Finite Group Databasebeta

← C4×#88447 #13140054

Subgroups — C4×#88447 #13140054

Order 1728.

1011 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 291 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×C4×#1084C211yesmaximal, characteristic
2C2×C4×(C3⋊Dic3)⋊C3C212yesmaximal
3C2×#88447C214yesmaximal
4C4×Dic32—32nomaximal
5C22×C4×H(3,𝔽3)C2211yescharacteristic
6C22×#1084C2212yes—
7C22×(C3⋊Dic3)⋊C3C2214yes—
8C4×#1084C414yescomplemented
9C4×(C3⋊Dic3)⋊C3C418yescomplemented
10#88447C4116yesdirect product factor
11C2×C4×C3⋊Dic3D312yes—
12C2×C3×C4×Dic3—32no—
13C2×C3×C4×Dic3—32no—
14C2×C3×C4×Dic3—61no—
15C2×Dic32—38no—
16C23×H(3,𝔽3)C2311yescharacteristic
17C2×C4×H(3,𝔽3)C2×C412yes—
18C2×C4×H(3,𝔽3)C2×C414yes—
19C2×#1084C2×C418yes—
20C2×#1084C2×C414yes—
21C2×(C3⋊Dic3)⋊C3C2×C4116yes—
22C2×(C3⋊Dic3)⋊C3C2×C418yes—
23C42×Dic3—91no—
24C42×Dic3—92no—
25C22×C32×C4—21no—
26C22×C32×C4C2×D312yes—
27C22×C3⋊Dic3C2×D314yes—
28C22×C3×Dic3—34no—
29C22×C3×Dic3—34no—
30C22×C3×Dic3—62no—
31C4×C3⋊Dic3Dic318yescomplemented
32C3×C4×Dic3—38no—
33C3×C4×Dic3—38no—
34C3×C4×Dic3—64no—
35Dic32—332no—
36C22×H(3,𝔽3)C22×C414yes—
37C22×H(3,𝔽3)C22×C411yescharacteristic
38C22×H(3,𝔽3)C22×C412yes—
39C4×H(3,𝔽3)C4214yescomplemented
40#1084C4218yescomplemented
41(C3⋊Dic3)⋊C3C42116yescomplemented
42C2×C3×C42—92no—
43C2×C3×C42—91no—
44C2×C4×Dic3—32no—
45C2×C4×Dic3—32no—
46C2×C4×Dic3—61no—
47C2×C4×Dic3—92no—
48C2×C4×Dic3—94no—
49C2×C4×Dic3—98no—
50C23×C32—21no—
51C23×C32C22×D312yes—
52C2×C32×C4—24no—
53C2×C32×C4—22no—
54C2×C32×C4C4×D314yes—
55C2×C32×C4C4×D314yes—
56C2×C32×C4C2×Dic314yes—
57C2×C3⋊Dic3C4×D3116yes—
58C2×C3⋊Dic3C2×Dic318yes—
59C2×C3×Dic3—316no—
60C2×C3×Dic3—38no—
61C2×C3×Dic3—316no—
62C2×C3×Dic3—38no—
63C2×C3×Dic3—64no—
64C2×C3×Dic3—68no—
65C43—271noSylow
66C2×H(3,𝔽3)C2×C4212yes—
67C2×H(3,𝔽3)C2×C4214yes—
68C2×H(3,𝔽3)C2×C4211yescharacteristic
69C22×C3×C4—32no—
70C22×C3×C4—61no—
71C22×C3×C4—94no—
72C22×C3×C4—92no—
73C22×C3×C4D3211yescharacteristic
74C22×Dic3—34no—
75C22×Dic3—34no—
76C22×Dic3—62no—
77C22×Dic3—94no—
78C3×C42—98no—
79C3×C42—94no—
80C4×Dic3—38no—
81C4×Dic3—38no—
82C4×Dic3—64no—
83C4×Dic3—98no—
84C4×Dic3—916no—
85C4×Dic3—932no—
86C22×C32—24no—
87C22×C32—22no—
88C22×C32—21no—
89C22×C32C2×C4×D318yes—
90C22×C32C2×C4×D312yes—
91C22×C32C2×C4×D312yes—
92C22×C32C22×Dic312yes—
93C32×C4—24no—
94C32×C4C4×Dic318yescomplemented
95C3⋊Dic3C4×Dic3116yescomplemented
96C3×Dic3—316no—
97C3×Dic3—316no—
98C3×Dic3—68no—
99C2×C42—91no—
100C2×C42—92no—
101C2×C42—274no—
102H(3,𝔽3)C4311yescharacteristic, complemented, Sylow
103C23×C3—32no—
104C23×C3—61no—
105C23×C3C2×D3211yescharacteristic
106C2×C3×C4—34no—
107C2×C3×C4—34no—
108C2×C3×C4—34no—
109C2×C3×C4—64no—
110C2×C3×C4—62no—
111C2×C3×C4—916no—
112C2×C3×C4—98no—
113C2×C3×C4—98no—
114C2×C3×C4—94no—
115C2×C3×C4D3×Dic314yes—
116C2×C3×C4#62412yes—
117C2×Dic3—38no—
118C2×Dic3—316no—
119C2×Dic3—316no—
120C2×Dic3—38no—
121C2×Dic3—64no—
122C2×Dic3—68no—
123C2×Dic3—916no—
124C2×Dic3—98no—
125C2×C32—24no—
126C2×C32—22no—
127C2×C32—21no—
128C2×C32C42×D314yes—
129C2×C32C42×D314yes—
130C2×C32C2×C4×Dic314yes—
131C2×C32C2×C4×Dic312yes—
132C22×C4—92no—
133C22×C4—94no—
134C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
135C42—98no—
136C42—94no—
137C42—2716no—
138C22×C3—38no—
139C22×C3—32no—
140C22×C3—32no—
141C22×C3—32no—
142C22×C3—64no—
143C22×C3—62no—
144C22×C3—61no—
145C22×C3C4×D3214yes—
146C22×C3C2×D3×Dic312yes—
147C22×C3C2×#62411yescharacteristic
148C3×C4—38no—
149C3×C4—64no—
150C3×C4—916no—
151C3×C4—98no—
152C3×C4Dic3214yes—
153Dic3—316no—
154Dic3—316no—
155Dic3—68no—
156Dic3—916no—
157C32—21no—
158C32C42×Dic312yescomplemented
159C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
160C2×C4—98no—
161C2×C4—916no—
162C2×C4—98no—
163C2×C4—94no—
164C2×C4((C3⋊D3)⋊C3)⋊C414yes—
165C2×C4#619912yes—
166C2×C3—32no—
167C2×C3—34no—
168C2×C3—34no—
169C2×C3—34no—
170C2×C3—61no—
171C2×C3—64no—
172C2×C3—62no—
173C2×C3C4×D3×Dic314yes—
174C2×C3C4×#62412yes—
175C2×C3C2×Dic3211yescharacteristic
176C22C4×((C3⋊D3)⋊C3)⋊C214yes—
177C22C2×((C3⋊D3)⋊C3)⋊C412yes—
178C22C2×#619911yescharacteristic
179C4—916no—
180C4—98no—
181C4#8844714yesdirect product factor
182C3—32no—
183C3—61no—
184C3C4×Dic3211yescharacteristic
185C2C4×((C3⋊D3)⋊C3)⋊C414yes—
186C2C4×#619912yes—
187C2C2×#8844711yescharacteristic
subgroupG/H
center, hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
commutator, hypocenterH(3,𝔽3)C43
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C4×#1084C4semidirect
9. C4×(C3⋊Dic3)⋊C3C4semidirect
10. #88447C4direct (N × H)
31. C4×C3⋊Dic3Dic3semidirect
39. C4×H(3,𝔽3)C42semidirect
40. #1084C42semidirect
41. (C3⋊Dic3)⋊C3C42semidirect
94. C32×C4C4×Dic3semidirect
95. C3⋊Dic3C4×Dic3semidirect
102. H(3,𝔽3)C43semidirect
158. C32C42×Dic3semidirect
181. C4#88447direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×C4×(C3⋊Dic3)⋊C32
C2×#884472
C4×Dic323