Finite Group Databasebeta

← #13144019

Subgroups — #13144019

Order 1728.

671 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 121 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
1Q8×#1084C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic
3#11721761C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic
6C8×#1084C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#4788—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88549C414yescomplemented
15#88542—21no—
16C4⋊#1084—21no—
17C4⋊#1084C2211yescharacteristic
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C3×Q8×Dic3—34no—
22C2×C3×#215—34no—
23C3×#893—34no—
24C32×#103D314yescomplemented
25C3×#821—34no—
26C3×C8×Dic3—34no—
27C3×#808—34no—
28C22×#272—21no—
29Q8×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)C2×C412yescomplemented
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084D411yescharacteristic
36C2×#1084D411yescharacteristic
37C4⋊#272—21no—
38C4⋊#272C2×C412yes—
39C4×#272—42no—
40C8×H(3,𝔽3)—21no—
41#6180C2×C412yescomplemented
42C3×C4×SD16—91no—
43#4788—34no—
44C2×C32×Q8C2×D314yes—
45C2×C3×D12—34no—
46C2×C3×C4×D3—64no—
47C2×C32×C8C2×D314yes—
48C32×C4⋊C4C2×D314yes—
49C3×#215—316no—
50C3×(C2×D3)⋊C4—64no—
51C3×C4⋊Dic3—34no—
52C3×C4⋊Dic3—64no—
53C3×C4×Dic3—34no—
54C3×C4×Dic3—64no—
55C2×C3×C3⋊C8—34no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C2×#272—22no—
58C2×#272—41no—
59C2×#272—41no—
60C4×H(3,𝔽3)—22no—
61C4×H(3,𝔽3)—41no—
62C4×H(3,𝔽3)C22×C411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—21no—
65#1084—41no—
66#1084SD1612yescomplemented
67C2×C3×SD16—91no—
68C3×C4×Q8—91no—
69C3×C4×D4—91no—
70Q8×Dic3—34no—
71C2×#215—34no—
72#893—34no—
73C3×#106—91no—
74C3×#103—34no—
75C3×#103C3⋊D311yescharacteristic
76C3×#102—91no—
77C3×C4×C8—91no—
78#821—34no—
79C8×Dic3—34no—
80#808—34no—
81C22×C3×D3—64no—
82C32×Q8—24no—
83C32×Q8C4×D318yescomplemented
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4C22×D314yes—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—64no—
90C3×D12—38no—
91C3×D12—64no—
92C3×C4×D3—128no—
93C32×C8—24no—
94C3×C3⋊C8—38no—
95C4×SD16—91noSylow
96C2×H(3,𝔽3)C4×D411yescharacteristic
97C2×H(3,𝔽3)C2×SD1611yescharacteristic
98C2×H(3,𝔽3)#13511yescharacteristic
99#272—42no—
100C2×C3×Q8—34no—
101C2×C3×Q8C2×C3⋊D311yescharacteristic
102C2×C3×D4—91no—
103C22×C3×C4—181no—
104C2×D12—34no—
105C2×C4×D3—64no—
106C3×SD16—94no—
107C2×C3×C8—34no—
108C2×C3×C8—91no—
109C2×C3×C8C2×C3⋊D311yescharacteristic
110C3×C4⋊C4—34no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—181no—
113C3×C4⋊C4C2×C3⋊D311yescharacteristic
114C3×C22⋊C4—181no—
115C3×C42—91no—
116C3×C42—181no—
117#215—316no—
118(C2×D3)⋊C4—64no—
119C4⋊Dic3—34no—
120C4⋊Dic3—64no—
121C4×Dic3—34no—
122C4×Dic3—64no—
123C2×C3⋊C8—34no—
124C22×C32D3×D414yes—
125C2×C3×D3—68no—
126C2×C3×D3—124no—
127C2×C3×D3—124no—
128C32×C4—28no—
129C32×C4—44no—
130C32×C4C2×C4×D314yes—
131C32×C4#23714yes—
132C3×Dic3—38no—
133C3×Dic3—64no—
134C3×Dic3—124no—
135C2×SD16—91no—
136C4×Q8—91no—
137C4×D4—91no—
138#106—91no—
139#103#27211yescharacteristic, complemented
140#102—91no—
141C4×C8—91no—
142H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
143C23×C3—181no—
144C22×D3—64no—
145C3×Q8—21no—
146C3×Q8—38no—
147C3×Q8—64no—
148C3×Q8C4×C3⋊D312yes—
149C3×D4—92no—
150C3×D4—181no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—34no—
154C2×C3×C4—64no—
155C2×C3×C4—64no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—181no—
159C2×C3×C4—362no—
160C2×C3×C4C22×C3⋊D311yescharacteristic
161C2×Dic3—34no—
162C2×Dic3—34no—
163C2×Dic3—64no—
164D12—38no—
165D12—64no—
166C4×D3—128no—
167C3×C8—21no—
168C3×C8—64no—
169C3×C8—92no—
170C3⋊C8—38no—
171C2×C32#67⋊C414yes—
172C2×C32D3×SD1614yes—
173C2×C32#91914yes—
174C3×D3—128no—
175C2×Q8C2×#27211yescharacteristic
176C2×D4—91no—
177C22×C4—181no—
178SD16—94no—
179C2×C8—91no—
180C2×C8C2×#27211yescharacteristic
181C4⋊C4—91no—
182C4⋊C4—181no—
183C4⋊C4C2×#27211yescharacteristic
184C22⋊C4—181no—
185C42—91no—
186C42—181no—
187C22×C3—34no—
188C22×C3—182no—
189C22×C3—361no—
190C22×C3—361no—
191C22×C3D4×C3⋊D311yescharacteristic
192C2×D3—68no—
193C2×D3—124no—
194C2×D3—124no—
195C3×C4—22no—
196C3×C4—34no—
197C3×C4—34no—
198C3×C4—41no—
199C3×C4—68no—
200C3×C4—92no—
201C3×C4—124no—
202C3×C4—181no—
203C3×C4—361no—
204C3×C4C2×C4×C3⋊D311yescharacteristic
205C3×C4#383011yescharacteristic
206Dic3—38no—
207Dic3—64no—
208Dic3—124no—
209C32#478814yescomplemented
210C23—181no—
211Q8—21no—
212Q8C4×#27212yescomplemented
213D4—92no—
214D4—181no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—91no—
218C2×C4—91no—
219C2×C4—181no—
220C2×C4—362no—
221C2×C4C22×#27211yescharacteristic
222C8—21no—
223C8—92no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3—362no—
228C2×C3#6409011yescharacteristic
229C2×C3SD16×C3⋊D311yescharacteristic
230C2×C3#6412811yescharacteristic
231D3—128no—
232C22—182no—
233C22—361no—
234C22—361no—
235C22D4×#27211yescharacteristic
236C4—22no—
237C4—41no—
238C4—92no—
239C4—181no—
240C4—361no—
241C4C2×C4×#27211yescharacteristic
242C4#8874411yescharacteristic
243C3—34no—
244C3#1059248711yescharacteristic
245C2—362no—
246C2#1172173711yescharacteristic
247C2SD16×#27211yescharacteristic
248C2#1172180811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)C4×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
1. Q8×#1084C2semidirect
4. H(3,𝔽3)×#103C2semidirect
6. C8×#1084C2semidirect
7. #11720530C2semidirect
14. #88549C4semidirect
24. C32×#103D3semidirect
30. Q8×H(3,𝔽3)C2×C4semidirect
41. #6180C2×C4semidirect
66. #1084SD16semidirect
83. C32×Q8C4×D3semidirect
139. #103#272semidirect
142. H(3,𝔽3)C4×SD16semidirect
209. C32#4788semidirect
212. Q8C4×#272semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
C2×#885492
#117217612
H(3,𝔽3)×#1032
#117205542
C8×#10842
#117205302
C3×#47883