Finite Group Databasebeta

← #13144002

Subgroups — #13144002

Order 1728.

616 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 99 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#11721864C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic
4#11721755C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic
8C3×#4798—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88550—22no—
15#88542—21no—
16#88542—21no—
17#88540—21no—
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88528—22no—
22C3×#947—34no—
23C2×C3×#216—34no—
24C2×C3×#204—34no—
25C3×#887—34no—
26C32×#103D314yescomplemented
27C3×Dic3⋊C8—34no—
28C3×#809—34no—
29C22×#272—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272—21no—
40#6229—21no—
41#6229D412yes—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44C3×#464—91no—
45#4798—34no—
46C2×C32×Q8C2×D314yes—
47C2×C3×D12—34no—
48C2×C3×Dic6—34no—
49C2×C32×C8C2×D314yes—
50C32×C4⋊C4C2×D314yes—
51C3×#216—68no—
52C3×(C2×D3)⋊C4—64no—
53C3×(C2×D3)⋊C4—64no—
54C3×Dic3⋊C4—64no—
55C3×C4×Dic3—34no—
56C3×C4×Dic3—64no—
57C2×C3×C3⋊C8—34no—
58C3×#204—68no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—41no—
61C2×#272—41no—
62C2×#272—41no—
63C4×H(3,𝔽3)—41no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)#4111yescharacteristic
67#1084—22no—
68#1084—21no—
69#1084—21no—
70C2×C3×Q16—91no—
71C2×C3×SD16—91no—
72C3×#124—91no—
73C3×C4×Q8—91no—
74#947—34no—
75C2×#216—34no—
76C2×#204—34no—
77#887—34no—
78C3×C4⋊C8—91no—
79C3×#103—34no—
80C3×#103C3⋊D311yescharacteristic
81C3×#102—91no—
82Dic3⋊C8—34no—
83#809—34no—
84C22×C3×D3—64no—
85C32×Q8—24no—
86C32×Q8—24no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4C22×D314yes—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—64no—
93C3×D12—64no—
94C3×D12—64no—
95C3×Dic6—38no—
96C3×Dic6—64no—
97C32×C8—24no—
98C3×C3⋊C8—64no—
99#464—91noSylow
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)#13511yescharacteristic
102C2×H(3,𝔽3)#13711yescharacteristic
103#272—81no—
104C2×C3×Q8—34no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C2×C3×D4—91no—
108C2×D12—34no—
109C2×Dic6—34no—
110C3×Q16—182no—
111C3×SD16—182no—
112C2×C3×C8—34no—
113C2×C3×C8—91no—
114C2×C3×C8C2×C3⋊D311yescharacteristic
115C3×C4⋊C4—34no—
116C3×C4⋊C4—181no—
117C3×C4⋊C4C2×C3⋊D311yescharacteristic
118C3×C22⋊C4—181no—
119C3×C22⋊C4—181no—
120C3×C42—91no—
121C3×C42—181no—
122#216—68no—
123(C2×D3)⋊C4—64no—
124(C2×D3)⋊C4—64no—
125Dic3⋊C4—64no—
126C4×Dic3—34no—
127C4×Dic3—64no—
128C2×C3⋊C8—34no—
129#204—68no—
130C22×C32D3×D414yes—
131C2×C3×D3—124no—
132C2×C3×D3—124no—
133C2×C3×D3—124no—
134C32×C4—44no—
135C32×C4—44no—
136C32×C4#23514yes—
137C32×C4D3×D414yes—
138C3×Dic3—68no—
139C3×Dic3—64no—
140C3×Dic3—64no—
141C2×Q16—91no—
142C2×SD16—91no—
143#124—91no—
144C4×Q8—91no—
145C4⋊C8—91no—
146#103#27211yescharacteristic, complemented
147#102—91no—
148H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
149C23×C3—181no—
150C22×D3—64no—
151C3×Q8—21no—
152C3×Q8—21no—
153C3×Q8—64no—
154C3×Q8—64no—
155C3×Q8—92no—
156C3×Q8—181no—
157C3×D4—181no—
158C3×D4—181no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—34no—
162C2×C3×C4—64no—
163C2×C3×C4—64no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—181no—
167C2×C3×C4C22×C3⋊D311yescharacteristic
168C2×Dic3—34no—
169C2×Dic3—34no—
170C2×Dic3—64no—
171D12—64no—
172D12—64no—
173Dic6—38no—
174Dic6—64no—
175C3×C8—21no—
176C3×C8—64no—
177C3×C8—181no—
178C3⋊C8—64no—
179C2×C32#88414yes—
180C2×C32#91614yes—
181C2×C32#91814yes—
182C3×D3—244no—
183C2×Q8—91no—
184C2×Q8C2×#27211yescharacteristic
185C2×D4—91no—
186Q16—182no—
187SD16—182no—
188C2×C8—91no—
189C2×C8C2×#27211yescharacteristic
190C4⋊C4—181no—
191C4⋊C4C2×#27211yescharacteristic
192C22⋊C4—181no—
193C22⋊C4—181no—
194C42—91no—
195C42—181no—
196C22×C3—34no—
197C22×C3—361no—
198C22×C3—361no—
199C22×C3—361no—
200C22×C3D4×C3⋊D311yescharacteristic
201C2×D3—124no—
202C2×D3—124no—
203C2×D3—124no—
204C3×C4—34no—
205C3×C4—34no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4—124no—
209C3×C4—124no—
210C3×C4—182no—
211C3×C4—181no—
212C3×C4—181no—
213C3×C4#382811yescharacteristic
214C3×C4D4×C3⋊D311yescharacteristic
215Dic3—68no—
216Dic3—64no—
217Dic3—64no—
218C32#479814yescomplemented
219C23—181no—
220Q8—21no—
221Q8—21no—
222Q8—92no—
223Q8—181no—
224D4—181no—
225D4—181no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—181no—
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—181no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—721no—
238C2×C3#6409311yescharacteristic
239C2×C3#6412511yescharacteristic
240C2×C3#6412711yescharacteristic
241D3—244no—
242C22—361no—
243C22—361no—
244C22—361no—
245C22D4×#27211yescharacteristic
246C4—41no—
247C4—41no—
248C4—182no—
249C4—181no—
250C4—181no—
251C4#8874011yescharacteristic
252C4D4×#27211yescharacteristic
253C3—34no—
254C3#1059249711yescharacteristic
255C2—721no—
256C2#1172174011yescharacteristic
257C2#1172180211yescharacteristic
258C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×#88550C2semidirect
4. #11721755C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #11720547C2semidirect
26. C32×#103D3semidirect
146. #103#272semidirect
148. H(3,𝔽3)#464semidirect
218. C32#4798semidirect

Maximal subgroups

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

subgroupindex
#117218642
C2×#885502
C2×#885282
#117217552
H(3,𝔽3)×#1032
#117205472
#117205312
C3×#47983