Finite Group Databasebeta

← #13135098

Subgroups — #13135098

Order 1728.

271 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 55 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×#88527C211yesmaximal, characteristic
2#63639⋊C3C212yesmaximal, complemented
3#10589109—32nomaximal
4C2×C4×#272C411yescharacteristic
5C2×C8×H(3,𝔽3)C2211yescharacteristic
6C2×#6180C411yescharacteristic
7#88527—22no—
8(C3⋊(C3⋊C16))⋊C3—22no—
9C2×C3×ΓL(1,𝔽25)—32no—
10C2×C3×ΓL(1,𝔽25)—61no—
11#63639D312yescomplemented
12C3×#812—32no—
13C22×#272C811yescharacteristic
14C2×C4×H(3,𝔽3)C2×C411yescharacteristic
15C2×#1084C811yescharacteristic
16C4×#272—21no—
17C4×#272—21no—
18C8×H(3,𝔽3)D412yes—
19#6180—21no—
20#4553—91no—
21#4514—92no—
22C2×C3×C4×D3—32no—
23C2×C3×C4×D3—61no—
24C2×C32×C8—21no—
25C2×C32×C8C2×D312yes—
26C2×C3×C3⋊C8—32no—
27C2×C3×C3⋊C8—61no—
28C3×ΓL(1,𝔽25)—64no—
29C3×ΓL(1,𝔽25)—62no—
30C3×ΓL(1,𝔽25)—62no—
31C3⋊(C3⋊C16)—22no—
32C3×C3⋊C16—62no—
33C22×H(3,𝔽3)C2×C811yescharacteristic
34C2×#272—21no—
35C2×#272—41no—
36C4×H(3,𝔽3)C22⋊C411yescharacteristic
37C4×H(3,𝔽3)M1611yescharacteristic
38#1084—21no—
39C2×C3×M16—91no—
40C2×ΓL(1,𝔽25)—32no—
41C2×ΓL(1,𝔽25)—61no—
42C3×M32—92no—
43#812—32no—
44#812—92no—
45C22×C3×D3—32no—
46C22×C3×D3—61no—
47C2×C32×C4—21no—
48C2×C32×C4C4×D312yes—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—61no—
51C3×C4×D3—62no—
52C3×C4×D3—62no—
53C3×C4×D3—62no—
54C3×C4×D3—62no—
55C32×C8—22no—
56C32×C8D1212yes—
57C32×C8#6712yes—
58C3×C3⋊C8—62no—
59C3×C3⋊C8—62no—
60#349—271noSylow
61C2×H(3,𝔽3)—21no—
62C2×H(3,𝔽3)C22⋊C811yescharacteristic
63#272—41no—
64C22×C3×C4—91no—
65C2×C4×D3—32no—
66C2×C4×D3—61no—
67C3×M16—182no—
68C2×C3×C8—32no—
69C2×C3×C8—61no—
70C2×C3×C8—91no—
71C2×C3×C8D3211yescharacteristic
72C2×C3⋊C8—32no—
73C2×C3⋊C8—61no—
74ΓL(1,𝔽25)—62no—
75ΓL(1,𝔽25)—62no—
76ΓL(1,𝔽25)—64no—
77C3×C16—182no—
78C3⋊C16—62no—
79C3⋊C16—182no—
80C22×C32—21no—
81C22×C32D3×C812yes—
82C2×C3×D3—62no—
83C2×C3×D3—62no—
84C2×C3×D3—122no—
85C2×C3×D3—122no—
86C32×C4—21no—
87C32×C4—21no—
88C32×C4ΓL(1,𝔽25)12yes—
89C32×C4(C2×D3)⋊C412yes—
90C3×Dic3—62no—
91C3×Dic3—62no—
92C2×M16—91no—
93M32—92no—
94H(3,𝔽3)#34911yescharacteristic, complemented, Sylow
95C23×C3—91no—
96C22×D3—32no—
97C22×D3—61no—
98C2×C3×C4—32no—
99C2×C3×C4—61no—
100C2×C3×C4—91no—
101C2×C3×C4—181no—
102C2×C3×C4—181no—
103C2×C3×C4#62411yescharacteristic
104C2×Dic3—32no—
105C2×Dic3—61no—
106C4×D3—62no—
107C4×D3—62no—
108C4×D3—62no—
109C4×D3—62no—
110C3×C8—32no—
111C3×C8—32no—
112C3×C8—62no—
113C3×C8—181no—
114C3×C8#62612yes—
115C3⋊C8—62no—
116C3⋊C8—62no—
117C2×C32—22no—
118C2×C32—22no—
119C2×C32—21no—
120C2×C32(C2×D3)⋊C812yes—
121C3×D3—122no—
122C3×D3—122no—
123C22×C4—91no—
124M16—182no—
125C2×C8—91no—
126C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
127C16—182no—
128C22×C3—32no—
129C22×C3—61no—
130C22×C3—181no—
131C22×C3—361no—
132C22×C3#371011yescharacteristic
133C2×D3—62no—
134C2×D3—62no—
135C2×D3—122no—
136C2×D3—122no—
137C3×C4—32no—
138C3×C4—32no—
139C3×C4—61no—
140C3×C4—61no—
141C3×C4—181no—
142C3×C4#371211yescharacteristic
143C3×C4#372211yescharacteristic
144Dic3—62no—
145Dic3—62no—
146C32—21no—
147C32#451412yescomplemented
148C23—91no—
149C2×C4—91no—
150C2×C4—181no—
151C2×C4—181no—
152C2×C4#619911yescharacteristic
153C8—181no—
154C8#620012yes—
155C2×C3—21no—
156C2×C3—32no—
157C2×C3—62no—
158C2×C3—62no—
159C2×C3—61no—
160C2×C3—361no—
161C2×C3#6355711yescharacteristic
162D3—122no—
163D3—122no—
164C22—181no—
165C22—361no—
166C22#8843411yescharacteristic
167C4—181no—
168C4#8843511yescharacteristic
169C4#8845111yescharacteristic
170C3—32no—
171C3—61no—
172C3#1058911011yescharacteristic
173C2—21no—
174C2—361no—
175C2#1172032311yescharacteristic
subgroupG/H
centerC4#88451
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63557
prime radicalC22×#272C8
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#349
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #63639⋊C3C2semidirect
11. #63639D3semidirect
94. H(3,𝔽3)#349semidirect
147. C32#4514semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#63639⋊C32
#105891093