Finite Group Databasebeta

← #13146602

Subgroups — #13146602

Order 1728.

305 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 49 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×#88589C211yesmaximal, characteristic
2#11721078C212yesmaximal
3C4×#6188C211yesmaximal, characteristic
4#11720704C211yesmaximal, characteristic
5#11720702C212yesmaximal
6C2×#6196C2212yes—
7C2×#6188C2211yescharacteristic
8C2×#6188C2211yescharacteristic
9#88589C414yescomplemented
10C4×#1084C2211yescharacteristic
11#3750⋊C3—22no—
12#88449C2212yes—
13#88447—22no—
14#63837—62no—
15C2×C4×H(3,𝔽3)D411yescharacteristic
16C2×#1084D411yescharacteristic
17C2×#1084C2311yescharacteristic
18C2×(C3⋊Dic3)⋊C3—22no—
19C2×(C3⋊Dic3)⋊C3—22no—
20#6196—22no—
21#6196C2×C414yes—
22#6188—21no—
23#6188C2×C412yes—
24C4×Dic12—91nomaximal
25C2×#627—62no—
26#3750—22no—
27C3×Dic3⋊C4—62no—
28C3×C4×Dic3—62no—
29Dic3⋊Dic3—62no—
30Dic32—62no—
31Dic32—62no—
32C22×H(3,𝔽3)C2×D411yescharacteristic
33C4×H(3,𝔽3)—21no—
34#1084Q1612yescomplemented
35#1084C22×C411yescharacteristic
36#1084#4111yescharacteristic
37(C3⋊Dic3)⋊C3—24no—
38(C3⋊Dic3)⋊C3—42no—
39C2×Dic12—91no—
40#887—182no—
41C4×Dic6—92no—
42C3×C4×C8—91no—
43#818—91no—
44#816—92no—
45C2×C32×C4—22no—
46C2×C3⋊Dic3—22no—
47C2×C3⋊Dic3—22no—
48C2×C3×Dic3—62no—
49C2×C3×Dic3—62no—
50C2×C3×Dic3—62no—
51C2×C3×Dic3—62no—
52#627—64no—
53#627—64no—
54C4×Q16—271noSylow
55C2×H(3,𝔽3)C4×D411yescharacteristic
56C2×H(3,𝔽3)C2×Q1611yescharacteristic
57C2×H(3,𝔽3)#13511yescharacteristic
58C2×Dic6—92no—
59C2×Dic6—182no—
60C2×C3×C8—91no—
61C2×C3×C8—91no—
62C3×C4⋊C4—182no—
63C3×C42—91no—
64C4⋊Dic3—92no—
65Dic3⋊C4—62no—
66Dic3⋊C4—182no—
67Dic3⋊C4—182no—
68C4×Dic3—62no—
69C4×Dic3—182no—
70C4×Dic3—182no—
71C4×Dic3—182no—
72Dic12—94no—
73C22×C32—22no—
74C32×C4—42no—
75C3⋊Dic3—24no—
76C3⋊Dic3—42no—
77C3×Dic3—64no—
78C3×Dic3—62no—
79C3×Dic3—62no—
80C3×Dic3—64no—
81C3×Dic3—122no—
82C2×Q16—271no—
83C4×Q8—272no—
84#107—271no—
85#103—272no—
86C4×C8—91no—
87H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
88C2×C3×C4—62no—
89C2×C3×C4—91no—
90C2×C3×C4—91no—
91C2×C3×C4—182no—
92C2×C3×C4—182no—
93C2×C3×C4GO+(4,𝔽2)11yescharacteristic
94C2×Dic3—62no—
95C2×Dic3—62no—
96C2×Dic3—62no—
97C2×Dic3—62no—
98C2×Dic3—182no—
99C2×Dic3—182no—
100Dic6—94no—
101Dic6—184no—
102Dic6—182no—
103Dic6—184no—
104C3×C8—92no—
105C3×C8—181no—
106C2×C32—22no—
107C2×C32—22no—
108C2×C32—22no—
109C2×Q8—272no—
110Q16—274no—
111C2×C8—91no—
112C2×C8—91no—
113C4⋊C4—182no—
114C4⋊C4—272no—
115C42—91no—
116C42—542no—
117C22×C3—62no—
118C22×C3C2×GO+(4,𝔽2)11yescharacteristic
119C3×C4—21no—
120C3×C4—92no—
121C3×C4—91no—
122C3×C4—91no—
123C3×C4—122no—
124C3×C4—184no—
125C3×C4—362no—
126Dic3—62no—
127Dic3—62no—
128Dic3—64no—
129Dic3—64no—
130Dic3—122no—
131Dic3—184no—
132Dic3—362no—
133C32—22no—
134Q8—274no—
135Q8—542no—
136C2×C4—91no—
137C2×C4—91no—
138C2×C4—182no—
139C2×C4—182no—
140C2×C4#1088⋊C211yescharacteristic
141C8—92no—
142C8—181no—
143C2×C3—62no—
144C2×C3—62no—
145C2×C3—62no—
146C2×C3#6422811yescharacteristic
147C2×C3C4×GO+(4,𝔽2)11yescharacteristic
148C2×C3#6423711yescharacteristic
149C22C2×#1088⋊C211yescharacteristic
150C4—21no—
151C4—92no—
152C4—91no—
153C4—91no—
154C4—184no—
155C4—362no—
156C3—62no—
157C3#1059344511yescharacteristic
158C2#1172229911yescharacteristic
159C2C4×#1088⋊C211yescharacteristic
160C2#1172231011yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator#1084C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×#1084C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
9. #88589C4semidirect
34. #1084Q16semidirect
87. H(3,𝔽3)C4×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#885892
#117210782
C4×#61882
#117207042
#117207022
C4×Dic129