Finite Group Databasebeta

← C2×C3×A4×Q16 #11919790

Subgroups — C2×C3×A4×Q16 #11919790

Order 1152.

808 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 176 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×C3×Q8×A4C212yesmaximal
2C3×A4×Q16C214yesmaximal, direct product factor
3C2×C3×C8×A4C211yesmaximal, characteristic
4C23×C3×Q16C311yesmaximal, characteristic, complemented
5C2×A4×Q16C313yesmaximal, direct product factor
6C3×Q8×A4—22no—
7C3×Q8×A4C2214yes—
8C2×C3×C4×A4—22no—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C32×Q16—41nomaximal
11C3×C8×A4C2212yes—
12C23×C3×Q8C2×C312yes—
13C2×Q8×A4C2×C316yes—
14C22×C3×Q16—31no—
15C22×C3×Q16—31no—
16C22×C3×Q16—32no—
17C22×C3×Q16—34no—
18C22×C3×Q16C2×C314yescomplemented
19C23×C3×C8C2×C311yescharacteristic
20A4×Q16C2×C3112yesdirect product factor
21C2×C8×A4C2×C313yes—
22C22×C3×A4D411yescharacteristic
23C2×C32×Q8—42no—
24C3×C4×A4—24no—
25C3×C4×A4D411yescharacteristic
26C3×C4×A4C2311yescharacteristic
27C32×Q16—44no—
28C2×C32×C8—41no—
29C23×Q16C3211yescharacteristic, complemented, Sylow
30C22×C3×Q8—22no—
31C22×C3×Q8—32no—
32C22×C3×Q8—34no—
33C22×C3×Q8—32no—
34C22×C3×Q8—62no—
35C22×C3×Q8—62no—
36C22×C3×Q8C22×C314yes—
37C23×C3×C4—22no—
38C23×C3×C4C22×C311yescharacteristic
39Q8×A4—26no—
40Q8×A4C22×C3112yes—
41C2×C4×A4—26no—
42C2×C4×A4C22×C313yes—
43C2×C3×Q16—38no—
44C2×C3×Q16—32no—
45C2×C3×Q16—34no—
46C2×C3×Q16—32no—
47C2×C3×Q16—34no—
48C2×C3×Q16—34no—
49C2×C3×Q16—34no—
50C2×C3×Q16—38no—
51C2×C3×Q16—31no—
52C2×C3×Q16—43no—
53C2×C3×Q16A411yescharacteristic, direct product factor
54C22×C3×C8—31no—
55C22×C3×C8—31no—
56C22×C3×C8—32no—
57C22×C3×C8C22×C312yes—
58C8×A4C22×C316yes—
59C2×C3×A4Q1612yesdirect product factor
60C2×C3×A4C2×D411yescharacteristic
61C32×Q8—44no—
62C32×Q8—82no—
63C2×C32×C4—41no—
64C2×C32×C4—82no—
65C32×C8—42no—
66C23×Q8C2×C3212yes—
67C22×Q16—31no—
68C22×Q16—31no—
69C22×Q16—32no—
70C22×Q16—34no—
71C22×Q16C2×C3214yescomplemented
72C23×C8C2×C3211yescharacteristic
73C24×C3C3×D411yescharacteristic
74C22×A4C3×D413yes—
75C2×C3×Q8—34no—
76C2×C3×Q8—34no—
77C2×C3×Q8—34no—
78C2×C3×Q8—34no—
79C2×C3×Q8—32no—
80C2×C3×Q8—46no—
81C2×C3×Q8—62no—
82C2×C3×Q8—62no—
83C2×C3×Q8—62no—
84C2×C3×Q8—64no—
85C2×C3×Q8—62no—
86C2×C3×Q8—64no—
87C2×C3×Q8—62no—
88C2×C3×Q8—68no—
89C2×C3×Q8—62no—
90C2×C3×Q8C2×A412yes—
91C22×C3×C4—24no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—62no—
97C22×C3×C4—64no—
98C22×C3×C4—62no—
99C22×C3×C4C3×D411yescharacteristic
100C22×C3×C4C23×C311yescharacteristic
101C4×A4—212no—
102C4×A4C3×D413yes—
103C4×A4C23×C313yes—
104C3×Q16—38no—
105C3×Q16—38no—
106C3×Q16—34no—
107C3×Q16—412no—
108C3×Q16C2×A414yesdirect product factor
109C2×C3×C8—32no—
110C2×C3×C8—31no—
111C2×C3×C8—32no—
112C2×C3×C8—32no—
113C2×C3×C8—32no—
114C2×C3×C8—43no—
115C2×C3×C8C2×A411yescharacteristic
116C22×C32—41no—
117C3×A4C2×Q1611yescharacteristic, direct product factor
118C32×C4—41no—
119C32×C4—41no—
120C32×C4—84no—
121C22×Q8—22no—
122C22×Q8—32no—
123C22×Q8—34no—
124C22×Q8—32no—
125C22×Q8—62no—
126C22×Q8—62no—
127C22×Q8C22×C3214yes—
128C23×C4—22no—
129C23×C4C22×C3211yescharacteristic
130C2×Q16—38no—
131C2×Q16—32no—
132C2×Q16—34no—
133C2×Q16—32no—
134C2×Q16—34no—
135C2×Q16—34no—
136C2×Q16—34no—
137C2×Q16—38no—
138C2×Q16—31no—
139C2×Q16C3×A411yescharacteristic, direct product factor
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—32no—
143C22×C8C22×C3212yes—
144C23×C3—31no—
145C23×C3—32no—
146C23×C3—31no—
147C23×C3C3×Q1612yescomplemented
148C23×C3C2×C3×D411yescharacteristic
149C2×A4C3×Q1616yesdirect product factor
150C2×A4C2×C3×D413yes—
151C3×Q8—22no—
152C3×Q8—34no—
153C3×Q8—412no—
154C3×Q8—64no—
155C3×Q8—64no—
156C3×Q8—64no—
157C3×Q8—64no—
158C3×Q8—62no—
159C3×Q8—86no—
160C3×Q8C22×A414yes—
161C2×C3×C4—22no—
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—43no—
171C2×C3×C4—64no—
172C2×C3×C4—64no—
173C2×C3×C4—64no—
174C2×C3×C4—62no—
175C2×C3×C4—64no—
176C2×C3×C4—86no—
177C2×C3×C4C22×A411yescharacteristic
178C3×C8—32no—
179C3×C8—46no—
180C3×C8C22×A412yes—
181C2×C32—41no—
182C2×C32—42no—
183C24C32×D411yescharacteristic
184C2×Q8—34no—
185C2×Q8—34no—
186C2×Q8—34no—
187C2×Q8—34no—
188C2×Q8—32no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×Q8—62no—
192C2×Q8—64no—
193C2×Q8—62no—
194C2×Q8—64no—
195C2×Q8—62no—
196C2×Q8—68no—
197C2×Q8—62no—
198C2×Q8C2×C3×A412yes—
199C22×C4—24no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—62no—
205C22×C4—64no—
206C22×C4—62no—
207C22×C4C32×D411yescharacteristic
208C22×C4C23×C3211yescharacteristic
209Q16—38no—
210Q16—38no—
211Q16—34no—
212Q16C2×C3×A414yesdirect product factor
213C2×C8—32no—
214C2×C8—31no—
215C2×C8—32no—
216C2×C8—32no—
217C2×C8—32no—
218C2×C8C2×C3×A411yescharacteristic
219C22×C3—32no—
220C22×C3—31no—
221C22×C3—32no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—32no—
225C22×C3—32no—
226C22×C3—43no—
227C22×C3C2×C3×Q1611yescharacteristic, complemented
228C22×C3D4×A411yescharacteristic
229A4C2×C3×Q1613yesdirect product factor
230C3×C4—24no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—43no—
234C3×C4—43no—
235C3×C4—64no—
236C3×C4—812no—
237C3×C4D4×A411yescharacteristic
238C3×C4C23×A411yescharacteristic
239C32—41noSylow
240C23—31no—
241C23—32no—
242C23—31no—
243C23C32×Q1612yescomplemented
244C23C2×C32×D411yescharacteristic
245Q8—22no—
246Q8—34no—
247Q8—64no—
248Q8—64no—
249Q8—64no—
250Q8—64no—
251Q8—62no—
252Q8C22×C3×A414yes—
253C2×C4—22no—
254C2×C4—32no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—64no—
263C2×C4—64no—
264C2×C4—64no—
265C2×C4—62no—
266C2×C4—64no—
267C2×C4C22×C3×A411yescharacteristic
268C8—32no—
269C8C22×C3×A412yes—
270C2×C3—32no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—43no—
274C2×C3—46no—
275C2×C3A4×Q1612yesdirect product factor
276C2×C3C2×D4×A411yescharacteristic
277C22—32no—
278C22—31no—
279C22—32no—
280C22—31no—
281C22—31no—
282C22—32no—
283C22—32no—
284C22C2×C32×Q1611yescharacteristic, complemented
285C22C3×D4×A411yescharacteristic
286C4—24no—
287C4—31no—
288C4—31no—
289C4—64no—
290C4C3×D4×A411yescharacteristic
291C4C23×C3×A411yescharacteristic
292C3—43no—
293C3C2×A4×Q1611yescharacteristic, direct product factor
294C2—31no—
295C2—32no—
296C2—31no—
297C2C3×A4×Q1612yesdirect product factor
298C2C2×C3×D4×A411yescharacteristic
subgroupG/H
centerC22×C3D4×A4
commutator, cosocleC22×C4C23×C32
FrattiniC4C23×C3×A4
FittingC23×C3×Q16C3
socleC24×C3C3×D4
prime radicalC22×C3×A4D4
hypercenterC2×C3×Q16A4
hypocenterC22C2×C32×Q16
LCSG ▷ C22×C4 ▷ C23 ▷ C22
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C3×A4×Q16C2direct (N × H)
4. C23×C3×Q16C3semidirect
5. C2×A4×Q16C3direct (N × H)
18. C22×C3×Q16C2×C3semidirect
20. A4×Q16C2×C3direct (N × H)
29. C23×Q16C32semidirect
53. C2×C3×Q16A4direct (N × H)
59. C2×C3×A4Q16direct (N × H)
71. C22×Q16C2×C32semidirect
108. C3×Q16C2×A4direct (N × H)
117. C3×A4C2×Q16direct (N × H)
139. C2×Q16C3×A4direct (N × H)
147. C23×C3C3×Q16semidirect
149. C2×A4C3×Q16direct (N × H)
212. Q16C2×C3×A4direct (N × H)
227. C22×C3C2×C3×Q16semidirect
229. A4C2×C3×Q16direct (N × H)
243. C23C32×Q16semidirect
275. C2×C3A4×Q16direct (N × H)
284. C22C2×C32×Q16semidirect
293. C3C2×A4×Q16direct (N × H)
297. C2C3×A4×Q16direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C3×A4×Q162
C2×C3×C8×A42
C23×C3×Q163
C2×A4×Q163
C2×C32×Q164