Finite Group Databasebeta

← #13146566

Subgroups — #13146566

Order 1728.

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

normal subgroups only

This group has 415 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×C4×#1088C211yesmaximal, characteristic
2#11721152C212yesmaximal
3#11720707C214yesmaximal
4C22×#1088C2212yes—
5C2×C4×#272C2211yescharacteristic
6C2×#6199C2214yes—
7C4×#1088C414yescomplemented
8C4⋊((C3⋊Dic3)⋊C3)—22no—
9#88449—22no—
10D3×C4⋊Dic3—62no—
11C22×#272C2311yescharacteristic
12C2×#1088—24no—
13C2×#1088C2×C414yes—
14C2×C4×H(3,𝔽3)D411yescharacteristic
15C2×#1084D411yescharacteristic
16C4×#272C2×C412yes—
17C4×#272Q812yes—
18C2×(C3⋊Dic3)⋊C3—24no—
19#6199—28no—
20#4936—91nomaximal
21C2×C3×C4×D3—62no—
22C2×D3×Dic3—64no—
23C4⋊(C3⋊Dic3)—22no—
24C3×C4⋊Dic3—62no—
25Dic3⋊Dic3—64no—
26C22×H(3,𝔽3)C2×D411yescharacteristic
27C2×#272C22×C411yescharacteristic
28C2×#272C2×Q811yescharacteristic
29C2×#272#4114yes—
30#1088—24no—
31C4×H(3,𝔽3)C4⋊C412yes—
32#1084C4⋊C412yes—
33(C3⋊Dic3)⋊C3—44no—
34C2×C3×C42—91no—
35C2×C4⋊Dic3—92no—
36D3×C4⋊C4—182no—
37C3⋊#95—94no—
38C22×C3×D3—62no—
39C2×C32×C4—22no—
40C2×C3⋊Dic3—24no—
41C2×C3×Dic3—62no—
42C2×C3×Dic3—64no—
43C3×C4×D3—64no—
44C3×C4×D3—64no—
45D3×Dic3—128no—
46#382—271noSylow
47C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
48C2×H(3,𝔽3)#12412yes—
49#272#11712yes—
50#272#12512yes—
51C22×C3×C4—91no—
52C22×C3×C4—92no—
53C22×Dic3—94no—
54C2×C4×D3—62no—
55C2×C4×D3—184no—
56C3×C4⋊C4—182no—
57C3×C42—94no—
58C4⋊Dic3—62no—
59C4⋊Dic3—182no—
60C4⋊Dic3—182no—
61Dic3⋊C4—184no—
62C22×C32—22no—
63C2×C3×D3—62no—
64C2×C3×D3—62no—
65C2×C3×D3—64no—
66C2×C3×D3—64no—
67C32×C4—24no—
68C3⋊Dic3—44no—
69C3×Dic3—64no—
70C3×Dic3—124no—
71C2×C4⋊C4—272no—
72C2×C42—91no—
73#95—274no—
74H(3,𝔽3)#38211yescharacteristic, complemented, Sylow
75C23×C3—91no—
76C22×D3—62no—
77C2×C3×C4—62no—
78C2×C3×C4—91no—
79C2×C3×C4—92no—
80C2×C3×C4—92no—
81C2×C3×C4—94no—
82C2×C3×C4—184no—
83C2×C3×C4—184no—
84C2×C3×C4GO+(4,𝔽2)11yescharacteristic
85C2×Dic3—62no—
86C2×Dic3—64no—
87C2×Dic3—184no—
88C2×Dic3—188no—
89C4×D3—64no—
90C4×D3—64no—
91C4×D3—368no—
92C2×C32—22no—
93C2×C32—22no—
94C2×C32—22no—
95C3×D3—64no—
96C3×D3—64no—
97C22×C4—91no—
98C22×C4—92no—
99C22×C4—274no—
100C4⋊C4—182no—
101C4⋊C4—542no—
102C42—94no—
103C22×C3—62no—
104C22×C3—94no—
105C22×C3—91no—
106C22×C3—91no—
107C22×C3C2×GO+(4,𝔽2)11yescharacteristic
108C2×D3—62no—
109C2×D3—62no—
110C2×D3—64no—
111C2×D3—64no—
112C3×C4—64no—
113C3×C4—92no—
114C3×C4—184no—
115C3×C4—364no—
116C3×C4#377312yes—
117Dic3—64no—
118Dic3—124no—
119Dic3—364no—
120C32—22no—
121C23—91no—
122C2×C4—91no—
123C2×C4—92no—
124C2×C4—92no—
125C2×C4—94no—
126C2×C4—184no—
127C2×C4—184no—
128C2×C4—548no—
129C2×C4#1088⋊C211yescharacteristic
130C2×C3—62no—
131C2×C3—62no—
132C2×C3—62no—
133C2×C3—92no—
134C2×C3—92no—
135C2×C3#6422112yes—
136C2×C3C2×#377311yescharacteristic
137D3—64no—
138D3—64no—
139C22—94no—
140C22—91no—
141C22—91no—
142C22C2×#1088⋊C211yescharacteristic
143C4—92no—
144C4—184no—
145C4—364no—
146C4#8859012yes—
147C3—62no—
148C3#1059334411yescharacteristic
149C2—92no—
150C2—92no—
151C2#1172230212yes—
152C2C2×#8859011yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutatorC2×#272C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radical, cosocleC22×#272C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#382
LCSG ▷ C2×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. C4×#1088C4semidirect
74. H(3,𝔽3)#382semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10882
#117211522
#117207072
#49369