Finite Group Databasebeta

← #13147329

Subgroups — #13147329

Order 1728.

323 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 63 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×#88631C211yesmaximal, characteristic
2C22×#6188C211yesmaximal, characteristic
3#11721733C211yesmaximal, characteristic
4#11720785C212yesmaximal, complemented
5C4⋊#6188C212yesmaximal, complemented
6C22×#1084C2211yescharacteristic
7#88631—22no—
8C2×#6188—22no—
9C2×#6188C2211yescharacteristic
10C2×#6188C2212yes—
11C2×#6188C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)C411yescharacteristic
13#88552C411yescharacteristic
14#88540C412yes—
15C4×#1084C2212yes—
16C3×#877—62no—
17C23×H(3,𝔽3)C2×C411yescharacteristic
18C2×C4×H(3,𝔽3)C2×C412yes—
19C2×#1084C2×C412yes—
20C2×#1084C2×C411yescharacteristic
21C2×#1084D412yes—
22C2×#1084Q812yes—
23C2×#1084C2311yescharacteristic
24#6188—22no—
25#6188—21no—
26#6188—21no—
27C3×#424—91nomaximal
28C22×C3×Dic3—62no—
29C32×C22⋊C4—22no—
30C3×#217—62no—
31C3×Dic3⋊C4—64no—
32C3×C4×Dic3—64no—
33C22×H(3,𝔽3)—22no—
34C22×H(3,𝔽3)C4⋊C412yes—
35C22×H(3,𝔽3)C22×C411yescharacteristic
36C4×H(3,𝔽3)—22no—
37#1084—22no—
38#1084C4⋊C412yes—
39#1084C2×D411yescharacteristic
40#1084C2×Q811yescharacteristic
41C2×C3×M16—91no—
42C22×C3×C8—91no—
43C3×#117—91no—
44#877—62no—
45C3×C4⋊C8—92no—
46C3×C4⋊C8—92no—
47C23×C32—22no—
48C2×C32×C4—24no—
49C2×C3×Dic3—62no—
50C2×C3×Dic3—64no—
51C2×C3×Dic3—64no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—64no—
54#424—91noSylow
55C2×H(3,𝔽3)—22no—
56C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
57C2×H(3,𝔽3)#13112yes—
58C22×C3×C4—91no—
59C22×Dic3—62no—
60C3×M16—182no—
61C2×C3×C8—92no—
62C2×C3×C8—91no—
63C2×C3×C8—91no—
64C2×C3×C8—182no—
65C3×C4⋊C4—92no—
66C3×C22⋊C4—62no—
67C3×C22⋊C4—91no—
68C3×C22⋊C4#15711yescharacteristic
69C3×C42—92no—
70#217—62no—
71Dic3⋊C4—64no—
72C4×Dic3—64no—
73C22×C32—22no—
74C22×C32—24no—
75C22×C32—44no—
76C32×C4—44no—
77C3×Dic3—64no—
78C3×Dic3—62no—
79C3×Dic3—62no—
80C3×Dic3—124no—
81C2×M16—91no—
82C22×C8—91no—
83#117—91no—
84C4⋊C8—92no—
85C4⋊C8—92no—
86H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
87C23×C3—62no—
88C23×C3C2×#15711yescharacteristic
89C2×C3×C4—64no—
90C2×C3×C4—91no—
91C2×C3×C4—91no—
92C2×C3×C4—92no—
93C2×C3×C4—92no—
94C2×C3×C4—92no—
95C2×C3×C4C2×#15712yes—
96C2×Dic3—62no—
97C2×Dic3—62no—
98C2×Dic3—64no—
99C2×Dic3—64no—
100C2×Dic3—64no—
101C3×C8—182no—
102C3×C8—181no—
103C3×C8—181no—
104C2×C32—22no—
105C2×C32—24no—
106C2×C32—44no—
107C22×C4—91no—
108M16—182no—
109C2×C8—92no—
110C2×C8—91no—
111C2×C8—91no—
112C2×C8—182no—
113C4⋊C4—92no—
114C22⋊C4—91no—
115C22⋊C4#108811yescharacteristic
116C42—92no—
117C22×C3—22no—
118C22×C3—64no—
119C22×C3—62no—
120C22×C3—124no—
121C22×C3C4⋊#15712yes—
122C22×C3C22×#15711yescharacteristic
123C3×C4—22no—
124C3×C4—91no—
125C3×C4—91no—
126C3×C4—92no—
127C3×C4—124no—
128C3×C4—182no—
129Dic3—62no—
130Dic3—62no—
131Dic3—64no—
132Dic3—124no—
133C32—22no—
134C23C2×#108811yescharacteristic
135C2×C4—92no—
136C2×C4—92no—
137C2×C4—91no—
138C2×C4—91no—
139C2×C4—92no—
140C2×C4C2×#108812yes—
141C8—182no—
142C8—181no—
143C8—181no—
144C2×C3—22no—
145C2×C3—62no—
146C2×C3—64no—
147C2×C3—124no—
148C2×C3C2×C4⋊#15711yescharacteristic
149C2×C3#6428712yes—
150C22—22no—
151C22C4⋊#108812yes—
152C22C22×#108811yescharacteristic
153C4—22no—
154C4—91no—
155C4—92no—
156C4—91no—
157C4—182no—
158C3—62no—
159C3#1059374311yescharacteristic
160C2—22no—
161C2C2×C4⋊#108811yescharacteristic
162C2#1172244912yes—
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22⋊C4×H(3,𝔽3)C4
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×#1084C23
hypercenterC3×C22⋊C4#157
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #11720785C2semidirect
5. C4⋊#6188C2semidirect
86. H(3,𝔽3)#424semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C22×#61882
#117217332
#117207852
C4⋊#61882
C3×#4249