Finite Group Databasebeta

← #13147362

Subgroups — #13147362

Order 1728.

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

normal subgroups only

This group has 289 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
1#11722445C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4#11720777C212yesmaximal, complemented
5#11720776C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7#88631—21no—
8C2×#6188—21no—
9#88627—21no—
10#88627C2211yescharacteristic
11#88627C2212yes—
12#88627C2211yescharacteristic
13#88626—21no—
14M16×H(3,𝔽3)C411yescharacteristic
15#88538C411yescharacteristic
16#88527C412yes—
17C8×#272C2212yes—
18C3×D3×M16—62no—
19C22×#272Q811yescharacteristic
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×#1084D411yescharacteristic
22C4×#272C2×C411yescharacteristic
23C4×#272D411yescharacteristic
24C4×#272Q811yescharacteristic
25C4×#272C2311yescharacteristic
26#6188—21no—
27#6188—21no—
28#6188—22no—
29C8×H(3,𝔽3)C2×C412yes—
30#6180C2×C412yes—
31C3×#430—91nomaximal
32C2×C3×C4×D3—62no—
33C32×M16—22no—
34C3×#208—62no—
35C3×ΓL(1,𝔽25)—64no—
36C3×D3×C8—64no—
37C22×H(3,𝔽3)C4⋊C411yescharacteristic
38C2×#272—41no—
39C2×#272C4⋊C411yescharacteristic
40C2×#272C2×Q811yescharacteristic
41C4×H(3,𝔽3)C4⋊C411yescharacteristic
42C4×H(3,𝔽3)C22×C411yescharacteristic
43#1084C4⋊C411yescharacteristic
44#1084C2×D411yescharacteristic
45C2×C3×M16—91no—
46C2×C3×M16—91no—
47C2×C3×M16—91no—
48D3×M16—62no—
49C3×#108—92no—
50C3×#108—92no—
51C22×C3×D3—62no—
52C2×C32×C4—22no—
53C2×C3×Dic3—62no—
54C3×C4×D3—62no—
55C3×C4×D3—62no—
56C3×C4×D3—62no—
57C3×C4×D3—62no—
58C32×C8—24no—
59C3×C3⋊C8—64no—
60#430—91noSylow
61C2×H(3,𝔽3)—21no—
62C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
63#272—21no—
64#272—21no—
65C22×C3×C4—91no—
66C2×C4×D3—62no—
67C3×M16—62no—
68C3×M16—91no—
69C3×M16—92no—
70C3×M16—91no—
71C3×M16—92no—
72C3×M16—91no—
73C3×M16—181no—
74C3×M16—181no—
75C3×M16#15711yescharacteristic
76C2×C3×C8—92no—
77C2×C3×C8—181no—
78C2×C3×C8—181no—
79#208—62no—
80ΓL(1,𝔽25)—64no—
81D3×C8—64no—
82C22×C32—22no—
83C2×C3×D3—62no—
84C2×C3×D3—62no—
85C2×C3×D3—124no—
86C32×C4—22no—
87C32×C4—22no—
88C3×Dic3—62no—
89C3×Dic3—62no—
90C2×M16—91no—
91C2×M16—91no—
92C2×M16—91no—
93#108—92no—
94#108—92no—
95H(3,𝔽3)#43011yescharacteristic, complemented, Sylow
96C23×C3—91no—
97C22×D3—62no—
98C2×C3×C4—62no—
99C2×C3×C4—91no—
100C2×C3×C4—91no—
101C2×C3×C4—91no—
102C2×C3×C4—91no—
103C2×C3×C4—91no—
104C2×C3×C4C2×#15711yescharacteristic
105C2×Dic3—62no—
106C4×D3—62no—
107C4×D3—62no—
108C4×D3—62no—
109C4×D3—62no—
110C3×C8—64no—
111C3×C8—92no—
112C3×C8—181no—
113C3×C8—181no—
114C3×C8—182no—
115C3×C8C2×#15712yes—
116C3⋊C8—64no—
117C2×C32—22no—
118C2×C32—42no—
119C3×D3—64no—
120C3×D3—122no—
121C22×C4—91no—
122M16—91no—
123M16—92no—
124M16—91no—
125M16—92no—
126M16—91no—
127M16—181no—
128M16—181no—
129M16#108811yescharacteristic
130C2×C8—92no—
131C2×C8—181no—
132C2×C8—181no—
133C22×C3—62no—
134C22×C3—91no—
135C22×C3—91no—
136C22×C3—361no—
137C22×C3C4⋊#15711yescharacteristic
138C2×D3—62no—
139C2×D3—62no—
140C2×D3—124no—
141C3×C4—62no—
142C3×C4—62no—
143C3×C4—91no—
144C3×C4—91no—
145C3×C4C4⋊#15711yescharacteristic
146C3×C4C22×#15711yescharacteristic
147Dic3—62no—
148Dic3—62no—
149C32—22no—
150C23—91no—
151C2×C4—91no—
152C2×C4—91no—
153C2×C4—91no—
154C2×C4—91no—
155C2×C4—91no—
156C2×C4C2×#108811yescharacteristic
157C8—92no—
158C8—182no—
159C8—181no—
160C8—181no—
161C8C2×#108812yes—
162C2×C3—21no—
163C2×C3—62no—
164C2×C3—122no—
165C2×C3—181no—
166C2×C3—181no—
167C2×C3C2×C4⋊#15711yescharacteristic
168D3—64no—
169D3—122no—
170C22—91no—
171C22—91no—
172C22—361no—
173C22C4⋊#108811yescharacteristic
174C4—91no—
175C4—91no—
176C4C4⋊#108811yescharacteristic
177C4C22×#108811yescharacteristic
178C3—62no—
179C3#1059377611yescharacteristic
180C2—21no—
181C2—181no—
182C2—181no—
183C2C2×C4⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C4⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingM16×H(3,𝔽3)C4
prime radicalC22×#272Q8
cosocleC4×#272C23
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#430
LCSG ▷ C4×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
4. #11720777C2semidirect
5. #11720776C2semidirect
95. H(3,𝔽3)#430semidirect

Maximal subgroups

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

subgroupindex
#117224452
C2×#886272
M16×#2722
#117207772
#117207762
C3×#4309