Finite Group Databasebeta

← #13144036

Subgroups — #13144036

Order 1728.

583 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 105 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#11721871C211yesmaximal, characteristic
2#11721869C211yesmaximal, characteristic, complemented
3#11721786C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
5#11720559C211yesmaximal, characteristic, complemented
6#11720522C211yesmaximal, characteristic, complemented
7C4×#6180C211yesmaximal, characteristic, complemented
8C3×#4824—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10#88740C2211yescharacteristic
11M16×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13#88550C411yescharacteristic, complemented
14#88549C411yescharacteristic, complemented
15#88546C411yescharacteristic, complemented
16#88545C411yescharacteristic, complemented
17#88538—21no—
18#88538C2211yescharacteristic
19C2×#6180—21no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22#88527—21no—
23C8×#272—21no—
24C3×#950—34no—
25C3×#948—34no—
26C3×#907—34no—
27C32×#104D314yescomplemented
28C3×#822—34no—
29C3×#805—34no—
30C3×C4×C3⋊C8—34no—
31Q8×H(3,𝔽3)C2×C411yescharacteristic
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#6235—21no—
38C4⋊#272C2×C411yescharacteristic
39C4×#272—21no—
40#6229C2×C411yescharacteristic
41C8×H(3,𝔽3)—21no—
42#6180—21no—
43#6180C2×C411yescharacteristic
44#6180C2×C411yescharacteristic
45#6180D411yescharacteristic
46#6180D411yescharacteristic
47C3×#443—91no—
48#4824—34no—
49C32×#41C2×D314yes—
50C3×#235—34no—
51C32×M16C2×D314yes—
52C32×C42C2×D314yes—
53C3×#216—34no—
54C3×#215—34no—
55C3×#214—34no—
56C3×#213—34no—
57C3×#208—34no—
58C3×#208—64no—
59C2×C3×C3⋊C8—34no—
60C2×C3×C3⋊C8—34no—
61C2×C3×C3⋊C8—64no—
62C3×ΓL(1,𝔽25)—64no—
63C3×D3×C8—64no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)#4111yescharacteristic
66C2×#272—21no—
67C4×H(3,𝔽3)—22no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)C22×C411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084—21no—
72C3×#135—91no—
73C3×#131—91no—
74C3×#131—91no—
75#950—34no—
76#948—34no—
77#907—34no—
78C3×#108—91no—
79C3×#104—34no—
80C3×#104—91no—
81C3×#104C3⋊D311yescharacteristic
82C3×C4×C8—91no—
83#822—34no—
84#805—34no—
85C4×C3⋊C8—34no—
86C32×Q8C4×D314yes—
87C32×D4—24no—
88C32×D4C4×D314yes—
89C2×C32×C4—24no—
90C2×C32×C4—24no—
91C2×C32×C4C22×D314yes—
92C3×#67—64no—
93C3×D12—34no—
94C3×C4×D3—64no—
95C3×Dic6—34no—
96C32×C8—24no—
97C3×C3⋊C8—34no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—34no—
101C3×C3⋊C8—64no—
102#443—91noSylow
103C2×H(3,𝔽3)—21no—
104C2×H(3,𝔽3)—41no—
105C2×H(3,𝔽3)C4×D411yescharacteristic
106#272—41no—
107C3×#41—34no—
108C3×#41—91no—
109C3×#41C2×C3⋊D311yescharacteristic
110#235—34no—
111C3×Q16—91no—
112C3×SD16—91no—
113C3×SD16—91no—
114C3×D8—91no—
115C3×M16—34no—
116C3×M16—91no—
117C3×M16—181no—
118C3×M16—181no—
119C3×M16C2×C3⋊D311yescharacteristic
120C2×C3×C8—91no—
121C2×C3×C8—91no—
122C2×C3×C8—181no—
123C2×C3×C8—181no—
124C3×C42—34no—
125C3×C42C2×C3⋊D311yescharacteristic
126#216—34no—
127#215—34no—
128#214—34no—
129#213—34no—
130#208—34no—
131#208—64no—
132C2×C3⋊C8—34no—
133C2×C3⋊C8—34no—
134C2×C3⋊C8—64no—
135ΓL(1,𝔽25)—64no—
136D3×C8—64no—
137C22×C32—24no—
138C22×C32#23714yes—
139C2×C3×D3—64no—
140C32×C4—28no—
141C32×C4—24no—
142C32×C4C2×C4×D314yes—
143C32×C4D3×D414yes—
144C3×Dic3—64no—
145#135—91no—
146#131—91no—
147#131—91no—
148#108—91no—
149#104—91no—
150#104#27211yescharacteristic, complemented
151C4×C8—91no—
152H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
153C3×Q8—34no—
154C3×Q8—91no—
155C3×Q8C4×C3⋊D311yescharacteristic
156C3×D4—21no—
157C3×D4—34no—
158C3×D4—64no—
159C3×D4—91no—
160C3×D4—181no—
161C3×D4C4×C3⋊D311yescharacteristic
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—34no—
165C2×C3×C4—64no—
166C2×C3×C4—64no—
167C2×C3×C4—181no—
168C2×C3×C4C22×C3⋊D311yescharacteristic
169#67—64no—
170D12—34no—
171C4×D3—64no—
172Dic6—34no—
173C3×C8—21no—
174C3×C8—64no—
175C3×C8—91no—
176C3×C8—91no—
177C3×C8—91no—
178C3×C8—91no—
179C3×C8—181no—
180C3⋊C8—34no—
181C3⋊C8—34no—
182C3⋊C8—34no—
183C3⋊C8—34no—
184C3⋊C8—64no—
185C2×C32—24no—
186C2×C32—44no—
187C2×C32#67⋊C414yes—
188C3×D3—124no—
189#41—91no—
190#41C2×#27211yescharacteristic
191Q16—91no—
192SD16—91no—
193SD16—91no—
194D8—91no—
195M16—91no—
196M16—181no—
197M16—181no—
198M16C2×#27211yescharacteristic
199C2×C8—91no—
200C2×C8—91no—
201C2×C8—181no—
202C2×C8—181no—
203C42C2×#27211yescharacteristic
204C22×C3—21no—
205C22×C3—34no—
206C22×C3—64no—
207C22×C3—181no—
208C22×C3#383011yescharacteristic
209C2×D3—64no—
210C3×C4—22no—
211C3×C4—21no—
212C3×C4—34no—
213C3×C4—34no—
214C3×C4—64no—
215C3×C4—68no—
216C3×C4—181no—
217C3×C4C2×C4×C3⋊D311yescharacteristic
218C3×C4D4×C3⋊D311yescharacteristic
219Dic3—64no—
220C32#482414yescomplemented
221Q8—91no—
222Q8C4×#27211yescharacteristic
223D4—21no—
224D4—91no—
225D4—181no—
226D4C4×#27211yescharacteristic
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—181no—
230C2×C4C22×#27211yescharacteristic
231C8—21no—
232C8—91no—
233C8—91no—
234C8—91no—
235C8—91no—
236C8—181no—
237C2×C3—21no—
238C2×C3—34no—
239C2×C3—41no—
240C2×C3—64no—
241C2×C3—124no—
242C2×C3—361no—
243C2×C3#6409011yescharacteristic
244D3—124no—
245C22—21no—
246C22—181no—
247C22#8874411yescharacteristic
248C4—22no—
249C4—21no—
250C4—181no—
251C4C2×C4×#27211yescharacteristic
252C4D4×#27211yescharacteristic
253C3—34no—
254C3#1059252311yescharacteristic
255C2—21no—
256C2—41no—
257C2—361no—
258C2#1172173711yescharacteristic
subgroupG/H
centerC3×C4D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#104C2
socleC2×C3#64090
prime radical#11721871C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#104C3⋊D3
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. #11721869C2semidirect
3. #11721786C2semidirect
4. H(3,𝔽3)×#104C2semidirect
5. #11720559C2semidirect
6. #11720522C2semidirect
7. C4×#6180C2semidirect
13. #88550C4semidirect
14. #88549C4semidirect
15. #88546C4semidirect
16. #88545C4semidirect
27. C32×#104D3semidirect
150. #104#272semidirect
152. H(3,𝔽3)#443semidirect
220. C32#4824semidirect

Maximal subgroups

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

subgroupindex
#117218712
#117218692
#117217862
H(3,𝔽3)×#1042
#117205592
#117205222
C4×#61802
C3×#48243