Finite Group Databasebeta

← #13144871

Subgroups — #13144871

Order 1728.

924 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 123 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
1H(3,𝔽3)×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#11721870C212yesmaximal, complemented
3#11721850C211yesmaximal, characteristic, complemented
4#11720586C212yesmaximal, complemented
5#11720579C211yesmaximal, characteristic, complemented
6C3×#5241—34nomaximal
7#41×H(3,𝔽3)—22no—
8#41×H(3,𝔽3)C2212yes—
9C2×D4×H(3,𝔽3)—22no—
10C2×D4×H(3,𝔽3)—22no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—22no—
13C2×C4⋊#272C2211yescharacteristic, complemented
14#88549—22no—
15#88545—22no—
16C4×#1084C2211yescharacteristic, complemented
17#88538C2212yescomplemented
18C32×H(5,𝔽2)D314yescomplemented
19C3×#949—38no—
20C3×#940—34no—
21C3×#837—38no—
22C3×#833—34no—
23C23×H(3,𝔽3)—22no—
24C23×H(3,𝔽3)D412yes—
25C22×#272—21no—
26Q8×H(3,𝔽3)D412yescomplemented
27D4×H(3,𝔽3)—24no—
28D4×H(3,𝔽3)—22no—
29D4×H(3,𝔽3)—22no—
30D4×H(3,𝔽3)D412yescomplemented
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34#6235—44no—
35C2×#1084—21no—
36C4⋊#272—22no—
37#6180—22no—
38C3×#453—91no—
39#5241—34no—
40C32×#41—28no—
41C32×#41C2×D318yes—
42C2×C32×D4—28no—
43C2×C32×D4—28no—
44C2×C32×D4C2×D314yes—
45C2×C3×#67—68no—
46C2×C3×D12—34no—
47C3×#215—68no—
48C3×#213—68no—
49C3×C4×Dic3—34no—
50C3×#208—38no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)—22no—
53C22×H(3,𝔽3)—42no—
54C22×H(3,𝔽3)—82no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C2×#272—42no—
57C2×#272—41no—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)C2×D412yes—
60#1084—22no—
61C3×H(5,𝔽2)—34no—
62C3×H(5,𝔽2)C3⋊D311yescharacteristic
63C3×AGL(1,ℤ/8)—92no—
64C3×#127—91no—
65#949—38no—
66#940—34no—
67C3×#104—92no—
68C3×#100—91no—
69#837—38no—
70#833—34no—
71C23×C32—28no—
72C23×C32#6718yes—
73C22×C3×D3—64no—
74C32×Q8#6718yescomplemented
75C32×D4—216no—
76C32×D4—28no—
77C32×D4—28no—
78C32×D4#6718yescomplemented
79C2×C32×C4—28no—
80C2×C32×C4—28no—
81C2×C32×C4C22×D314yes—
82C3×#67—1216no—
83C2×C3×Dic3—64no—
84C3×D12—68no—
85C3×C3⋊C8—68no—
86#453—91noSylow
87C2×H(3,𝔽3)—21no—
88C2×H(3,𝔽3)—42no—
89C2×H(3,𝔽3)—42no—
90C2×H(3,𝔽3)#12011yescharacteristic
91#272—81no—
92C3×#41—22no—
93C3×#41—38no—
94C3×#41—68no—
95C3×#41C2×C3⋊D312yes—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C2×C3×D4—34no—
99C2×C3×D4—68no—
100C2×C3×D4—68no—
101C2×C3×D4—91no—
102C2×C3×D4—182no—
103C2×C3×D4C2×C3⋊D311yescharacteristic
104C2×#67—68no—
105C2×D12—34no—
106C3×SD16—182no—
107C3×D8—182no—
108C3×M16—92no—
109C3×C42—91no—
110#215—68no—
111#213—68no—
112C4×Dic3—34no—
113#208—38no—
114C22×C32—28no—
115C22×C32—28no—
116C22×C32—48no—
117C22×C32—88no—
118C22×C32C2×#6714yes—
119C2×C3×D3—128no—
120C2×C3×D3—124no—
121C32×C4—28no—
122C32×C4C2×#6718yes—
123C3×Dic3—68no—
124H(5,𝔽2)#27211yescharacteristic, complemented
125AGL(1,ℤ/8)—92no—
126#127—91no—
127#104—92no—
128#100—91no—
129H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
130C23×C3—22no—
131C23×C3—38no—
132C23×C3—68no—
133C23×C3—181no—
134C23×C3#63812yes—
135C22×D3—64no—
136C3×Q8—38no—
137C3×Q8#63812yes—
138C3×D4—24no—
139C3×D4—22no—
140C3×D4—22no—
141C3×D4—38no—
142C3×D4—68no—
143C3×D4—68no—
144C3×D4—616no—
145C3×D4—182no—
146C3×D4—364no—
147C3×D4#63812yes—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—34no—
151C2×C3×C4—68no—
152C2×C3×C4—68no—
153C2×C3×C4—181no—
154C2×C3×C4C22×C3⋊D311yescharacteristic
155#67—1216no—
156C2×Dic3—64no—
157D12—68no—
158C3×C8—182no—
159C3⋊C8—68no—
160C2×C32—24no—
161C2×C32—48no—
162C2×C32—48no—
163C2×C32#95314yes—
164C3×D3—244no—
165#41—22no—
166#41C2×#27212yes—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—91no—
170C2×D4—182no—
171C2×D4C2×#27211yescharacteristic
172SD16—182no—
173D8—182no—
174M16—92no—
175C42—91no—
176C22×C3—22no—
177C22×C3—22no—
178C22×C3—34no—
179C22×C3—42no—
180C22×C3—68no—
181C22×C3—68no—
182C22×C3—82no—
183C22×C3—128no—
184C22×C3—248no—
185C22×C3—362no—
186C22×C3—361no—
187C22×C3C2×#63811yescharacteristic
188C2×D3—124no—
189C2×D3—128no—
190C3×C4—22no—
191C3×C4—38no—
192C3×C4—68no—
193C3×C4—182no—
194C3×C4C2×#63812yes—
195Dic3—68no—
196C32#524114yescomplemented
197C23—22no—
198C23—181no—
199C23#623512yes—
200Q8#623512yescomplemented
201D4—24no—
202D4—22no—
203D4—22no—
204D4—182no—
205D4—364no—
206D4#623512yescomplemented
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—181no—
210C2×C4C22×#27211yescharacteristic
211C8—182no—
212C2×C3—21no—
213C2×C3—34no—
214C2×C3—42no—
215C2×C3—42no—
216C2×C3—64no—
217C2×C3—128no—
218C2×C3—128no—
219C2×C3—721no—
220C2×C3#6416211yescharacteristic
221D3—244no—
222C22—22no—
223C22—22no—
224C22—42no—
225C22—82no—
226C22—362no—
227C22—361no—
228C22C2×#623511yescharacteristic
229C4—22no—
230C4—182no—
231C4C2×#623512yes—
232C3—34no—
233C3#1059294011yescharacteristic
234C2—21no—
235C2—42no—
236C2—42no—
237C2—721no—
238C2#1172187611yescharacteristic
subgroupG/H
center, socleC2×C3#64162
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×H(5,𝔽2)C2
hypercenterC3×H(5,𝔽2)C3⋊D3
hypocenterH(3,𝔽3)#453
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×H(5,𝔽2)C2semidirect
2. #11721870C2semidirect
3. #11721850C2semidirect
4. #11720586C2semidirect
5. #11720579C2semidirect
13. C2×C4⋊#272C22semidirect
16. C4×#1084C22semidirect
17. #88538C22semidirect
18. C32×H(5,𝔽2)D3semidirect
26. Q8×H(3,𝔽3)D4semidirect
30. D4×H(3,𝔽3)D4semidirect
74. C32×Q8#67semidirect
78. C32×D4#67semidirect
124. H(5,𝔽2)#272semidirect
129. H(3,𝔽3)#453semidirect
196. C32#5241semidirect
200. Q8#6235semidirect
206. D4#6235semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×H(5,𝔽2)2
#117218702
#117218502
#117205862
#117205792
C3×#52413