Finite Group Databasebeta

← #13146618

Subgroups — #13146618

Order 1728.

463 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 81 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
2C2×#88590C214yesmaximal
3#11721152C212yesmaximal
4C22×#1088C2212yes—
5C2×C4×#272C2211yescharacteristic
6C2×#6199C2214yes—
7C4×#1088C414yescomplemented
8#88590—28no—
9C4⋊((C3⋊Dic3)⋊C3)—22no—
10#88449—22no—
11D3×C4⋊Dic3—62no—
12C22×#272C2311yescharacteristic
13C2×#1088C2×C414yes—
14C2×#1088D414yes—
15C2×#1088Q814yes—
16C2×C4×H(3,𝔽3)D411yescharacteristic
17C2×#1084D411yescharacteristic
18C4×#272C2×C412yes—
19C4×#272Q812yes—
20C2×(C3⋊Dic3)⋊C3—24no—
21#6199—28no—
22#4935—91nomaximal
23C2×C3×C4×D3—62no—
24C2×D3×Dic3—64no—
25C4⋊(C3⋊Dic3)—22no—
26C3×C4⋊Dic3—62no—
27Dic3⋊Dic3—64no—
28C22×H(3,𝔽3)C2×D411yescharacteristic
29C2×#272C22×C411yescharacteristic
30C2×#272C2×D412yes—
31C2×#272C2×Q812yes—
32C2×#272C2×Q811yescharacteristic
33#1088C4⋊C418yescomplemented
34C4×H(3,𝔽3)C4⋊C412yescomplemented
35#1084C4⋊C412yescomplemented
36(C3⋊Dic3)⋊C3—44no—
37C2×C3×C42—91no—
38C2×C4⋊Dic3—92no—
39C2×C4⋊Dic3—94no—
40D3×C4⋊C4—182no—
41C22×C3×D3—62no—
42C2×C32×C4—22no—
43C2×C3⋊Dic3—24no—
44C2×C3×Dic3—62no—
45C2×C3×Dic3—64no—
46C3×C4×D3—64no—
47C3×C4×D3—64no—
48D3×Dic3—128no—
49#384—271noSylow
50C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
51C2×H(3,𝔽3)#12711yescharacteristic
52C2×H(3,𝔽3)C4⋊Q811yescharacteristic
53#272C2×C4⋊C412yes—
54#272C4⋊Q812yes—
55C22×C3×C4—92no—
56C22×C3×C4—91no—
57C22×Dic3—94no—
58C2×C4×D3—62no—
59C2×C4×D3—184no—
60C3×C4⋊C4—182no—
61C3×C42—94no—
62C4⋊Dic3—62no—
63C4⋊Dic3—182no—
64C4⋊Dic3—182no—
65C4⋊Dic3—188no—
66Dic3⋊C4—184no—
67C22×C32—22no—
68C2×C3×D3—62no—
69C2×C3×D3—64no—
70C2×C3×D3—62no—
71C2×C3×D3—64no—
72C32×C4—24no—
73C3⋊Dic3—44no—
74C3×Dic3—64no—
75C3×Dic3—124no—
76C2×C4⋊C4—272no—
77C2×C4⋊C4—274no—
78C2×C42—91no—
79H(3,𝔽3)#38411yescharacteristic, complemented, Sylow
80C23×C3—91no—
81C22×D3—62no—
82C2×C3×C4—62no—
83C2×C3×C4—94no—
84C2×C3×C4—94no—
85C2×C3×C4—94no—
86C2×C3×C4—91no—
87C2×C3×C4—92no—
88C2×C3×C4—92no—
89C2×C3×C4—184no—
90C2×C3×C4GO+(4,𝔽2)11yescharacteristic
91C2×Dic3—62no—
92C2×Dic3—64no—
93C2×Dic3—184no—
94C2×Dic3—188no—
95C4×D3—64no—
96C4×D3—64no—
97C4×D3—368no—
98C2×C32—22no—
99C2×C32—22no—
100C2×C32—22no—
101C3×D3—64no—
102C3×D3—64no—
103C22×C4—92no—
104C22×C4—91no—
105C22×C4—274no—
106C4⋊C4—182no—
107C4⋊C4—542no—
108C4⋊C4—548no—
109C42—94no—
110C22×C3—62no—
111C22×C3—92no—
112C22×C3—92no—
113C22×C3—91no—
114C22×C3—91no—
115C22×C3C2×GO+(4,𝔽2)11yescharacteristic
116C2×D3—62no—
117C2×D3—62no—
118C2×D3—64no—
119C2×D3—64no—
120C3×C4—64no—
121C3×C4—98no—
122C3×C4—92no—
123C3×C4—364no—
124C3×C4#377312yes—
125Dic3—64no—
126Dic3—124no—
127Dic3—364no—
128C32—22no—
129C23—91no—
130C2×C4—94no—
131C2×C4—94no—
132C2×C4—94no—
133C2×C4—91no—
134C2×C4—92no—
135C2×C4—92no—
136C2×C4—184no—
137C2×C4—548no—
138C2×C4#1088⋊C211yescharacteristic
139C2×C3—62no—
140C2×C3—62no—
141C2×C3—62no—
142C2×C3—92no—
143C2×C3—92no—
144C2×C3#6422211yescharacteristic
145C2×C3#6423111yescharacteristic
146C2×C3C2×#377311yescharacteristic
147D3—64no—
148D3—64no—
149C22—92no—
150C22—92no—
151C22—91no—
152C22—91no—
153C22C2×#1088⋊C211yescharacteristic
154C4—98no—
155C4—92no—
156C4—364no—
157C4#8859012yescomplemented
158C3—62no—
159C3#1059341311yescharacteristic
160C2—92no—
161C2—92no—
162C2#1172230011yescharacteristic
163C2#1172230511yescharacteristic
164C2C2×#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)#384
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
33. #1088C4⋊C4semidirect
34. C4×H(3,𝔽3)C4⋊C4semidirect
35. #1084C4⋊C4semidirect
79. H(3,𝔽3)#384semidirect
157. C4#88590semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10882
C2×#885902
#117211522
#49359