Finite Group Databasebeta

← C3×A4×2−1+4 #11919876

Subgroups — C3×A4×2−1+4 #11919876

Order 1152.

2736 conjugacy classes of subgroups; only the 544 normal subgroups are shown.

normal subgroups only

This group has 2736 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×#41C2110yesmaximal, complemented
2C2×C3×Q8×A4C215yesmaximal, complemented
3C22×C3×2−1+4C311yesmaximal, characteristic, complemented
4A4×2−1+4C313yesmaximal, direct product factor
5C32×2−1+4—41nomaximal
6C3×Q8×A4C22110yes—
7C3×D4×A4C22110yes—
8C2×C3×C4×A4C22115yes—
9C2×C3×2−1+4—310no—
10C2×C3×2−1+4—35no—
11C2×C3×2−1+4—31no—
12C22×C3×#41C2×C3110yescomplemented
13C23×C3×Q8C2×C315yescomplemented
14A4×#41C2×C3130yescomplemented
15C2×Q8×A4C2×C3115yescomplemented
16C22×C3×A4C2315yes—
17C32×#41—410no—
18C2×C32×Q8—45no—
19C3×C4×A4C23110yes—
20C22×2−1+4C3211yescharacteristic, complemented, Sylow
21C3×2−1+4—320no—
22C3×2−1+4—330no—
23C3×2−1+4—320no—
24C3×2−1+4—310no—
25C3×2−1+4—35no—
26C3×2−1+4—43no—
27C3×2−1+4A411yescharacteristic, direct product factor
28C2×C3×#41—330no—
29C2×C3×#41—330no—
30C2×C3×#41—310no—
31C2×C3×#41—310no—
32C22×C3×Q8—315no—
33C22×C3×Q8—320no—
34C22×C3×Q8—35no—
35C22×C3×Q8C22×C3110yes—
36C22×C3×D4C22×C3110yes—
37C23×C3×C4C22×C3115yes—
38Q8×A4C22×C3130yes—
39D4×A4C22×C3130yes—
40C2×C4×A4C22×C3145yes—
41C2×C3×A4—25no—
42C2×C3×A4C2411yescharacteristic
43C32×Q8—410no—
44C32×D4—410no—
45C2×C32×C4—415no—
46C2×2−1+4—310no—
47C2×2−1+4—35no—
48C2×2−1+4—31no—
49C22×#41C2×C32110yescomplemented
50C23×Q8C2×C3215yescomplemented
51C24×C3C23×C315yes—
52C22×A4C23×C3115yes—
53C3×#41—360no—
54C3×#41—360no—
55C3×#41—320no—
56C3×#41—310no—
57C3×#41—330no—
58C3×#41—330no—
59C3×#41—430no—
60C3×#41C2×A4110yescomplemented
61C2×C3×Q8—330no—
62C2×C3×Q8—360no—
63C2×C3×Q8—310no—
64C2×C3×Q8—310no—
65C2×C3×Q8—320no—
66C2×C3×Q8—315no—
67C2×C3×Q8—415no—
68C2×C3×Q8C2×A415yescomplemented
69C2×C3×D4—310no—
70C2×C3×D4—310no—
71C2×C3×D4—320no—
72C22×C3×C4—315no—
73C22×C3×C4—315no—
74C22×C3×C4—330no—
75C22×C3×C4C23×C3110yes—
76C4×A4C23×C3130yes—
77C22×C32—45no—
78C3×A42−1+411yescharacteristic, direct product factor
79C32×C4—410no—
802−1+4—320no—
812−1+4—330no—
822−1+4—320no—
832−1+4—310no—
842−1+4—35no—
852−1+4C3×A411yescharacteristic, direct product factor
86C2×#41—330no—
87C2×#41—330no—
88C2×#41—310no—
89C2×#41—310no—
90C22×Q8—315no—
91C22×Q8—320no—
92C22×Q8—35no—
93C22×Q8C22×C32110yes—
94C22×D4C22×C32110yes—
95C23×C4C22×C32115yes—
96C23×C3—25no—
97C23×C3—35no—
98C23×C3—35no—
99C23×C3—65no—
100C23×C3C24×C311yescharacteristic
101C2×A4—215no—
102C2×A4C24×C313yes—
103C3×Q8—320no—
104C3×Q8—330no—
105C3×Q8—430no—
106C3×Q8C22×A4110yes—
107C3×D4—320no—
108C3×D4—320no—
109C3×D4—310no—
110C3×D4—430no—
111C3×D4C22×A4110yes—
112C2×C3×C4—330no—
113C2×C3×C4—315no—
114C2×C3×C4—310no—
115C2×C3×C4—330no—
116C2×C3×C4—310no—
117C2×C3×C4—445no—
118C2×C3×C4C22×A4115yes—
119C2×C32—41no—
120C2×C32—85no—
121C24C23×C3215yes—
122#41—360no—
123#41—360no—
124#41—320no—
125#41—310no—
126#41—330no—
127#41—330no—
128#41C2×C3×A4110yescomplemented
129C2×Q8—330no—
130C2×Q8—360no—
131C2×Q8—310no—
132C2×Q8—310no—
133C2×Q8—320no—
134C2×Q8—315no—
135C2×Q8C2×C3×A415yescomplemented
136C2×D4—310no—
137C2×D4—310no—
138C2×D4—320no—
139C22×C4—315no—
140C22×C4—315no—
141C22×C4—330no—
142C22×C4C23×C32110yes—
143C22×C3—31no—
144C22×C3—31no—
145C22×C3—35no—
146C22×C3—415no—
147C22×C3—65no—
148C22×C3—65no—
149C22×C3—65no—
150C22×C3—65no—
151C22×C3C3×2−1+411yescharacteristic, complemented
152C22×C3C23×A415yes—
153A4C3×2−1+413yesdirect product factor
154C3×C4—310no—
155C3×C4—430no—
156C3×C4C23×A4110yes—
157C32—41noSylow
158C23—25no—
159C23—35no—
160C23—35no—
161C23—65no—
162C23C24×C3211yescharacteristic
163Q8—320no—
164Q8—330no—
165Q8C22×C3×A4110yes—
166D4—320no—
167D4—320no—
168D4—310no—
169D4C22×C3×A4110yes—
170C2×C4—330no—
171C2×C4—315no—
172C2×C4—310no—
173C2×C4—330no—
174C2×C4—310no—
175C2×C4C22×C3×A4115yes—
176C2×C3—25no—
177C2×C3—31no—
178C2×C3—31no—
179C2×C3—43no—
180C2×C3—65no—
181C2×C3—815no—
182C2×C3C24×A411yescharacteristic
183C22—31no—
184C22—31no—
185C22—35no—
186C22—65no—
187C22—65no—
188C22—65no—
189C22—65no—
190C22C32×2−1+411yescharacteristic, complemented
191C22C23×C3×A415yes—
192C4—310no—
193C4C23×C3×A4110yes—
194C3—43no—
195C3A4×2−1+411yescharacteristic, direct product factor
196C2—25no—
197C2—31no—
198C2—31no—
199C2—65no—
200C2C24×C3×A411yescharacteristic
subgroupG/H
centerC2×C3C24×A4
commutator, cosocleC23C24×C32
FrattiniC2C24×C3×A4
FittingC22×C3×2−1+4C3
socleC23×C3C24×C3
hypercenterC3×2−1+4A4
hypocenterC22C32×2−1+4
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3 ◁ C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×A4×#41C2semidirect
2. C2×C3×Q8×A4C2semidirect
3. C22×C3×2−1+4C3semidirect
4. A4×2−1+4C3direct (N × H)
12. C22×C3×#41C2×C3semidirect
13. C23×C3×Q8C2×C3semidirect
14. A4×#41C2×C3semidirect
15. C2×Q8×A4C2×C3semidirect
20. C22×2−1+4C32semidirect
27. C3×2−1+4A4direct (N × H)
49. C22×#41C2×C32semidirect
50. C23×Q8C2×C32semidirect
60. C3×#41C2×A4semidirect
68. C2×C3×Q8C2×A4semidirect
78. C3×A42−1+4direct (N × H)
85. 2−1+4C3×A4direct (N × H)
128. #41C2×C3×A4semidirect
135. C2×Q8C2×C3×A4semidirect
151. C22×C3C3×2−1+4semidirect
153. A4C3×2−1+4direct (N × H)
190. C22C32×2−1+4semidirect
195. C3A4×2−1+4direct (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×Q8×A42
C22×C3×2−1+43
A4×2−1+43
C32×2−1+44