Finite Group Databasebeta

← #13138722

Subgroups — #13138722

Order 1728.

341 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 65 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#64113⋊C3C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11720545C211yesmaximal, characteristic, complemented
4#11720249C212yesmaximal, complemented
5#11720248C212yesmaximal, complemented
6#10590634—31nomaximal
7#10590633—31nomaximal
8#3828⋊C3—21no—
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#3745⋊C3—21no—
12#3745⋊C3C2212yes—
13#3745⋊C3C2211yescharacteristic, complemented
14#3744⋊C3—21no—
15#3746⋊C3C2211yescharacteristic, complemented
16#88386C2212yescomplemented
17#64113D311yescharacteristic, complemented
18C2×#3745D311yescharacteristic, complemented
19C3×#904—31no—
20C2×C3×D24—31no—
21C3×#812—31no—
22C3×#812—31no—
23C3×#812—61no—
24#63549—32no—
25#63548—32no—
26#63546—32no—
27#63546—32no—
28C22×(C3⋊D3)⋊C3—41no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30#638⋊C3—41no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)—21no—
33C4⋊((C3⋊D3)⋊C3)—41no—
34C4×(C3⋊D3)⋊C3—41no—
35#634⋊C3—21no—
36C8×H(3,𝔽3)D411yescharacteristic
37C8×H(3,𝔽3)C2311yescharacteristic
38#5193—91no—
39#5147—91no—
40#4908—91no—
41#3828—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C3×#235—61no—
44C2×C3×D12—61no—
45C2×C32×C8—21no—
46C2×C32×C8C2×D311yescharacteristic
47C2×C32×C8C2×D311yescharacteristic
48#3746C2×D311yescharacteristic, complemented
49#3745—21no—
50#3745C2×D312yes—
51#3745C2×D311yescharacteristic, complemented
52#3744—21no—
53C3×Dic12—31no—
54C3×D24—32no—
55C3×D24—31no—
56C3×D24—61no—
57C3×#204—61no—
58C3×C3⋊C16—32no—
59C3×C3⋊C16—32no—
60C3×C3⋊C16—62no—
61C22×H(3,𝔽3)D811yescharacteristic
62C2×(C3⋊D3)⋊C3—41no—
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3—82no—
65C4×H(3,𝔽3)D811yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67(C3⋊Dic3)⋊C3—41no—
68C3×#135—91no—
69C2×C3×D8—91no—
70#904—31no—
71#904—91no—
72C2×D24—31no—
73C2×D24—91no—
74C3×M32—91no—
75#828—92no—
76#827—92no—
77#826—92no—
78#826—92no—
79#812—31no—
80#812—31no—
81#812—61no—
82#800—92no—
83D48—92no—
84C22×C3⋊D3—41no—
85C22×C3×D3—121no—
86C2×C32×C4—21no—
87C2×C32×C4#6711yescharacteristic
88C2×C32×C4#6711yescharacteristic
89#638—41no—
90C4⋊(C3⋊D3)—21no—
91C4⋊(C3⋊D3)—22no—
92C4⋊(C3⋊D3)—41no—
93C4×C3⋊D3—41no—
94#634—21no—
95C3×#67—121no—
96C3×D12—61no—
97C3×D12—62no—
98C3×D12—121no—
99C3×C4×D3—121no—
100C3×Dic6—61no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8#6711yescharacteristic
104C32×C8#6711yescharacteristic
105C32×C8C22×D311yescharacteristic
106C32×C8C22×D311yescharacteristic
107#509—271noSylow
108C2×H(3,𝔽3)—21no—
109C2×H(3,𝔽3)C2×D811yescharacteristic
110(C3⋊D3)⋊C3—81no—
111(C3⋊D3)⋊C3—82no—
112C3×#41—181no—
113C2×C3×D4—181no—
114#235—61no—
115#235—181no—
116C2×D12—61no—
117C2×D12—181no—
118C3×Q16—91no—
119C3×SD16—181no—
120C3×D8—91no—
121C3×D8—92no—
122C3×D8—181no—
123C2×C3×C8—31no—
124C2×C3×C8—31no—
125C2×C3×C8—61no—
126C2×C3×C8D3211yescharacteristic
127Dic12—31no—
128Dic12—91no—
129D24—31no—
130D24—32no—
131D24—61no—
132D24—91no—
133D24—92no—
134D24—181no—
135#204—61no—
136#204—181no—
137C3×C16—92no—
138C3⋊C16—32no—
139C3⋊C16—32no—
140C3⋊C16—62no—
141C22×C32—21no—
142C22×C32#21311yescharacteristic
143C22×C32#21311yescharacteristic
144C2×C3⋊D3—41no—
145C2×C3⋊D3—42no—
146C2×C3⋊D3—82no—
147C2×C3×D3—121no—
148C2×C3×D3—122no—
149C2×C3×D3—242no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4#21311yescharacteristic
153C32×C4#21311yescharacteristic
154C32×C4C2×#6711yescharacteristic
155C32×C4C2×#6711yescharacteristic
156C3⋊Dic3—41no—
157C3×Dic3—121no—
158#135—91no—
159C2×D8—91no—
160SD32—272no—
161D16—272no—
162M32—91no—
163H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
164C23×C3—361no—
165C22×D3—121no—
166C22×D3—361no—
167C3×Q8—181no—
168C3×D4—182no—
169C3×D4—181no—
170C3×D4—361no—
171C3×D4—361no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—61no—
175C2×C3×C4—361no—
176C2×C3×C4#62511yescharacteristic
177#67—121no—
178#67—361no—
179D12—61no—
180D12—62no—
181D12—121no—
182D12—181no—
183D12—182no—
184D12—361no—
185C4×D3—121no—
186C4×D3—361no—
187Dic6—61no—
188Dic6—181no—
189C3×C8—31no—
190C3×C8—31no—
191C3×C8—31no—
192C3×C8—31no—
193C3×C8—61no—
194C3×C8—61no—
195C3×C8#62511yescharacteristic
196C3×C8C2×D3211yescharacteristic
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—41no—
201C2×C32C2×#21311yescharacteristic
202C2×C32C2×#21311yescharacteristic
203C3⋊D3—82no—
204C3⋊D3—81no—
205C3×D3—241no—
206C3×D3—242no—
207#41—181no—
208C2×D4—181no—
209Q16—91no—
210SD16—181no—
211D8—91no—
212D8—92no—
213D8—181no—
214C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
215C16—92no—
216C22×C3—31no—
217C22×C3—31no—
218C22×C3—61no—
219C22×C3—361no—
220C22×C3—362no—
221C22×C3—722no—
222C22×C3#371311yescharacteristic
223C2×D3—121no—
224C2×D3—122no—
225C2×D3—242no—
226C2×D3—361no—
227C2×D3—362no—
228C2×D3—722no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—361no—
236C3×C4#371311yescharacteristic
237C3×C4C2×#62511yescharacteristic
238Dic3—121no—
239Dic3—361no—
240C32—21no—
241C32#514711yescharacteristic, complemented
242C32#519311yescharacteristic, complemented
243C23—361no—
244Q8—181no—
245D4—182no—
246D4—181no—
247D4—361no—
248D4—361no—
249C2×C4—361no—
250C2×C4#619811yescharacteristic
251C8#619811yescharacteristic
252C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C2×C3—21no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3—61no—
259C2×C3—121no—
260C2×C3—722no—
261C2×C3—721no—
262C2×C3C2×#371311yescharacteristic
263D3—242no—
264D3—241no—
265D3—722no—
266D3—721no—
267C22—361no—
268C22—362no—
269C22—722no—
270C22#8843211yescharacteristic
271C4—361no—
272C4#8843211yescharacteristic
273C4C2×#619811yescharacteristic
274C3—31no—
275C3—31no—
276C3—61no—
277C3#1059063011yescharacteristic
278C2—21no—
279C2—722no—
280C2—721no—
281C2C2×#8843211yescharacteristic
subgroupG/H
centerC2C2×#88432
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3713
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#509
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #64113⋊C3C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11720545C2semidirect
4. #11720249C2semidirect
5. #11720248C2semidirect
13. #3745⋊C3C22semidirect
15. #3746⋊C3C22semidirect
16. #88386C22semidirect
17. #64113D3semidirect
18. C2×#3745D3semidirect
48. #3746C2×D3semidirect
51. #3745C2×D3semidirect
163. H(3,𝔽3)#509semidirect
241. C32#5147semidirect
242. C32#5193semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
C2×#3745⋊C32
#117205452
#117202492
#117202482
#105906343
#105906333