Finite Group Databasebeta

← C2×A4×Dic12 #11923304

Subgroups — C2×A4×Dic12 #11923304

Order 1152.

710 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×A4×Dic6C212yesmaximal
2C2×C3×C8×A4C211yesmaximal, characteristic
3A4×Dic12C214yesmaximal, direct product factor
4C23×Dic12C311yesmaximal, characteristic, complemented
5C2×A4×Q16—31nomaximal
6C2×C3×C4×A4C2211yescharacteristic
7C2×Dic3×A4—22no—
8A4×Dic6—22no—
9A4×Dic6C2214yes—
10C2×C3×Dic12—41nomaximal
11C3×C8×A4C2212yes—
12C23×Dic6C2×C312yes—
13C2×Q8×A4—32no—
14C23×C3×C8C2×C311yescharacteristic
15C22×Dic12—31no—
16C22×Dic12—32no—
17C22×Dic12—31no—
18C22×Dic12—34no—
19C22×Dic12C2×C314yescomplemented
20A4×Q16—34no—
21C2×C8×A4—21no—
22C2×C8×A4D311yescharacteristic
23C22×C3×A4D411yescharacteristic
24C2×C3×Dic6—42no—
25C3×C4×A4D411yescharacteristic
26C3×C4×A4C2311yescharacteristic
27Dic3×A4—24no—
28C2×C32×C8—41no—
29C3×Dic12—44no—
30C23×Q16—31noSylow
31C23×C3×C4C22×C311yescharacteristic
32C23×Dic3—22no—
33C22×Dic6—22no—
34C22×Dic6—32no—
35C22×Dic6—32no—
36C22×Dic6—34no—
37C22×Dic6—62no—
38C22×Dic6—62no—
39C22×Dic6C22×C314yes—
40Q8×A4—34no—
41Q8×A4—62no—
42C2×C4×A4—21no—
43C2×C4×A4—62no—
44C2×C4×A4C2×D311yescharacteristic
45C2×C3×Q16—121no—
46C22×C3×C8—31no—
47C22×C3×C8—31no—
48C22×C3×C8—32no—
49C22×C3×C8C22×C312yes—
50C2×Dic12—32no—
51C2×Dic12—38no—
52C2×Dic12—34no—
53C2×Dic12—32no—
54C2×Dic12—31no—
55C2×Dic12—34no—
56C2×Dic12—34no—
57C2×Dic12—38no—
58C2×Dic12—34no—
59C2×Dic12A411yescharacteristic, direct product factor
60C8×A4—22no—
61C8×A4C2×D312yes—
62C2×C3×A4Q1612yescomplemented
63C2×C3×A4C2×D411yescharacteristic
64C2×C32×C4—41no—
65C2×C3×Dic3—82no—
66C3×Dic6—44no—
67C3×Dic6—82no—
68C32×C8—42no—
69C23×Q8—32no—
70C22×Q16—34no—
71C22×Q16—91no—
72C22×Q16—91no—
73C22×Q16—94no—
74C22×Q16—92no—
75C23×C8C3×D311yescharacteristic
76C24×C3C3×D411yescharacteristic
77C22×A4—21no—
78C22×A4D1211yescharacteristic
79C2×C3×Q8—122no—
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4C3×D411yescharacteristic
85C22×C3×C4C23×C311yescharacteristic
86C22×Dic3—24no—
87C22×Dic3—62no—
88C22×Dic3—64no—
89C22×Dic3—62no—
90C2×Dic6—34no—
91C2×Dic6—32no—
92C2×Dic6—34no—
93C2×Dic6—34no—
94C2×Dic6—34no—
95C2×Dic6—62no—
96C2×Dic6—62no—
97C2×Dic6—62no—
98C2×Dic6—68no—
99C2×Dic6—62no—
100C2×Dic6—64no—
101C2×Dic6—62no—
102C2×Dic6—62no—
103C2×Dic6—64no—
104C2×Dic6C2×A412yes—
105C4×A4—21no—
106C4×A4—21no—
107C4×A4—64no—
108C4×A4D1211yescharacteristic
109C4×A4C22×D311yescharacteristic
110C3×Q16—124no—
111C2×C3×C8—32no—
112C2×C3×C8—31no—
113C2×C3×C8—32no—
114C2×C3×C8—32no—
115C2×C3×C8—32no—
116C2×C3×C8—41no—
117C2×C3×C8—81no—
118C2×C3×C8C2×A411yescharacteristic
119Dic12—38no—
120Dic12—38no—
121Dic12—34no—
122Dic12C2×A414yesdirect product factor
123C22×C32—41no—
124C3×A4C2×Q1611yescharacteristic, complemented
125C32×C4—41no—
126C32×C4—41no—
127C3×Dic3—84no—
128C22×Q8—34no—
129C22×Q8—62no—
130C22×Q8—92no—
131C22×Q8—92no—
132C22×Q8—94no—
133C22×Q8—182no—
134C22×Q8—182no—
135C23×C4—62no—
136C23×C4C2×C3×D311yescharacteristic
137C2×Q16—31no—
138C2×Q16—94no—
139C2×Q16—94no—
140C2×Q16—92no—
141C2×Q16—98no—
142C2×Q16—92no—
143C2×Q16—94no—
144C2×Q16—94no—
145C2×Q16—98no—
146C2×Q16—91no—
147C22×C8—31no—
148C22×C8—31no—
149C22×C8—32no—
150C22×C8C2×C3×D312yes—
151C23×C3—31no—
152C23×C3—32no—
153C23×C3—31no—
154C23×C3C3×Q1612yescomplemented
155C23×C3C2×C3×D411yescharacteristic
156C2×A4—22no—
157C2×A4—21no—
158C2×A4Dic1212yesdirect product factor
159C2×A4C2×D1211yescharacteristic
160C3×Q8—124no—
161C3×Q8—242no—
162C2×C3×C4—32no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—41no—
171C2×C3×C4—81no—
172C2×C3×C4—242no—
173C2×C3×C4C22×A411yescharacteristic
174C2×Dic3—22no—
175C2×Dic3—64no—
176C2×Dic3—64no—
177C2×Dic3—64no—
178C2×Dic3—62no—
179C2×Dic3—64no—
180Dic6—22no—
181Dic6—34no—
182Dic6—64no—
183Dic6—64no—
184Dic6—64no—
185Dic6—62no—
186Dic6—64no—
187Dic6C22×A414yes—
188C3×C8—32no—
189C3×C8—42no—
190C3×C8—82no—
191C3×C8C22×A412yes—
192C2×C32—41no—
193C2×C32—42no—
194C24C3×D1211yescharacteristic
195C2×Q8—32no—
196C2×Q8—94no—
197C2×Q8—94no—
198C2×Q8—94no—
199C2×Q8—94no—
200C2×Q8—92no—
201C2×Q8—182no—
202C2×Q8—182no—
203C2×Q8—182no—
204C2×Q8—184no—
205C2×Q8—182no—
206C2×Q8—184no—
207C2×Q8—182no—
208C2×Q8—188no—
209C2×Q8—182no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—64no—
215C22×C4—182no—
216C22×C4—184no—
217C22×C4—182no—
218C22×C4C3×D1211yescharacteristic
219C22×C4C22×C3×D311yescharacteristic
220Q16—34no—
221Q16—98no—
222Q16—98no—
223Q16—94no—
224C2×C8—32no—
225C2×C8—31no—
226C2×C8—32no—
227C2×C8—32no—
228C2×C8—32no—
229C2×C8D3×A411yescharacteristic
230C22×C3—32no—
231C22×C3—31no—
232C22×C3—32no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—41no—
238C22×C3—81no—
239C22×C3C2×C3×Q1611yescharacteristic, complemented
240C22×C3D4×A411yescharacteristic
241A4—21no—
242A4C2×Dic1211yescharacteristic, direct product factor
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—81no—
248C3×C4—81no—
249C3×C4—244no—
250C3×C4D4×A411yescharacteristic
251C3×C4C23×A411yescharacteristic
252Dic3—24no—
253Dic3—64no—
254C32—41noSylow
255C23—31no—
256C23—32no—
257C23—31no—
258C23C3×Dic1212yescomplemented
259C23C2×C3×D1211yescharacteristic
260Q8—34no—
261Q8—62no—
262Q8—94no—
263Q8—184no—
264Q8—184no—
265Q8—184no—
266Q8—184no—
267Q8—182no—
268C2×C4—32no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—62no—
277C2×C4—184no—
278C2×C4—184no—
279C2×C4—184no—
280C2×C4—182no—
281C2×C4—184no—
282C2×C4C2×D3×A411yescharacteristic
283C8—32no—
284C8C2×D3×A412yes—
285C2×C3—32no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—41no—
289C2×C3—42no—
290C2×C3—81no—
291C2×C3—82no—
292C2×C3A4×Q1612yescomplemented
293C2×C3C2×D4×A411yescharacteristic
294C22—32no—
295C22—31no—
296C22—32no—
297C22—31no—
298C22—31no—
299C22—32no—
300C22—32no—
301C22C2×C3×Dic1211yescharacteristic, complemented
302C22A4×D1211yescharacteristic
303C4—31no—
304C4—31no—
305C4—64no—
306C4—184no—
307C4A4×D1211yescharacteristic
308C4C22×D3×A411yescharacteristic
309C3—41no—
310C3—81no—
311C3C2×A4×Q1611yescharacteristic, complemented
312C2—31no—
313C2—32no—
314C2—31no—
315C2A4×Dic1212yesdirect product factor
316C2C2×A4×D1211yescharacteristic
subgroupG/H
centerC22A4×D12
commutator, cosocleC22×C3×C4C23×C3
FrattiniC4C22×D3×A4
FittingC23×C3×C8C2×C3
socleC24×C3C3×D4
prime radicalC22×C3×A4D4
hypercenterC2×C8D3×A4
hypocenterC22×C3C2×C3×Q16
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. A4×Dic12C2direct (N × H)
4. C23×Dic12C3semidirect
19. C22×Dic12C2×C3semidirect
59. C2×Dic12A4direct (N × H)
62. C2×C3×A4Q16semidirect
122. Dic12C2×A4direct (N × H)
124. C3×A4C2×Q16semidirect
154. C23×C3C3×Q16semidirect
158. C2×A4Dic12direct (N × H)
239. C22×C3C2×C3×Q16semidirect
242. A4C2×Dic12direct (N × H)
258. C23C3×Dic12semidirect
292. C2×C3A4×Q16semidirect
301. C22C2×C3×Dic12semidirect
311. C3C2×A4×Q16semidirect
315. C2A4×Dic12direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×Dic62
C2×C3×C8×A42
A4×Dic122
C23×Dic123
C2×A4×Q163
C2×C3×Dic124