Finite Group Databasebeta

← #13138731

Subgroups — #13138731

Order 1728.

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

normal subgroups only

This group has 391 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
1C2×#88529C211yesmaximal, characteristic, complemented
2#64113⋊C3C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#11720254C212yesmaximal, complemented
5#11720253C212yesmaximal, complemented
6#10590633—31nomaximal
7#10590630—31nomaximal
8C2×C4⋊#272—21no—
9#3828⋊C3—21no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88529—21no—
12#88529C2212yes—
13#3745⋊C3C2211yescharacteristic, complemented
14#3744⋊C3—21no—
15#3746⋊C3C2211yescharacteristic, complemented
16(C3⋊(C3⋊C16))⋊C3C2212yescomplemented
17#64113D311yescharacteristic, complemented
18C3×#904—31no—
19C2×C3×D24—31no—
20C2×C3×D24—31no—
21C2×C3×D24—61no—
22#63639D311yescharacteristic, complemented
23C3×#812—31no—
24#63548—32no—
25#63547—32no—
26#63546—32no—
27#63545—32no—
28C22×#272—41no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C4⋊#272—22no—
31C4⋊#272—41no—
32#638⋊C3—41no—
33C4⋊((C3⋊D3)⋊C3)—21no—
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—
42C3×#235—61no—
43C2×C3×D12—62no—
44C2×C3×D12—61no—
45C2×C3×D12—61no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49#3746C2×D311yescharacteristic, complemented
50#3745C2×D311yescharacteristic, complemented
51#3744—21no—
52C3×Dic12—31no—
53C3×D24—32no—
54C3×D24—32no—
55C3×D24—31no—
56C3×D24—62no—
57C3×D24—62no—
58C3×D24—61no—
59C3×D24—61no—
60C3×#204—61no—
61C3⋊(C3⋊C16)C2×D312yescomplemented
62C3×C3⋊C16—32no—
63C22×H(3,𝔽3)D811yescharacteristic
64C2×#272—42no—
65C2×#272—82no—
66C2×(C3⋊D3)⋊C3—41no—
67C4×H(3,𝔽3)D811yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69(C3⋊Dic3)⋊C3—41no—
70C3×#135—91no—
71C2×C3×D8—91no—
72#904—31no—
73#904—91no—
74C2×D24—31no—
75C2×D24—31no—
76C2×D24—61no—
77C3×M32—91no—
78#828—92no—
79#827—92no—
80#826—92no—
81#826—92no—
82#812—31no—
83#812—91no—
84#800—92no—
85D48—92no—
86C22×C3×D3—122no—
87C22×C3×D3—121no—
88C22×C3×D3—121no—
89C2×C32×C4—21no—
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4#6711yescharacteristic
92#638—41no—
93C4⋊(C3⋊D3)—21no—
94C4×C3⋊D3—41no—
95#634—21no—
96C3×#67—121no—
97C3×D12—64no—
98C3×D12—61no—
99C3×D12—62no—
100C3×D12—62no—
101C3×D12—122no—
102C3×D12—121no—
103C3×D12—121no—
104C3×C4×D3—121no—
105C3×Dic6—61no—
106C32×C8—21no—
107C32×C8—21no—
108C32×C8D1211yescharacteristic
109C32×C8#6711yescharacteristic
110C32×C8C22×D311yescharacteristic
111C32×C8C22×D311yescharacteristic
112#509—271noSylow
113C2×H(3,𝔽3)—21no—
114C2×H(3,𝔽3)C2×D811yescharacteristic
115#272—82no—
116(C3⋊D3)⋊C3—81no—
117C3×#41—181no—
118C2×C3×D4—181no—
119#235—61no—
120#235—181no—
121C2×D12—61no—
122C2×D12—61no—
123C2×D12—62no—
124C3×Q16—91no—
125C3×SD16—181no—
126C3×D8—91no—
127C3×D8—92no—
128C3×D8—181no—
129C2×C3×C8—31no—
130C2×C3×C8—31no—
131C2×C3×C8—61no—
132C2×C3×C8D3211yescharacteristic
133Dic12—31no—
134Dic12—91no—
135D24—32no—
136D24—32no—
137D24—31no—
138D24—62no—
139D24—61no—
140D24—61no—
141D24—62no—
142D24—91no—
143#204—61no—
144#204—181no—
145C3×C16—92no—
146C3⋊C16—32no—
147C3⋊C16—92no—
148C22×C32—21no—
149C22×C32D2411yescharacteristic
150C22×C32#21311yescharacteristic
151C2×C3⋊D3—41no—
152C2×C3×D3—124no—
153C2×C3×D3—121no—
154C2×C3×D3—122no—
155C2×C3×D3—122no—
156C2×C3×D3—244no—
157C2×C3×D3—242no—
158C2×C3×D3—242no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4D2411yescharacteristic
162C32×C4#21311yescharacteristic
163C32×C4C2×D1211yescharacteristic
164C32×C4C2×#6711yescharacteristic
165C3⋊Dic3—41no—
166C3×Dic3—121no—
167#135—91no—
168C2×D8—91no—
169SD32—272no—
170D16—272no—
171M32—91no—
172H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
173C23×C3—361no—
174C22×D3—121no—
175C22×D3—121no—
176C22×D3—122no—
177C3×Q8—181no—
178C3×D4—181no—
179C3×D4—182no—
180C3×D4—361no—
181C3×D4—361no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—361no—
186C2×C3×C4#62611yescharacteristic
187#67—121no—
188#67—361no—
189D12—62no—
190D12—62no—
191D12—64no—
192D12—61no—
193D12—121no—
194D12—121no—
195D12—122no—
196D12—181no—
197C4×D3—121no—
198C4×D3—361no—
199Dic6—61no—
200Dic6—181no—
201C3×C8—31no—
202C3×C8—31no—
203C3×C8—31no—
204C3×C8—31no—
205C3×C8—61no—
206C3×C8—61no—
207C3×C8#62611yescharacteristic
208C3×C8C2×D3211yescharacteristic
209C2×C32—22no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32C2×D2411yescharacteristic
214C2×C32C2×#21311yescharacteristic
215C3⋊D3—81no—
216C3×D3—244no—
217C3×D3—241no—
218C3×D3—242no—
219C3×D3—242no—
220#41—181no—
221C2×D4—181no—
222Q16—91no—
223SD16—181no—
224D8—91no—
225D8—92no—
226D8—181no—
227C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C16—92no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—61no—
232C22×C3—361no—
233C22×C3—362no—
234C22×C3—722no—
235C22×C3#371411yescharacteristic
236C2×D3—122no—
237C2×D3—122no—
238C2×D3—124no—
239C2×D3—121no—
240C2×D3—242no—
241C2×D3—242no—
242C2×D3—244no—
243C2×D3—361no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—361no—
251C3×C4#371411yescharacteristic
252C3×C4C2×#62611yescharacteristic
253Dic3—121no—
254Dic3—361no—
255C32—21no—
256C32#490811yescharacteristic, complemented
257C32#519311yescharacteristic, complemented
258C23—361no—
259Q8—181no—
260D4—182no—
261D4—181no—
262D4—361no—
263D4—361no—
264C2×C4—361no—
265C2×C4#620011yescharacteristic
266C8#620011yescharacteristic
267C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
268C2×C3—21no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—62no—
274C2×C3—61no—
275C2×C3—721no—
276C2×C3—722no—
277C2×C3C2×#371411yescharacteristic
278D3—242no—
279D3—242no—
280D3—244no—
281D3—241no—
282D3—721no—
283C22—361no—
284C22—362no—
285C22—722no—
286C22#8843611yescharacteristic
287C4—361no—
288C4#8843611yescharacteristic
289C4C2×#620011yescharacteristic
290C3—31no—
291C3—31no—
292C3—61no—
293C3#1059063411yescharacteristic
294C2—21no—
295C2—722no—
296C2—721no—
297C2C2×#8843611yescharacteristic
subgroupG/H
centerC2C2×#88436
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3714
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. C2×#88529C2semidirect
2. #64113⋊C3C2semidirect
3. #63639⋊C3C2semidirect
4. #11720254C2semidirect
5. #11720253C2semidirect
13. #3745⋊C3C22semidirect
15. #3746⋊C3C22semidirect
16. (C3⋊(C3⋊C16))⋊C3C22semidirect
17. #64113D3semidirect
22. #63639D3semidirect
49. #3746C2×D3semidirect
50. #3745C2×D3semidirect
61. C3⋊(C3⋊C16)C2×D3semidirect
172. H(3,𝔽3)#509semidirect
256. C32#4908semidirect
257. C32#5193semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#64113⋊C32
#63639⋊C32
#117202542
#117202532
#105906333
#105906303