no—
122C3×C42—68no—
123C3×C42—64no—
124C4×Dic3—44no—
125C22×C32—44no—
126C22×C32—42no—
127C22×C32—41no—
128C3×A4C2×C4211yescharacteristic, complemented
129C32×C4—44no—
130C3×Dic3—48no—
131C25C2×C3×D311yescharacteristic
132C23×C4—31no—
133C23×C4—31no—
134C23×C4—32no—
135C23×C4—34no—
136C23×C4—32no—
137C23×C4—32no—
138C23×C4C3×Dic314yescomplemented
139C23×C4C2×C3×D312yes—
140C2×C22⋊C4—34no—
141C2×C22⋊C4—38no—
142C2×C22⋊C4—38no—
143C2×C22⋊C4—34no—
144C2×C42—34no—
145C2×C42—34no—
146C2×C42—61no—
147C2×C42—64no—
148C2×C42—62no—
149C2×C42—61no—
150#95—38no—
151C23×C3—34no—
152C23×C3—34no—
153C23×C3—32no—
154C23×C3—32no—
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—34no—
158C23×C3—32no—
159C23×C3—32no—
160C23×C3—31no—
161C23×C3—38no—
162C23×C3—34no—
163C23×C3—41no—
164C23×C3—61no—
165C23×C3—62no—
166C23×C3—64no—
167C23×C3—81no—
168C23×C3C4×Dic314yescomplemented
169C23×C3C2×C4×D312yes—
170C23×C3C22×Dic311yescharacteristic
171C23×C3C2×S411yescharacteristic
172C2×A4—24no—
173C2×A4—22no—
174C2×A4—21no—
175C2×A4C3×C4214yescomplemented
176C2×A4C22×C3×C412yes—
177C2×A4C22×C3×C411yescharacteristic
178C2×C3×C4—34no—
179C2×C3×C4—32no—
180C2×C3×C4—34no—
181C2×C3×C4—34no—
182C2×C3×C4—38no—
183C2×C3×C4—34no—
184C2×C3×C4—38no—
185C2×C3×C4—38no—
186C2×C3×C4—42no—
187C2×C3×C4—44no—
188C2×C3×C4—62no—
189C2×C3×C4—64no—
190C2×C3×C4—64no—
191C2×C3×C4—64no—
192C2×C3×C4—616no—
193C2×C3×C4—64no—
194C2×C3×C4—68no—
195C2×C3×C4—68no—
196C2×C3×C4—64no—
197C2×C3×C4—82no—
198C2×C3×C4—84no—
199C2×C3×C4SL(2,ℤ/4)14yesdirect product factor
200C2×C3×C4C2×S412yes—
201C2×Dic3—44no—
202C2×Dic3—48no—
203C2×C32—42no—
204C2×C32—44no—
205C2×C32—41no—
206C24—31no—
207C24—31no—
208C24—34no—
209C24—32no—
210C24C3×C4×D314yes—
211C24C2×C3×Dic312yes—
212C24C22×C3×D311yescharacteristic
213C22×C4—31no—
214C22×C4—32no—
215C22×C4—32no—
216C22×C4—34no—
217C22×C4—34no—
218C22×C4—32no—
219C22×C4—34no—
220C22×C4—38no—
221C22×C4—34no—
222C22×C4—34no—
223C22×C4—34no—
224C22×C4—38no—
225C22×C4—38no—
226C22×C4—64no—
227C22×C4—62no—
228C22×C4—61no—
229C22×C4—64no—
230C22×C4—62no—
231C22×C4—64no—
232C22×C4—64no—
233C22×C4—62no—
234C22×C4C2×C3×Dic314yescomplemented
235C22×C4C3×S411yescharacteristic
236C22⋊C4—316no—
237C22⋊C4—38no—
238C22⋊C4—38no—
239C42—64no—
240C42—68no—
241C42—64no—
242C22×C3—34no—
243C22×C3—32no—
244C22×C3—31no—
245C22×C3—32no—
246C22×C3—32no—
247C22×C3—34no—
248C22×C3—38no—
249C22×C3—34no—
250C22×C3—34no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—44no—
257C22×C3—61no—
258C22×C3—64no—
259C22×C3—62no—
260C22×C3—82no—
261C22×C3—81no—
262C22×C3—84no—
263C22×C3C2×C4×Dic311yescharacteristic, complemented
264C22×C3C4×S414yes—
265C22×C3C2×SL(2,ℤ/4)12yes—
266C22×C3C22×S411yescharacteristic
267A4—21no—
268A4C2×C3×C4211yescharacteristic, complemented
269C3×C4—34no—
270C3×C4—44no—
271C3×C4—68no—
272C3×C4—68no—
273C3×C4—84no—
274C3×C4C2×SL(2,ℤ/4)14yesdirect product factor
275Dic3—48no—
276C32—41noSylow
277C23—32no—
278C23—32no—
279C23—31no—
280C23—31no—
281C23—34no—
282C23—34no—
283C23—34no—
284C23—32no—
285C23—32no—
286C23—31no—
287C23—38no—
288C23—34no—
289C23—61no—
290C23—64no—
291C23—62no—
292C23C3×C4×Dic314yescomplemented
293C23C2×C3×C4×D312yes—
294C23C22×C3×Dic311yescharacteristic
295C23C2×C3×S411yescharacteristic
296C2×C4—34no—
297C2×C4—32no—
298C2×C4—34no—
299C2×C4—34no—
300C2×C4—38no—
301C2×C4—34no—
302C2×C4—38no—
303C2×C4—38no—
304C2×C4—62no—
305C2×C4—64no—
306C2×C4—64no—
307C2×C4—616no—
308C2×C4—64no—
309C2×C4—64no—
310C2×C4—68no—
311C2×C4—64no—
312C2×C4—68no—
313C2×C4C3×SL(2,ℤ/4)14yesdirect product factor
314C2×C4C2×C3×S412yes—
315C2×C3—32no—
316C2×C3—31no—
317C2×C3—34no—
318C2×C3—31no—
319C2×C3—44no—
320C2×C3—42no—
321C2×C3—41no—
322C2×C3—84no—
323C2×C3—82no—
324C2×C3—81no—
325C2×C3C4×SL(2,ℤ/4)14yesdirect product factor
326C2×C3C2×C4×S412yes—
327C2×C3C22×SL(2,ℤ/4)11yescharacteristic
328C22—34no—
329C22—32no—
330C22—31no—
331C22—38no—
332C22—34no—
333C22—34no—
334C22—32no—
335C22—32no—
336C22—34no—
337C22—31no—
338C22—31no—
339C22—32no—
340C22—61no—
341C22—64no—
342C22—62no—
343C22C2×C3×C4×Dic311yescharacteristic, complemented
344C22C3×C4×S414yes—
345C22C2×C3×SL(2,ℤ/4)12yes—
346C22C22×C3×S411yescharacteristic
347C4—34no—
348C4—68no—
349C4—68no—
350C4C2×C3×SL(2,ℤ/4)14yesdirect product factor
351C3—41no—
352C3—81no—
353C3C2×C4×SL(2,ℤ/4)11yescharacteristic, direct product factor
354C2—32no—
355C2—31no—
356C2—34no—
357C2—31no—
358C2C3×C4×SL(2,ℤ/4)14yesdirect product factor
359C2C2×C3×C4×S412yes—
360C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4S4
commutator, hypocenterA4C2×C3×C42
FrattiniC22C22×C3×S4
FittingC24×C3×C4D3
socleC25×C3C2×D3
prime radicalC23×C3×A4C22
cosocleC22×A4C23×C3
LCSG ▷ A4
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×C4×SL(2,ℤ/4)C2direct (N × H)
4. C2×C4×SL(2,ℤ/4)C3direct (N × H)
7. C2×C3×C4×A4C4semidirect
9. C2×C3×SL(2,ℤ/4)C4direct (N × H)
19. C4×SL(2,ℤ/4)C2×C3direct (N × H)
28. C3×C4×A4C2×C4semidirect
29. C3×SL(2,ℤ/4)C2×C4direct (N × H)
41. C23×C3×C4Dic3semidirect
45. C2×C4×A4C3×C4semidirect
47. C2×SL(2,ℤ/4)C3×C4direct (N × H)
62. C2×C3×A4C42semidirect
112. C22×C3×C4C2×Dic3semidirect
116. C4×A4C2×C3×C4semidirect
117. SL(2,ℤ/4)C2×C3×C4direct (N × H)
128. C3×A4C2×C42semidirect
138. C23×C4C3×Dic3semidirect
168. C23×C3C4×Dic3semidirect
175. C2×A4C3×C42semidirect
199. C2×C3×C4SL(2,ℤ/4)direct (N × H)
234. C22×C4C2×C3×Dic3semidirect
263. C22×C3C2×C4×Dic3semidirect
268. A4C2×C3×C42semidirect
274. C3×C4C2×SL(2,ℤ/4)direct (N × H)
292. C23C3×C4×Dic3semidirect
313. C2×C4C3×SL(2,ℤ/4)direct (N × H)
325. C2×C3C4×SL(2,ℤ/4)direct (N × H)
343. C22C2×C3×C4×Dic3semidirect
350. C4C2×C3×SL(2,ℤ/4)direct (N × H)
353. C3C2×C4×SL(2,ℤ/4)direct (N × H)
358. C2C3×C4×SL(2,ℤ/4)direct (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
C22×C3×SL(2,ℤ/4)2
C3×C4×SL(2,ℤ/4)2
C2×C4×SL(2,ℤ/4)3
C2×C3×C4×C22⋊C43
C2×C3×C4×Dic34