Finite Group Databasebeta

← #13144208

Subgroups — #13144208

Order 1728.

495 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721794C211yesmaximal, characteristic, complemented
2D8×#272C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×D16C211yesmaximal, characteristic, complemented
4#11720566C211yesmaximal, characteristic, complemented
5#11720565C211yesmaximal, characteristic, complemented
6#11720513C211yesmaximal, characteristic, complemented
7#11720511C211yesmaximal, characteristic, complemented
8C3×#4911—34nomaximal
9#88744—21no—
10D4×#272—21no—
11D8×H(3,𝔽3)C2211yescharacteristic
12D8×H(3,𝔽3)C2211yescharacteristic, complemented
13#88546—21no—
14#88545—21no—
15#88530C2211yescharacteristic, complemented
16#88529C2211yescharacteristic
17C8×#272C2211yescharacteristic
18C16×H(3,𝔽3)C2211yescharacteristic, complemented
19#88386C2211yescharacteristic, complemented
20C3×#912—34no—
21C3×D3×D8—34no—
22C32×D16D314yescomplemented
23C3×#827—34no—
24C3×#826—34no—
25C3×#800—34no—
26C3×#798—34no—
27C22×#272—41no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—21no—
30#6235—41no—
31#6235—41no—
32C2×#1084—41no—
33C4⋊#272—21no—
34C4×#272D411yescharacteristic
35#6229—21no—
36C8×H(3,𝔽3)C2311yescharacteristic
37#6180D411yescharacteristic
38C3×#509—91no—
39#4911—34no—
40C3×#237—64no—
41C3×D3×D4—64no—
42C32×D8C2×D314yes—
43C32×D8C2×D314yescomplemented
44C3×#214—64no—
45C3×#213—64no—
46C3×Dic12—34no—
47C3×D24—34no—
48C3×D3×C8—34no—
49C32×C16C2×D314yescomplemented
50C3×C3⋊C16—34no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)—41no—
53C2×#272—41no—
54C2×#272—82no—
55C2×#272D811yescharacteristic
56C4×H(3,𝔽3)C2×D411yescharacteristic
57#1084—41no—
58#1084D811yescharacteristic
59C3×#135—91no—
60C2×C3×D8—91no—
61#912—34no—
62D3×D8—34no—
63C3×SD32—91no—
64C3×SD32—91no—
65C3×D16—34no—
66C3×D16—91no—
67C3×D16C3⋊D311yescharacteristic
68C3×M32—91no—
69#827—34no—
70#826—34no—
71#800—34no—
72#798—34no—
73C22×C3×D3—124no—
74C32×D4—24no—
75C32×D4—24no—
76C3×#67—124no—
77C3×#67—124no—
78C2×C3×Dic3—124no—
79C3×D12—64no—
80C3×C4×D3—34no—
81C3×Dic6—64no—
82C32×C8C22×D314yes—
83C3×C3⋊C8—34no—
84#509—91noSylow
85C2×H(3,𝔽3)—81no—
86C2×H(3,𝔽3)—81no—
87C2×H(3,𝔽3)C2×D811yescharacteristic
88#272—21no—
89#272—81no—
90C3×#41—181no—
91C2×C3×D4—181no—
92#237—64no—
93D3×D4—64no—
94C3×Q16—91no—
95C3×SD16—181no—
96C3×D8—34no—
97C3×D8—34no—
98C3×D8—91no—
99C3×D8—181no—
100C3×D8C2×C3⋊D311yescharacteristic
101C3×D8C2×C3⋊D311yescharacteristic
102C2×C3×C8—91no—
103#214—64no—
104#213—64no—
105Dic12—34no—
106D24—34no—
107D3×C8—34no—
108C3×C16—34no—
109C3×C16—91no—
110C3×C16C2×C3⋊D311yescharacteristic
111C3⋊C16—34no—
112C22×C32—44no—
113C22×C32—44no—
114C2×C3×D3—34no—
115C2×C3×D3—124no—
116C2×C3×D3—248no—
117C32×C4D3×D414yes—
118C3×Dic3—34no—
119C3×Dic3—124no—
120#135—91no—
121C2×D8—91no—
122SD32—91no—
123SD32—91no—
124D16—91no—
125D16#27211yescharacteristic, complemented
126M32—91no—
127H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
128C23×C3—361no—
129C22×D3—124no—
130C3×Q8—181no—
131C3×D4—21no—
132C3×D4—21no—
133C3×D4—64no—
134C3×D4—64no—
135C3×D4—181no—
136C3×D4—361no—
137C3×D4—361no—
138C2×C3×C4—91no—
139C2×C3×C4—361no—
140#67—124no—
141#67—124no—
142C2×Dic3—124no—
143D12—64no—
144C4×D3—34no—
145Dic6—64no—
146C3×C8—34no—
147C3×C8—91no—
148C3×C8C22×C3⋊D311yescharacteristic
149C3⋊C8—34no—
150C2×C32—84no—
151C2×C32—84no—
152C2×C32D3×D814yes—
153C3×D3—64no—
154C3×D3—244no—
155#41—181no—
156C2×D4—181no—
157Q16—91no—
158SD16—181no—
159D8—91no—
160D8—181no—
161D8C2×#27211yescharacteristic
162D8C2×#27211yescharacteristic, complemented
163C2×C8—91no—
164C16—91no—
165C16C2×#27211yescharacteristic, complemented
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—91no—
169C22×C3—124no—
170C22×C3—124no—
171C22×C3—361no—
172C22×C3—722no—
173C2×D3—34no—
174C2×D3—124no—
175C2×D3—248no—
176C3×C4—34no—
177C3×C4—91no—
178C3×C4—361no—
179C3×C4D4×C3⋊D311yescharacteristic
180Dic3—34no—
181Dic3—124no—
182C32#491114yescomplemented
183C23—361no—
184Q8—181no—
185D4—21no—
186D4—21no—
187D4—181no—
188D4—361no—
189D4—361no—
190C2×C4—91no—
191C2×C4—361no—
192C8—91no—
193C8C22×#27211yescharacteristic
194C2×C3—34no—
195C2×C3—81no—
196C2×C3—81no—
197C2×C3—181no—
198C2×C3—244no—
199C2×C3—244no—
200C2×C3—721no—
201C2×C3D8×C3⋊D311yescharacteristic
202D3—64no—
203D3—244no—
204C22—41no—
205C22—41no—
206C22—91no—
207C22—361no—
208C22—722no—
209C4—91no—
210C4—361no—
211C4D4×#27211yescharacteristic
212C3—34no—
213C3#1059261011yescharacteristic
214C2—81no—
215C2—81no—
216C2—181no—
217C2—721no—
218C2D8×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D8×C3⋊D3
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×D16C2
hypercenterC3×D16C3⋊D3
hypocenterH(3,𝔽3)#509
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×D16

Complements

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

NH ≅ G/Ntype
1. #11721794C2semidirect
2. D8×#272C2semidirect
3. H(3,𝔽3)×D16C2semidirect
4. #11720566C2semidirect
5. #11720565C2semidirect
6. #11720513C2semidirect
7. #11720511C2semidirect
12. D8×H(3,𝔽3)C22semidirect
15. #88530C22semidirect
18. C16×H(3,𝔽3)C22semidirect
19. #88386C22semidirect
22. C32×D16D3semidirect
43. C32×D8C2×D3semidirect
49. C32×C16C2×D3semidirect
125. D16#272semidirect
127. H(3,𝔽3)#509semidirect
162. D8C2×#272semidirect
165. C16C2×#272semidirect
182. C32#4911semidirect

Maximal subgroups

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

subgroupindex
#117217942
D8×#2722
H(3,𝔽3)×D162
#117205662
#117205652
#117205132
#117205112
C3×#49113