Finite Group Databasebeta

← #11721099

Subgroups — #11721099

Order 864.

268 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 64 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×#634⋊C3C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic
3C4×#1084C211yesmaximal, characteristic
4#88449C214yesmaximal
5#63866—31nomaximal
6#63866—31nomaximal
7C2×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#1084C2212yes—
9C2×(C3⋊Dic3)⋊C3C2212yes—
10C2×(C3⋊Dic3)⋊C3C2212yes—
11#634⋊C3—22no—
12#634⋊C3—22no—
13C2×#634D311yescharacteristic
14C2×#634D311yescharacteristic
15C2×C3×Dic6—31no—
16C2×C3×Dic6—31no—
17C3×C4×Dic3—31no—
18C3×C4×Dic3—31no—
19C3×C4×Dic3—61no—
20Dic3⋊Dic3—34no—
21Dic3⋊Dic3—34no—
22C22×H(3,𝔽3)C2311yescharacteristic
23C4×H(3,𝔽3)D412yes—
24#1084Q814yescomplemented
25(C3⋊Dic3)⋊C3—22no—
26(C3⋊Dic3)⋊C3—22no—
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—
36#634—22no—
37C2×C3×Dic3—32no—
38C2×C3×Dic3—32no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—32no—
41C2×C3×Dic3—62no—
42C3×Dic6—62no—
43C3×Dic6—62no—
44C2×H(3,𝔽3)C2×D411yescharacteristic
45C2×H(3,𝔽3)C2×Q812yes—
46C2×C3×Q8—91no—
47C2×C3×Q8—91no—
48C2×Dic6—31no—
49C2×Dic6—31no—
50C2×Dic6—91no—
51C2×Dic6—181no—
52C3×C42—91no—
53C4⋊Dic3—94no—
54Dic3⋊C4—94no—
55Dic3⋊C4—94no—
56C4×Dic3—31no—
57C4×Dic3—31no—
58C4×Dic3—61no—
59C22×C32—21no—
60C22×C32C22×D311yescharacteristic
61C22×C32C22×D311yescharacteristic
62C32×C4—22no—
63C32×C4#6712yes—
64C32×C4#6712yes—
65C3⋊Dic3—22no—
66C3⋊Dic3—22no—
67C3×Dic3—34no—
68C3×Dic3—34no—
69C3×Dic3—64no—
70C3×Dic3—62no—
71C3×Dic3—62no—
72C4⋊Q8—271noSylow
73H(3,𝔽3)C4⋊Q811yescharacteristic, complemented, Sylow
74C3×Q8—182no—
75C3×Q8—182no—
76C2×C3×C4—31no—
77C2×C3×C4—61no—
78C2×C3×C4—61no—
79C2×C3×C4—92no—
80C2×C3×C4—92no—
81C2×C3×C4—92no—
82C2×C3×C4D3211yescharacteristic
83C2×Dic3—32no—
84C2×Dic3—32no—
85C2×Dic3—32no—
86C2×Dic3—32no—
87C2×Dic3—62no—
88C2×Dic3—92no—
89C2×Dic3—182no—
90Dic6—62no—
91Dic6—62no—
92Dic6—182no—
93Dic6—182no—
94C2×C32—22no—
95C2×C32—21no—
96C2×C32D3×Q812yes—
97C2×C32D3×Q812yes—
98C2×C32C2×#6711yescharacteristic
99C2×C32C2×#6711yescharacteristic
100C2×Q8—91no—
101C2×Q8—91no—
102C4⋊C4—274no—
103C42—91no—
104C22×C3—31no—
105C22×C3—61no—
106C22×C3—61no—
107C22×C3C2×D3211yescharacteristic
108C3×C4—32no—
109C3×C4—62no—
110C3×C4—62no—
111C3×C4—94no—
112C3×C4—182no—
113C3×C4—182no—
114C3×C4#62512yes—
115Dic3—34no—
116Dic3—34no—
117Dic3—64no—
118Dic3—62no—
119Dic3—62no—
120Dic3—182no—
121Dic3—182no—
122C32—21no—
123C32#94411yescharacteristic, complemented
124C32#94411yescharacteristic, complemented
125Q8—182no—
126Q8—182no—
127C2×C4—92no—
128C2×C4—92no—
129C2×C4—92no—
130C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
131C2×C3—32no—
132C2×C3—31no—
133C2×C3—62no—
134C2×C3—62no—
135C2×C3—61no—
136C2×C3—61no—
137C2×C3#379712yes—
138C2×C3C2×#62511yescharacteristic
139C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
140C4—94no—
141C4—182no—
142C4—182no—
143C4#619812yes—
144C3—31no—
145C3—61no—
146C3—61no—
147C3#6387311yescharacteristic
148C2#8865112yes—
149C2C2×#619811yescharacteristic
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
24. #1084Q8semidirect
73. H(3,𝔽3)C4⋊Q8semidirect
123. C32#944semidirect
124. C32#944semidirect

Maximal subgroups

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

subgroupindex
C2×#634⋊C32
C4×#10842
#884492
#638663