Finite Group Databasebeta

← #13138727

Subgroups — #13138727

Order 1728.

356 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721776C211yesmaximal, characteristic
2#63639⋊C3C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic, complemented
4#11720252C212yesmaximal, complemented
5#11720251C212yesmaximal, complemented
6#10590622—31nomaximal
7#10590621—31nomaximal
8#88740C411yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10#88538C411yescharacteristic
11#88527C412yes—
12C8×#272C2212yes—
13(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
14(C3⋊(C3⋊C16))⋊C3C2212yes—
15(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
16C3×#901—31no—
17C3×#901—31no—
18C3×#901—61no—
19#63639D311yescharacteristic, complemented
20C2×C3⋊(C3⋊C16)D311yescharacteristic, complemented
21C3×#812—31no—
22C2×C3×C3⋊C16—31no—
23#63543—32no—
24#63543—32no—
25D3×C3⋊C16—32no—
26D3×C3⋊C16—32no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28#6235C812yes—
29C4⋊#272C811yescharacteristic
30C4×#272C2×C412yes—
31#6229C811yescharacteristic
32C8×H(3,𝔽3)C2×C411yescharacteristic
33C8×H(3,𝔽3)C2311yescharacteristic
34#6180C2×C412yes—
35#5140—91no—
36#4907—91no—
37#4901—91no—
38C3×#235—31no—
39C3×#235—31no—
40C3×#235—61no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D311yescharacteristic
43C2×C32×C8C2×D311yescharacteristic
44C3×#208—31no—
45C3×#208—31no—
46C3×#208—61no—
47C3×ΓL(1,𝔽25)—32no—
48C3×ΓL(1,𝔽25)—32no—
49C3×ΓL(1,𝔽25)—62no—
50C3×D3×C8—32no—
51C3×D3×C8—32no—
52C3×D3×C8—62no—
53C3⋊(C3⋊C16)C2×D311yescharacteristic
54C3⋊(C3⋊C16)C2×D312yes—
55C3⋊(C3⋊C16)C2×D311yescharacteristic
56C3×C3⋊C16—31no—
57C3×C3⋊C16—32no—
58C3×C3⋊C16—31no—
59C22×H(3,𝔽3)C2×C811yescharacteristic
60C2×#272C2×C812yes—
61C4×H(3,𝔽3)C2×C811yescharacteristic
62C4×H(3,𝔽3)C22×C411yescharacteristic
63#1084C2×C812yes—
64C3×#131—91no—
65#901—31no—
66#901—31no—
67#901—61no—
68C3×M32—91no—
69C2×C3×C16—91no—
70#812—31no—
71#812—91no—
72#812—92no—
73C2×C3⋊C16—31no—
74C2×C3⋊C16—91no—
75C2×C3⋊C16—92no—
76#798—92no—
77#798—92no—
78D3×C16—92no—
79D3×C16—92no—
80C2×C32×C4—21no—
81C2×C32×C4C4×D311yescharacteristic
82C2×C32×C4C4×D311yescharacteristic
83C3×#67—32no—
84C3×#67—32no—
85C3×#67—62no—
86C3×D12—31no—
87C3×D12—31no—
88C3×D12—61no—
89C3×C4×D3—32no—
90C3×C4×D3—32no—
91C3×C4×D3—62no—
92C3×Dic6—31no—
93C3×Dic6—31no—
94C3×Dic6—61no—
95C32×C8—21no—
96C32×C8—21no—
97C32×C8C4×D311yescharacteristic
98C32×C8C4×D311yescharacteristic
99C32×C8C22×D311yescharacteristic
100C32×C8C22×D311yescharacteristic
101C3×C3⋊C8—32no—
102C3×C3⋊C8—32no—
103C3×C3⋊C8—62no—
104#504—271noSylow
105C2×H(3,𝔽3)—21no—
106C2×H(3,𝔽3)C22×C811yescharacteristic
107#272—22no—
108C3×#41—91no—
109#235—31no—
110#235—31no—
111#235—61no—
112C3×M16—91no—
113C3×M16—92no—
114C2×C3×C8—31no—
115C2×C3×C8—31no—
116C2×C3×C8—61no—
117C2×C3×C8—92no—
118C2×C3×C8D3211yescharacteristic
119#208—31no—
120#208—31no—
121#208—61no—
122ΓL(1,𝔽25)—32no—
123ΓL(1,𝔽25)—32no—
124ΓL(1,𝔽25)—62no—
125D3×C8—32no—
126D3×C8—32no—
127D3×C8—62no—
128C3×C16—91no—
129C3×C16—92no—
130C3×C16—91no—
131C3⋊C16—31no—
132C3⋊C16—31no—
133C3⋊C16—32no—
134C3⋊C16—91no—
135C3⋊C16—92no—
136C3⋊C16—91no—
137C22×C32—21no—
138C22×C32D3×C811yescharacteristic
139C22×C32D3×C811yescharacteristic
140C2×C3×D3—32no—
141C2×C3×D3—32no—
142C2×C3×D3—62no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4D3×C811yescharacteristic
146C32×C4D3×C811yescharacteristic
147C32×C4C2×C4×D311yescharacteristic
148C32×C4C2×C4×D311yescharacteristic
149C3×Dic3—32no—
150C3×Dic3—32no—
151C3×Dic3—62no—
152#131—91no—
153M32—91no—
154M32—272no—
155C2×C16—91no—
156C2×C16—272no—
157H(3,𝔽3)#50411yescharacteristic, complemented, Sylow
158C3×Q8—91no—
159C3×D4—92no—
160C3×D4—91no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—61no—
164C2×C3×C4—92no—
165C2×C3×C4#62411yescharacteristic
166#67—32no—
167#67—32no—
168#67—62no—
169D12—31no—
170D12—31no—
171D12—61no—
172C4×D3—32no—
173C4×D3—32no—
174C4×D3—62no—
175Dic6—31no—
176Dic6—31no—
177Dic6—61no—
178C3×C8—31no—
179C3×C8—31no—
180C3×C8—31no—
181C3×C8—31no—
182C3×C8—61no—
183C3×C8—61no—
184C3×C8—92no—
185C3×C8#62411yescharacteristic
186C3×C8C2×D3211yescharacteristic
187C3⋊C8—32no—
188C3⋊C8—32no—
189C3⋊C8—62no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—41no—
194C2×C32C2×D3×C811yescharacteristic
195C2×C32C2×D3×C811yescharacteristic
196C3×D3—62no—
197C3×D3—62no—
198C3×D3—122no—
199#41—91no—
200M16—91no—
201M16—92no—
202C2×C8—92no—
203C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
204C16—91no—
205C16—91no—
206C16—92no—
207C22×C3—31no—
208C22×C3—31no—
209C22×C3—61no—
210C22×C3—92no—
211C22×C3#371011yescharacteristic
212C2×D3—32no—
213C2×D3—32no—
214C2×D3—62no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—92no—
222C3×C4#371011yescharacteristic
223C3×C4C2×#62411yescharacteristic
224Dic3—32no—
225Dic3—32no—
226Dic3—62no—
227C32—21no—
228C32#490111yescharacteristic, complemented
229C32#490711yescharacteristic, complemented
230Q8—91no—
231D4—92no—
232D4—91no—
233C2×C4—92no—
234C2×C4#619911yescharacteristic
235C8—92no—
236C8#619911yescharacteristic
237C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C2×C3—21no—
239C2×C3—31no—
240C2×C3—31no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—121no—
245C2×C3—182no—
246C2×C3C2×#371011yescharacteristic
247D3—62no—
248D3—62no—
249D3—122no—
250C22—92no—
251C22#8843411yescharacteristic
252C4—92no—
253C4#8843411yescharacteristic
254C4C2×#619911yescharacteristic
255C3—31no—
256C3—31no—
257C3—61no—
258C3#1059062411yescharacteristic
259C2—21no—
260C2—182no—
261C2C2×#8843411yescharacteristic
subgroupG/H
centerC8C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3710
prime radical#88740C4
cosocleC8×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#504
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #63639⋊C3C2semidirect
3. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
4. #11720252C2semidirect
5. #11720251C2semidirect
19. #63639D3semidirect
20. C2×C3⋊(C3⋊C16)D3semidirect
157. H(3,𝔽3)#504semidirect
228. C32#4901semidirect
229. C32#4907semidirect

Maximal subgroups

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

subgroupindex
#117217762
#63639⋊C32
C2×(C3⋊(C3⋊C16))⋊C32
#117202522
#117202512
#105906223
#105906213