Finite Group Databasebeta

← C3×A4×H(5,𝔽2) #11919874

Subgroups — C3×A4×H(5,𝔽2) #11919874

Order 1152.

3072 conjugacy classes of subgroups; only the 544 normal subgroups are shown.

normal subgroups only

This group has 3072 conjugacy classes of subgroups — too many to draw. Only its 544 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
1C3×A4×#41C216yesmaximal, complemented
2C2×C3×D4×A4C219yesmaximal, complemented
3C22×C3×H(5,𝔽2)C311yesmaximal, characteristic, complemented
4A4×H(5,𝔽2)C313yesmaximal, direct product factor
5C23×C3×A4C2216yescomplemented
6C32×H(5,𝔽2)—41nomaximal
7C3×Q8×A4C2212yescomplemented
8C3×D4×A4C22118yescomplemented
9C2×C3×C4×A4C2219yescomplemented
10C2×C3×H(5,𝔽2)—39no—
11C2×C3×H(5,𝔽2)—36no—
12C2×C3×H(5,𝔽2)—31no—
13C22×C3×#41C2×C316yescomplemented
14C23×C3×D4C2×C319yescomplemented
15A4×#41C2×C3118yescomplemented
16C2×D4×A4C2×C3127yescomplemented
17C22×C3×A4—46no—
18C22×C3×A4C2319yes—
19C32×#41—46no—
20C2×C32×D4—49no—
21C3×C4×A4C2316yes—
22C22×H(5,𝔽2)C3211yescharacteristic, complemented, Sylow
23C25×C3C22×C316yescomplemented
24C23×A4C22×C3118yescomplemented
25C3×H(5,𝔽2)—34no—
26C3×H(5,𝔽2)—318no—
27C3×H(5,𝔽2)—336no—
28C3×H(5,𝔽2)—312no—
29C3×H(5,𝔽2)—36no—
30C3×H(5,𝔽2)—39no—
31C3×H(5,𝔽2)—43no—
32C3×H(5,𝔽2)A411yescharacteristic, direct product factor
33C2×C3×#41—318no—
34C2×C3×#41—318no—
35C2×C3×#41—36no—
36C2×C3×#41—36no—
37C22×C3×Q8C22×C312yescomplemented
38C22×C3×D4—39no—
39C22×C3×D4—39no—
40C22×C3×D4—318no—
41C22×C3×D4—336no—
42C22×C3×D4C22×C3118yescomplemented
43C23×C3×C4C22×C319yescomplemented
44Q8×A4C22×C316yescomplemented
45D4×A4C22×C3154yescomplemented
46C2×C4×A4C22×C3127yescomplemented
47C23×C32—46no—
48C2×C3×A4—29no—
49C2×C3×A4C2411yescharacteristic
50C32×Q8—42no—
51C32×D4—418no—
52C2×C32×C4—49no—
53C2×H(5,𝔽2)—39no—
54C2×H(5,𝔽2)—36no—
55C2×H(5,𝔽2)—31no—
56C22×#41C2×C3216yescomplemented
57C23×D4C2×C3219yescomplemented
58C24×C3—318no—
59C24×C3—36no—
60C24×C3—46no—
61C24×C3—126no—
62C24×C3C23×C319yes—
63C22×A4—418no—
64C22×A4C23×C3127yes—
65C3×#41—336no—
66C3×#41—336no—
67C3×#41—312no—
68C3×#41—36no—
69C3×#41—318no—
70C3×#41—318no—
71C3×#41—418no—
72C3×#41C2×A416yescomplemented
73C2×C3×Q8—36no—
74C2×C3×Q8—32no—
75C2×C3×D4—372no—
76C2×C3×D4—318no—
77C2×C3×D4—336no—
78C2×C3×D4—318no—
79C2×C3×D4—318no—
80C2×C3×D4—336no—
81C2×C3×D4—318no—
82C2×C3×D4—336no—
83C2×C3×D4—39no—
84C2×C3×D4—427no—
85C2×C3×D4C2×A419yescomplemented
86C22×C3×C4—39no—
87C22×C3×C4—39no—
88C22×C3×C4—318no—
89C22×C3×C4C23×C316yes—
90C4×A4C23×C3118yes—
91C22×C32—49no—
92C22×C32—166no—
93C3×A4H(5,𝔽2)11yescharacteristic, direct product factor
94C32×C4—46no—
95C25C22×C3216yescomplemented
96H(5,𝔽2)—34no—
97H(5,𝔽2)—318no—
98H(5,𝔽2)—336no—
99H(5,𝔽2)—312no—
100H(5,𝔽2)—36no—
101H(5,𝔽2)—39no—
102H(5,𝔽2)C3×A411yescharacteristic, direct product factor
103C2×#41—318no—
104C2×#41—318no—
105C2×#41—36no—
106C2×#41—36no—
107C22×Q8C22×C3212yescomplemented
108C22×D4—39no—
109C22×D4—39no—
110C22×D4—318no—
111C22×D4—336no—
112C22×D4C22×C32118yescomplemented
113C23×C4C22×C3219yescomplemented
114C23×C3—29no—
115C23×C3—312no—
116C23×C3—39no—
117C23×C3—39no—
118C23×C3—318no—
119C23×C3—418no—
120C23×C3—69no—
121C23×C3—126no—
122C23×C3—126no—
123C23×C3—1218no—
124C23×C3—1218no—
125C23×C3C22×A416yescomplemented
126C23×C3C24×C311yescharacteristic
127C2×A4—227no—
128C2×A4C24×C313yes—
129C3×Q8—34no—
130C3×Q8—36no—
131C3×Q8—46no—
132C3×Q8C22×A412yescomplemented
133C3×D4—336no—
134C3×D4—336no—
135C3×D4—318no—
136C3×D4—454no—
137C3×D4C22×A4118yescomplemented
138C2×C3×C4—318no—
139C2×C3×C4—39no—
140C2×C3×C4—36no—
141C2×C3×C4—318no—
142C2×C3×C4—36no—
143C2×C3×C4—427no—
144C2×C3×C4C22×A419yescomplemented
145C2×C32—41no—
146C2×C32—89no—
147C24—318no—
148C24—36no—
149C24—46no—
150C24—126no—
151C24C23×C3219yes—
152#41—336no—
153#41—336no—
154#41—312no—
155#41—36no—
156#41—318no—
157#41—318no—
158#41C2×C3×A416yescomplemented
159C2×Q8—36no—
160C2×Q8—32no—
161C2×D4—372no—
162C2×D4—318no—
163C2×D4—336no—
164C2×D4—318no—
165C2×D4—318no—
166C2×D4—336no—
167C2×D4—318no—
168C2×D4—336no—
169C2×D4—39no—
170C2×D4C2×C3×A419yescomplemented
171C22×C4—39no—
172C22×C4—39no—
173C22×C4—318no—
174C22×C4C23×C3216yes—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3—39no—
178C22×C3—46no—
179C22×C3—427no—
180C22×C3—69no—
181C22×C3—69no—
182C22×C3—69no—
183C22×C3—69no—
184C22×C3—1218no—
185C22×C3—1212no—
186C22×C3—1618no—
187C22×C3C3×H(5,𝔽2)11yescharacteristic, complemented
188C22×C3C23×A419yes—
189A4C3×H(5,𝔽2)13yesdirect product factor
190C3×C4—36no—
191C3×C4—418no—
192C3×C4C23×A416yes—
193C32—41noSylow
194C23—29no—
195C23—312no—
196C23—39no—
197C23—39no—
198C23—318no—
199C23—69no—
200C23—126no—
201C23—126no—
202C23—1218no—
203C23—1218no—
204C23C22×C3×A416yescomplemented
205C23C24×C3211yescharacteristic
206Q8—34no—
207Q8—36no—
208Q8C22×C3×A412yescomplemented
209D4—336no—
210D4—336no—
211D4—318no—
212D4C22×C3×A4118yescomplemented
213C2×C4—318no—
214C2×C4—39no—
215C2×C4—36no—
216C2×C4—318no—
217C2×C4—36no—
218C2×C4C22×C3×A419yescomplemented
219C2×C3—29no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—43no—
223C2×C3—69no—
224C2×C3—827no—
225C2×C3C24×A411yescharacteristic
226C22—31no—
227C22—31no—
228C22—39no—
229C22—46no—
230C22—69no—
231C22—69no—
232C22—69no—
233C22—69no—
234C22—1218no—
235C22—1212no—
236C22C32×H(5,𝔽2)11yescharacteristic, complemented
237C22C23×C3×A419yes—
238C4—36no—
239C4C23×C3×A416yes—
240C3—43no—
241C3A4×H(5,𝔽2)11yescharacteristic, direct product factor
242C2—29no—
243C2—31no—
244C2—31no—
245C2—69no—
246C2C24×C3×A411yescharacteristic
subgroupG/H
centerC2×C3C24×A4
commutator, cosocleC23C24×C32
FrattiniC2C24×C3×A4
FittingC22×C3×H(5,𝔽2)C3
socleC23×C3C24×C3
hypercenterC3×H(5,𝔽2)A4
hypocenterC22C32×H(5,𝔽2)
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×A4×#41C2semidirect
2. C2×C3×D4×A4C2semidirect
3. C22×C3×H(5,𝔽2)C3semidirect
4. A4×H(5,𝔽2)C3direct (N × H)
5. C23×C3×A4C22semidirect
7. C3×Q8×A4C22semidirect
8. C3×D4×A4C22semidirect
9. C2×C3×C4×A4C22semidirect
13. C22×C3×#41C2×C3semidirect
14. C23×C3×D4C2×C3semidirect
15. A4×#41C2×C3semidirect
16. C2×D4×A4C2×C3semidirect
22. C22×H(5,𝔽2)C32semidirect
23. C25×C3C22×C3semidirect
24. C23×A4C22×C3semidirect
32. C3×H(5,𝔽2)A4direct (N × H)
37. C22×C3×Q8C22×C3semidirect
42. C22×C3×D4C22×C3semidirect
43. C23×C3×C4C22×C3semidirect
44. Q8×A4C22×C3semidirect
45. D4×A4C22×C3semidirect
46. C2×C4×A4C22×C3semidirect
56. C22×#41C2×C32semidirect
57. C23×D4C2×C32semidirect
72. C3×#41C2×A4semidirect
85. C2×C3×D4C2×A4semidirect
93. C3×A4H(5,𝔽2)direct (N × H)
95. C25C22×C32semidirect
102. H(5,𝔽2)C3×A4direct (N × H)
107. C22×Q8C22×C32semidirect
112. C22×D4C22×C32semidirect
113. C23×C4C22×C32semidirect
125. C23×C3C22×A4semidirect
132. C3×Q8C22×A4semidirect
137. C3×D4C22×A4semidirect
144. C2×C3×C4C22×A4semidirect
158. #41C2×C3×A4semidirect
170. C2×D4C2×C3×A4semidirect
187. C22×C3C3×H(5,𝔽2)semidirect
189. A4C3×H(5,𝔽2)direct (N × H)
204. C23C22×C3×A4semidirect
208. Q8C22×C3×A4semidirect
212. D4C22×C3×A4semidirect
218. C2×C4C22×C3×A4semidirect
236. C22C32×H(5,𝔽2)semidirect
241. C3A4×H(5,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×#412
C2×C3×D4×A42
C22×C3×H(5,𝔽2)3
A4×H(5,𝔽2)3
C32×H(5,𝔽2)4