Finite Group Databasebeta

← #157×C22⋊A4 #13181829

Subgroups — #157×C22⋊A4 #13181829

Order 1728.

848 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 32 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
1C3⋊D3×C22⋊A4C211yesmaximal, characteristic
2C24×#157C311yesmaximal, characteristic, complemented
3C32×C22⋊A4C411yescharacteristic, complemented
4A4×#157—45nomaximal
5C24×C3⋊D3C2×C311yescharacteristic
6C23×#157—35no—
7C23×#157—35no—
8D3×C22⋊A4—62no—
9A4×C3⋊D3—45no—
10C4×C22⋊A4—91nomaximal
11C24×C32C3×C411yescharacteristic, complemented
12C23×C3⋊D3—35no—
13C23×C3⋊D3—35no—
14C3×C22⋊A4—22no—
15C3×C22⋊A4—42no—
16C22×#157—330no—
17C22×#157—310no—
18C22×#157—35no—
19C22×#157A415yescomplemented
20C32×A4—45no—
21C3×#157—161no—
22C24×D3—62no—
23C2×C22⋊A4—91no—
24C23×C32—35no—
25C22×C3⋊D3—310no—
26C22×C3⋊D3—330no—
27C22×C3⋊D3—35no—
28C22×C3⋊D3C2×A415yes—
29C2×#157—330no—
30C2×#157—35no—
31C2×#157—35no—
32D3×A4—2410no—
33C24×C4—91noSylow
34C3×C3⋊D3—161no—
35C24×C3—22no—
36C23×D3—1810no—
37C23×D3—1810no—
38C22⋊A4—42no—
39C22⋊A4#15711yescharacteristic, direct product factor
40C4×A4—365no—
41C22×C32—310no—
42C22×C32C4×A415yescomplemented
43C2×C3⋊D3—330no—
44C2×C3⋊D3—35no—
45C2×C3⋊D3—35no—
46C3×A4—810no—
47C3×A4—1610no—
48#157—35no—
49#157C22⋊A411yescharacteristic, direct product factor
50C25—91no—
51C23×C4—275no—
52C23×C4—275no—
53C33—161noSylow
54C23×C3—610no—
55C22×D3—610no—
56C22×D3—1810no—
57C22×D3—1820no—
58C22×D3—1860no—
59C2×A4—365no—
60C2×C32—35no—
61C3⋊D3—35no—
62C3⋊D3C2×C22⋊A411yescharacteristic
63C3×D3—962no—
64C24—275no—
65C24—275no—
66C24C3×#15711yescharacteristic, complemented
67C22×C4—95no—
68C22×C4—275no—
69C22×C4—2710no—
70C22×C4—2730no—
71C22×C3—210no—
72C22×C3—620no—
73C2×D3—1810no—
74C2×D3—1860no—
75C2×D3—1810no—
76A4—45no—
77A4—1610no—
78C3×C4—1441no—
79C32—322no—
80C32—642no—
81C32C4×C22⋊A411yescharacteristic, complemented
82C23—35no—
83C23—95no—
84C23—2710no—
85C23—2730no—
86C23—275no—
87C2×C4—275no—
88C2×C4—2730no—
89C2×C4—275no—
90C2×C3—610no—
91C2×C3—1441no—
92D3—62no—
93D3—1810no—
94C22—310no—
95C22—275no—
96C22—275no—
97C22—2730no—
98C22A4×#15715yescomplemented
99C4—91no—
100C4—275no—
101C3—22no—
102C3—161no—
103C3—642no—
104C2—35no—
105C2—91no—
106C2—275no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC24×C32C3×C4
prime radicalC3⋊D3×C22⋊A4C2
cosocleC24×C3⋊D3C2×C3
LCSG ▷ C24×C32
UCSC1

Complements

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

NH ≅ G/Ntype
2. C24×#157C3semidirect
3. C32×C22⋊A4C4semidirect
11. C24×C32C3×C4semidirect
19. C22×#157A4semidirect
39. C22⋊A4#157direct (N × H)
42. C22×C32C4×A4semidirect
49. #157C22⋊A4direct (N × H)
66. C24C3×#157semidirect
81. C32C4×C22⋊A4semidirect
98. C22A4×#157semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C22⋊A42
C24×#1573
A4×#1574
C4×C22⋊A49