Finite Group Databasebeta

← #13144767

Subgroups — #13144767

Order 1728.

605 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 111 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×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721863C212yesmaximal
3C2×#88527C211yesmaximal, characteristic
4#11720583C212yesmaximal, complemented
5#11720550C211yesmaximal, characteristic, complemented
6C3×#5189—34nomaximal
7C2×Q8×H(3,𝔽3)C2212yes—
8C2×C4×#272C2211yescharacteristic
9Q16×H(3,𝔽3)—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88542—22no—
12C4⋊#1084C2212yes—
13#88540—22no—
14C2×#6180C2211yescharacteristic
15#88527—22no—
16C2×C32×Q16D314yescomplemented
17C3×#946—38no—
18C2×C3×ΓL(1,𝔽25)—34no—
19C3×#835—38no—
20C3×#817—34no—
21C22×#272D411yescharacteristic
22Q8×H(3,𝔽3)—22no—
23Q8×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—22no—
27C2×#1084D411yescharacteristic
28C4×#272—21no—
29C4×#272—21no—
30C8×H(3,𝔽3)D412yes—
31#6180—21no—
32C3×#470—91no—
33#5189—34no—
34C2×C32×Q8C2×D318yes—
35C2×C3×C4×D3—34no—
36C32×Q16—28no—
37C2×C32×C8C2×D314yes—
38C3×(C2×D3)⋊C4—68no—
39C3×C4⋊Dic3—38no—
40C3×Dic3⋊C4—68no—
41C2×C3×C3⋊C8—34no—
42C3×ΓL(1,𝔽25)—68no—
43C22×H(3,𝔽3)C2×D411yescharacteristic
44C2×#272—21no—
45C2×#272—22no—
46C4×H(3,𝔽3)—42no—
47C4×H(3,𝔽3)C2×D411yescharacteristic
48C4×H(3,𝔽3)#4111yescharacteristic
49#1084—21no—
50#1084—42no—
51C2×C3×Q16—34no—
52C2×C3×Q16C3⋊D311yescharacteristic
53C2×C3×M16—91no—
54C3×C22⋊Q8—92no—
55#946—38no—
56C2×ΓL(1,𝔽25)—34no—
57C3×#106—91no—
58C3×#103—92no—
59#835—38no—
60#817—34no—
61C22×C3×D3—34no—
62C32×Q8—28no—
63C32×Q8—28no—
64C2×C32×C4—28no—
65C2×C32×C4C22×D314yes—
66C2×C3×Dic3—34no—
67C2×C3×Dic3—68no—
68C3×C4×D3—64no—
69C3×C4×D3—64no—
70C32×C8#6718yes—
71C3×C3⋊C8—64no—
72#470—91noSylow
73C2×H(3,𝔽3)#12111yescharacteristic
74C2×H(3,𝔽3)#13712yes—
75#272—41no—
76C2×C3×Q8—38no—
77C2×C3×Q8C2×C3⋊D312yes—
78C22×C3×C4—91no—
79C2×C4×D3—34no—
80C3×Q16—22no—
81C3×Q16—68no—
82C3×M16—182no—
83C2×C3×C8—34no—
84C2×C3×C8—91no—
85C2×C3×C8C2×C3⋊D311yescharacteristic
86C3×C4⋊C4—92no—
87C3×C4⋊C4—182no—
88C3×C22⋊C4—182no—
89(C2×D3)⋊C4—68no—
90C4⋊Dic3—38no—
91Dic3⋊C4—68no—
92C2×C3⋊C8—34no—
93ΓL(1,𝔽25)—68no—
94C22×C32D3×D414yes—
95C2×C3×D3—68no—
96C2×C3×D3—64no—
97C32×C4—48no—
98C32×C4#23714yes—
99C32×C4C2×#6714yes—
100C3×Dic3—64no—
101C3×Dic3—128no—
102C2×Q16#27211yescharacteristic, complemented
103C2×M16—91no—
104C22⋊Q8—92no—
105#106—91no—
106#103—92no—
107H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—34no—
110C3×Q8—22no—
111C3×Q8—22no—
112C3×Q8—68no—
113C3×Q8—68no—
114C2×C3×C4—22no—
115C2×C3×C4—34no—
116C2×C3×C4—68no—
117C2×C3×C4—91no—
118C2×C3×C4—181no—
119C2×C3×C4—181no—
120C2×C3×C4—182no—
121C2×C3×C4C22×C3⋊D311yescharacteristic
122C2×Dic3—34no—
123C2×Dic3—68no—
124C4×D3—64no—
125C4×D3—64no—
126C3×C8—38no—
127C3×C8—181no—
128C3×C8#63812yes—
129C3⋊C8—64no—
130C2×C32#91818yes—
131C2×C32#93814yes—
132C3×D3—124no—
133C2×Q8C2×#27212yes—
134C22×C4—91no—
135Q16—22no—
136M16—182no—
137C2×C8—91no—
138C2×C8C2×#27211yescharacteristic
139C4⋊C4—92no—
140C4⋊C4—182no—
141C22⋊C4—182no—
142C22×C3—34no—
143C22×C3—182no—
144C22×C3—181no—
145C22×C3D4×C3⋊D311yescharacteristic
146C2×D3—64no—
147C2×D3—68no—
148C3×C4—34no—
149C3×C4—34no—
150C3×C4—42no—
151C3×C4—128no—
152C3×C4—181no—
153C3×C4—362no—
154C3×C4#383011yescharacteristic
155C3×C4C2×#63811yescharacteristic
156Dic3—64no—
157Dic3—128no—
158C32#518914yescomplemented
159C23—91no—
160Q8—22no—
161Q8—22no—
162C2×C4—22no—
163C2×C4—91no—
164C2×C4—181no—
165C2×C4—181no—
166C2×C4—182no—
167C2×C4C22×#27211yescharacteristic
168C8—181no—
169C8#623512yes—
170C2×C3—38no—
171C2×C3—34no—
172C2×C3—361no—
173C2×C3#6412712yes—
174C2×C3#6414711yescharacteristic
175D3—124no—
176C22—182no—
177C22—181no—
178C22D4×#27211yescharacteristic
179C4—42no—
180C4—181no—
181C4—362no—
182C4#8874411yescharacteristic
183C4C2×#623511yescharacteristic
184C3—34no—
185C3#1059288811yescharacteristic
186C2—361no—
187C2#1172180712yes—
188C2#1172184811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×Q16×H(3,𝔽3)C2
prime radicalC22×#272D4
hypercenterC2×C3×Q16C3⋊D3
hypocenterH(3,𝔽3)#470
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×H(3,𝔽3)C2semidirect
4. #11720583C2semidirect
5. #11720550C2semidirect
16. C2×C32×Q16D3semidirect
102. C2×Q16#272semidirect
107. H(3,𝔽3)#470semidirect
158. C32#5189semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×H(3,𝔽3)2
#117218632
C2×#885272
#117205832
#117205502
C3×#51893