Finite Group Databasebeta

← #13143881

Subgroups — #13143881

Order 1728.

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

normal subgroups only

This group has 924 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
1#11722664C211yesmaximal, characteristic, complemented
2#11721850C211yesmaximal, characteristic, complemented
3#11721787C212yesmaximal, complemented
4H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
5#11720562C212yesmaximal, complemented
6C3×#4747—34nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—22no—
9C2×#6235—22no—
10#88744—22no—
11D4×#272—22no—
12#88740C2212yes—
13C2×C4⋊#272C2211yescharacteristic, complemented
14M16×H(3,𝔽3)C2212yescomplemented
15C4×#1084C2211yescharacteristic, complemented
16#88529—22no—
17#88528—22no—
18C3×#1004—34no—
19C3×#940—34no—
20C3×#908—38no—
21C32×#100D314yescomplemented
22C3×#825—38no—
23C23×H(3,𝔽3)D412yes—
24C22×#272—22no—
25C22×#272—21no—
26D4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28#6235—24no—
29#6235—22no—
30#6235—44no—
31C2×#1084—22no—
32C2×#1084—21no—
33C4⋊#272—22no—
34C4⋊#272D412yescomplemented
35C4×#272—22no—
36#6229D412yescomplemented
37C8×H(3,𝔽3)—22no—
38C3×#453—91no—
39#4747—34no—
40C2×C32×D4C2×D314yes—
41C2×C3×#67—68no—
42C2×C3×#67—68no—
43C3×#237—68no—
44C3×D3×D4—68no—
45C3×#235—38no—
46C2×C3×D12—34no—
47C32×M16C2×D318yescomplemented
48C3×C4×Dic3—34no—
49C3×D24—68no—
50C3×#204—68no—
51C22×H(3,𝔽3)—22no—
52C22×H(3,𝔽3)—42no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×#272—22no—
55C2×#272—42no—
56C2×#272—41no—
57C2×#272—82no—
58C4×H(3,𝔽3)C2×D412yes—
59#1084—22no—
60#1084—22no—
61C3×H(5,𝔽2)—91no—
62#1004—34no—
63C3×AGL(1,ℤ/8)—92no—
64C3×#127—91no—
65#940—34no—
66#908—38no—
67C3×#104—92no—
68C3×#100—34no—
69C3×#100C3⋊D311yescharacteristic
70#825—38no—
71C23×C32D1218yes—
72C22×C3×D3—68no—
73C22×C3×D3—64no—
74C32×D4—28no—
75C2×C32×C4C22×D314yes—
76C3×#67—616no—
77C3×#67—68no—
78C3×#67—1216no—
79C2×C3×Dic3—68no—
80C2×C3×Dic3—64no—
81C3×D12—38no—
82C3×D12—68no—
83C3×C4×D3—68no—
84C3×Dic6—38no—
85C32×C8—28no—
86#453—91noSylow
87C2×H(3,𝔽3)—21no—
88C2×H(3,𝔽3)—42no—
89C2×H(3,𝔽3)#12011yescharacteristic
90#272—42no—
91#272—81no—
92C3×#41—92no—
93C3×#41—182no—
94C2×C3×D4—34no—
95C2×C3×D4—91no—
96C2×C3×D4—182no—
97C2×C3×D4—182no—
98C2×C3×D4—182no—
99C2×C3×D4C2×C3⋊D311yescharacteristic
100C2×#67—68no—
101C2×#67—68no—
102#237—68no—
103D3×D4—68no—
104#235—38no—
105C2×D12—34no—
106C3×SD16—182no—
107C3×D8—182no—
108C3×M16—38no—
109C3×M16C2×C3⋊D312yes—
110C3×C42—91no—
111C4×Dic3—34no—
112D24—68no—
113#204—68no—
114C22×C32—28no—
115C22×C32—48no—
116C22×C32C2×D1214yes—
117C2×C3×D3—68no—
118C2×C3×D3—128no—
119C2×C3×D3—124no—
120C2×C3×D3—248no—
121C32×C4D3×D418yes—
122C3×Dic3—68no—
123C3×Dic3—68no—
124H(5,𝔽2)—91no—
125AGL(1,ℤ/8)—92no—
126#127—91no—
127#104—92no—
128#100#27211yescharacteristic, complemented
129H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
130C23×C3—38no—
131C23×C3—182no—
132C23×C3—181no—
133C23×C3C4⋊(C3⋊D3)12yes—
134C22×D3—68no—
135C22×D3—64no—
136C3×Q8—92no—
137C3×D4—22no—
138C3×D4—68no—
139C3×D4—92no—
140C3×D4—184no—
141C3×D4—182no—
142C3×D4—182no—
143C3×D4—364no—
144C2×C3×C4—34no—
145C2×C3×C4—182no—
146C2×C3×C4—181no—
147C2×C3×C4—182no—
148C2×C3×C4C22×C3⋊D311yescharacteristic
149#67—68no—
150#67—616no—
151#67—1216no—
152C2×Dic3—64no—
153C2×Dic3—68no—
154D12—38no—
155D12—68no—
156C4×D3—68no—
157Dic6—38no—
158C3×C8—22no—
159C3×C8—68no—
160C2×C32—24no—
161C2×C32—48no—
162C2×C32#88214yes—
163C3×D3—128no—
164C3×D3—244no—
165#41—92no—
166#41—182no—
167C2×D4—91no—
168C2×D4—182no—
169C2×D4—182no—
170C2×D4—182no—
171C2×D4C2×#27211yescharacteristic
172SD16—182no—
173D8—182no—
174M16C2×#27212yescomplemented
175C42—91no—
176C22×C3—22no—
177C22×C3—34no—
178C22×C3—42no—
179C22×C3—68no—
180C22×C3—128no—
181C22×C3—182no—
182C22×C3—362no—
183C22×C3—361no—
184C22×C3—722no—
185C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
186C2×D3—68no—
187C2×D3—124no—
188C2×D3—128no—
189C2×D3—248no—
190C3×C4—38no—
191C3×C4—182no—
192C3×C4—182no—
193C3×C4D4×C3⋊D312yes—
194Dic3—68no—
195Dic3—68no—
196C32#474714yescomplemented
197C23—182no—
198C23—181no—
199C23C4⋊#27212yes—
200Q8—92no—
201D4—22no—
202D4—92no—
203D4—184no—
204D4—182no—
205D4—182no—
206D4—364no—
207C2×C4—182no—
208C2×C4—181no—
209C2×C4—182no—
210C2×C4C22×#27211yescharacteristic
211C8—22no—
212C2×C3—21no—
213C2×C3—34no—
214C2×C3—42no—
215C2×C3—64no—
216C2×C3—128no—
217C2×C3—362no—
218C2×C3—721no—
219C2×C3#6409111yescharacteristic
220D3—128no—
221D3—244no—
222C22—22no—
223C22—42no—
224C22—182no—
225C22—362no—
226C22—361no—
227C22—722no—
228C22C2×C4⋊#27211yescharacteristic
229C4—182no—
230C4—182no—
231C4D4×#27212yes—
232C3—34no—
233C3#1059244611yescharacteristic
234C2—21no—
235C2—42no—
236C2—362no—
237C2—721no—
238C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#100C2
hypercenterC3×#100C3⋊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×#100

Complements

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

NH ≅ G/Ntype
1. #11722664C2semidirect
2. #11721850C2semidirect
3. #11721787C2semidirect
4. H(3,𝔽3)×#100C2semidirect
5. #11720562C2semidirect
13. C2×C4⋊#272C22semidirect
14. M16×H(3,𝔽3)C22semidirect
15. C4×#1084C22semidirect
21. C32×#100D3semidirect
34. C4⋊#272D4semidirect
36. #6229D4semidirect
47. C32×M16C2×D3semidirect
128. #100#272semidirect
129. H(3,𝔽3)#453semidirect
174. M16C2×#272semidirect
196. C32#4747semidirect

Maximal subgroups

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

subgroupindex
#117226642
#117218502
#117217872
H(3,𝔽3)×#1002
#117205622
C3×#47473