Finite Group Databasebeta

← C22×#88455 #13152158

Subgroups — C22×#88455 #13152158

Order 1728.

588 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 65 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×#88455C213yesmaximal, direct product factor
2C22×C32×C42C311yesmaximal, characteristic
3C22×C42⋊C9C313yesmaximal, complemented
4C22×#1094—41nomaximal
5#88455C2211yescharacteristic, direct product factor
6C23×C32×C4—31no—
7C2×C32×C42—33no—
8C2×C32×C42C2×C313yes—
9C2×C42⋊C9C2×C319yescomplemented
10C2×#1094—43no—
11C22×C3×C42—31no—
12C22×C3×C42C3211yescharacteristic
13C24×C32A411yescharacteristic
14C22×C32×C4—36no—
15C22×C32×C4—32no—
16C22×C32×C4—33no—
17C22×C32×C4—33no—
18C22×C22⋊C9—43no—
19C32×C42—32no—
20C32×C42—33no—
21C32×C42C22×C311yescharacteristic
22C42⋊C9C22×C313yescomplemented
23C22×3−1+2—161no—
24#1094—41no—
25C23×C3×C4—33no—
26C23×C3×C4—31no—
27C2×C3×C42—33no—
28C2×C3×C42—39no—
29C2×C3×C42—33no—
30C2×C3×C42C2×C3213yes—
31C23×C32—33no—
32C23×C32—31no—
33C23×C32C2×A413yes—
34C2×C32×C4—36no—
35C2×C32×C4—36no—
36C2×C32×C4—36no—
37C2×C32×C4—36no—
38C2×C32×C4—31no—
39C2×C32×C4—33no—
40C2×C22⋊C9—49no—
41C22×C423−1+211yescharacteristic, complemented, Sylow
42C2×3−1+2—163no—
43C24×C3—31no—
44C24×C3C3×A411yescharacteristic
45C22×C3×C4—39no—
46C22×C3×C4—39no—
47C22×C3×C4—36no—
48C22×C3×C4—318no—
49C22×C3×C4—36no—
50C22×C3×C4—32no—
51C22×C3×C4—33no—
52C22×C3×C4—33no—
53C3×C42—33no—
54C3×C42—39no—
55C3×C42—36no—
56C3×C42—32no—
57C3×C42—31no—
58C3×C42C22×C3211yescharacteristic
59C22×C32—33no—
60C22×C32—32no—
61C22×C32—33no—
62C22×C32—33no—
63C22×C32C42⋊C311yescharacteristic
64C22×C32C22×A411yescharacteristic
65C32×C4—32no—
66C32×C4—36no—
67C22×C9—163no—
68C22⋊C9—43no—
69C23×C4—31no—
70C2×C42—33no—
71C2×C42C2×3−1+213yescomplemented
723−1+2—161noSylow
73C23×C3—33no—
74C23×C3—39no—
75C23×C3—33no—
76C23×C3—31no—
77C23×C3—33no—
78C23×C3C2×C3×A413yes—
79C2×C3×C4—33no—
80C2×C3×C4—39no—
81C2×C3×C4—36no—
82C2×C3×C4—36no—
83C2×C3×C4—318no—
84C2×C3×C4—318no—
85C2×C3×C4—318no—
86C2×C3×C4—318no—
87C2×C3×C4—36no—
88C2×C3×C4—36no—
89C2×C3×C4—31no—
90C2×C3×C4—33no—
91C2×C32—31no—
92C2×C32—33no—
93C2×C32C2×C42⋊C313yes—
94C2×C9—169no—
95C24#109411yescharacteristic
96C22×C4—36no—
97C22×C4—32no—
98C22×C4—33no—
99C22×C4—33no—
100C42—33no—
101C42—32no—
102C42C22×3−1+211yescharacteristic, complemented
103C22×C3—33no—
104C22×C3—39no—
105C22×C3—39no—
106C22×C3—36no—
107C22×C3—32no—
108C22×C3—39no—
109C22×C3—33no—
110C22×C3—33no—
111C22×C3—31no—
112C22×C3—31no—
113C22×C3C3×C42⋊C311yescharacteristic
114C22×C3C22×C3×A411yescharacteristic
115C3×C4—36no—
116C3×C4—318no—
117C3×C4—32no—
118C3×C4—36no—
119C32C22×C42⋊C311yescharacteristic
120C9—163no—
121C23—33no—
122C23—31no—
123C23C2×#109413yes—
124C2×C4—36no—
125C2×C4—36no—
126C2×C4—36no—
127C2×C4—36no—
128C2×C4—31no—
129C2×C4—33no—
130C2×C3—33no—
131C2×C3—39no—
132C2×C3—31no—
133C2×C3—33no—
134C2×C3—33no—
135C2×C3C2×C3×C42⋊C313yes—
136C22—33no—
137C22—32no—
138C22—33no—
139C22—33no—
140C22#8845511yescharacteristic, direct product factor
141C22C22×#109411yescharacteristic
142C4—32no—
143C4—36no—
144C3—31no—
145C3C22×C3×C42⋊C311yescharacteristic
146C2—31no—
147C2—33no—
148C2C2×#8845513yesdirect product factor
subgroupG/H
centerC22×C3C3×C42⋊C3
commutator, cosocleC3×C42C22×C32
FrattiniC22×C3C22×C3×A4
FittingC22×C32×C42C3
socleC24×C3C3×A4
prime radicalC24×C32A4
hypercenterC22×C32C42⋊C3
hypocenterC42C22×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C22×C3 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C2×#88455C2direct (N × H)
3. C22×C42⋊C9C3semidirect
5. #88455C22direct (N × H)
9. C2×C42⋊C9C2×C3semidirect
22. C42⋊C9C22×C3semidirect
41. C22×C423−1+2semidirect
71. C2×C42C2×3−1+2semidirect
102. C42C22×3−1+2semidirect
140. C22#88455direct (N × H)
148. C2C2×#88455direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#884552
C22×C32×C423
C22×C42⋊C93
C22×#10944