Finite Group Databasebeta

← C3×A4×C4⋊Q8 #11919712

Subgroups — C3×A4×C4⋊Q8 #11919712

Order 1152.

904 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 208 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×C4⋊C4C214yesmaximal
3C3×C42×A4C211yesmaximal, characteristic
4C22×C3×C4⋊Q8C311yesmaximal, characteristic, complemented
5A4×C4⋊Q8C313yesmaximal, direct product factor
6C3×Q8×A4—24no—
7C2×C3×C4×A4C2214yes—
8C2×C3×C4×A4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C32×C4⋊Q8—41nomaximal
11C23×C3×Q8C2×C312yes—
12C2×Q8×A4C2×C316yes—
13C2×C3×C4⋊Q8—31no—
14C2×C3×C4⋊Q8—31no—
15C2×C3×C4⋊Q8—34no—
16C2×C3×C4⋊Q8—32no—
17C22×C3×C4⋊C4C2×C314yes—
18C22×C3×C42C2×C311yescharacteristic
19A4×C4⋊C4C2×C3112yes—
20C42×A4C2×C313yes—
21C22×C3×A4C2311yescharacteristic
22C2×C32×Q8—42no—
23C3×C4×A4—24no—
24C3×C4×A4D412yes—
25C3×C4×A4Q814yescomplemented
26C32×C4⋊C4—44no—
27C32×C42—41no—
28C22×C4⋊Q8C3211yescharacteristic, complemented, Sylow
29C22×C3×Q8—24no—
30C22×C3×Q8—32no—
31C22×C3×Q8—34no—
32C22×C3×Q8—32no—
33C22×C3×Q8—64no—
34C23×C3×C4C22×C314yes—
35C23×C3×C4C22×C311yescharacteristic
36C23×C3×C4C22×C312yes—
37Q8×A4—212no—
38C2×C4×A4C22×C3112yes—
39C2×C4×A4C22×C313yes—
40C2×C4×A4C22×C316yes—
41C3×C4⋊Q8—38no—
42C3×C4⋊Q8—32no—
43C3×C4⋊Q8—34no—
44C3×C4⋊Q8—32no—
45C3×C4⋊Q8—34no—
46C3×C4⋊Q8—31no—
47C3×C4⋊Q8—43no—
48C3×C4⋊Q8A411yescharacteristic, direct product factor
49C2×C3×C4⋊C4—34no—
50C2×C3×C4⋊C4—34no—
51C2×C3×C4⋊C4—38no—
52C2×C3×C42—31no—
53C2×C3×C42—32no—
54C2×C3×C42—31no—
55C2×C3×A4C2×D411yescharacteristic
56C2×C3×A4C2×Q812yes—
57C32×Q8—84no—
58C2×C32×C4—42no—
59C2×C32×C4—41no—
60C2×C32×C4—44no—
61C23×Q8C2×C3212yes—
62C2×C4⋊Q8—31no—
63C2×C4⋊Q8—31no—
64C2×C4⋊Q8—34no—
65C2×C4⋊Q8—32no—
66C22×C4⋊C4C2×C3214yes—
67C22×C42C2×C3211yescharacteristic
68C24×C3C23×C311yescharacteristic
69C22×A4C23×C313yes—
70C2×C3×Q8—34no—
71C2×C3×Q8—34no—
72C2×C3×Q8—32no—
73C2×C3×Q8—46no—
74C2×C3×Q8—64no—
75C2×C3×Q8—64no—
76C2×C3×Q8—64no—
77C2×C3×Q8—68no—
78C2×C3×Q8—64no—
79C2×C3×Q8—68no—
80C2×C3×Q8C2×A412yes—
81C22×C3×C4—24no—
82C22×C3×C4—32no—
83C22×C3×C4—31no—
84C22×C3×C4—32no—
85C22×C3×C4—31no—
86C22×C3×C4—34no—
87C22×C3×C4—32no—
88C22×C3×C4—34no—
89C22×C3×C4—34no—
90C22×C3×C4—64no—
91C22×C3×C4C3×D412yes—
92C22×C3×C4C3×Q814yescomplemented
93C4×A4—212no—
94C4×A4C3×D416yes—
95C4×A4C3×Q8112yescomplemented
96C3×C4⋊C4—38no—
97C3×C4⋊C4—38no—
98C3×C4⋊C4—34no—
99C3×C4⋊C4—412no—
100C3×C4⋊C4C2×A414yes—
101C3×C42—32no—
102C3×C42—31no—
103C3×C42—32no—
104C3×C42—43no—
105C3×C42C2×A411yescharacteristic
106C22×C32—41no—
107C3×A4C4⋊Q811yescharacteristic, direct product factor
108C32×C4—42no—
109C32×C4—44no—
110C32×C4—84no—
111C22×Q8—24no—
112C22×Q8—32no—
113C22×Q8—34no—
114C22×Q8—32no—
115C22×Q8—64no—
116C23×C4C22×C3214yes—
117C23×C4C22×C3211yescharacteristic
118C23×C4C22×C3212yes—
119C4⋊Q8—38no—
120C4⋊Q8—32no—
121C4⋊Q8—34no—
122C4⋊Q8—32no—
123C4⋊Q8—34no—
124C4⋊Q8—31no—
125C4⋊Q8C3×A411yescharacteristic, direct product factor
126C2×C4⋊C4—34no—
127C2×C4⋊C4—34no—
128C2×C4⋊C4—38no—
129C2×C42—31no—
130C2×C42—32no—
131C2×C42—31no—
132C23×C3—31no—
133C23×C3—32no—
134C23×C3—31no—
135C23×C3C2×C3×D411yescharacteristic
136C23×C3C2×C3×Q812yes—
137C2×A4C2×C3×D413yes—
138C2×A4C2×C3×Q816yes—
139C3×Q8—24no—
140C3×Q8—68no—
141C3×Q8—68no—
142C3×Q8—64no—
143C3×Q8—812no—
144C2×C3×C4—34no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—31no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—34no—
151C2×C3×C4—34no—
152C2×C3×C4—32no—
153C2×C3×C4—34no—
154C2×C3×C4—34no—
155C2×C3×C4—46no—
156C2×C3×C4—43no—
157C2×C3×C4—412no—
158C2×C3×C4—64no—
159C2×C3×C4—64no—
160C2×C3×C4—64no—
161C2×C3×C4—64no—
162C2×C3×C4C22×A414yes—
163C2×C3×C4C22×A411yescharacteristic
164C2×C3×C4C22×A412yes—
165C2×C32—41no—
166C2×C32—42no—
167C24C23×C3211yescharacteristic
168C2×Q8—34no—
169C2×Q8—34no—
170C2×Q8—32no—
171C2×Q8—64no—
172C2×Q8—64no—
173C2×Q8—64no—
174C2×Q8—68no—
175C2×Q8—64no—
176C2×Q8—68no—
177C2×Q8C2×C3×A412yes—
178C22×C4—24no—
179C22×C4—32no—
180C22×C4—31no—
181C22×C4—32no—
182C22×C4—31no—
183C22×C4—32no—
184C22×C4—34no—
185C22×C4—34no—
186C22×C4—34no—
187C22×C4—64no—
188C22×C4C32×D412yes—
189C22×C4C32×Q814yescomplemented
190C4⋊C4—38no—
191C4⋊C4—38no—
192C4⋊C4—34no—
193C4⋊C4C2×C3×A414yes—
194C42—32no—
195C42—31no—
196C42—32no—
197C42C2×C3×A411yescharacteristic
198C22×C3—31no—
199C22×C3—32no—
200C22×C3—32no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—32no—
204C22×C3—32no—
205C22×C3—43no—
206C22×C3C3×C4⋊Q811yescharacteristic, complemented
207C22×C3C23×A411yescharacteristic
208A4C3×C4⋊Q813yesdirect product factor
209C3×C4—24no—
210C3×C4—32no—
211C3×C4—34no—
212C3×C4—46no—
213C3×C4—412no—
214C3×C4—64no—
215C3×C4—812no—
216C3×C4D4×A412yes—
217C3×C4Q8×A414yescomplemented
218C32—41noSylow
219C23—31no—
220C23—32no—
221C23—31no—
222C23C2×C32×D411yescharacteristic
223C23C2×C32×Q812yes—
224Q8—24no—
225Q8—68no—
226Q8—68no—
227Q8—64no—
228C2×C4—32no—
229C2×C4—34no—
230C2×C4—32no—
231C2×C4—31no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—34no—
235C2×C4—34no—
236C2×C4—32no—
237C2×C4—34no—
238C2×C4—34no—
239C2×C4—64no—
240C2×C4—64no—
241C2×C4—64no—
242C2×C4—64no—
243C2×C4C22×C3×A414yes—
244C2×C4C22×C3×A411yescharacteristic
245C2×C4C22×C3×A412yes—
246C2×C3—31no—
247C2×C3—32no—
248C2×C3—31no—
249C2×C3—43no—
250C2×C3—46no—
251C2×C3C2×D4×A411yescharacteristic
252C2×C3C2×Q8×A412yes—
253C22—31no—
254C22—32no—
255C22—32no—
256C22—31no—
257C22—31no—
258C22—32no—
259C22—32no—
260C22C32×C4⋊Q811yescharacteristic, complemented
261C22C23×C3×A411yescharacteristic
262C4—24no—
263C4—32no—
264C4—34no—
265C4—64no—
266C4C3×D4×A412yes—
267C4C3×Q8×A414yescomplemented
268C3—43no—
269C3A4×C4⋊Q811yescharacteristic, direct product factor
270C2—31no—
271C2—31no—
272C2—32no—
273C2C2×C3×D4×A411yescharacteristic
274C2C2×C3×Q8×A412yes—
subgroupG/H
centerC22×C3C23×A4
commutator, cosocleC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×C4⋊Q8C3
socleC24×C3C23×C3
prime radicalC22×C3×A4C23
hypercenterC3×C4⋊Q8A4
hypocenterC22C32×C4⋊Q8
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C22×C3×C4⋊Q8C3semidirect
5. A4×C4⋊Q8C3direct (N × H)
25. C3×C4×A4Q8semidirect
28. C22×C4⋊Q8C32semidirect
48. C3×C4⋊Q8A4direct (N × H)
92. C22×C3×C4C3×Q8semidirect
95. C4×A4C3×Q8semidirect
107. C3×A4C4⋊Q8direct (N × H)
125. C4⋊Q8C3×A4direct (N × H)
189. C22×C4C32×Q8semidirect
206. C22×C3C3×C4⋊Q8semidirect
208. A4C3×C4⋊Q8direct (N × H)
217. C3×C4Q8×A4semidirect
260. C22C32×C4⋊Q8semidirect
267. C4C3×Q8×A4semidirect
269. C3A4×C4⋊Q8direct (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×C4⋊C42
C3×C42×A42
C22×C3×C4⋊Q83
A4×C4⋊Q83
C32×C4⋊Q84