Finite Group Databasebeta

← #13136942

Subgroups — #13136942

Order 1728.

331 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 47 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×#88631C212yesmaximal
2C2×#88540C211yesmaximal, characteristic
3C22×C4×H(3,𝔽3)C411yescharacteristic
4C22×#1084C411yescharacteristic
5C22×#1084C2211yescharacteristic
6#88631—22no—
7#88631C414yescomplemented
8C2×#6188—22no—
9#88540—22no—
10C2×C3×Dic3⋊C4—62no—
11C23×H(3,𝔽3)C2×C411yescharacteristic
12C2×C4×H(3,𝔽3)—21no—
13C2×C4×H(3,𝔽3)—41no—
14C2×#1084—21no—
15C2×#1084—41no—
16C2×#1084C2×C412yes—
17C2×#1084D411yescharacteristic
18C2×#1084D412yes—
19C2×#1084Q811yescharacteristic
20#6188—24no—
21C3×#343—91nomaximal
22C22×C32×C4—22no—
23C22×C3×Dic3—62no—
24C22×C3×Dic3—62no—
25C3×Dic3⋊C4—68no—
26C22×H(3,𝔽3)—21no—
27C22×H(3,𝔽3)—21no—
28C22×H(3,𝔽3)C4211yescharacteristic
29C22×H(3,𝔽3)C22⋊C411yescharacteristic
30C22×H(3,𝔽3)C4⋊C411yescharacteristic
31C4×H(3,𝔽3)—41no—
32#1084—41no—
33#1084C22⋊C412yes—
34#1084C4⋊C412yes—
35C2×C3×M16—92no—
36C2×C3×C4⋊C4—91no—
37C2×Dic3⋊C4—62no—
38C23×C32—22no—
39C2×C32×C4—24no—
40C2×C32×C4—44no—
41C2×C3×Dic3—64no—
42C2×C3×Dic3—64no—
43C2×C3×Dic3—62no—
44C2×C3×Dic3—62no—
45C2×C3×Dic3—64no—
46C2×C3×Dic3—124no—
47#343—91noSylow
48C2×H(3,𝔽3)—21no—
49C2×H(3,𝔽3)—21no—
50C2×H(3,𝔽3)#9511yescharacteristic
51C2×H(3,𝔽3)#10011yescharacteristic
52C2×H(3,𝔽3)#10111yescharacteristic
53C22×C3×C4—62no—
54C22×C3×C4—91no—
55C22×C3×C4—91no—
56C22×C3×C4#15711yescharacteristic
57C22×Dic3—62no—
58C22×Dic3—62no—
59C3×M16—94no—
60C3×M16—182no—
61C2×C3×C8—182no—
62C3×C4⋊C4—182no—
63Dic3⋊C4—68no—
64C22×C32—24no—
65C22×C32—24no—
66C22×C32—22no—
67C22×C32—22no—
68C22×C32—22no—
69C32×C4—44no—
70C3×Dic3—64no—
71C3×Dic3—64no—
72C3×Dic3—124no—
73C2×M16—92no—
74C2×C4⋊C4—91no—
75H(3,𝔽3)#34311yescharacteristic, complemented, Sylow
76C23×C3—62no—
77C23×C3C2×#15711yescharacteristic
78C2×C3×C4—21no—
79C2×C3×C4—41no—
80C2×C3×C4—64no—
81C2×C3×C4—92no—
82C2×C3×C4—91no—
83C2×C3×C4—91no—
84C2×C3×C4—92no—
85C2×C3×C4—124no—
86C2×C3×C4—181no—
87C2×C3×C4—361no—
88C2×Dic3—64no—
89C2×Dic3—64no—
90C2×Dic3—62no—
91C2×Dic3—62no—
92C2×Dic3—64no—
93C2×Dic3—124no—
94C3×C8—184no—
95C2×C32—24no—
96C2×C32—24no—
97C2×C32—22no—
98C2×C32—22no—
99C2×C32—22no—
100C22×C4—91no—
101C22×C4—91no—
102C22×C4#108811yescharacteristic
103M16—94no—
104M16—182no—
105C2×C8—182no—
106C4⋊C4—182no—
107C22×C3—21no—
108C22×C3—21no—
109C22×C3—64no—
110C22×C3—64no—
111C22×C3—62no—
112C22×C3—62no—
113C22×C3—62no—
114C22×C3C4×#15711yescharacteristic
115C22×C3C4⋊#15711yescharacteristic
116C22×C3C22⋊#15711yescharacteristic
117C3×C4—41no—
118C3×C4—92no—
119C3×C4—92no—
120C3×C4—124no—
121C3×C4—361no—
122Dic3—64no—
123Dic3—64no—
124Dic3—124no—
125C32—22no—
126C23C2×#108811yescharacteristic
127C2×C4—21no—
128C2×C4—41no—
129C2×C4—92no—
130C2×C4—91no—
131C2×C4—91no—
132C2×C4—92no—
133C2×C4—181no—
134C2×C4—361no—
135C8—184no—
136C2×C3—21no—
137C2×C3—21no—
138C2×C3—64no—
139C2×C3—64no—
140C2×C3—62no—
141C2×C3—62no—
142C2×C3—62no—
143C2×C3#6378011yescharacteristic
144C2×C3#6378111yescharacteristic
145C2×C3#6378811yescharacteristic
146C22—21no—
147C22—21no—
148C22C4×#108811yescharacteristic
149C22C4⋊#108811yescharacteristic
150C22C22⋊#108811yescharacteristic
151C4—41no—
152C4—92no—
153C4—92no—
154C4—361no—
155C3—62no—
156C3#1059020811yescharacteristic
157C2—21no—
158C2—21no—
159C2#1172078611yescharacteristic
160C2#1172078711yescharacteristic
161C2#1172079411yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC22×H(3,𝔽3)C42
FrattiniC23×C3C2×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC22×#1084C22
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#343
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #88631C4semidirect
75. H(3,𝔽3)#343semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C2×#885402
C3×#3439