Finite Group Databasebeta

← #13147322

Subgroups — #13147322

Order 1728.

447 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 71 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×C4⋊#1088C212yesmaximal
2C42×#272C211yesmaximal, characteristic
3#11720787C214yesmaximal
4C22×#1088C2214yes—
5C2×C4×#272C2212yes—
6C2×C4×#272C2211yescharacteristic
7C4⋊#1088—24no—
8C42×H(3,𝔽3)C411yescharacteristic, complemented
9C4×#1084C411yescharacteristic, complemented
10C4×#1084C412yescomplemented
11C3×C42×D3—62no—
12C22×#272C2311yescharacteristic
13C2×#1088—24no—
14C2×#1088—28no—
15C2×C4×H(3,𝔽3)C2×C412yes—
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17C2×#1084C2×C412yes—
18C2×#1084C2×C411yescharacteristic
19C4×#272—24no—
20C4×#272D412yes—
21C4×#272Q812yes—
22C3×#382—91nomaximal
23C2×C3×C4×D3—62no—
24C2×C3×C4×D3—64no—
25C32×C42—22no—
26C3×C4×Dic3—64no—
27C3×C4×Dic3—62no—
28C22×H(3,𝔽3)C22×C411yescharacteristic
29C2×#272C2×D411yescharacteristic
30C2×#272C2×Q811yescharacteristic
31C2×#272#4114yes—
32#1088—44no—
33C4×H(3,𝔽3)—22no—
34C4×H(3,𝔽3)C4⋊C412yes—
35#1084—22no—
36#1084C4⋊C412yes—
37C2×C3×C4⋊C4—92no—
38C2×C3×C42—91no—
39C42×D3—62no—
40C3×#95—94no—
41C22×C3×D3—62no—
42C2×C32×C4—24no—
43C2×C32×C4—22no—
44C2×C3×Dic3—62no—
45C2×C3×Dic3—64no—
46C3×C4×D3—64no—
47C3×C4×D3—64no—
48C3×C4×D3—616no—
49#382—91noSylow
50C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
51C2×H(3,𝔽3)#11712yes—
52#272#12412yes—
53#272#12512yes—
54C22×C3×C4—91no—
55C22×C3×C4—92no—
56C22×C3×C4—94no—
57C2×C4×D3—62no—
58C2×C4×D3—64no—
59C3×C4⋊C4—184no—
60C3×C42—62no—
61C3×C42—92no—
62C3×C42—91no—
63C3×C42#15711yescharacteristic
64C4×Dic3—64no—
65C4×Dic3—62no—
66C22×C32—22no—
67C2×C3×D3—62no—
68C2×C3×D3—62no—
69C2×C3×D3—68no—
70C32×C4—28no—
71C32×C4—24no—
72C3×Dic3—64no—
73C3×Dic3—68no—
74C2×C4⋊C4—92no—
75C2×C42—91no—
76#95—94no—
77H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
78C23×C3—91no—
79C22×D3—62no—
80C2×C3×C4—64no—
81C2×C3×C4—62no—
82C2×C3×C4—91no—
83C2×C3×C4—92no—
84C2×C3×C4—92no—
85C2×C3×C4—92no—
86C2×C3×C4—184no—
87C2×C3×C4—184no—
88C2×C3×C4—188no—
89C2×C3×C4C2×#15712yes—
90C2×C3×C4C2×#15711yescharacteristic
91C2×Dic3—62no—
92C2×Dic3—64no—
93C4×D3—64no—
94C4×D3—64no—
95C4×D3—616no—
96C2×C32—22no—
97C2×C32—24no—
98C3×D3—64no—
99C3×D3—64no—
100C22×C4—91no—
101C22×C4—92no—
102C22×C4—94no—
103C4⋊C4—184no—
104C42—92no—
105C42—91no—
106C42#108811yescharacteristic, complemented
107C22×C3—62no—
108C22×C3—91no—
109C22×C3—91no—
110C22×C3—94no—
111C22×C3C22×#15711yescharacteristic
112C2×D3—62no—
113C2×D3—62no—
114C2×D3—68no—
115C3×C4—22no—
116C3×C4—68no—
117C3×C4—64no—
118C3×C4—92no—
119C3×C4—182no—
120C3×C4—364no—
121C3×C4C4⋊#15712yes—
122Dic3—64no—
123Dic3—68no—
124C32—22no—
125C23—91no—
126C2×C4—91no—
127C2×C4—92no—
128C2×C4—92no—
129C2×C4—92no—
130C2×C4—184no—
131C2×C4—184no—
132C2×C4—188no—
133C2×C4C2×#108812yes—
134C2×C4C2×#108811yescharacteristic
135C2×C3—62no—
136C2×C3—64no—
137C2×C3—92no—
138C2×C3—92no—
139C2×C3C2×C4⋊#15711yescharacteristic
140C2×C3#6428612yes—
141D3—64no—
142D3—64no—
143C22—91no—
144C22—91no—
145C22—94no—
146C22C22×#108811yescharacteristic
147C4—22no—
148C4—92no—
149C4—182no—
150C4—364no—
151C4C4⋊#108812yes—
152C3—62no—
153C3#1059373611yescharacteristic
154C2—92no—
155C2—92no—
156C2C2×C4⋊#108811yescharacteristic
157C2#1172244812yes—
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC42×H(3,𝔽3)C4
prime radical, cosocleC22×#272C23
hypercenterC3×C42#157
hypocenterH(3,𝔽3)#382
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
8. C42×H(3,𝔽3)C4semidirect
9. C4×#1084C4semidirect
10. C4×#1084C4semidirect
77. H(3,𝔽3)#382semidirect
106. C42#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C42×#2722
#117207872
C3×#3829