Finite Group Databasebeta

← #13144215

Subgroups — #13144215

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#11721808C211yesmaximal, characteristic, complemented
2D8×#272C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×SD32C211yesmaximal, characteristic, complemented
4#11720569C211yesmaximal, characteristic, complemented
5#11720565C211yesmaximal, characteristic, complemented
6#11720512C211yesmaximal, characteristic, complemented
7#11720511C211yesmaximal, characteristic, complemented
8C3×#4914—34nomaximal
9#88748—21no—
10D4×#272—21no—
11Q16×H(3,𝔽3)C2211yescharacteristic, complemented
12D8×H(3,𝔽3)C2211yescharacteristic
13#88549—21no—
14#88545—21no—
15#88529C2211yescharacteristic
16#88529C2211yescharacteristic, complemented
17C8×#272C2211yescharacteristic
18C16×H(3,𝔽3)C2211yescharacteristic, complemented
19#88386C2211yescharacteristic, complemented
20C3×#919—34no—
21C3×D3×D8—34no—
22C32×SD32D314yescomplemented
23C3×#828—34no—
24C3×#826—34no—
25C3×D48—34no—
26C3×#798—34no—
27C22×#272—41no—
28Q8×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—21no—
30#6235—41no—
31C4⋊#272—21no—
32C4⋊#272—21no—
33C4⋊#272—41no—
34C4×#272—41no—
35C4×#272D411yescharacteristic
36C8×H(3,𝔽3)C2311yescharacteristic
37#6180D411yescharacteristic
38C3×#509—91no—
39#4914—34no—
40C3×#239—64no—
41C3×D3×D4—64no—
42C32×Q16C2×D314yescomplemented
43C32×D8C2×D314yes—
44C3×#215—64no—
45C3×#213—64no—
46C3×D24—34no—
47C3×D24—34no—
48C3×D3×C8—34no—
49C32×C16C2×D314yescomplemented
50C3×C3⋊C16—34no—
51C22×H(3,𝔽3)—41no—
52C2×#272—41no—
53C2×#272—41no—
54C2×#272—82no—
55C2×#272D811yescharacteristic
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)C2×D411yescharacteristic
58#1084D811yescharacteristic
59C3×#135—91no—
60C2×C3×D8—91no—
61#919—34no—
62D3×D8—34no—
63C3×SD32—34no—
64C3×SD32—91no—
65C3×SD32C3⋊D311yescharacteristic
66C3×D16—91no—
67C3×D16—91no—
68C3×M32—91no—
69#828—34no—
70#826—34no—
71D48—34no—
72#798—34no—
73C22×C3×D3—124no—
74C32×Q8—24no—
75C32×D4—24no—
76C3×#67—124no—
77C3×D12—64no—
78C3×D12—64no—
79C3×D12—124no—
80C3×C4×D3—34no—
81C3×C4×D3—124no—
82C32×C8C22×D314yes—
83C3×C3⋊C8—34no—
84#509—91noSylow
85C2×H(3,𝔽3)—81no—
86C2×H(3,𝔽3)C2×D811yescharacteristic
87#272—21no—
88#272—81no—
89#272—81no—
90C3×#41—181no—
91C2×C3×D4—181no—
92#239—64no—
93D3×D4—64no—
94C3×Q16—34no—
95C3×Q16C2×C3⋊D311yescharacteristic
96C3×SD16—181no—
97C3×D8—34no—
98C3×D8—91no—
99C3×D8—91no—
100C3×D8—181no—
101C3×D8C2×C3⋊D311yescharacteristic
102C2×C3×C8—91no—
103#215—64no—
104#213—64no—
105D24—34no—
106D24—34no—
107D3×C8—34no—
108C3×C16—34no—
109C3×C16—91no—
110C3×C16C2×C3⋊D311yescharacteristic
111C3⋊C16—34no—
112C22×C32—44no—
113C2×C3×D3—34no—
114C2×C3×D3—124no—
115C2×C3×D3—124no—
116C2×C3×D3—248no—
117C32×C4—44no—
118C32×C4D3×D414yes—
119C3×Dic3—34no—
120#135—91no—
121C2×D8—91no—
122SD32—91no—
123SD32#27211yescharacteristic, complemented
124D16—91no—
125D16—91no—
126M32—91no—
127H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
128C23×C3—361no—
129C22×D3—124no—
130C3×Q8—21no—
131C3×Q8—64no—
132C3×D4—21no—
133C3×D4—64no—
134C3×D4—181no—
135C3×D4—181no—
136C3×D4—361no—
137C3×D4—361no—
138C2×C3×C4—91no—
139C2×C3×C4—361no—
140#67—124no—
141D12—64no—
142D12—64no—
143D12—124no—
144C4×D3—34no—
145C4×D3—124no—
146C3×C8—34no—
147C3×C8—91no—
148C3×C8C22×C3⋊D311yescharacteristic
149C3⋊C8—34no—
150C2×C32—84no—
151C2×C32D3×D814yes—
152C3×D3—64no—
153C3×D3—244no—
154C3×D3—244no—
155#41—181no—
156C2×D4—181no—
157Q16C2×#27211yescharacteristic, complemented
158SD16—181no—
159D8—91no—
160D8—91no—
161D8—181no—
162D8C2×#27211yescharacteristic
163C2×C8—91no—
164C16—91no—
165C16C2×#27211yescharacteristic, complemented
166C22×C3—41no—
167C22×C3—91no—
168C22×C3—124no—
169C22×C3—361no—
170C22×C3—361no—
171C22×C3—722no—
172C2×D3—34no—
173C2×D3—124no—
174C2×D3—124no—
175C2×D3—248no—
176C3×C4—34no—
177C3×C4—41no—
178C3×C4—91no—
179C3×C4—124no—
180C3×C4D4×C3⋊D311yescharacteristic
181Dic3—34no—
182C32#491414yescomplemented
183C23—361no—
184Q8—21no—
185D4—21no—
186D4—181no—
187D4—181no—
188D4—361no—
189D4—361no—
190C2×C4—91no—
191C2×C4—361no—
192C8—91no—
193C8C22×#27211yescharacteristic
194C2×C3—34no—
195C2×C3—81no—
196C2×C3—181no—
197C2×C3—244no—
198C2×C3—721no—
199C2×C3—721no—
200C2×C3D8×C3⋊D311yescharacteristic
201D3—64no—
202D3—244no—
203D3—244no—
204C22—41no—
205C22—91no—
206C22—361no—
207C22—361no—
208C22—722no—
209C4—41no—
210C4—91no—
211C4D4×#27211yescharacteristic
212C3—34no—
213C3#1059261311yescharacteristic
214C2—81no—
215C2—181no—
216C2—721no—
217C2—721no—
218C2D8×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D8×C3⋊D3
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×SD32C2
hypercenterC3×SD32C3⋊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×SD32

Complements

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

NH ≅ G/Ntype
1. #11721808C2semidirect
2. D8×#272C2semidirect
3. H(3,𝔽3)×SD32C2semidirect
4. #11720569C2semidirect
5. #11720565C2semidirect
6. #11720512C2semidirect
7. #11720511C2semidirect
11. Q16×H(3,𝔽3)C22semidirect
16. #88529C22semidirect
18. C16×H(3,𝔽3)C22semidirect
19. #88386C22semidirect
22. C32×SD32D3semidirect
42. C32×Q16C2×D3semidirect
49. C32×C16C2×D3semidirect
123. SD32#272semidirect
127. H(3,𝔽3)#509semidirect
157. Q16C2×#272semidirect
165. C16C2×#272semidirect
182. C32#4914semidirect

Maximal subgroups

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

subgroupindex
#117218082
D8×#2722
H(3,𝔽3)×SD322
#117205692
#117205652
#117205122
#117205112
C3×#49143