Finite Group Databasebeta

← #13146672

Subgroups — #13146672

Order 1728.

323 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 57 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
2#11721298C212yesmaximal
3#11720704C212yesmaximal, complemented
4#11720703C212yesmaximal, complemented
5C22×#1084C2211yescharacteristic
6C2×#6188C414yescomplemented
7C2×#6188C2211yescharacteristic
8C2×#6188C2211yescharacteristic
9#3757⋊C3—22no—
10#88449C2214yes—
11#88447—22no—
12#63955—62no—
13C23×H(3,𝔽3)D411yescharacteristic
14C2×#1084C2×C412yes—
15C2×#1084D411yescharacteristic
16C2×#1084Q812yes—
17C2×#1084C2311yescharacteristic
18C2×(C3⋊Dic3)⋊C3—24no—
19#6188C2×C412yes—
20#6188C2×C412yes—
21#5106—91nomaximal
22C22×C3×Dic3—62no—
23#3757—22no—
24C3×#217—62no—
25Dic3⋊Dic3—64no—
26Dic32—64no—
27C22×H(3,𝔽3)—22no—
28C22×H(3,𝔽3)C4⋊C412yes—
29C22×H(3,𝔽3)C2×D411yescharacteristic
30#1084C4⋊C412yes—
31#1084C22×C411yescharacteristic
32#1084C2×Q811yescharacteristic
33(C3⋊Dic3)⋊C3—44no—
34C22×C3×C8—91no—
35#926—92no—
36#877—182no—
37#818—92no—
38#817—92no—
39C23×C32—22no—
40C2×C3⋊Dic3—24no—
41C2×C3×Dic3—64no—
42C2×C3×Dic3—62no—
43C2×C3×Dic3—64no—
44C2×C3×Dic3—62no—
45C2×C3×Dic3—64no—
46#427—271noSylow
47C2×H(3,𝔽3)—22no—
48C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
49C2×H(3,𝔽3)#13512yes—
50C22×C3×C4—91no—
51C22×Dic3—62no—
52C2×C3×C8—91no—
53C2×C3×C8—91no—
54C2×C3×C8—94no—
55C3×C22⋊C4—182no—
56#217—62no—
57#217—182no—
58C4⋊Dic3—94no—
59Dic3⋊C4—184no—
60C4×Dic3—182no—
61C4×Dic3—184no—
62C22×C32—22no—
63C22×C32—24no—
64C22×C32—44no—
65C3⋊Dic3—44no—
66C3×Dic3—64no—
67C3×Dic3—62no—
68C3×Dic3—62no—
69C3×Dic3—124no—
70C22×C8—91no—
71#117—272no—
72#107—272no—
73#106—272no—
74H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
75C23×C3—62no—
76C23×C3GO+(4,𝔽2)11yescharacteristic
77C2×C3×C4—91no—
78C2×C3×C4—91no—
79C2×C3×C4—92no—
80C2×C3×C4—92no—
81C2×C3×C4—184no—
82C2×Dic3—62no—
83C2×Dic3—62no—
84C2×Dic3—64no—
85C2×Dic3—64no—
86C2×Dic3—64no—
87C2×Dic3—184no—
88C3×C8—92no—
89C3×C8—92no—
90C2×C32—22no—
91C2×C32—24no—
92C2×C32—44no—
93C22×C4—91no—
94C2×C8—91no—
95C2×C8—91no—
96C2×C8—94no—
97C4⋊C4—274no—
98C22⋊C4—182no—
99C42—542no—
100C22×C3—22no—
101C22×C3—64no—
102C22×C3—62no—
103C22×C3—124no—
104C22×C3#377312yes—
105C22×C3C2×GO+(4,𝔽2)11yescharacteristic
106C3×C4—91no—
107C3×C4—92no—
108C3×C4—91no—
109C3×C4—364no—
110Dic3—62no—
111Dic3—64no—
112Dic3—62no—
113Dic3—124no—
114Dic3—364no—
115C32—22no—
116C23#1088⋊C211yescharacteristic
117C2×C4—92no—
118C2×C4—92no—
119C2×C4—91no—
120C2×C4—91no—
121C2×C4—184no—
122C8—92no—
123C8—92no—
124C2×C3—22no—
125C2×C3—64no—
126C2×C3—62no—
127C2×C3—124no—
128C2×C3C2×#377311yescharacteristic
129C2×C3#6423712yes—
130C22—22no—
131C22#8859012yes—
132C22C2×#1088⋊C211yescharacteristic
133C4—91no—
134C4—92no—
135C4—91no—
136C4—364no—
137C3—62no—
138C3#1059351111yescharacteristic
139C2—22no—
140C2#1172231012yes—
141C2C2×#8859011yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator#1084C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
cosocleC2×#1084C23
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#427
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11720704C2semidirect
4. #11720703C2semidirect
6. C2×#6188C4semidirect
74. H(3,𝔽3)#427semidirect

Maximal subgroups

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

subgroupindex
C22×#61882
#117212982
#117207042
#117207032
#51069