Finite Group Databasebeta

← #13147313

Subgroups — #13147313

Order 1728.

405 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 67 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×#88627C212yesmaximal
2C42×#272C211yesmaximal, characteristic
3#11720785C214yesmaximal, complemented
4C2×C4×#272C412yes—
5C2×C4×#272C2211yescharacteristic
6C2×#6188C2214yes—
7#88627—24no—
8C42×H(3,𝔽3)C411yescharacteristic
9C4×#1084C411yescharacteristic
10C4×#1084C2212yes—
11C3×C42×D3—62no—
12C22×#272C2×C411yescharacteristic
13C2×C4×H(3,𝔽3)C2×C412yes—
14C2×C4×H(3,𝔽3)C2×C411yescharacteristic
15C2×#1084C2×C412yes—
16C2×#1084C2311yescharacteristic
17C4×#272—24no—
18C4×#272D412yes—
19C4×#272Q812yes—
20#6188—24no—
21C3×#423—91nomaximal
22C2×C3×C4×D3—64no—
23C2×C3×C4×D3—62no—
24C32×C42—22no—
25C3×C4×Dic3—62no—
26C3×C4×Dic3—64no—
27C22×H(3,𝔽3)C22×C411yescharacteristic
28C2×#272—22no—
29C2×#272C4⋊C412yes—
30C4×H(3,𝔽3)—22no—
31C4×H(3,𝔽3)C4⋊C412yes—
32#1084M1614yescomplemented
33#1084C2×D411yescharacteristic
34#1084C2×Q811yescharacteristic
35C2×C3×M16—92no—
36C2×C3×C42—91no—
37C42×D3—62no—
38C3×C4⋊C8—94no—
39C22×C3×D3—62no—
40C2×C32×C4—24no—
41C2×C32×C4—22no—
42C2×C3×Dic3—64no—
43C2×C3×Dic3—62no—
44C3×C4×D3—64no—
45C3×C4×D3—64no—
46C3×C4×D3—616no—
47#423—91noSylow
48C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
49C2×H(3,𝔽3)C2×M1612yes—
50#272—22no—
51C22×C3×C4—91no—
52C22×C3×C4—92no—
53C2×C4×D3—62no—
54C2×C4×D3—64no—
55C3×M16—184no—
56C2×C3×C8—94no—
57C3×C42—62no—
58C3×C42—92no—
59C3×C42—91no—
60C3×C42#15711yescharacteristic
61C4×Dic3—64no—
62C4×Dic3—62no—
63C22×C32—22no—
64C2×C3×D3—64no—
65C2×C3×D3—68no—
66C32×C4—28no—
67C32×C4—24no—
68C3×Dic3—68no—
69C3×Dic3—62no—
70C3×Dic3—62no—
71C2×M16—92no—
72C2×C42—91no—
73C4⋊C8—94no—
74H(3,𝔽3)#42311yescharacteristic, complemented, Sylow
75C23×C3—91no—
76C22×D3—62no—
77C2×C3×C4—64no—
78C2×C3×C4—62no—
79C2×C3×C4—91no—
80C2×C3×C4—92no—
81C2×C3×C4—92no—
82C2×C3×C4—92no—
83C2×C3×C4—184no—
84C2×C3×C4C2×#15712yes—
85C2×C3×C4C2×#15711yescharacteristic
86C2×Dic3—62no—
87C2×Dic3—64no—
88C4×D3—64no—
89C4×D3—64no—
90C4×D3—616no—
91C3×C8—184no—
92C2×C32—24no—
93C2×C32—22no—
94C3×D3—68no—
95C22×C4—91no—
96C22×C4—92no—
97M16—184no—
98C2×C8—94no—
99C42—92no—
100C42—91no—
101C42#108811yescharacteristic
102C22×C3—62no—
103C22×C3—92no—
104C22×C3—182no—
105C22×C3C22×#15711yescharacteristic
106C2×D3—64no—
107C2×D3—68no—
108C3×C4—22no—
109C3×C4—68no—
110C3×C4—64no—
111C3×C4—94no—
112C3×C4—91no—
113C3×C4—91no—
114C3×C4C4⋊#15712yes—
115Dic3—68no—
116Dic3—62no—
117Dic3—62no—
118C32—22no—
119C23—91no—
120C2×C4—91no—
121C2×C4—92no—
122C2×C4—92no—
123C2×C4—92no—
124C2×C4—184no—
125C2×C4C2×#108812yes—
126C2×C4C2×#108811yescharacteristic
127C8—184no—
128C2×C3—64no—
129C2×C3—62no—
130C2×C3—182no—
131C2×C3#6428312yes—
132C2×C3C2×C4⋊#15711yescharacteristic
133D3—68no—
134C22—92no—
135C22—182no—
136C22C22×#108811yescharacteristic
137C4—22no—
138C4—94no—
139C4—91no—
140C4—91no—
141C4C4⋊#108812yes—
142C3—62no—
143C3#1059372911yescharacteristic
144C2—182no—
145C2#1172244512yes—
146C2C2×C4⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC42×H(3,𝔽3)C4
prime radicalC22×#272C2×C4
cosocleC2×#1084C23
hypercenterC3×C42#157
hypocenterH(3,𝔽3)#423
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. #11720785C2semidirect
32. #1084M16semidirect
74. H(3,𝔽3)#423semidirect

Maximal subgroups

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

subgroupindex
C2×#886272
C42×#2722
#117207852
C3×#4239