Finite Group Databasebeta

← C2×A4×#634 #13181456

Subgroups — C2×A4×#634 #13181456

Order 1728.

1358 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 177 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×C32×C4×A4C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C212yesmaximal
3A4×#634C214yesmaximal, direct product factor
4C23×#634C311yesmaximal, characteristic, complemented
5C2×A4×Dic6—34nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3×#634—41nomaximal
8C32×C4×A4C2212yes—
9A4×C3⋊Dic3C2214yes—
10C23×C32×C4C2×C311yescharacteristic
11C23×C3⋊Dic3C2×C312yes—
12C22×#634—31no—
13C22×#634—31no—
14C22×#634—32no—
15C22×#634—34no—
16C22×#634C2×C314yescomplemented
17C2×C3×C4×A4—24no—
18C2×C3×C4×A4D314yes—
19C2×Dic3×A4—38no—
20A4×Dic6—316no—
21C2×C32×A4Q812yescomplemented
22C2×C32×A4C2311yescharacteristic
23C2×C33×C4—41no—
24C2×C3×C3⋊Dic3—42no—
25C3×#634—44no—
26C23×Dic6—34no—
27C2×Q8×A4—91no—
28C24×C32C22×C311yescharacteristic
29C22×C3×A4—24no—
30C22×C3×A4C2×D314yes—
31C22×C32×C4—31no—
32C22×C32×C4—31no—
33C22×C32×C4—32no—
34C22×C32×C4C22×C312yes—
35C22×C3⋊Dic3—32no—
36C22×C3⋊Dic3—32no—
37C22×C3⋊Dic3—34no—
38C22×C3⋊Dic3C22×C314yes—
39C2×#634—32no—
40C2×#634—34no—
41C2×#634—38no—
42C2×#634—31no—
43C2×#634—32no—
44C2×#634—34no—
45C2×#634—34no—
46C2×#634—34no—
47C2×#634—38no—
48C2×#634A411yescharacteristic, direct product factor
49C2×C3×Dic6—124no—
50C3×C4×A4—28no—
51C3×C4×A4C2×D318yes—
52Dic3×A4—316no—
53C22×C33—41no—
54C32×A4C2×Q811yescharacteristic, complemented
55C33×C4—42no—
56C3×C3⋊Dic3—44no—
57C23×C3×C4C3×D314yes—
58C23×Dic3—38no—
59C22×Dic6—316no—
60C22×Dic6—94no—
61C22×Dic6—916no—
62C22×Dic6—98no—
63C22×Dic6—94no—
64Q8×A4—94no—
65C2×C4×A4—24no—
66C2×C4×A4—92no—
67C2×C4×A4C3⋊D311yescharacteristic
68C23×C32—32no—
69C23×C32—31no—
70C23×C32—31no—
71C23×C32C3×Q812yescomplemented
72C23×C32C23×C311yescharacteristic
73C2×C3×A4—28no—
74C2×C3×A4—24no—
75C2×C3×A4Dic618yescomplemented
76C2×C3×A4C22×D314yes—
77C2×C32×C4—32no—
78C2×C32×C4—31no—
79C2×C32×C4—32no—
80C2×C32×C4—32no—
81C2×C32×C4—32no—
82C2×C32×C4—44no—
83C2×C32×C4—84no—
84C2×C32×C4C2×A411yescharacteristic
85C2×C3⋊Dic3—34no—
86C2×C3⋊Dic3—32no—
87C2×C3⋊Dic3—34no—
88C2×C3⋊Dic3—34no—
89C2×C3⋊Dic3—34no—
90C2×C3⋊Dic3C2×A412yes—
91#634—38no—
92#634—34no—
93#634—38no—
94#634C2×A414yesdirect product factor
95C2×C3×Dic3—128no—
96C3×Dic6—1216no—
97C23×Q8—91noSylow
98C2×C33—42no—
99C2×C33—41no—
100C24×C3C2×C3×D314yes—
101C22×A4—24no—
102C22×A4C2×C3⋊D311yescharacteristic
103C2×C3×Q8—361no—
104C22×C3×C4—34no—
105C22×C3×C4—34no—
106C22×C3×C4—38no—
107C22×C3×C4C2×C3×D318yes—
108C22×Dic3—316no—
109C22×Dic3—98no—
110C22×Dic3—916no—
111C22×Dic3—98no—
112C2×Dic6—34no—
113C2×Dic6—916no—
114C2×Dic6—916no—
115C2×Dic6—98no—
116C2×Dic6—932no—
117C2×Dic6—98no—
118C2×Dic6—916no—
119C2×Dic6—94no—
120C2×Dic6—932no—
121C2×Dic6—916no—
122C4×A4—28no—
123C4×A4—94no—
124C4×A4C2×C3⋊D312yes—
125C22×C32—32no—
126C22×C32—31no—
127C22×C32—32no—
128C22×C32—32no—
129C22×C32—32no—
130C22×C32—31no—
131C22×C32—31no—
132C22×C32—44no—
133C22×C32—84no—
134C22×C32C2×C3×Q811yescharacteristic, complemented
135C22×C32C22×A411yescharacteristic
136C3×A4—24no—
137C3×A4C2×Dic614yescomplemented
138C32×C4—32no—
139C32×C4—48no—
140C32×C4—88no—
141C32×C4C22×A412yes—
142C3⋊Dic3—34no—
143C3⋊Dic3C22×A414yes—
144C3×Dic3—1216no—
145C22×Q8—94no—
146C22×Q8—271no—
147C22×Q8—271no—
148C22×Q8—272no—
149C22×Q8—274no—
150C23×C4—92no—
151C23×C4C3×C3⋊D311yescharacteristic
152C33—41noSylow
153C23×C3—34no—
154C23×C3—38no—
155C23×C3—34no—
156C23×C3C3×Dic618yescomplemented
157C23×C3C22×C3×D314yes—
158C2×A4—28no—
159C2×A4—24no—
160C2×A4#63412yesdirect product factor
161C2×A4C22×C3⋊D311yescharacteristic
162C3×Q8—364no—
163C2×C3×C4—38no—
164C2×C3×C4—34no—
165C2×C3×C4—38no—
166C2×C3×C4—38no—
167C2×C3×C4—38no—
168C2×C3×C4—41no—
169C2×C3×C4—84no—
170C2×C3×C4—362no—
171C2×C3×C4D3×A414yes—
172C2×Dic3—38no—
173C2×Dic3—916no—
174C2×Dic3—916no—
175C2×Dic3—916no—
176C2×Dic3—98no—
177C2×Dic3—916no—
178Dic6—316no—
179Dic6—932no—
180Dic6—932no—
181Dic6—916no—
182C2×C32—32no—
183C2×C32—31no—
184C2×C32—31no—
185C2×C32—48no—
186C2×C32—44no—
187C2×C32—88no—
188C2×C32—84no—
189C2×C32Q8×A412yescomplemented
190C2×C32C23×A411yescharacteristic
191C24C2×C3×C3⋊D311yescharacteristic
192C2×Q8—91no—
193C2×Q8—272no—
194C2×Q8—274no—
195C2×Q8—274no—
196C2×Q8—274no—
197C2×Q8—278no—
198C2×Q8—278no—
199C2×Q8—272no—
200C2×Q8—274no—
201C2×Q8—271no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—32no—
205C22×C4—94no—
206C22×C4—272no—
207C22×C4—274no—
208C22×C4—272no—
209C22×C4C2×C3×C3⋊D312yes—
210C22×C3—38no—
211C22×C3—34no—
212C22×C3—38no—
213C22×C3—38no—
214C22×C3—38no—
215C22×C3—34no—
216C22×C3—34no—
217C22×C3—41no—
218C22×C3—84no—
219C22×C3C2×C3×Dic614yescomplemented
220C22×C3C2×D3×A414yes—
221A4—24no—
222A4C2×#63411yescharacteristic, direct product factor
223C3×C4—38no—
224C3×C4—42no—
225C3×C4—88no—
226C3×C4—364no—
227C3×C4C2×D3×A418yes—
228Dic3—316no—
229Dic3—916no—
230C32—44no—
231C32—84no—
232C32C2×Q8×A411yescharacteristic, complemented
233C23—31no—
234C23—32no—
235C23—31no—
236C23C3×#63412yescomplemented
237C23C22×C3×C3⋊D311yescharacteristic
238Q8—94no—
239Q8—278no—
240Q8—278no—
241Q8—274no—
242C2×C4—32no—
243C2×C4—31no—
244C2×C4—32no—
245C2×C4—32no—
246C2×C4—32no—
247C2×C4—92no—
248C2×C4—274no—
249C2×C4—274no—
250C2×C4—274no—
251C2×C4—272no—
252C2×C4—274no—
253C2×C4A4×C3⋊D311yescharacteristic
254C2×C3—34no—
255C2×C3—38no—
256C2×C3—34no—
257C2×C3—42no—
258C2×C3—41no—
259C2×C3—88no—
260C2×C3—84no—
261C2×C3A4×Dic618yescomplemented
262C2×C3C22×D3×A414yes—
263C22—32no—
264C22—31no—
265C22—32no—
266C22—32no—
267C22—32no—
268C22—31no—
269C22—31no—
270C22C2×C3×#63411yescharacteristic, complemented
271C22C2×A4×C3⋊D311yescharacteristic
272C4—32no—
273C4—94no—
274C4—274no—
275C4C2×A4×C3⋊D312yes—
276C3—41no—
277C3—84no—
278C3C2×A4×Dic614yescomplemented
279C2—31no—
280C2—32no—
281C2—31no—
282C2A4×#63412yesdirect product factor
283C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC22C2×A4×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC23×C32×C4C2×C3
socleC24×C32C22×C3
prime radicalC22×C32×A4C22
hypercenterC2×C4A4×C3⋊D3
hypocenterC22×C32C2×C3×Q8
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. A4×#634C2direct (N × H)
4. C23×#634C3semidirect
16. C22×#634C2×C3semidirect
21. C2×C32×A4Q8semidirect
48. C2×#634A4direct (N × H)
54. C32×A4C2×Q8semidirect
71. C23×C32C3×Q8semidirect
75. C2×C3×A4Dic6semidirect
94. #634C2×A4direct (N × H)
134. C22×C32C2×C3×Q8semidirect
137. C3×A4C2×Dic6semidirect
156. C23×C3C3×Dic6semidirect
160. C2×A4#634direct (N × H)
189. C2×C32Q8×A4semidirect
219. C22×C3C2×C3×Dic6semidirect
222. A4C2×#634direct (N × H)
232. C32C2×Q8×A4semidirect
236. C23C3×#634semidirect
261. C2×C3A4×Dic6semidirect
270. C22C2×C3×#634semidirect
278. C3C2×A4×Dic6semidirect
282. C2A4×#634direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C2×A4×C3⋊Dic32
A4×#6342
C23×#6343
C2×A4×Dic63
C2×C3×#6344