Finite Group Databasebeta

← SD32×#272 #13144214

Subgroups — SD32×#272 #13144214

Order 1728.

495 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 73 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
1Q16×#272C211yesmaximal, characteristic, complemented
2D8×#272C211yesmaximal, characteristic
3H(3,𝔽3)×SD32C211yesmaximal, characteristic, complemented
4#11720569C211yesmaximal, characteristic, complemented
5#11720566C211yesmaximal, characteristic, complemented
6#11720513C211yesmaximal, characteristic, complemented
7C16×#272C211yesmaximal, characteristic, complemented
8C3×D3×SD32—34nomaximal
9Q8×#272—21no—
10D4×#272—21no—
11Q16×H(3,𝔽3)C2211yescharacteristic, complemented
12D8×H(3,𝔽3)C2211yescharacteristic
13#88550—21no—
14#88545—21no—
15#88530C2211yescharacteristic, complemented
16#88529C2211yescharacteristic
17C8×#272C2211yescharacteristic
18C16×H(3,𝔽3)C2211yescharacteristic, complemented
19#88386C2211yescharacteristic, complemented
20C3×D3×Q16—34no—
21C3×D3×D8—34no—
22C32×SD32D314yescomplemented
23C3×#828—34no—
24C3×#827—34no—
25C3×#800—34no—
26C3×D3×C16—34no—
27C22×#272—41no—
28Q8×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—21no—
30#6235—41no—
31C4⋊#272—21no—
32C4×#272—41no—
33C4×#272D411yescharacteristic
34#6229—21no—
35#6229—41no—
36C8×H(3,𝔽3)C2311yescharacteristic
37#6180D411yescharacteristic
38C2×C3×SD32—91no—
39D3×SD32—34no—
40C3×D3×Q8—64no—
41C3×D3×D4—64no—
42C32×Q16C2×D314yescomplemented
43C32×D8C2×D314yes—
44C3×#216—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—
52C2×#272—41no—
53C2×#272—82no—
54C2×#272D811yescharacteristic
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)C2×D411yescharacteristic
57#1084—41no—
58#1084D811yescharacteristic
59C2×C3×Q16—91no—
60C2×C3×D8—91no—
61D3×Q16—34no—
62D3×D8—34no—
63C3×SD32—34no—
64C3×SD32—91no—
65C3×SD32—91no—
66C3×SD32—91no—
67C3×SD32C3⋊D311yescharacteristic
68C2×C3×C16—91no—
69#828—34no—
70#827—34no—
71#800—34no—
72D3×C16—34no—
73C22×C3×D3—124no—
74C32×Q8—24no—
75C32×D4—24no—
76C3×#67—124no—
77C3×D12—64no—
78C3×C4×D3—34no—
79C3×C4×D3—124no—
80C3×Dic6—64no—
81C3×Dic6—124no—
82C32×C8C22×D314yes—
83C3×C3⋊C8—34no—
84C2×SD32—91noSylow
85C2×H(3,𝔽3)—81no—
86C2×H(3,𝔽3)C2×D811yescharacteristic
87#272—81no—
88#272SD3212yesdirect product factor
89C2×C3×Q8—181no—
90C2×C3×D4—181no—
91D3×Q8—64no—
92D3×D4—64no—
93C3×Q16—34no—
94C3×Q16—91no—
95C3×Q16—181no—
96C3×Q16C2×C3⋊D311yescharacteristic
97C3×D8—34no—
98C3×D8—91no—
99C3×D8—181no—
100C3×D8C2×C3⋊D311yescharacteristic
101C2×C3×C8—91no—
102#216—64no—
103#213—64no—
104Dic12—34no—
105D24—34no—
106D3×C8—34no—
107C3×C16—34no—
108C3×C16—91no—
109C3×C16C2×C3⋊D311yescharacteristic
110C3⋊C16—34no—
111C22×C32—44no—
112C2×C3×D3—34no—
113C2×C3×D3—124no—
114C2×C3×D3—248no—
115C32×C4—44no—
116C32×C4D3×D414yes—
117C3×Dic3—34no—
118C3×Dic3—124no—
119C2×Q16—91no—
120C2×D8—91no—
121SD32—91no—
122SD32—91no—
123SD32—91no—
124SD32#27211yescharacteristic, direct product factor
125C2×C16—91no—
126H(3,𝔽3)C2×SD3211yescharacteristic, complemented, Sylow
127C23×C3—361no—
128C22×D3—124no—
129C3×Q8—21no—
130C3×Q8—64no—
131C3×Q8—181no—
132C3×Q8—361no—
133C3×D4—21no—
134C3×D4—64no—
135C3×D4—181no—
136C3×D4—361no—
137C2×C3×C4—91no—
138C2×C3×C4—361no—
139#67—124no—
140D12—64no—
141C4×D3—34no—
142C4×D3—124no—
143Dic6—64no—
144Dic6—124no—
145C3×C8—34no—
146C3×C8—91no—
147C3×C8C22×C3⋊D311yescharacteristic
148C3⋊C8—34no—
149C2×C32—84no—
150C2×C32D3×D814yes—
151C3×D3—38no—
152C3×D3—244no—
153C2×Q8—181no—
154C2×D4—181no—
155Q16—91no—
156Q16—181no—
157Q16C2×#27211yescharacteristic, complemented
158D8—91no—
159D8—181no—
160D8C2×#27211yescharacteristic
161C2×C8—91no—
162C16—91no—
163C16C2×#27211yescharacteristic, complemented
164C22×C3—41no—
165C22×C3—91no—
166C22×C3—124no—
167C22×C3—361no—
168C22×C3—722no—
169C2×D3—34no—
170C2×D3—124no—
171C2×D3—248no—
172C3×C4—34no—
173C3×C4—41no—
174C3×C4—91no—
175C3×C4—124no—
176C3×C4—361no—
177C3×C4D4×C3⋊D311yescharacteristic
178Dic3—34no—
179Dic3—124no—
180C32D3×SD3214yescomplemented
181C23—361no—
182Q8—21no—
183Q8—181no—
184Q8—361no—
185D4—21no—
186D4—181no—
187D4—361no—
188C2×C4—91no—
189C2×C4—361no—
190C8—91no—
191C8C22×#27211yescharacteristic
192C2×C3—34no—
193C2×C3—81no—
194C2×C3—92no—
195C2×C3—244no—
196C2×C3—721no—
197C2×C3D8×C3⋊D311yescharacteristic
198D3—38no—
199D3—244no—
200C22—41no—
201C22—91no—
202C22—361no—
203C22—722no—
204C4—41no—
205C4—91no—
206C4—361no—
207C4D4×#27211yescharacteristic
208C3—34no—
209C3C3⋊D3×SD3211yescharacteristic
210C2—81no—
211C2—92no—
212C2—721no—
213C2D8×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D8×C3⋊D3
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×SD32C2
prime radicalD8×#272C2
hypercenterC3×SD32C3⋊D3
hypocenterH(3,𝔽3)C2×SD32
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. Q16×#272C2semidirect
3. H(3,𝔽3)×SD32C2semidirect
4. #11720569C2semidirect
5. #11720566C2semidirect
6. #11720513C2semidirect
7. C16×#272C2semidirect
11. Q16×H(3,𝔽3)C22semidirect
15. #88530C22semidirect
18. C16×H(3,𝔽3)C22semidirect
19. #88386C22semidirect
22. C32×SD32D3semidirect
42. C32×Q16C2×D3semidirect
49. C32×C16C2×D3semidirect
88. #272SD32direct (N × H)
124. SD32#272direct (N × H)
126. H(3,𝔽3)C2×SD32semidirect
157. Q16C2×#272semidirect
163. C16C2×#272semidirect
180. C32D3×SD32semidirect

Maximal subgroups

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

subgroupindex
Q16×#2722
D8×#2722
H(3,𝔽3)×SD322
#117205692
#117205662
#117205132
C16×#2722
C3×D3×SD323