Finite Group Databasebeta

← C4×#88455 #13141218

Subgroups — C4×#88455 #13141218

Order 1728.

312 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 39 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×#88455C211yesmaximal, characteristic
2C32×C43C311yesmaximal, characteristic
3C4×C42⋊C9C313yesmaximal, complemented
4C4×#1094—41nomaximal
5#88455C411yescharacteristic, direct product factor
6C2×C32×C42—31no—
7C2×C32×C42—31no—
8C2×C32×C42C2×C311yescharacteristic
9C2×C42⋊C9C2×C313yes—
10C2×#1094—41no—
11C3×C43—31no—
12C3×C43C3211yescharacteristic
13C22×C32×C4—31no—
14C22×C32×C4—31no—
15C22×C32×C4A411yescharacteristic
16C32×C42—34no—
17C32×C42—32no—
18C32×C42—32no—
19C32×C42—31no—
20C32×C42C3×C411yescharacteristic
21C4×C22⋊C9—43no—
22C42⋊C9C3×C413yescomplemented
23#1094—41no—
24C4×3−1+2—161no—
25C2×C3×C42—31no—
26C2×C3×C42—33no—
27C2×C3×C42—33no—
28C2×C3×C42—31no—
29C2×C3×C42—31no—
30C2×C3×C42C2×C3211yescharacteristic
31C23×C32C2×A411yescharacteristic
32C2×C32×C4—34no—
33C2×C32×C4—32no—
34C2×C32×C4—32no—
35C2×C32×C4—32no—
36C2×C32×C4—31no—
37C2×C32×C4—31no—
38C2×C32×C4—31no—
39C2×C32×C4—31no—
40C2×C22⋊C9—43no—
41C433−1+211yescharacteristic, complemented, Sylow
42C2×3−1+2—161no—
43C22×C3×C4—33no—
44C22×C3×C4—33no—
45C22×C3×C4—31no—
46C22×C3×C4—31no—
47C22×C3×C4—31no—
48C22×C3×C4C3×A411yescharacteristic
49C3×C42—31no—
50C3×C42—33no—
51C3×C42—34no—
52C3×C42—312no—
53C3×C42—36no—
54C3×C42—36no—
55C3×C42—32no—
56C3×C42—32no—
57C3×C42—31no—
58C3×C42C32×C411yescharacteristic
59C22×C32—31no—
60C22×C32—31no—
61C22×C32C4×A411yescharacteristic
62C32×C4—34no—
63C32×C4—32no—
64C32×C4—32no—
65C32×C4—31no—
66C32×C4C42⋊C311yescharacteristic
67C22⋊C9—43no—
68C4×C9—163no—
69C2×C42—31no—
70C2×C42—31no—
71C2×C42C2×3−1+211yescharacteristic
723−1+2—161noSylow
73C23×C3—31no—
74C23×C3C2×C3×A411yescharacteristic
75C2×C3×C4—32no—
76C2×C3×C4—34no—
77C2×C3×C4—36no—
78C2×C3×C4—312no—
79C2×C3×C4—36no—
80C2×C3×C4—36no—
81C2×C3×C4—32no—
82C2×C3×C4—32no—
83C2×C3×C4—33no—
84C2×C3×C4—33no—
85C2×C3×C4—33no—
86C2×C3×C4—33no—
87C2×C3×C4—31no—
88C2×C3×C4—31no—
89C2×C3×C4—31no—
90C2×C3×C4—31no—
91C2×C32—31no—
92C2×C32—31no—
93C2×C32C2×C42⋊C311yescharacteristic
94C2×C9—163no—
95C22×C4—31no—
96C22×C4—31no—
97C22×C4#109411yescharacteristic
98C42—31no—
99C42—34no—
100C42—32no—
101C42—32no—
102C42C4×3−1+211yescharacteristic, complemented
103C22×C3—31no—
104C22×C3—33no—
105C22×C3—33no—
106C22×C3—31no—
107C22×C3—31no—
108C22×C3C3×C4×A411yescharacteristic
109C3×C4—36no—
110C3×C4—36no—
111C3×C4—312no—
112C3×C4—32no—
113C3×C4—32no—
114C3×C4—34no—
115C3×C4—33no—
116C3×C4—31no—
117C3×C4—31no—
118C3×C4C3×C42⋊C311yescharacteristic
119C32C4×C42⋊C311yescharacteristic
120C9—163no—
121C23C2×#109411yescharacteristic
122C2×C4—32no—
123C2×C4—34no—
124C2×C4—32no—
125C2×C4—32no—
126C2×C4—31no—
127C2×C4—31no—
128C2×C4—31no—
129C2×C4—31no—
130C2×C3—33no—
131C2×C3—33no—
132C2×C3—31no—
133C2×C3—31no—
134C2×C3—31no—
135C2×C3C2×C3×C42⋊C311yescharacteristic
136C22—31no—
137C22—31no—
138C22C4×#109411yescharacteristic
139C4—32no—
140C4—32no—
141C4—34no—
142C4—31no—
143C4#8845511yescharacteristic, direct product factor
144C3—31no—
145C3C3×C4×C42⋊C311yescharacteristic
146C2—31no—
147C2—31no—
148C2C2×#8845511yescharacteristic
subgroupG/H
centerC3×C4C3×C42⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×C43C3
prime radicalC23×C32C2×A4
cosocleC2×C3×C42C2×C32
hypercenterC32×C4C42⋊C3
hypocenterC42C4×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3×C4 ◁ C32×C4

Complements

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

NH ≅ G/Ntype
3. C4×C42⋊C9C3semidirect
5. #88455C4direct (N × H)
22. C42⋊C9C3×C4semidirect
41. C433−1+2semidirect
102. C42C4×3−1+2semidirect
143. C4#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
C32×C433
C4×C42⋊C93
C4×#10944