Finite Group Databasebeta

← #13136916

Subgroups — #13136916

Order 1728.

323 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 55 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×C4⋊#1088C212yesmaximal
2C2×C8×#272C211yesmaximal, characteristic
3C22×#1088—22no—
4C2×C4×#272C2211yescharacteristic
5C4⋊#1088—22no—
6C4⋊#1088C414yescomplemented
7C2×C8×H(3,𝔽3)C411yescharacteristic, complemented
8C2×#6180C411yescharacteristic, complemented
9C8×#272—22no—
10C2×C3×D3×C8—62no—
11C22×#272D411yescharacteristic
12C2×#1088—24no—
13C2×#1088—42no—
14C2×#1088—42no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×#1084Q811yescharacteristic
17C4×#272C2×C412yes—
18C4×#272D412yes—
19C8×H(3,𝔽3)—21no—
20#6180—21no—
21C3×#340—91nomaximal
22C2×C3×C4×D3—62no—
23C2×C32×C8—22no—
24C2×C3×C3⋊C8—62no—
25C3×D3×C8—68no—
26C22×H(3,𝔽3)C4⋊C411yescharacteristic
27C2×#272C22⋊C412yes—
28C2×#272D811yescharacteristic
29C2×#272SD1612yes—
30C2×#272Q1611yescharacteristic
31#1088—44no—
32C4×H(3,𝔽3)C4211yescharacteristic
33C4×H(3,𝔽3)C22⋊C411yescharacteristic
34#1084C4⋊C412yes—
35C22×C3×C8—91no—
36C2×C3×C4⋊C4—92no—
37C2×D3×C8—62no—
38C22×C3×D3—62no—
39C2×C32×C4—22no—
40C2×C3×Dic3—62no—
41C3×C4×D3—64no—
42C3×C4×D3—64no—
43C32×C8—24no—
44C3×C3⋊C8—64no—
45#340—91noSylow
46C2×H(3,𝔽3)#9511yescharacteristic
47C2×H(3,𝔽3)#10611yescharacteristic
48C2×H(3,𝔽3)#10711yescharacteristic
49#272#10212yes—
50#272#10312yes—
51C22×C3×C4—91no—
52C22×C3×C4—182no—
53C2×C4×D3—62no—
54C2×C3×C8—62no—
55C2×C3×C8—91no—
56C2×C3×C8—182no—
57C2×C3×C8#15711yescharacteristic
58C3×C4⋊C4—94no—
59C3×C4⋊C4—182no—
60C2×C3⋊C8—62no—
61D3×C8—68no—
62C22×C32—22no—
63C2×C3×D3—64no—
64C2×C3×D3—62no—
65C2×C3×D3—64no—
66C2×C3×D3—62no—
67C32×C4—22no—
68C32×C4—22no—
69C3×Dic3—64no—
70C22×C8—91no—
71C2×C4⋊C4—92no—
72H(3,𝔽3)#34011yescharacteristic, complemented, Sylow
73C23×C3—91no—
74C22×D3—62no—
75C2×C3×C4—62no—
76C2×C3×C4—91no—
77C2×C3×C4—92no—
78C2×C3×C4—92no—
79C2×C3×C4—184no—
80C2×C3×C4—362no—
81C2×C3×C4—362no—
82C2×C3×C4C2×#15711yescharacteristic
83C2×Dic3—62no—
84C4×D3—64no—
85C4×D3—64no—
86C3×C8—21no—
87C3×C8—64no—
88C3×C8—181no—
89C3⋊C8—64no—
90C2×C32—22no—
91C2×C32—22no—
92C2×C32—22no—
93C3×D3—64no—
94C3×D3—64no—
95C22×C4—91no—
96C22×C4—182no—
97C2×C8—91no—
98C2×C8—182no—
99C2×C8#108811yescharacteristic, complemented
100C4⋊C4—94no—
101C4⋊C4—182no—
102C22×C3—62no—
103C22×C3—92no—
104C22×C3—91no—
105C22×C3—91no—
106C22×C3—92no—
107C22×C3C4⋊#15711yescharacteristic
108C2×D3—64no—
109C2×D3—64no—
110C2×D3—62no—
111C2×D3—62no—
112C3×C4—62no—
113C3×C4—62no—
114C3×C4—92no—
115C3×C4—364no—
116C3×C4C4×#15711yescharacteristic
117C3×C4C22⋊#15711yescharacteristic
118Dic3—64no—
119C32—22no—
120C23—91no—
121C2×C4—91no—
122C2×C4—92no—
123C2×C4—92no—
124C2×C4—184no—
125C2×C4—362no—
126C2×C4—362no—
127C2×C4C2×#108811yescharacteristic
128C8—21no—
129C8—181no—
130C2×C3—62no—
131C2×C3—62no—
132C2×C3—62no—
133C2×C3—92no—
134C2×C3—92no—
135C2×C3#6376811yescharacteristic
136C2×C3#6376911yescharacteristic
137C2×C3#6378111yescharacteristic
138D3—64no—
139D3—64no—
140C22—92no—
141C22—91no—
142C22—91no—
143C22—92no—
144C22C4⋊#108811yescharacteristic
145C4—92no—
146C4—364no—
147C4C4×#108811yescharacteristic
148C4C22⋊#108811yescharacteristic
149C3—62no—
150C3#1059018211yescharacteristic
151C2—92no—
152C2—92no—
153C2#1172077411yescharacteristic
154C2#1172077511yescharacteristic
155C2#1172078711yescharacteristic
subgroupG/H
center, socleC22×C3C4⋊#157
commutatorC4×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingC2×C8×H(3,𝔽3)C4
prime radicalC22×#272D4
cosocleC2×C4×#272C22
hypercenterC2×C3×C8#157
hypocenterH(3,𝔽3)#340
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
6. C4⋊#1088C4semidirect
7. C2×C8×H(3,𝔽3)C4semidirect
8. C2×#6180C4semidirect
72. H(3,𝔽3)#340semidirect
99. C2×C8#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
C2×C8×#2722
C3×#3409