Finite Group Databasebeta

← #13136922

Subgroups — #13136922

Order 1728.

283 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 43 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⋊#1088C211yesmaximal, characteristic
2#11722445C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4C22×#1088C412yes—
5C2×C4×#272C2211yescharacteristic
6#88631C412yescomplemented
7C4⋊#1088—21no—
8C4⋊#1088—21no—
9#88627—21no—
10#88626—21no—
11M16×H(3,𝔽3)C411yescharacteristic, complemented
12#88538C411yescharacteristic, complemented
13#88527—21no—
14C8×#272—21no—
15C3×D3×M16—62no—
16C22×#272C2×C411yescharacteristic
17C2×#1088—22no—
18C2×#1088—42no—
19C2×C4×H(3,𝔽3)C2×C411yescharacteristic
20C2×#1084C2×C411yescharacteristic
21C4×#272D411yescharacteristic
22C4×#272D411yescharacteristic
23C4×#272D411yescharacteristic
24C4×#272Q811yescharacteristic
25#6188—22no—
26C8×H(3,𝔽3)—21no—
27#6180—21no—
28C3×#343—91nomaximal
29C2×C3×C4×D3—62no—
30C32×M16—22no—
31C3×#208—62no—
32C3×ΓL(1,𝔽25)—64no—
33C3×D3×C8—64no—
34C22×H(3,𝔽3)C4211yescharacteristic
35C2×#272—21no—
36C2×#272—21no—
37C2×#272C22⋊C411yescharacteristic
38C2×#272C4⋊C411yescharacteristic
39#1088—42no—
40C4×H(3,𝔽3)C22⋊C411yescharacteristic
41C4×H(3,𝔽3)C4⋊C411yescharacteristic
42#1084C22⋊C411yescharacteristic
43#1084C4⋊C411yescharacteristic
44C2×C3×M16—91no—
45C2×C3×M16—91no—
46C2×C3×C4⋊C4—91no—
47D3×M16—62no—
48C22×C3×D3—62no—
49C2×C32×C4—22no—
50C2×C3×Dic3—62no—
51C3×C4×D3—62no—
52C3×C4×D3—62no—
53C3×C4×D3—62no—
54C3×C4×D3—62no—
55C32×C8—24no—
56C3×C3⋊C8—64no—
57#343—91noSylow
58C2×H(3,𝔽3)—21no—
59C2×H(3,𝔽3)#9511yescharacteristic
60#272—21no—
61#272#10011yescharacteristic
62#272#10111yescharacteristic
63C22×C3×C4—91no—
64C22×C3×C4—92no—
65C2×C4×D3—62no—
66C3×M16—62no—
67C3×M16—92no—
68C3×M16—91no—
69C3×M16—181no—
70C3×M16—181no—
71C3×M16#15711yescharacteristic
72C2×C3×C8—181no—
73C2×C3×C8—181no—
74C3×C4⋊C4—181no—
75C3×C4⋊C4—181no—
76#208—62no—
77ΓL(1,𝔽25)—64no—
78D3×C8—64no—
79C22×C32—22no—
80C2×C3×D3—62no—
81C2×C3×D3—62no—
82C2×C3×D3—122no—
83C2×C3×D3—122no—
84C32×C4—22no—
85C32×C4—22no—
86C3×Dic3—62no—
87C3×Dic3—62no—
88C2×M16—91no—
89C2×M16—91no—
90C2×C4⋊C4—91no—
91H(3,𝔽3)#34311yescharacteristic, complemented, Sylow
92C23×C3—91no—
93C22×D3—62no—
94C2×C3×C4—62no—
95C2×C3×C4—91no—
96C2×C3×C4—91no—
97C2×C3×C4—91no—
98C2×C3×C4—91no—
99C2×C3×C4—91no—
100C2×C3×C4—182no—
101C2×C3×C4—362no—
102C2×C3×C4C2×#15711yescharacteristic
103C2×Dic3—62no—
104C4×D3—62no—
105C4×D3—62no—
106C4×D3—62no—
107C4×D3—62no—
108C3×C8—21no—
109C3×C8—64no—
110C3×C8—182no—
111C3×C8—181no—
112C3⋊C8—64no—
113C2×C32—22no—
114C2×C32—42no—
115C3×D3—62no—
116C3×D3—62no—
117C3×D3—122no—
118C22×C4—91no—
119C22×C4—92no—
120M16—92no—
121M16—91no—
122M16—181no—
123M16—181no—
124M16#108811yescharacteristic, complemented
125C2×C8—181no—
126C2×C8—181no—
127C4⋊C4—181no—
128C4⋊C4—181no—
129C22×C3—62no—
130C22×C3—91no—
131C22×C3—91no—
132C22×C3—181no—
133C22×C3—181no—
134C22×C3C4×#15711yescharacteristic
135C2×D3—62no—
136C2×D3—62no—
137C2×D3—122no—
138C2×D3—122no—
139C3×C4—62no—
140C3×C4—62no—
141C3×C4—91no—
142C3×C4—91no—
143C3×C4—362no—
144C3×C4C4⋊#15711yescharacteristic
145C3×C4C22⋊#15711yescharacteristic
146Dic3—62no—
147Dic3—62no—
148C32—22no—
149C23—91no—
150C2×C4—91no—
151C2×C4—91no—
152C2×C4—91no—
153C2×C4—91no—
154C2×C4—91no—
155C2×C4—182no—
156C2×C4—362no—
157C2×C4C2×#108811yescharacteristic
158C8—21no—
159C8—182no—
160C8—181no—
161C2×C3—21no—
162C2×C3—62no—
163C2×C3—91no—
164C2×C3—91no—
165C2×C3—122no—
166C2×C3—181no—
167C2×C3#6378111yescharacteristic
168D3—62no—
169D3—62no—
170D3—122no—
171C22—91no—
172C22—91no—
173C22—181no—
174C22—181no—
175C22C4×#108811yescharacteristic
176C4—91no—
177C4—91no—
178C4—362no—
179C4C4⋊#108811yescharacteristic
180C4C22⋊#108811yescharacteristic
181C3—62no—
182C3#1059018811yescharacteristic
183C2—21no—
184C2—91no—
185C2—91no—
186C2—181no—
187C2#1172078711yescharacteristic
subgroupG/H
center, socleC2×C3#63781
commutatorC22×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingM16×H(3,𝔽3)C4
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#343
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
6. #88631C4semidirect
11. M16×H(3,𝔽3)C4semidirect
12. #88538C4semidirect
91. H(3,𝔽3)#343semidirect
124. M16#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
#117224452
M16×#2722
C3×#3439