Finite Group Databasebeta

← #11721090

Subgroups — #11721090

Order 864.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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
2C2×#634⋊C3C211yesmaximal, characteristic
3C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4#88448C214yesmaximal
5#63873—31nomaximal
6#63866—31nomaximal
7C2×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#1084C2212yes—
9#6229—22no—
10C2×(C3⋊Dic3)⋊C3C2212yes—
11C2×(C3⋊Dic3)⋊C3C2212yes—
12#634⋊C3—22no—
13C2×#634D311yescharacteristic
14C2×C3×Dic6—31no—
15C2×C3×Dic6—31no—
16C2×C3×Dic6—31no—
17C2×C3×Dic6—61no—
18C4×C3⋊Dic3D311yescharacteristic
19C3×C4×Dic3—31no—
20#3724—34no—
21Dic3⋊Dic3—34no—
22C22×H(3,𝔽3)C2311yescharacteristic
23C4×H(3,𝔽3)D412yes—
24#1084—22no—
25(C3⋊Dic3)⋊C3—22no—
26(C3⋊Dic3)⋊C3Q814yescomplemented
27#944—91no—
28#944—91no—
29#869—91no—
30C2×C32×C4—21no—
31C2×C32×C4C2×D311yescharacteristic
32C2×C32×C4C2×D311yescharacteristic
33C2×C3⋊Dic3C2×D312yes—
34C2×C3⋊Dic3C2×D312yes—
35#634—22no—
36C2×C3×Dic3—32no—
37C2×C3×Dic3—32no—
38C2×C3×Dic3—32no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—62no—
41C3×Dic6—64no—
42C3×Dic6—62no—
43C3×Dic6—62no—
44C3×Dic6—62no—
45C2×H(3,𝔽3)C2×D411yescharacteristic
46C2×H(3,𝔽3)C2×Q812yes—
47C2×C3×Q8—91no—
48C2×C3×Q8—91no—
49C2×Dic6—31no—
50C2×Dic6—31no—
51C2×Dic6—31no—
52C2×Dic6—61no—
53C2×Dic6—91no—
54C3×C42—91no—
55C4⋊Dic3—94no—
56Dic3⋊C4—94no—
57Dic3⋊C4—94no—
58C4×Dic3—31no—
59C4×Dic3—181no—
60C22×C32—21no—
61C22×C32C22×D311yescharacteristic
62C22×C32C22×D311yescharacteristic
63C32×C4—22no—
64C32×C4D1212yes—
65C32×C4#6712yes—
66C3⋊Dic3—22no—
67C3⋊Dic3Dic614yescomplemented
68C3×Dic3—34no—
69C3×Dic3—64no—
70C3×Dic3—62no—
71C3×Dic3—62no—
72C3×Dic3—62no—
73C4⋊Q8—271noSylow
74H(3,𝔽3)C4⋊Q811yescharacteristic, complemented, Sylow
75C3×Q8—182no—
76C3×Q8—182no—
77C2×C3×C4—31no—
78C2×C3×C4—61no—
79C2×C3×C4—92no—
80C2×C3×C4—92no—
81C2×C3×C4—92no—
82C2×C3×C4—121no—
83C2×C3×C4D3211yescharacteristic
84C2×Dic3—32no—
85C2×Dic3—32no—
86C2×Dic3—32no—
87C2×Dic3—32no—
88C2×Dic3—62no—
89C2×Dic3—92no—
90C2×Dic3—182no—
91Dic6—62no—
92Dic6—62no—
93Dic6—64no—
94Dic6—62no—
95Dic6—182no—
96C2×C32—22no—
97C2×C32—21no—
98C2×C32C2×Dic612yes—
99C2×C32C2×D1211yescharacteristic
100C2×C32D3×Q812yes—
101C2×C32C2×#6711yescharacteristic
102C2×Q8—91no—
103C2×Q8—91no—
104C4⋊C4—274no—
105C42—91no—
106C22×C3—31no—
107C22×C3—61no—
108C22×C3—121no—
109C22×C3C2×D3211yescharacteristic
110C3×C4—32no—
111C3×C4—62no—
112C3×C4—94no—
113C3×C4—122no—
114C3×C4—182no—
115C3×C4—182no—
116C3×C4#62612yes—
117Dic3—34no—
118Dic3—62no—
119Dic3—62no—
120Dic3—64no—
121Dic3—62no—
122Dic3—184no—
123Dic3—182no—
124C32—21no—
125C32#86911yescharacteristic, complemented
126C32#94411yescharacteristic, complemented
127Q8—182no—
128Q8—182no—
129C2×C4—92no—
130C2×C4—92no—
131C2×C4—92no—
132C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
133C2×C3—32no—
134C2×C3—31no—
135C2×C3—62no—
136C2×C3—61no—
137C2×C3—122no—
138C2×C3—121no—
139C2×C3D3×Dic612yes—
140C2×C3C2×#62611yescharacteristic
141C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
142C4—94no—
143C4—182no—
144C4—182no—
145C4#620012yes—
146C3—31no—
147C3—61no—
148C3—121no—
149C3#6386611yescharacteristic
150C2((C3⋊D3)⋊C3)⋊Q812yes—
151C2C2×#620011yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
26. (C3⋊Dic3)⋊C3Q8semidirect
67. C3⋊Dic3Dic6semidirect
74. H(3,𝔽3)C4⋊Q8semidirect
125. C32#869semidirect
126. C32#944semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
C2×#634⋊C32
C4×(C3⋊Dic3)⋊C32
#884482
#638733
#638663