Finite Group Databasebeta

← #11720327

Subgroups — #11720327

Order 864.

171 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 38 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×#6229C211yesmaximal, characteristic
2#3748⋊C3C211yesmaximal, characteristic
3#3748⋊C3C211yesmaximal, characteristic
4#63560—31nomaximal
5#63560—31nomaximal
6C2×C4×H(3,𝔽3)C2211yescharacteristic
7C2×#1084C412yes—
8#6229—21no—
9#6229—21no—
10(C3⋊(C3⋊C8))⋊C3—21no—
11(C3⋊(C3⋊C8))⋊C3—21no—
12C2×C3×Dic6—31no—
13C2×C3×Dic6—31no—
14C2×C3×Dic6—61no—
15#3748D311yescharacteristic
16#3748D311yescharacteristic
17C3×#208—31no—
18C3×#208—31no—
19C22×H(3,𝔽3)C2×C411yescharacteristic
20C4×H(3,𝔽3)D411yescharacteristic
21C4×H(3,𝔽3)D411yescharacteristic
22#1084—22no—
23#836—91no—
24#824—91no—
25#824—91no—
26C2×C32×C4—21no—
27C2×C32×C4C2×D311yescharacteristic
28C2×C32×C4C2×D311yescharacteristic
29C2×C3×Dic3—32no—
30C2×C3×Dic3—32no—
31C2×C3×Dic3—62no—
32C3×Dic6—62no—
33C3×Dic6—62no—
34C3×Dic6—61no—
35C3×Dic6—61no—
36C3×Dic6—61no—
37C3×Dic6—61no—
38C3⋊(C3⋊C8)—21no—
39C3⋊(C3⋊C8)—21no—
40C3×C3⋊C8—61no—
41C3×C3⋊C8—61no—
42C2×H(3,𝔽3)—21no—
43C2×H(3,𝔽3)C22⋊C411yescharacteristic
44C2×C3×Q8—91no—
45C2×Dic6—31no—
46C2×Dic6—31no—
47C2×Dic6—61no—
48C3×M16—91no—
49C3×M16—91no—
50#208—31no—
51#208—31no—
52#208—91no—
53#208—181no—
54C22×C32—21no—
55C22×C32C4×D311yescharacteristic
56C22×C32C4×D311yescharacteristic
57C32×C4—21no—
58C32×C4—21no—
59C32×C4D1211yescharacteristic
60C32×C4D1211yescharacteristic
61C32×C4#6711yescharacteristic
62C32×C4#6711yescharacteristic
63C3×Dic3—64no—
64C3×Dic3—62no—
65C3×Dic3—62no—
66#101—271noSylow
67H(3,𝔽3)#10111yescharacteristic, complemented, Sylow
68C3×Q8—181no—
69C3×Q8—181no—
70C2×C3×C4—31no—
71C2×C3×C4—61no—
72C2×C3×C4—92no—
73C2×C3×C4—121no—
74C2×C3×C4D3211yescharacteristic
75C2×Dic3—32no—
76C2×Dic3—32no—
77C2×Dic3—62no—
78Dic6—61no—
79Dic6—61no—
80Dic6—61no—
81Dic6—61no—
82Dic6—62no—
83Dic6—62no—
84C3×C8—181no—
85C3×C8—181no—
86C3⋊C8—61no—
87C3⋊C8—61no—
88C3⋊C8—181no—
89C3⋊C8—181no—
90C2×C32—21no—
91C2×C32—22no—
92C2×C32—21no—
93C2×C32—21no—
94C2×C32(C2×D3)⋊C411yescharacteristic
95C2×C32(C2×D3)⋊C411yescharacteristic
96C2×Q8—91no—
97M16—91no—
98M16—91no—
99C22×C3—31no—
100C22×C3—61no—
101C22×C3—121no—
102C22×C3#62411yescharacteristic
103C3×C4—31no—
104C3×C4—31no—
105C3×C4—61no—
106C3×C4—61no—
107C3×C4—121no—
108C3×C4—121no—
109C3×C4—182no—
110C3×C4#62611yescharacteristic
111C3×C4#62611yescharacteristic
112Dic3—62no—
113Dic3—62no—
114Dic3—64no—
115C32—21no—
116C32#82411yescharacteristic, complemented
117C32#82411yescharacteristic, complemented
118Q8—181no—
119Q8—181no—
120C2×C4—92no—
121C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
122C8—181no—
123C8—181no—
124C2×C3—21no—
125C2×C3—31no—
126C2×C3—61no—
127C2×C3—61no—
128C2×C3—121no—
129C2×C3—122no—
130C2×C3—121no—
131C2×C3#372211yescharacteristic
132C22#619911yescharacteristic
133C4—182no—
134C4#620011yescharacteristic
135C4#620011yescharacteristic
136C3—31no—
137C3—61no—
138C3—121no—
139C3#6356111yescharacteristic
140C2—21no—
141C2#8845111yescharacteristic
subgroupG/H
centerC2#88451
commutator, prime radicalC22×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC2×C3#3722
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#101
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
67. H(3,𝔽3)#101semidirect
116. C32#824semidirect
117. C32#824semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
#3748⋊C32
#635603