Finite Group Databasebeta

← #13147029

Subgroups — #13147029

Order 1728.

295 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 36 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
1#11722392C212yesmaximal
2#131⋊H(3,𝔽3)C211yesmaximal, characteristic
3C3×#5403—33nomaximal
4C3×#5143—31nomaximal
5#41⋊H(3,𝔽3)C2211yescharacteristic
6#88610—22no—
7#88530—41nomaximal
8C3×#984—36no—
9C32×#131D311yescharacteristic
10C3×#951—32no—
11C2×C3×Dic12—31no—
12C3×#131⋊C3D313yes—
13C3×#824—32no—
14C3×#819—31no—
15#6229—42no—
16Q8⋊H(3,𝔽3)D411yescharacteristic
17C8×H(3,𝔽3)—41no—
18#5403—33no—
19C3×#473—91no—
20#5143—31no—
21C32×#41C2×D311yescharacteristic
22C2×C3×Dic6—32no—
23C3×#41⋊C3C2×D313yes—
24C3×#226—66no—
25C32×M16—31no—
26C2×C32×C8—31no—
27C3×#216—62no—
28C3×#214—62no—
29C3×#208—32no—
30C3×Dic12—61no—
31C3×Dic12—61no—
32C3×Dic12—123no—
33C4×H(3,𝔽3)—41no—
34#1084—82no—
35#984—36no—
36C3×#137—92no—
37C2×C3×Q16—91no—
38C3×#131—31no—
39C3×#131C3⋊D311yescharacteristic
40#951—32no—
41C2×Dic12—31no—
42#131⋊C3—33no—
43C3×#108—91no—
44C3×#101—92no—
45#824—32no—
46#819—31no—
47C32×Q8D1211yescharacteristic
48C32×D4—31no—
49C2×C32×C4—31no—
50C2×C3×Dic3—62no—
51C3×Dic6—62no—
52C3×Dic6—62no—
53C3×Dic6—126no—
54C3×SL(2,𝔽3)D1213yes—
55C32×C8—31no—
56C32×C8—43no—
57C32×C8S411yescharacteristic
58C3×C3⋊C8—62no—
59#473—91noSylow
60C2×H(3,𝔽3)—41no—
61C3×#41—31no—
62C3×#41C2×C3⋊D311yescharacteristic
63C2×C3×Q8—92no—
64C2×Dic6—32no—
65#41⋊C3—33no—
66#226—66no—
67C3×Q16—181no—
68C3×Q16—182no—
69C3×Q16—181no—
70C3×SD16—182no—
71C3×M16—31no—
72C3×M16—31no—
73C3×M16—61no—
74C3×M16—92no—
75C2×C3×C8—31no—
76C2×C3×C8—31no—
77C2×C3×C8—61no—
78#216—62no—
79#214—62no—
80#208—32no—
81Dic12—61no—
82Dic12—61no—
83Dic12—123no—
84C22×C32—31no—
85C32×C4—31no—
86C32×C4—43no—
87C32×C4C2×S411yescharacteristic
88C3×Dic3—122no—
89C3×Dic3—246no—
90#137—92no—
91C2×Q16—91no—
92#131#27211yescharacteristic
93#108—91no—
94#101—92no—
95H(3,𝔽3)—41noSylow
96C3×Q8—31no—
97C3×Q8—182no—
98C3×Q8—182no—
99C3×Q8C4⋊(C3⋊D3)11yescharacteristic
100C3×D4—31no—
101C3×D4—31no—
102C3×D4—61no—
103C2×C3×C4—31no—
104C2×C3×C4—31no—
105C2×C3×C4—61no—
106C2×C3×C4—182no—
107C2×Dic3—62no—
108Dic6—62no—
109Dic6—62no—
110Dic6—126no—
111SL(2,𝔽3)—33no—
112C3×C8—31no—
113C3×C8—31no—
114C3×C8—31no—
115C3×C8—61no—
116C3×C8—123no—
117C3×C8—182no—
118C3×C8C3⋊S411yescharacteristic
119C3⋊C8—62no—
120C2×C32—43no—
121C2×C32—61no—
122C2×C32C4⋊S411yescharacteristic
123#41C2×#27211yescharacteristic
124C2×Q8—92no—
125Q16—181no—
126Q16—182no—
127Q16—181no—
128SD16—182no—
129M16—31no—
130M16—92no—
131C2×C8—31no—
132C22×C3—31no—
133C22×C3—31no—
134C22×C3—61no—
135C3×C4—31no—
136C3×C4—31no—
137C3×C4—31no—
138C3×C4—61no—
139C3×C4—123no—
140C3×C4—362no—
141C3×C4C2×C3⋊S411yescharacteristic
142Dic3—122no—
143Dic3—246no—
144C32—43no—
145C32#540311yescharacteristic, complemented
146Q8—182no—
147Q8—182no—
148Q8C4⋊#27211yescharacteristic
149D4—31no—
150C2×C4—31no—
151C2×C4—182no—
152C8—31no—
153C8—182no—
154C8#625811yescharacteristic
155C2×C3—31no—
156C2×C3—61no—
157C2×C3—61no—
158C2×C3—121no—
159C2×C3—123no—
160C2×C3C4⋊(C3⋊S4)11yescharacteristic
161C22—31no—
162C4—31no—
163C4—362no—
164C4C2×#625811yescharacteristic
165C3—31no—
166C3—123no—
167C3#1059363711yescharacteristic
168C2—61no—
169C2C4⋊#625811yescharacteristic
subgroupG/H
center, socleC2×C3C4⋊(C3⋊S4)
commutator, prime radical, cosocle#41⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC32×#131D3
hypercenterC3×C8C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ #41⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
145. C32#5403semidirect

Maximal subgroups

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

subgroupindex
#117223922
#131⋊H(3,𝔽3)2
C3×#54033
C3×#51433
#885304