Finite Group Databasebeta

← C22×#64259 #11923713

Subgroups — C22×#64259 #11923713

Order 1152.

2046 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 267 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×C3⋊SL(2,ℤ/4)C212yesmaximal
3C2×#64259C2112yesmaximal, direct product factor
4C22×A4⋊Q8—33nomaximal
5C22×#924—31nomaximal
6C23×C3×A4C2211yescharacteristic
7C22×#634—41nomaximal
8C2×C3×C4×A4C2216yes—
9C2×C3⋊SL(2,ℤ/4)C22112yes—
10#64259C22116yesdirect product factor
11C24×C3×C4D311yescharacteristic
12C23×Dic6—31no—
13C22×C4×A4D313yes—
14C22×SL(2,ℤ/4)—36no—
15C2×A4⋊Q8—336no—
16C22×#217—32no—
17C22×C4⋊Dic3—31no—
18C2×#924—312no—
19C2×#924—312no—
20C22×Dic3⋊C4—32no—
21C22×C3×A4Q814yescomplemented
22C22×C3×A4C2313yes—
23C22×C32×C4—41no—
24C22×C3⋊Dic3—42no—
25C2×#634—412no—
26C3×C4×A4C2314yes—
27C3⋊SL(2,ℤ/4)C2318yes—
28C22×C22⋊Q8—91noSylow
29C25×C3C2×D311yescharacteristic
30C23×A4C2×D313yes—
31C23×C3×C4—31no—
32C23×C3×C4—31no—
33C23×C3×C4—36no—
34C23×C3×C4C2×D316yes—
35C23×Dic3—32no—
36C23×Dic3—32no—
37C22×Dic6—312no—
38C22×Dic6—61no—
39C22×Dic6—66no—
40C22×Dic6—61no—
41C22×Dic6—123no—
42C2×C4×A4C2×D3118yes—
43C2×SL(2,ℤ/4)—336no—
44A4⋊Q8—348no—
45C2×#217—312no—
46C2×#217—312no—
47C2×C4⋊Dic3—312no—
48#924—348no—
49#924—316no—
50C2×Dic3⋊C4—324no—
51C23×C32—41no—
52C2×C3×A4C2×Q816yescomplemented
53C2×C3×A4C2411yescharacteristic
54C2×C32×C4—46no—
55C2×C3⋊Dic3—412no—
56#634—416no—
57C23×Q8—91no—
58C24×C4C3⋊D311yescharacteristic
59C2×C22⋊Q8—912no—
60C2×C22⋊Q8—912no—
61C22×C4⋊C4—92no—
62C22×C4⋊C4—91no—
63C22×C22⋊C4—92no—
64C24×C3—31no—
65C24×C3—34no—
66C24×C3—33no—
67C24×C3Dic614yescomplemented
68C24×C3C22×D313yes—
69C22×A4Dic6112yescomplemented
70C22×A4C22×D319yes—
71C22×C3×C4—31no—
72C22×C3×C4—36no—
73C22×C3×C4—36no—
74C22×C3×C4—36no—
75C22×C3×C4—312no—
76C22×C3×C4—43no—
77C22×C3×C4—64no—
78C22×C3×C4—63no—
79C22×C3×C4S411yescharacteristic
80C22×C3×C4C22×D314yes—
81C22×Dic3—312no—
82C22×Dic3—312no—
83C22×Dic3—66no—
84C22×Dic3—68no—
85C22×Dic3—62no—
86C22×Dic3—126no—
87C2×Dic6—316no—
88C2×Dic6—612no—
89C2×Dic6—624no—
90C2×Dic6—612no—
91C2×Dic6—1236no—
92C4×A4C22×D3112yes—
93SL(2,ℤ/4)—324no—
94#217—324no—
95#217—38no—
96C4⋊Dic3—316no—
97Dic3⋊C4—332no—
98C22×C32—44no—
99C22×C32—43no—
100C3×A4C22×Q811yescharacteristic, complemented
101C32×C4—44no—
102C3⋊Dic3—48no—
103C25C2×C3⋊D311yescharacteristic
104C22×Q8—912no—
105C22×Q8—181no—
106C22×Q8—186no—
107C22×Q8—181no—
108C23×C4—31no—
109C23×C4—31no—
110C23×C4—36no—
111C23×C4—92no—
112C23×C4—92no—
113C23×C4C2×C3⋊D316yes—
114C22⋊Q8—948no—
115C22⋊Q8—916no—
116C2×C4⋊C4—924no—
117C2×C4⋊C4—912no—
118C2×C22⋊C4—912no—
119C2×C22⋊C4—912no—
120C23×C3—34no—
121C23×C3—34no—
122C23×C3—33no—
123C23×C3—33no—
124C23×C3—33no—
125C23×C3—312no—
126C23×C3—36no—
127C23×C3—43no—
128C23×C3—61no—
129C23×C3—66no—
130C23×C3C2×Dic616yescomplemented
131C23×C3C2×S411yescharacteristic
132C23×C3C23×D311yescharacteristic
133C2×A4C2×Dic6118yescomplemented
134C2×A4C23×D313yes—
135C2×C3×C4—36no—
136C2×C3×C4—34no—
137C2×C3×C4—312no—
138C2×C3×C4—34no—
139C2×C3×C4—418no—
140C2×C3×C4—66no—
141C2×C3×C4C2×S416yes—
142C2×Dic3—38no—
143C2×Dic3—38no—
144C2×Dic3—612no—
145C2×Dic3—624no—
146C2×Dic3—612no—
147C2×Dic3—1236no—
148Dic6—616no—
149Dic6—616no—
150Dic6—1248no—
151C2×C32—46no—
152C2×C32—41no—
153C24—31no—
154C24—33no—
155C24—34no—
156C24#63414yescomplemented
157C24C22×C3⋊D313yes—
158C2×Q8—916no—
159C2×Q8—1812no—
160C2×Q8—1824no—
161C2×Q8—1812no—
162C22×C4—31no—
163C22×C4—36no—
164C22×C4—36no—
165C22×C4—36no—
166C22×C4—312no—
167C22×C4—64no—
168C22×C4—63no—
169C22×C4—912no—
170C22×C4—912no—
171C22×C4—186no—
172C22×C4—188no—
173C22×C4—182no—
174C22×C4C3⋊S411yescharacteristic
175C22×C4C22×C3⋊D314yes—
176C4⋊C4—932no—
177C4⋊C4—916no—
178C22⋊C4—924no—
179C22⋊C4—98no—
180C22×C3—36no—
181C22×C3—31no—
182C22×C3—36no—
183C22×C3—36no—
184C22×C3—312no—
185C22×C3—31no—
186C22×C3—33no—
187C22×C3—412no—
188C22×C3—49no—
189C22×C3—63no—
190C22×C3—64no—
191C22×C3A4⋊Q814yescomplemented
192C22×C3C22×Dic611yescharacteristic, complemented
193C22×C3C22×S413yes—
194A4C22×Dic613yescomplemented
195C3×C4—34no—
196C3×C4—412no—
197C3×C4C22×S414yes—
198Dic3—68no—
199Dic3—68no—
200Dic3—1224no—
201C32—41noSylow
202C23—34no—
203C23—34no—
204C23—33no—
205C23—33no—
206C23—33no—
207C23—312no—
208C23—36no—
209C23—61no—
210C23—66no—
211C23C2×#63416yescomplemented
212C23C2×C3⋊S411yescharacteristic
213C23C23×C3⋊D311yescharacteristic
214Q8—1816no—
215Q8—1816no—
216C2×C4—36no—
217C2×C4—34no—
218C2×C4—312no—
219C2×C4—34no—
220C2×C4—66no—
221C2×C4—98no—
222C2×C4—98no—
223C2×C4—1812no—
224C2×C4—1824no—
225C2×C4—1812no—
226C2×C4C2×C3⋊S416yes—
227C2×C3—31no—
228C2×C3—31no—
229C2×C3—36no—
230C2×C3—43no—
231C2×C3—418no—
232C2×C3C2×A4⋊Q816yescomplemented
233C2×C3C23×S411yescharacteristic
234C22—31no—
235C22—36no—
236C22—31no—
237C22—33no—
238C22—36no—
239C22—36no—
240C22—312no—
241C22—63no—
242C22—64no—
243C22#6425914yesdirect product factor
244C22C22×#63411yescharacteristic, complemented
245C22C22×C3⋊S413yes—
246C4—34no—
247C4—188no—
248C4—188no—
249C4C22×C3⋊S414yes—
250C3—43no—
251C3C22×A4⋊Q811yescharacteristic, complemented
252C2—31no—
253C2—31no—
254C2—36no—
255C2C2×#6425916yesdirect product factor
256C2C23×C3⋊S411yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocleC2×C3×A4C24
FrattiniC2C23×C3⋊S4
FittingC24×C3×C4D3
socleC25×C3C2×D3
prime radicalC23×C3×A4C22
hypercenterC22×C4C3⋊S4
hypocenterC3×A4C22×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#64259C2direct (N × H)
10. #64259C22direct (N × H)
21. C22×C3×A4Q8semidirect
52. C2×C3×A4C2×Q8semidirect
67. C24×C3Dic6semidirect
69. C22×A4Dic6semidirect
100. C3×A4C22×Q8semidirect
130. C23×C3C2×Dic6semidirect
133. C2×A4C2×Dic6semidirect
156. C24#634semidirect
191. C22×C3A4⋊Q8semidirect
192. C22×C3C22×Dic6semidirect
194. A4C22×Dic6semidirect
211. C23C2×#634semidirect
232. C2×C3C2×A4⋊Q8semidirect
243. C22#64259direct (N × H)
244. C22C22×#634semidirect
251. C3C22×A4⋊Q8semidirect
255. C2C2×#64259direct (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×C3⋊SL(2,ℤ/4)2
C2×#642592
C22×A4⋊Q83
C22×#9243
C22×#6344