Finite Group Databasebeta

← #13138916

Subgroups — #13138916

Order 1728.

475 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 91 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×C8×#272C211yesmaximal, characteristic
2#11721089C212yesmaximal
3#63644⋊C3C212yesmaximal, complemented
4#11720315C212yesmaximal, complemented
5#10590706—32nomaximal
6C2×C4×#272C2211yescharacteristic
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C2×#6180C2211yescharacteristic
9C8×#272C414yescomplemented
10C4⋊((C3⋊Dic3)⋊C3)C2214yes—
11#88451—22no—
12#88447—22no—
13C2×C3×D3×C8—32no—
14C2×C3×D3×C8—61no—
15#63841—34no—
16#63644D312yescomplemented
17C3×#817—32no—
18#63575—34no—
19C22×#272D411yescharacteristic
20C2×C4×H(3,𝔽3)C2311yescharacteristic
21C2×#1084D411yescharacteristic
22C4×#272C2×C412yes—
23C4×#272Q812yes—
24C2×(C3⋊Dic3)⋊C3—24no—
25C8×H(3,𝔽3)C2×C412yes—
26#6180C2×C412yes—
27#5106—91no—
28#4860—92no—
29C2×C3×C4×D3—32no—
30C2×C3×C4×D3—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D312yes—
33C4⋊(C3⋊Dic3)C2×D314yes—
34C3×C4⋊Dic3—34no—
35C2×C3×C3⋊C8—32no—
36C2×C3×C3⋊C8—61no—
37C3×D3×C8—38no—
38C3×D3×C8—64no—
39(C2×D3)⋊Dic3—64no—
40Dic32—64no—
41C22×H(3,𝔽3)C2×D411yescharacteristic
42C2×#272—22no—
43C2×#272C4⋊C412yes—
44C4×H(3,𝔽3)C22×C411yescharacteristic
45C4×H(3,𝔽3)C2×Q811yescharacteristic
46#1084C4⋊C412yes—
47(C3⋊Dic3)⋊C3—44no—
48C22×C3×C8—91no—
49#926—92no—
50C2×D3×C8—32no—
51C2×D3×C8—61no—
52#892—94no—
53C3×#106—92no—
54#818—92no—
55#817—32no—
56#817—92no—
57#807—94no—
58C22×C3×D3—32no—
59C22×C3×D3—61no—
60C2×C32×C4—21no—
61C2×C32×C4C22×D312yes—
62C2×C3⋊Dic3—24no—
63C2×C3×Dic3—32no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—64no—
66C3×C4×D3—34no—
67C3×C4×D3—34no—
68C3×C4×D3—62no—
69C3×C4×D3—62no—
70C32×C8—22no—
71C32×C8C4×D314yes—
72C3×C3⋊C8—34no—
73C3×C3⋊C8—62no—
74#427—271noSylow
75C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
76C2×H(3,𝔽3)#13512yes—
77#272—22no—
78C22×C3×C4—91no—
79C2×C4×D3—32no—
80C2×C4×D3—61no—
81C2×C3×C8—32no—
82C2×C3×C8—61no—
83C2×C3×C8—91no—
84C2×C3×C8—94no—
85C2×C3×C8D3211yescharacteristic
86C3×C4⋊C4—94no—
87#217—182no—
88(C2×D3)⋊C4—184no—
89C4⋊Dic3—34no—
90C4⋊Dic3—94no—
91C4×Dic3—184no—
92C4×Dic3—182no—
93C2×C3⋊C8—32no—
94C2×C3⋊C8—61no—
95D3×C8—38no—
96D3×C8—64no—
97C22×C32—21no—
98C22×C32D3×D412yes—
99C2×C3×D3—34no—
100C2×C3×D3—62no—
101C2×C3×D3—64no—
102C2×C3×D3—64no—
103C32×C4—21no—
104C32×C4—21no—
105C32×C4C2×C4×D312yes—
106C32×C4D3×Q812yes—
107C3⋊Dic3—44no—
108C3×Dic3—34no—
109C3×Dic3—62no—
110C3×Dic3—124no—
111C22×C8—91no—
112#117—272no—
113#107—272no—
114#106—92no—
115H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
116C23×C3—91no—
117C22×D3—32no—
118C22×D3—61no—
119C2×C3×C4—32no—
120C2×C3×C4—61no—
121C2×C3×C4—91no—
122C2×C3×C4—92no—
123C2×C3×C4—92no—
124C2×C3×C4—184no—
125C2×C3×C4C2×D3211yescharacteristic
126C2×Dic3—32no—
127C2×Dic3—61no—
128C2×Dic3—64no—
129C2×Dic3—184no—
130C4×D3—34no—
131C4×D3—34no—
132C4×D3—62no—
133C4×D3—62no—
134C3×C8—34no—
135C3×C8—62no—
136C3×C8—92no—
137C3×C8#62412yes—
138C3⋊C8—34no—
139C3⋊C8—62no—
140C2×C32—22no—
141C2×C32—21no—
142C2×C32D3×C4⋊C412yes—
143C2×C32#91614yes—
144C3×D3—64no—
145C3×D3—64no—
146C22×C4—91no—
147C2×C8—91no—
148C2×C8—94no—
149C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
150C4⋊C4—94no—
151C22⋊C4—542no—
152C42—542no—
153C22×C3—32no—
154C22×C3—61no—
155C22×C3—92no—
156C22×C3—182no—
157C22×C3#380211yescharacteristic
158C2×D3—34no—
159C2×D3—62no—
160C2×D3—64no—
161C2×D3—64no—
162C3×C4—32no—
163C3×C4—32no—
164C3×C4—61no—
165C3×C4—61no—
166C3×C4—92no—
167C3×C4—364no—
168C3×C4#379711yescharacteristic
169C3×C4C2×#62411yescharacteristic
170Dic3—34no—
171Dic3—62no—
172Dic3—124no—
173Dic3—364no—
174C32—21no—
175C32#486012yescomplemented
176C23—91no—
177C2×C4—91no—
178C2×C4—92no—
179C2×C4—92no—
180C2×C4—184no—
181C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
182C8—92no—
183C8#619912yes—
184C2×C3—34no—
185C2×C3—32no—
186C2×C3—62no—
187C2×C3—61no—
188C2×C3—182no—
189C2×C3#6380912yes—
190C2×C3#6390011yescharacteristic
191D3—64no—
192D3—64no—
193C22—92no—
194C22—182no—
195C22#8865511yescharacteristic
196C4—92no—
197C4—364no—
198C4#8865111yescharacteristic
199C4C2×#619911yescharacteristic
200C3—32no—
201C3—61no—
202C3#1059071211yescharacteristic
203C2—182no—
204C2#1172084312yes—
205C2#1172115211yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#427
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #63644⋊C3C2semidirect
4. #11720315C2semidirect
9. C8×#272C4semidirect
16. #63644D3semidirect
115. H(3,𝔽3)#427semidirect
175. C32#4860semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117210892
#63644⋊C32
#117203152
#105907063