Finite Group Databasebeta

← C2×C4×Dic3×A4 #11923342

Subgroups — C2×C4×Dic3×A4 #11923342

Order 1152.

1414 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1414 conjugacy classes of subgroups — too many to draw. Only its 243 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×Dic3×A4C212yesmaximal
3C4×Dic3×A4C214yesmaximal, direct product factor
4C23×C4×Dic3C311yesmaximal, characteristic, complemented
5C2×C42×A4—31nomaximal
6C23×C3×A4C2211yescharacteristic
7C2×C3×C4×A4C414yescomplemented
8C2×C3×C4×A4C2212yes—
9C2×Dic3×A4C418yesdirect product factor
10C2×Dic3×A4C2214yes—
11C2×C3×C4×Dic3—41nomaximal
12C24×C3×C4C2×C311yescharacteristic
13C24×Dic3C2×C312yes—
14C22×C4×A4—21no—
15C22×C4×A4—32no—
16C22×C4×A4D311yescharacteristic
17C22×C4×Dic3—31no—
18C22×C4×Dic3—31no—
19C22×C4×Dic3—32no—
20C22×C4×Dic3—34no—
21C22×C4×Dic3C2×C314yescomplemented
22C42×A4—34no—
23C22×C3×A4C2×C414yes—
24C22×C3×A4C2×C412yes—
25C22×C3×A4C2311yescharacteristic
26C22×C32×C4—41no—
27C22×C3×Dic3—42no—
28C3×C4×A4C2×C414yescomplemented
29Dic3×A4C2×C418yesdirect product factor
30C3×C4×Dic3—44no—
31C23×C42—31noSylow
32C25×C3C22×C311yescharacteristic
33C23×A4—21no—
34C23×A4C2×D311yescharacteristic
35C23×C3×C4—31no—
36C23×C3×C4—34no—
37C23×C3×C4—31no—
38C23×C3×C4—32no—
39C23×C3×C4C3×C414yescomplemented
40C23×C3×C4C22×C312yes—
41C23×Dic3—32no—
42C23×Dic3—38no—
43C23×Dic3—32no—
44C23×Dic3—34no—
45C23×Dic3C3×C418yescomplemented
46C23×Dic3C22×C314yes—
47C2×C4×A4—24no—
48C2×C4×A4—22no—
49C2×C4×A4—38no—
50C2×C4×A4—34no—
51C2×C4×A4Dic314yesdirect product factor
52C2×C4×A4C2×D312yes—
53C2×C3×C42—121no—
54C2×C4×Dic3—32no—
55C2×C4×Dic3—34no—
56C2×C4×Dic3—38no—
57C2×C4×Dic3—31no—
58C2×C4×Dic3—32no—
59C2×C4×Dic3—34no—
60C2×C4×Dic3—34no—
61C2×C4×Dic3—34no—
62C2×C4×Dic3—38no—
63C2×C4×Dic3A411yescharacteristic, direct product factor
64C23×C32—41no—
65C2×C3×A4C4214yescomplemented
66C2×C3×A4C22×C412yes—
67C2×C3×A4C22×C411yescharacteristic
68C2×C32×C4—44no—
69C2×C32×C4—42no—
70C2×C3×Dic3—48no—
71C2×C3×Dic3—44no—
72C24×C4—32no—
73C24×C4C3×D311yescharacteristic
74C22×C42—34no—
75C22×C42—91no—
76C22×C42—94no—
77C22×C42—92no—
78C22×C42—91no—
79C24×C3—34no—
80C24×C3—31no—
81C24×C3—32no—
82C24×C3—31no—
83C24×C3C2×C3×C414yes—
84C24×C3C2×C3×C412yes—
85C24×C3C23×C311yescharacteristic
86C22×A4—24no—
87C22×A4—22no—
88C22×A4—21no—
89C22×A4C4×D314yes—
90C22×A4C2×Dic312yes—
91C22×A4C22×D311yescharacteristic
92C22×C3×C4—32no—
93C22×C3×C4—34no—
94C22×C3×C4—38no—
95C22×C3×C4—31no—
96C22×C3×C4—34no—
97C22×C3×C4—34no—
98C22×C3×C4—32no—
99C22×C3×C4—32no—
100C22×C3×C4—32no—
101C22×C3×C4—34no—
102C22×C3×C4—38no—
103C22×C3×C4—34no—
104C22×C3×C4—41no—
105C22×C3×C4—81no—
106C22×C3×C4—122no—
107C22×C3×C4C2×C3×C414yescomplemented
108C22×C3×C4C2×A411yescharacteristic
109C22×Dic3—34no—
110C22×Dic3—38no—
111C22×Dic3—316no—
112C22×Dic3—32no—
113C22×Dic3—38no—
114C22×Dic3—38no—
115C22×Dic3—34no—
116C22×Dic3—34no—
117C22×Dic3—34no—
118C22×Dic3—38no—
119C22×Dic3—316no—
120C22×Dic3—38no—
121C22×Dic3C2×C3×C418yescomplemented
122C22×Dic3C2×A412yes—
123C4×A4—24no—
124C4×A4—38no—
125C4×A4C2×Dic314yesdirect product factor
126C3×C42—124no—
127C4×Dic3—38no—
128C4×Dic3—34no—
129C4×Dic3—38no—
130C4×Dic3C2×A414yesdirect product factor
131C22×C32—44no—
132C22×C32—42no—
133C22×C32—41no—
134C3×A4C2×C4211yescharacteristic, complemented
135C32×C4—44no—
136C3×Dic3—48no—
137C25C2×C3×D311yescharacteristic
138C23×C4—31no—
139C23×C4—34no—
140C23×C4—31no—
141C23×C4—32no—
142C23×C4—38no—
143C23×C4—34no—
144C23×C4—92no—
145C23×C4—98no—
146C23×C4—94no—
147C23×C4—92no—
148C23×C4C3×Dic314yescomplemented
149C23×C4C2×C3×D312yes—
150C2×C42—31no—
151C2×C42—94no—
152C2×C42—94no—
153C2×C42—92no—
154C2×C42—98no—
155C2×C42—92no—
156C2×C42—94no—
157C2×C42—91no—
158C2×C42—98no—
159C2×C42—94no—
160C23×C3—34no—
161C23×C3—32no—
162C23×C3—32no—
163C23×C3—31no—
164C23×C3—38no—
165C23×C3—34no—
166C23×C3—34no—
167C23×C3—32no—
168C23×C3—38no—
169C23×C3—34no—
170C23×C3—34no—
171C23×C3—32no—
172C23×C3—32no—
173C23×C3—31no—
174C23×C3—31no—
175C23×C3—41no—
176C23×C3—81no—
177C23×C3C3×C4214yescomplemented
178C23×C3C22×C3×C412yes—
179C23×C3C22×C3×C411yescharacteristic
180C23×C3C22×A411yescharacteristic
181C2×A4—24no—
182C2×A4—22no—
183C2×A4—21no—
184C2×A4C4×Dic314yesdirect product factor
185C2×A4C2×C4×D312yes—
186C2×A4C22×Dic311yescharacteristic
187C2×C3×C4—38no—
188C2×C3×C4—34no—
189C2×C3×C4—34no—
190C2×C3×C4—32no—
191C2×C3×C4—34no—
192C2×C3×C4—38no—
193C2×C3×C4—34no—
194C2×C3×C4—34no—
195C2×C3×C4—42no—
196C2×C3×C4—44no—
197C2×C3×C4—82no—
198C2×C3×C4—84no—
199C2×C3×C4—124no—
200C2×C3×C4—128no—
201C2×C3×C4C4×A414yescomplemented
202C2×C3×C4C22×A412yes—
203C2×Dic3—316no—
204C2×Dic3—38no—
205C2×Dic3—38no—
206C2×Dic3—34no—
207C2×Dic3—38no—
208C2×Dic3—316no—
209C2×Dic3—38no—
210C2×Dic3—38no—
211C2×Dic3C4×A418yesdirect product factor
212C2×Dic3C22×A414yes—
213C2×C32—42no—
214C2×C32—41no—
215C2×C32—44no—
216C24—31no—
217C24—32no—
218C24—34no—
219C24—31no—
220C24C3×C4×D314yes—
221C24C2×C3×Dic312yes—
222C24C22×C3×D311yescharacteristic
223C22×C4—32no—
224C22×C4—34no—
225C22×C4—38no—
226C22×C4—32no—
227C22×C4—31no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—34no—
231C22×C4—34no—
232C22×C4—32no—
233C22×C4—34no—
234C22×C4—38no—
235C22×C4—34no—
236C22×C4—38no—
237C22×C4—94no—
238C22×C4—94no—
239C22×C4—98no—
240C22×C4—98no—
241C22×C4—94no—
242C22×C4—916no—
243C22×C4—98no—
244C22×C4—92no—
245C22×C4—94no—
246C22×C4—916no—
247C22×C4—98no—
248C22×C4—98no—
249C22×C4C2×C3×Dic314yescomplemented
250C22×C4D3×A411yescharacteristic
251C42—34no—
252C42—98no—
253C42—98no—
254C42—94no—
255C22×C3—34no—
256C22×C3—32no—
257C22×C3—31no—
258C22×C3—38no—
259C22×C3—32no—
260C22×C3—34no—
261C22×C3—34no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—32no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—34no—
268C22×C3—38no—
269C22×C3—34no—
270C22×C3—44no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—84no—
274C22×C3—82no—
275C22×C3—81no—
276C22×C3C2×C3×C4211yescharacteristic, complemented
277C22×C3C2×C4×A414yes—
278C22×C3C2×C4×A412yes—
279C22×C3C23×A411yescharacteristic
280A4—21no—
281A4C2×C4×Dic311yescharacteristic, direct product factor
282C3×C4—34no—
283C3×C4—44no—
284C3×C4—84no—
285C3×C4—128no—
286C3×C4C2×C4×A414yescomplemented
287Dic3—38no—
288Dic3C2×C4×A418yesdirect product factor
289C32—41noSylow
290C23—32no—
291C23—31no—
292C23—34no—
293C23—32no—
294C23—38no—
295C23—34no—
296C23—32no—
297C23—34no—
298C23—38no—
299C23—34no—
300C23—34no—
301C23—32no—
302C23—32no—
303C23—31no—
304C23—31no—
305C23C3×C4×Dic314yescomplemented
306C23C2×C3×C4×D312yes—
307C23C22×C3×Dic311yescharacteristic
308C23C2×D3×A411yescharacteristic
309C2×C4—38no—
310C2×C4—34no—
311C2×C4—38no—
312C2×C4—34no—
313C2×C4—34no—
314C2×C4—32no—
315C2×C4—34no—
316C2×C4—34no—
317C2×C4—38no—
318C2×C4—34no—
319C2×C4—98no—
320C2×C4—98no—
321C2×C4—916no—
322C2×C4—916no—
323C2×C4—98no—
324C2×C4—98no—
325C2×C4—94no—
326C2×C4—98no—
327C2×C4Dic3×A414yesdirect product factor
328C2×C4C2×D3×A412yes—
329C2×C3—34no—
330C2×C3—31no—
331C2×C3—32no—
332C2×C3—31no—
333C2×C3—42no—
334C2×C3—41no—
335C2×C3—44no—
336C2×C3—82no—
337C2×C3—81no—
338C2×C3—84no—
339C2×C3C42×A414yescomplemented
340C2×C3C22×C4×A412yes—
341C2×C3C22×C4×A411yescharacteristic
342C22—34no—
343C22—32no—
344C22—31no—
345C22—34no—
346C22—32no—
347C22—38no—
348C22—32no—
349C22—34no—
350C22—32no—
351C22—31no—
352C22—32no—
353C22—31no—
354C22—34no—
355C22—38no—
356C22—34no—
357C22C2×C3×C4×Dic311yescharacteristic, complemented
358C22C4×D3×A414yes—
359C22C2×Dic3×A412yes—
360C22C22×D3×A411yescharacteristic
361C4—38no—
362C4—34no—
363C4—98no—
364C4C2×Dic3×A414yesdirect product factor
365C3—41no—
366C3—81no—
367C3C2×C42×A411yescharacteristic, complemented
368C2—31no—
369C2—34no—
370C2—32no—
371C2—31no—
372C2C4×Dic3×A414yesdirect product factor
373C2C2×C4×D3×A412yes—
374C2C22×Dic3×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×A4
commutator, hypocenterC22×C3C2×C3×C42
FrattiniC22C22×D3×A4
FittingC24×C3×C4C2×C3
socleC25×C3C22×C3
prime radicalC23×C3×A4C22
cosocleC24×C3C23×C3
LCSG ▷ C22×C3
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C4×Dic3×A4C2direct (N × H)
4. C23×C4×Dic3C3semidirect
7. C2×C3×C4×A4C4semidirect
9. C2×Dic3×A4C4direct (N × H)
21. C22×C4×Dic3C2×C3semidirect
28. C3×C4×A4C2×C4semidirect
29. Dic3×A4C2×C4direct (N × H)
39. C23×C3×C4C3×C4semidirect
45. C23×Dic3C3×C4semidirect
51. C2×C4×A4Dic3direct (N × H)
63. C2×C4×Dic3A4direct (N × H)
65. C2×C3×A4C42semidirect
107. C22×C3×C4C2×C3×C4semidirect
121. C22×Dic3C2×C3×C4semidirect
125. C4×A4C2×Dic3direct (N × H)
130. C4×Dic3C2×A4direct (N × H)
134. C3×A4C2×C42semidirect
148. C23×C4C3×Dic3semidirect
177. C23×C3C3×C42semidirect
184. C2×A4C4×Dic3direct (N × H)
201. C2×C3×C4C4×A4semidirect
211. C2×Dic3C4×A4direct (N × H)
249. C22×C4C2×C3×Dic3semidirect
276. C22×C3C2×C3×C42semidirect
281. A4C2×C4×Dic3direct (N × H)
286. C3×C4C2×C4×A4semidirect
288. Dic3C2×C4×A4direct (N × H)
305. C23C3×C4×Dic3semidirect
327. C2×C4Dic3×A4direct (N × H)
339. C2×C3C42×A4semidirect
357. C22C2×C3×C4×Dic3semidirect
364. C4C2×Dic3×A4direct (N × H)
367. C3C2×C42×A4semidirect
372. C2C4×Dic3×A4direct (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×Dic3×A42
C4×Dic3×A42
C23×C4×Dic33
C2×C42×A43
C2×C3×C4×Dic34