Finite Group Databasebeta

← #13138904

Subgroups — #13138904

Order 1728.

531 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 99 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#11721152C212yesmaximal
3#63644⋊C3C212yesmaximal, complemented
4#11720314C212yesmaximal, complemented
5D3×#817—32nomaximal
6C2×C4×#272C2211yescharacteristic
7C2×#6199—22no—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C8×#272C414yescomplemented
11C4⋊((C3⋊Dic3)⋊C3)C2214yes—
12#88449—22no—
13C2×C3×D3×C8—32no—
14C2×C3×D3×C8—61no—
15D3×C4⋊Dic3—34no—
16#63644D312yescomplemented
17C3×#817—32no—
18#63573—34no—
19C22×#272D411yescharacteristic
20C2×C4×H(3,𝔽3)C2311yescharacteristic
21C2×#1084D411yescharacteristic
22C4×#272C2×C412yes—
23C4×#272Q812yes—
24C2×(C3⋊Dic3)⋊C3—24no—
25#6199—44no—
26C8×H(3,𝔽3)C2×C412yescomplemented
27#6180C2×C412yescomplemented
28C2×#817—91no—
29D3×#106—92no—
30C2×C3×C4×D3—32no—
31C2×C3×C4×D3—61no—
32C2×D3×Dic3—64no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D312yes—
35C4⋊(C3⋊Dic3)C2×D314yes—
36C3×C4⋊Dic3—34no—
37C2×C3×C3⋊C8—32no—
38C2×C3×C3⋊C8—61no—
39C3×D3×C8—38no—
40C3×D3×C8—64no—
41Dic3⋊Dic3—64no—
42C22×H(3,𝔽3)C2×D411yescharacteristic
43C2×#272C4⋊C412yes—
44C2×#272SD1614yes—
45C4×H(3,𝔽3)C22×C411yescharacteristic
46C4×H(3,𝔽3)C2×Q811yescharacteristic
47#1084C4⋊C412yes—
48(C3⋊Dic3)⋊C3—44no—
49C22×C3×C8—91no—
50C2×C4⋊Dic3—92no—
51C2×D3×C8—32no—
52C2×D3×C8—61no—
53D3×C4⋊C4—94no—
54C3×#106—92no—
55#817—32no—
56#817—92no—
57#817—92no—
58#808—94no—
59C22×C3×D3—32no—
60C22×C3×D3—61no—
61C2×C32×C4—21no—
62C2×C32×C4C22×D312yes—
63C2×C3⋊Dic3—24no—
64C2×C3×Dic3—32no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—64no—
67C3×C4×D3—34no—
68C3×C4×D3—34no—
69C3×C4×D3—62no—
70C3×C4×D3—62no—
71D3×Dic3—128no—
72C32×C8—22no—
73C32×C8C4×D314yescomplemented
74C3×C3⋊C8—34no—
75C3×C3⋊C8—62no—
76C2×#106—271noSylow
77C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
78C2×H(3,𝔽3)C2×SD1612yes—
79#272#10614yescomplemented
80C22×C3×C4—91no—
81C22×Dic3—182no—
82C2×C4×D3—32no—
83C2×C4×D3—61no—
84C2×C4×D3—184no—
85C2×C3×C8—32no—
86C2×C3×C8—61no—
87C2×C3×C8—91no—
88C2×C3×C8—94no—
89C2×C3×C8D3211yescharacteristic
90C3×C4⋊C4—94no—
91C4⋊Dic3—34no—
92C4⋊Dic3—94no—
93C4⋊Dic3—182no—
94Dic3⋊C4—184no—
95C2×C3⋊C8—32no—
96C2×C3⋊C8—61no—
97D3×C8—38no—
98D3×C8—64no—
99C22×C32—21no—
100C22×C32D3×D412yes—
101C2×C3×D3—34no—
102C2×C3×D3—38no—
103C2×C3×D3—62no—
104C2×C3×D3—64no—
105C32×C4—21no—
106C32×C4—21no—
107C32×C4C2×C4×D312yes—
108C32×C4D3×Q812yes—
109C3⋊Dic3—44no—
110C3×Dic3—34no—
111C3×Dic3—62no—
112C3×Dic3—124no—
113C22×C8—91no—
114C2×C4⋊C4—272no—
115#106—92no—
116#106—272no—
117H(3,𝔽3)C2×#10611yescharacteristic, complemented, Sylow
118C23×C3—91no—
119C22×D3—32no—
120C22×D3—61no—
121C2×C3×C4—32no—
122C2×C3×C4—61no—
123C2×C3×C4—91no—
124C2×C3×C4—92no—
125C2×C3×C4—92no—
126C2×C3×C4—184no—
127C2×C3×C4C2×D3211yescharacteristic
128C2×Dic3—32no—
129C2×Dic3—61no—
130C2×Dic3—64no—
131C2×Dic3—184no—
132C2×Dic3—364no—
133C4×D3—34no—
134C4×D3—34no—
135C4×D3—62no—
136C4×D3—62no—
137C4×D3—368no—
138C3×C8—34no—
139C3×C8—62no—
140C3×C8—92no—
141C3×C8#62412yes—
142C3⋊C8—34no—
143C3⋊C8—62no—
144C2×C32—22no—
145C2×C32—21no—
146C2×C32D3×C4⋊C412yes—
147C2×C32D3×SD1614yes—
148C3×D3—38no—
149C3×D3—64no—
150C22×C4—91no—
151C22×C4—542no—
152C2×C8—91no—
153C2×C8—94no—
154C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
155C4⋊C4—94no—
156C4⋊C4—542no—
157C22×C3—32no—
158C22×C3—61no—
159C22×C3—94no—
160C22×C3—92no—
161C22×C3#380211yescharacteristic
162C2×D3—34no—
163C2×D3—38no—
164C2×D3—62no—
165C2×D3—64no—
166C3×C4—32no—
167C3×C4—32no—
168C3×C4—61no—
169C3×C4—61no—
170C3×C4—92no—
171C3×C4—364no—
172C3×C4#379711yescharacteristic
173C3×C4C2×#62411yescharacteristic
174Dic3—34no—
175Dic3—62no—
176Dic3—124no—
177Dic3—364no—
178C32—21no—
179C32D3×#10612yescomplemented
180C23—91no—
181C2×C4—91no—
182C2×C4—92no—
183C2×C4—92no—
184C2×C4—184no—
185C2×C4—1084no—
186C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C8—92no—
188C8#619912yescomplemented
189C2×C3—34no—
190C2×C3—32no—
191C2×C3—62no—
192C2×C3—61no—
193C2×C3—94no—
194C2×C3#6379612yes—
195C2×C3#6390011yescharacteristic
196D3—38no—
197D3—64no—
198C22—94no—
199C22—92no—
200C22#8865511yescharacteristic
201C4—92no—
202C4—364no—
203C4#8865111yescharacteristic
204C4C2×#619911yescharacteristic
205C3—32no—
206C3—61no—
207C3#1059068511yescharacteristic
208C2—94no—
209C2#1172083712yes—
210C2#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)C2×#106
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. #11720314C2semidirect
10. C8×#272C4semidirect
16. #63644D3semidirect
26. C8×H(3,𝔽3)C2×C4semidirect
27. #6180C2×C4semidirect
73. C32×C8C4×D3semidirect
79. #272#106semidirect
117. H(3,𝔽3)C2×#106semidirect
179. C32D3×#106semidirect
188. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117211522
#63644⋊C32
#117203142
D3×#8173