Finite Group Databasebeta

← #13144086

Subgroups — #13144086

Order 1728.

616 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#88530C211yesmaximal, characteristic, complemented
2C2×#88528C211yesmaximal, characteristic
3#11721721C211yesmaximal, characteristic
4C4×#6229C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
6#11720554C211yesmaximal, characteristic
7#11720549C211yesmaximal, characteristic, complemented
8C3×#4845—34nomaximal
9C2×C4⋊#272C2211yescharacteristic
10C2×#6229C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542—21no—
16#88542—21no—
17C4⋊#1084C2211yescharacteristic
18#88540—21no—
19C4×#1084—21no—
20#88530—22no—
21#88528—22no—
22C2×C3×Dic12—34no—
23C2×C3×#204—34no—
24C3×#875—34no—
25C3×C4×Dic6—34no—
26C32×C4⋊C8D314yescomplemented
27C3×#821—34no—
28C3×#816—34no—
29C22×#272—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084—21no—
35C2×#1084—21no—
36C4⋊#272—21no—
37C4⋊#272—21no—
38#6229—21no—
39#6229—21no—
40#6229—21no—
41#6229D412yes—
42C8×H(3,𝔽3)—21no—
43C8×H(3,𝔽3)—21no—
44C3×#464—91no—
45#4845—34no—
46C2×C3×D12—34no—
47C2×C3×Dic6—34no—
48C2×C3×Dic6—34no—
49C2×C32×C8C2×D314yes—
50C2×C32×C8C2×D314yes—
51C32×C42C2×D314yes—
52C3×(C2×D3)⋊C4—64no—
53C3×(C2×D3)⋊C4—64no—
54C3×C4⋊Dic3—34no—
55C3×Dic3⋊C4—64no—
56C3×C4×Dic3—64no—
57C3×Dic12—68no—
58C3×#204—68no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—41no—
61C2×#272—41no—
62C2×#272—41no—
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)#4111yescharacteristic
67#1084—22no—
68#1084—41no—
69#1084—41no—
70C2×C3×Q16—91no—
71C2×C3×SD16—91no—
72C3×#124—91no—
73C3×C4×Q8—91no—
74C2×Dic12—34no—
75C2×#204—34no—
76#875—34no—
77C4×Dic6—34no—
78C3×C4⋊C8—34no—
79C3×C4⋊C8C3⋊D311yescharacteristic
80C3×#103—91no—
81C3×#102—91no—
82#821—34no—
83#816—34no—
84C22×C3×D3—64no—
85C2×C32×C4D1214yes—
86C2×C32×C4D1214yes—
87C2×C32×C4C22×D314yes—
88C2×C3×Dic3—64no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—64no—
91C3×D12—64no—
92C3×D12—64no—
93C3×Dic6—38no—
94C3×Dic6—64no—
95C3×Dic6—64no—
96C3×Dic6—64no—
97C32×C8—24no—
98C32×C8—24no—
99#464—91noSylow
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)#13511yescharacteristic
102C2×H(3,𝔽3)#13711yescharacteristic
103#272—81no—
104C2×C3×Q8—91no—
105C2×C3×Q8—91no—
106C2×C3×D4—91no—
107C2×D12—34no—
108C2×Dic6—34no—
109C2×Dic6—34no—
110C3×Q16—182no—
111C3×SD16—182no—
112C2×C3×C8—34no—
113C2×C3×C8—34no—
114C2×C3×C8C2×C3⋊D311yescharacteristic
115C2×C3×C8C2×C3⋊D311yescharacteristic
116C3×C4⋊C4—91no—
117C3×C4⋊C4—181no—
118C3×C22⋊C4—181no—
119C3×C22⋊C4—181no—
120C3×C42—34no—
121C3×C42—181no—
122C3×C42C2×C3⋊D311yescharacteristic
123(C2×D3)⋊C4—64no—
124(C2×D3)⋊C4—64no—
125C4⋊Dic3—34no—
126Dic3⋊C4—64no—
127C4×Dic3—64no—
128Dic12—68no—
129#204—68no—
130C22×C32C2×D1214yes—
131C2×C3×D3—124no—
132C2×C3×D3—124no—
133C2×C3×D3—124no—
134C32×C4—24no—
135C32×C4—24no—
136C32×C4D3×D414yes—
137C32×C4#23914yes—
138C3×Dic3—68no—
139C3×Dic3—124no—
140C3×Dic3—124no—
141C2×Q16—91no—
142C2×SD16—91no—
143#124—91no—
144C4×Q8—91no—
145C4⋊C8#27211yescharacteristic, complemented
146#103—91no—
147#102—91no—
148H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
149C23×C3—181no—
150C22×D3—64no—
151C3×Q8—92no—
152C3×Q8—181no—
153C3×Q8—181no—
154C3×Q8—181no—
155C3×D4—181no—
156C3×D4—181no—
157C2×C3×C4—34no—
158C2×C3×C4—34no—
159C2×C3×C4—34no—
160C2×C3×C4—181no—
161C2×C3×C4—181no—
162C2×C3×C4—181no—
163C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
164C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
165C2×C3×C4C22×C3⋊D311yescharacteristic
166C2×Dic3—64no—
167C2×Dic3—64no—
168C2×Dic3—64no—
169D12—64no—
170D12—64no—
171Dic6—38no—
172Dic6—64no—
173Dic6—64no—
174Dic6—64no—
175C3×C8—21no—
176C3×C8—21no—
177C3×C8—64no—
178C3×C8—64no—
179C2×C32#89514yes—
180C2×C32#90414yes—
181C2×C32#90914yes—
182C3×D3—244no—
183C2×Q8—91no—
184C2×Q8—91no—
185C2×D4—91no—
186Q16—182no—
187SD16—182no—
188C2×C8C2×#27211yescharacteristic
189C2×C8C2×#27211yescharacteristic
190C4⋊C4—91no—
191C4⋊C4—181no—
192C22⋊C4—181no—
193C22⋊C4—181no—
194C42—181no—
195C42C2×#27211yescharacteristic
196C22×C3—34no—
197C22×C3—361no—
198C22×C3—361no—
199C22×C3—361no—
200C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
201C2×D3—124no—
202C2×D3—124no—
203C2×D3—124no—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—34no—
207C3×C4—34no—
208C3×C4—64no—
209C3×C4—64no—
210C3×C4—182no—
211C3×C4—361no—
212C3×C4—361no—
213C3×C4D4×C3⋊D311yescharacteristic
214C3×C4#383211yescharacteristic
215Dic3—68no—
216Dic3—124no—
217Dic3—124no—
218C32#484514yescomplemented
219C23—181no—
220Q8—92no—
221Q8—181no—
222Q8—181no—
223Q8—181no—
224D4—181no—
225D4—181no—
226C2×C4—181no—
227C2×C4—181no—
228C2×C4—181no—
229C2×C4C4⋊#27211yescharacteristic
230C2×C4C4⋊#27211yescharacteristic
231C2×C4C22×#27211yescharacteristic
232C8—21no—
233C8—21no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—34no—
237C2×C3—721no—
238C2×C3#6410411yescharacteristic
239C2×C3#6411311yescharacteristic
240C2×C3#6411811yescharacteristic
241D3—244no—
242C22—361no—
243C22—361no—
244C22—361no—
245C22C2×C4⋊#27211yescharacteristic
246C4—21no—
247C4—21no—
248C4—182no—
249C4—361no—
250C4—361no—
251C4D4×#27211yescharacteristic
252C4#8874811yescharacteristic
253C3—34no—
254C3#1059254411yescharacteristic
255C2—721no—
256C2#1172176311yescharacteristic
257C2#1172177911yescharacteristic
258C2#1172178811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C4⋊C8C2
prime radicalC2×C4⋊#272C22
hypercenterC3×C4⋊C8C3⋊D3
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#88530C2semidirect
4. C4×#6229C2semidirect
5. H(3,𝔽3)×C4⋊C8C2semidirect
7. #11720549C2semidirect
26. C32×C4⋊C8D3semidirect
145. C4⋊C8#272semidirect
148. H(3,𝔽3)#464semidirect
218. C32#4845semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
C2×#885282
#117217212
C4×#62292
H(3,𝔽3)×C4⋊C82
#117205542
#117205492
C3×#48453