Finite Group Databasebeta

← C22×#88943 #13181170

Subgroups — C22×#88943 #13181170

Order 1728.

936 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 159 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×#6296C212yesmaximal
2C22×C3×#622C211yesmaximal, characteristic
3C2×#88943C2112yesmaximal, direct product factor
4C22×#3776—31nomaximal
5C22×C3×C3⋊Dic3—22no—
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×#6296—26no—
8C2×#6296C22112yes—
9C2×C3×#622C2216yes—
10#88943C22116yesdirect product factor
11C22×#627—32no—
12C22×#627—62no—
13C22×#622D311yescharacteristic
14C2×#3776—312no—
15C23×C33D411yescharacteristic
16C2×C3×C3⋊Dic3—212no—
17C2×C3×C3⋊Dic3D413yes—
18C2×C3×C3⋊Dic3C2313yes—
19#6296—212no—
20#6296C2318yes—
21C3×#622C2314yes—
22C22×Dic12—91nomaximal
23C22×C3⋊Dic3—22no—
24C22×C3⋊Dic3C2×D311yescharacteristic
25C22×C3×Dic3—62no—
26C22×C3×Dic3—62no—
27C22×C3×Dic3—62no—
28C22×C3×Dic3—122no—
29C2×#627—312no—
30C2×#627—612no—
31C2×#627—612no—
32C2×#627—66no—
33C2×#622C2×D316yes—
34#3776—316no—
35C22×C33Q1614yescomplemented
36C22×C33C2×D413yes—
37C3×C3⋊Dic3—28no—
38C3×C3⋊Dic3C2×D413yes—
39C3×C3⋊Dic3C2411yescharacteristic
40C22×Dic6—92no—
41C22×Dic6—182no—
42C22×C3×C8—91no—
43C2×Dic12—912no—
44C23×C32—22no—
45C23×C32—42no—
46C23×C32D1211yescharacteristic
47C2×C3⋊Dic3—212no—
48C2×C3⋊Dic3D1213yes—
49C2×C3⋊Dic3C22×D313yes—
50C2×C3×Dic3—66no—
51C2×C3×Dic3—66no—
52C2×C3×Dic3—612no—
53C2×C3×Dic3—612no—
54C2×C3×Dic3—1212no—
55#627—38no—
56#627—68no—
57#627—624no—
58#627—612no—
59#622C22×D314yes—
60C22×Q16—271noSylow
61C2×C33C2×Q1616yescomplemented
62C2×C33C22×D411yescharacteristic
63C22×C3×C4—91no—
64C22×C3×C4—182no—
65C22×Dic3—62no—
66C22×Dic3—62no—
67C22×Dic3—62no—
68C22×Dic3—122no—
69C2×Dic6—912no—
70C2×Dic6—1812no—
71C2×Dic6—186no—
72C2×Dic6—1812no—
73C2×C3×C8—96no—
74Dic12—916no—
75C22×C32—26no—
76C22×C32—28no—
77C22×C32—48no—
78C22×C32—46no—
79C22×C32Dic1214yescomplemented
80C22×C32C2×D1213yes—
81C3⋊Dic3—28no—
82C3⋊Dic3C2×D1213yes—
83C3⋊Dic3C23×D311yescharacteristic
84C3×Dic3—66no—
85C3×Dic3—62no—
86C3×Dic3—68no—
87C3×Dic3—68no—
88C3×Dic3—128no—
89C22×Q8—272no—
90C2×Q16—2712no—
91C22×C8—91no—
92C33C22×Q1611yescharacteristic, complemented, Sylow
93C23×C3—22no—
94C23×C3—42no—
95C23×C3GO+(4,𝔽2)11yescharacteristic
96C2×C3×C4—93no—
97C2×C3×C4—93no—
98C2×C3×C4—1812no—
99C2×Dic3—66no—
100C2×Dic3—66no—
101C2×Dic3—612no—
102C2×Dic3—612no—
103C2×Dic3—1212no—
104Dic6—98no—
105Dic6—188no—
106Dic6—1812no—
107Dic6—1824no—
108C3×C8—94no—
109C2×C32—22no—
110C2×C32—212no—
111C2×C32—412no—
112C2×C32—42no—
113C2×C32C2×Dic1216yescomplemented
114C2×C32C22×D1211yescharacteristic
115C2×Q8—2712no—
116C2×Q8—546no—
117C22×C4—91no—
118C22×C4—182no—
119Q16—2716no—
120C2×C8—96no—
121C22×C3—26no—
122C22×C3—28no—
123C22×C3—46no—
124C22×C3—48no—
125C22×C3#377614yescomplemented
126C22×C3C2×GO+(4,𝔽2)13yes—
127C3×C4—91no—
128C3×C4—93no—
129C3×C4—188no—
130Dic3—66no—
131Dic3—62no—
132Dic3—68no—
133Dic3—68no—
134Dic3—128no—
135C32—22no—
136C32—42no—
137C32C22×Dic1211yescharacteristic, complemented
138C23#157⋊D311yescharacteristic
139Q8—278no—
140Q8—5412no—
141C2×C4—93no—
142C2×C4—93no—
143C2×C4—1812no—
144C8—94no—
145C2×C3—212no—
146C2×C3—22no—
147C2×C3—412no—
148C2×C3—42no—
149C2×C3C2×#377616yescomplemented
150C2×C3C22×GO+(4,𝔽2)11yescharacteristic
151C22#8894314yesdirect product factor
152C22C2×#157⋊D313yes—
153C4—91no—
154C4—93no—
155C4—188no—
156C3—22no—
157C3—42no—
158C3C22×#377611yescharacteristic, complemented
159C2C2×#8894316yesdirect product factor
160C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
Fitting, socle, prime radicalC23×C33D4
hypocenterC33C22×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×#88943C2direct (N × H)
10. #88943C22direct (N × H)
35. C22×C33Q16semidirect
61. C2×C33C2×Q16semidirect
79. C22×C32Dic12semidirect
92. C33C22×Q16semidirect
113. C2×C32C2×Dic12semidirect
125. C22×C3#3776semidirect
137. C32C22×Dic12semidirect
149. C2×C3C2×#3776semidirect
151. C22#88943direct (N × H)
158. C3C22×#3776semidirect
159. C2C2×#88943direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62962
C22×C3×#6222
C2×#889432
C22×#37763
C22×Dic129