Finite Group Databasebeta

← #13147033

Subgroups — #13147033

Order 1728.

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

normal subgroups only

This group has 325 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#11722394C212yesmaximal, complemented
2#131⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#5407—33nomaximal
4C3×#5142—31nomaximal
5#41⋊H(3,𝔽3)C2211yescharacteristic
6#88611—22no—
7#88529—41nomaximal
8C3×#986—36no—
9C32×#131D311yescharacteristic, complemented
10C3×#949—32no—
11C2×C3×D24—31no—
12C3×#131⋊C3D313yescomplemented
13C3×#823—32no—
14C3×#819—31no—
15C4⋊#272—42no—
16Q8⋊H(3,𝔽3)D411yescharacteristic
17C8×H(3,𝔽3)—41no—
18#5407—33no—
19C3×#472—91no—
20#5142—31no—
21C32×#41C2×D311yescharacteristic
22C2×C3×D12—32no—
23C3×#41⋊C3C2×D313yes—
24C3×GL(2,𝔽3)—66no—
25C32×M16—31no—
26C2×C32×C8—31no—
27C3×#215—62no—
28C3×#213—62no—
29C3×#208—32no—
30C3×D24—61no—
31C3×D24—61no—
32C3×D24—123no—
33C2×#272—82no—
34C4×H(3,𝔽3)—41no—
35#986—36no—
36C3×AGL(1,ℤ/8)—92no—
37C2×C3×D8—91no—
38C3×#131—31no—
39C3×#131C3⋊D311yescharacteristic
40#949—32no—
41C2×D24—31no—
42#131⋊C3—33no—
43C3×#108—91no—
44C3×#100—92no—
45#823—32no—
46#819—31no—
47C22×C3×D3—62no—
48C32×Q8D1211yescharacteristic
49C32×D4—31no—
50C2×C32×C4—31no—
51C3×D12—62no—
52C3×D12—62no—
53C3×D12—126no—
54C3×SL(2,𝔽3)D1213yes—
55C32×C8—31no—
56C32×C8—43no—
57C32×C8S411yescharacteristic
58C3×C3⋊C8—62no—
59#472—91noSylow
60C2×H(3,𝔽3)—41no—
61#272—162no—
62C3×#41—31no—
63C3×#41C2×C3⋊D311yescharacteristic
64C2×C3×D4—92no—
65C2×D12—32no—
66#41⋊C3—33no—
67GL(2,𝔽3)—66no—
68C3×SD16—182no—
69C3×D8—181no—
70C3×D8—182no—
71C3×D8—181no—
72C3×M16—31no—
73C3×M16—31no—
74C3×M16—61no—
75C3×M16—92no—
76C2×C3×C8—31no—
77C2×C3×C8—31no—
78C2×C3×C8—61no—
79#215—62no—
80#213—62no—
81#208—32no—
82D24—61no—
83D24—61no—
84D24—123no—
85C22×C32—31no—
86C2×C3×D3—122no—
87C2×C3×D3—246no—
88C2×C3×D3—242no—
89C32×C4—31no—
90C32×C4—43no—
91C32×C4C2×S411yescharacteristic
92AGL(1,ℤ/8)—92no—
93C2×D8—91no—
94#131#27211yescharacteristic, complemented
95#108—91no—
96#100—92no—
97H(3,𝔽3)—41noSylow
98C23×C3—182no—
99C22×D3—62no—
100C3×Q8—31no—
101C3×Q8C4⋊(C3⋊D3)11yescharacteristic
102C3×D4—31no—
103C3×D4—31no—
104C3×D4—61no—
105C3×D4—182no—
106C3×D4—182no—
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—61no—
110D12—62no—
111D12—62no—
112D12—126no—
113SL(2,𝔽3)—33no—
114C3×C8—31no—
115C3×C8—31no—
116C3×C8—31no—
117C3×C8—61no—
118C3×C8—123no—
119C3×C8—182no—
120C3×C8C3⋊S411yescharacteristic
121C3⋊C8—62no—
122C2×C32—43no—
123C2×C32—61no—
124C2×C32C4⋊S411yescharacteristic
125C3×D3—242no—
126C3×D3—486no—
127#41C2×#27211yescharacteristic
128C2×D4—92no—
129SD16—182no—
130D8—181no—
131D8—182no—
132D8—181no—
133M16—31no—
134M16—92no—
135C2×C8—31no—
136C22×C3—31no—
137C22×C3—31no—
138C22×C3—61no—
139C22×C3—362no—
140C22×C3—722no—
141C2×D3—122no—
142C2×D3—246no—
143C2×D3—242no—
144C3×C4—31no—
145C3×C4—31no—
146C3×C4—31no—
147C3×C4—61no—
148C3×C4—123no—
149C3×C4C2×C3⋊S411yescharacteristic
150C32—43no—
151C32#540711yescharacteristic, complemented
152C23—182no—
153Q8C4⋊#27211yescharacteristic
154D4—31no—
155D4—182no—
156D4—182no—
157C2×C4—31no—
158C8—31no—
159C8—182no—
160C8#625811yescharacteristic
161C2×C3—31no—
162C2×C3—61no—
163C2×C3—61no—
164C2×C3—121no—
165C2×C3—123no—
166C2×C3—722no—
167C2×C3C4⋊(C3⋊S4)11yescharacteristic
168D3—242no—
169D3—486no—
170C22—31no—
171C22—362no—
172C22—722no—
173C4—31no—
174C4C2×#625811yescharacteristic
175C3—31no—
176C3—123no—
177C3#1059364111yescharacteristic
178C2—61no—
179C2—722no—
180C2C4⋊#625811yescharacteristic
subgroupG/H
center, socleC2×C3C4⋊(C3⋊S4)
commutator, 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
1. #11722394C2semidirect
2. #131⋊H(3,𝔽3)C2semidirect
9. C32×#131D3semidirect
12. C3×#131⋊C3D3semidirect
94. #131#272semidirect
151. C32#5407semidirect

Maximal subgroups

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

subgroupindex
#117223942
#131⋊H(3,𝔽3)2
C3×#54073
C3×#51423
#885294