Finite Group Databasebeta

← C2×#11720703 #13146668

Subgroups — C2×#11720703 #13146668

Order 1728.

379 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×#6188C211yesmaximal, characteristic
2C2×#88449C212yesmaximal
3#11720703C214yesmaximal, direct product factor
4C22×#1084C2211yescharacteristic
5C22×(C3⋊Dic3)⋊C3—22no—
6C2×#6188C414yescomplemented
7C2×#6188C2212yes—
8#88449—22no—
9#88449C2214yes—
10C2×Dic3⋊Dic3—62no—
11C23×H(3,𝔽3)D411yescharacteristic
12C2×#1084C2×C412yes—
13C2×#1084D411yescharacteristic
14C2×#1084Q812yes—
15C2×#1084C2311yescharacteristic
16C2×(C3⋊Dic3)⋊C3—24no—
17C2×(C3⋊Dic3)⋊C3—44no—
18#6188C2×C414yescomplemented
19C2×#817—91nomaximal
20C22×C3⋊Dic3—22no—
21C22×C3×Dic3—62no—
22C22×C3×Dic3—62no—
23Dic3⋊Dic3—64no—
24Dic3⋊Dic3—64no—
25C22×H(3,𝔽3)C4⋊C412yes—
26C22×H(3,𝔽3)SD1614yes—
27C22×H(3,𝔽3)C2×D411yescharacteristic
28#1084C4⋊C412yes—
29#1084C22×C411yescharacteristic
30#1084C2×Q811yescharacteristic
31(C3⋊Dic3)⋊C3—44no—
32C22×C3×C8—91no—
33C2×C4⋊Dic3—92no—
34C2×Dic3⋊C4—182no—
35#817—94no—
36C23×C32—22no—
37C2×C3⋊Dic3—24no—
38C2×C3⋊Dic3—44no—
39C2×C3×Dic3—64no—
40C2×C3×Dic3—62no—
41C2×C3×Dic3—64no—
42C2×C3×Dic3—62no—
43C2×C3×Dic3—64no—
44C2×C3×Dic3—124no—
45C2×#106—271noSylow
46C2×H(3,𝔽3)#10614yescomplemented
47C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
48C2×H(3,𝔽3)C2×SD1612yes—
49C22×C3×C4—91no—
50C22×C3×C4—182no—
51C22×Dic3—62no—
52C22×Dic3—62no—
53C22×Dic3—182no—
54C2×C3×C8—92no—
55C2×C3×C8—94no—
56C4⋊Dic3—94no—
57C4⋊Dic3—182no—
58Dic3⋊C4—184no—
59Dic3⋊C4—184no—
60C22×C32—22no—
61C22×C32—24no—
62C22×C32—24no—
63C22×C32—24no—
64C3⋊Dic3—44no—
65C3×Dic3—64no—
66C3×Dic3—62no—
67C3×Dic3—62no—
68C3×Dic3—124no—
69C22×C8—91no—
70C2×C4⋊C4—272no—
71#106—274no—
72H(3,𝔽3)C2×#10611yescharacteristic, complemented, Sylow
73C23×C3—62no—
74C23×C3GO+(4,𝔽2)11yescharacteristic
75C2×C3×C4—91no—
76C2×C3×C4—91no—
77C2×C3×C4—92no—
78C2×C3×C4—92no—
79C2×C3×C4—184no—
80C2×C3×C4—364no—
81C2×Dic3—62no—
82C2×Dic3—62no—
83C2×Dic3—64no—
84C2×Dic3—64no—
85C2×Dic3—64no—
86C2×Dic3—124no—
87C2×Dic3—184no—
88C2×Dic3—364no—
89C3×C8—94no—
90C2×C32—22no—
91C2×C32—22no—
92C2×C32—22no—
93C2×C32—28no—
94C22×C4—91no—
95C22×C4—182no—
96C2×C8—92no—
97C2×C8—94no—
98C4⋊C4—274no—
99C4⋊C4—542no—
100C22×C3—64no—
101C22×C3—64no—
102C22×C3—64no—
103C22×C3—62no—
104C22×C3#377312yes—
105C22×C3#377514yes—
106C22×C3C2×GO+(4,𝔽2)11yescharacteristic
107C3×C4—91no—
108C3×C4—91no—
109C3×C4—92no—
110C3×C4—364no—
111Dic3—62no—
112Dic3—64no—
113Dic3—62no—
114Dic3—124no—
115Dic3—364no—
116C32—22no—
117C23#1088⋊C211yescharacteristic
118C2×C4—92no—
119C2×C4—92no—
120C2×C4—91no—
121C2×C4—91no—
122C2×C4—184no—
123C2×C4—364no—
124C8—94no—
125C2×C3—62no—
126C2×C3—62no—
127C2×C3—62no—
128C2×C3—68no—
129C2×C3#6374214yes—
130C2×C3C2×#377311yescharacteristic
131C2×C3C2×#377512yes—
132C22#8858714yes—
133C22#8859012yes—
134C22C2×#1088⋊C211yescharacteristic
135C4—91no—
136C4—92no—
137C4—91no—
138C4—364no—
139C3—62no—
140C3C2×#6374211yescharacteristic
141C2#1172070314yesdirect product factor
142C2C2×#8858712yes—
143C2C2×#8859011yescharacteristic
subgroupG/H
center, hypercenterC23#1088⋊C2
commutator#1084C22×C4
FrattiniC22×C3C2×GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
socleC23×C3GO+(4,𝔽2)
cosocleC2×#1084C23
hypocenterH(3,𝔽3)C2×#106
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11720703C2direct (N × H)
6. C2×#6188C4semidirect
18. #6188C2×C4semidirect
46. C2×H(3,𝔽3)#106semidirect
72. H(3,𝔽3)C2×#106semidirect
141. C2#11720703direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61882
C2×#884492
#117207032
C2×#8179