Finite Group Databasebeta

← C4⋊(C3⋊(#126⋊C3)) #11921122

Subgroups — C4⋊(C3⋊(#126⋊C3)) #11921122

Order 1152.

261 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 33 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⋊(#126⋊C3)C212yesmaximal, complemented
2C3×C4×C42⋊C3C211yesmaximal, characteristic, complemented
3#73314C311yesmaximal, characteristic, complemented
4C4⋊(#126⋊C3)—31nomaximal
5A4×D12—41nomaximal
6C2×C3×C42⋊C3C2211yescharacteristic
7C3⋊(#126⋊C3)—22no—
8C2×#876C2×C312yescomplemented
9C2×#126⋊C3—32no—
10C3×C43C2×C311yescharacteristic, complemented
11#4940—32no—
12#4939—31no—
13#4928—31no—
14C4×C42⋊C3—21no—
15C4×C42⋊C3D311yescharacteristic, complemented
16C2×D3×A4—42no—
17C3×C4×A4—41no—
18C3×C42⋊C3D411yescharacteristic, complemented
19#2860—31noSylow
20C22×D12A411yescharacteristic
21D4×A4—121no—
22C2×C3×C42—31no—
23C2×C3×C42—31no—
24C2×C3×C42C22×C311yescharacteristic
25C2×(C2×D3)⋊C4—32no—
26C2×(C2×D3)⋊C4—32no—
27C2×C4⋊Dic3—31no—
28C2×Dic3⋊C4—32no—
29#876—22no—
30#876—62no—
31#126⋊C3—62no—
32C2×C42⋊C3—21no—
33C2×C42⋊C3C2×D311yescharacteristic
34C3⋊#95—32no—
35C2×C3×A4—41no—
36D3×A4—82no—
37C3×D12—161no—
38C2×#126—32no—
39#390—91no—
40#389—91no—
41#388—92no—
42C43C3×D311yescharacteristic, complemented
43C23×D3C2×A412yes—
44C22×A4—122no—
45C22×C3×C4—31no—
46C22×C3×C4—31no—
47C22×C3×C4C2×A411yescharacteristic
48C22×Dic3—32no—
49C2×D12—61no—
50C2×D12—61no—
51C4×A4—41no—
52C4×A4—81no—
53C3×C42—31no—
54C3×C42—61no—
55C3×C42—61no—
56C3×C42—62no—
57C3×C42C3×D411yescharacteristic, complemented
58(C2×D3)⋊C4—62no—
59(C2×D3)⋊C4—62no—
60(C2×D3)⋊C4—64no—
61C4⋊Dic3—61no—
62C4⋊Dic3—61no—
63Dic3⋊C4—62no—
64Dic3⋊C4—62no—
65C42⋊C3—21no—
66C42⋊C3D1211yescharacteristic, complemented
67C2×C3×D3—162no—
68C3×A4—41no—
69C32×C4—161no—
70C22×D4—31no—
71#126—62no—
72#126—182no—
73C2×C4⋊C4—92no—
74C2×C4⋊C4—91no—
75C2×C22⋊C4—92no—
76C2×C22⋊C4—92no—
77C2×C42—31no—
78C2×C42—31no—
79C2×C42C2×C3×D311yescharacteristic
80#95—92no—
81C23×C3C22×A411yescharacteristic
82C22×D3—22no—
83C22×D3—62no—
84C22×D3—62no—
85C2×A4—41no—
86C2×A4—81no—
87C2×A4—242no—
88C3×D4—481no—
89C2×C3×C4—31no—
90C2×C3×C4—31no—
91C2×C3×C4—32no—
92C2×C3×C4—31no—
93C2×C3×C4—31no—
94C2×C3×C4—61no—
95C2×C3×C4—61no—
96C2×C3×C4—61no—
97C2×C3×C4—61no—
98C2×Dic3—62no—
99C2×Dic3—62no—
100C2×Dic3—62no—
101D12—41no—
102D12—121no—
103C2×C32—161no—
104C3×D3—322no—
105C24—32no—
106C2×D4—181no—
107C2×D4—181no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—92no—
111C22×C4D3×A411yescharacteristic
112C4⋊C4—182no—
113C4⋊C4—182no—
114C4⋊C4—181no—
115C4⋊C4—181no—
116C22⋊C4—182no—
117C22⋊C4—182no—
118C22⋊C4—184no—
119C42—31no—
120C42—61no—
121C42—61no—
122C42—62no—
123C42C3×D1211yescharacteristic, complemented
124C22×C3—31no—
125C22×C3—31no—
126C22×C3—482no—
127C22×C3D4×A411yescharacteristic
128C2×D3—42no—
129C2×D3—122no—
130C2×D3—122no—
131A4—41no—
132A4—81no—
133C3×C4—31no—
134C3×C4—61no—
135C3×C4—61no—
136C3×C4—62no—
137C3×C4—161no—
138C3×C4—321no—
139C3×C4#126⋊C311yescharacteristic, complemented
140Dic3—122no—
141C32—161noSylow
142C23—62no—
143C23—182no—
144C23—182no—
145C23C2×D3×A411yescharacteristic
146D4—121no—
147D4—361no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—32no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—182no—
158C2×C4—182no—
159C2×C4—182no—
160C2×C3—31no—
161C2×C3—31no—
162C2×C3—161no—
163C2×C3—321no—
164C2×C3—962no—
165C2×C3C2×#126⋊C311yescharacteristic
166D3—82no—
167C22—31no—
168C22—31no—
169C22—122no—
170C22—362no—
171C22—362no—
172C22A4×D1211yescharacteristic
173C4—31no—
174C4—61no—
175C4—61no—
176C4—62no—
177C4—362no—
178C4C3⋊(#126⋊C3)11yescharacteristic, complemented
179C3—161no—
180C3—321no—
181C3C4⋊(#126⋊C3)11yescharacteristic, complemented
182C2—31no—
183C2—31no—
184C2—242no—
185C2C2×C3⋊(#126⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#126⋊C3)
commutator, cosocleC2×C3×C42C22×C3
FrattiniC23C2×D3×A4
FittingC3×C43C2×C3
socleC23×C3C22×A4
hypercenterC4C3⋊(#126⋊C3)
hypocenterC3×C42C3×D4
LCSG ▷ C2×C3×C42 ▷ C3×C42
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(#126⋊C3)C2semidirect
2. C3×C4×C42⋊C3C2semidirect
3. #73314C3semidirect
8. C2×#876C2×C3semidirect
10. C3×C43C2×C3semidirect
15. C4×C42⋊C3D3semidirect
18. C3×C42⋊C3D4semidirect
42. C43C3×D3semidirect
57. C3×C42C3×D4semidirect
66. C42⋊C3D12semidirect
123. C42C3×D12semidirect
139. C3×C4#126⋊C3semidirect
178. C4C3⋊(#126⋊C3)semidirect
181. C3C4⋊(#126⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#126⋊C3)2
C3×C4×C42⋊C32
#733143
C4⋊(#126⋊C3)3
A4×D124