Finite Group Databasebeta

← C3×#10592858 #13163197

Subgroups — C3×#10592858 #13163197

Order 1728.

1084 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 194 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×C33×D8C211yesmaximal, characteristic, complemented
2C3×#64147C212yesmaximal, complemented
3C2×C3×#3743C211yesmaximal, characteristic, complemented
4C3×#63659C212yesmaximal, complemented
5C3×#63644C211yesmaximal, characteristic, complemented
6#10592858C311yesmaximal, characteristic, direct product factor
7C3×#5159—34nomaximal
8C2×C33×D4C2212yes—
9C2×C3×#638—22no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C33×D8—22no—
12C2×C33×C8C2211yescharacteristic
13C3×#3757—22no—
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#3743—22no—
17C2×C32×D8—24no—
18C2×C32×D8D314yescomplemented
19C2×C32×D8C2×C311yescharacteristic, complemented
20#64147C2×C312yescomplemented
21C2×#3743C2×C311yescharacteristic, complemented
22C3×#938—38no—
23C2×C3×ΓL(1,𝔽25)—34no—
24#63659C2×C312yescomplemented
25#63644C2×C311yescharacteristic, complemented
26C3×#832—38no—
27C3×#817—34no—
28C23×C33—22no—
29C22×C3×C3⋊D3D411yescharacteristic
30C33×D4—22no—
31C33×D4—22no—
32C2×C33×C4C2311yescharacteristic
33C3×#638—44no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—21no—
37C3×C4×C3⋊D3—21no—
38C33×C8D412yescomplemented
39C3×C3⋊(C3⋊C8)—21no—
40C3×#469—91no—
41#5159—34no—
42C2×C32×D4—28no—
43C2×C32×D4C2×D318yes—
44C2×C32×D4C22×C312yes—
45C2×#638—22no—
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C2×C3×#67—68no—
48C2×C3×C4×D3—34no—
49C32×D8—216no—
50C32×D8—28no—
51C32×D8—22no—
52C2×C32×C8—24no—
53C2×C32×C8C2×D314yes—
54C2×C32×C8C22×C311yescharacteristic
55#3757—22no—
56C4⋊(C3⋊Dic3)C22×C312yes—
57C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
58#3743—22no—
59C3×#217—68no—
60C3×C4⋊Dic3—38no—
61C2×C3×C3⋊C8—34no—
62C3×ΓL(1,𝔽25)—68no—
63C22×C33—42no—
64C22×C33—42no—
65C22×C33—42no—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—22no—
69C33×C4C2×D411yescharacteristic
70C33×C4#4111yescharacteristic
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3—42no—
73C2×C3×D8—24no—
74C2×C3×D8C3×D314yescomplemented
75C2×C3×D8C3⋊D311yescharacteristic, complemented
76C2×C3×M16—91no—
77C3×#121—92no—
78#938—38no—
79C2×ΓL(1,𝔽25)—34no—
80C3×#106—91no—
81C3×#102—92no—
82#832—38no—
83#817—34no—
84C23×C32—22no—
85C23×C32—28no—
86C23×C32—48no—
87C22×C3⋊D3C3×D411yescharacteristic
88C22×C3×D3—34no—
89C32×D4—216no—
90C32×D4—28no—
91C32×D4—22no—
92C32×D4—22no—
93C32×D4—28no—
94C32×D4—48no—
95C2×C32×C4—24no—
96C2×C32×C4C22×D314yes—
97C2×C32×C4C23×C311yescharacteristic
98#638—44no—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3C3×D411yescharacteristic
101C4×C3⋊D3—21no—
102C4×C3⋊D3—21no—
103C3×#67—1216no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—68no—
106C3×C4×D3—64no—
107C3×C4×D3—64no—
108C32×C8—28no—
109C32×C8#6718yescomplemented
110C32×C8C3×D412yescomplemented
111C3⋊(C3⋊C8)—21no—
112C3×C3⋊C8—64no—
113#469—91noSylow
114C2×C33—82no—
115C2×C33#12111yescharacteristic
116C2×C33AGL(1,ℤ/8)12yes—
117C3×C3⋊D3—41no—
118C2×C3×D4—28no—
119C2×C3×D4—182no—
120C2×C3×D4C2×C3×D318yes—
121C2×C3×D4C2×C3⋊D312yes—
122C22×C3×C4—91no—
123C2×#67—68no—
124C2×C4×D3—34no—
125C3×D8—216no—
126C3×D8—28no—
127C3×D8—22no—
128C3×M16—182no—
129C2×C3×C8—24no—
130C2×C3×C8—91no—
131C2×C3×C8C2×C3×D314yes—
132C2×C3×C8C2×C3⋊D311yescharacteristic
133C3×C4⋊C4—92no—
134C3×C22⋊C4—182no—
135#217—68no—
136C4⋊Dic3—38no—
137C2×C3⋊C8—34no—
138ΓL(1,𝔽25)—68no—
139C22×C32—24no—
140C22×C32—416no—
141C22×C32—48no—
142C22×C32—42no—
143C22×C32—42no—
144C22×C32—42no—
145C22×C32—48no—
146C22×C32—48no—
147C22×C32—88no—
148C22×C32—88no—
149C22×C32D3×D414yes—
150C22×C32C2×C3×D411yescharacteristic
151C2×C3⋊D3—21no—
152C2×C3⋊D3—22no—
153C2×C3×D3—68no—
154C2×C3×D3—64no—
155C32×C4—24no—
156C32×C4—24no—
157C32×C4#23714yes—
158C32×C4C2×#6714yes—
159C32×C4C2×C3×D411yescharacteristic
160C32×C4C3×#4111yescharacteristic
161C3⋊Dic3—21no—
162C3⋊Dic3—42no—
163C3×Dic3—64no—
164C3×Dic3—128no—
165C2×D8C3×C3⋊D311yescharacteristic, complemented
166C2×M16—91no—
167#121—92no—
168#106—91no—
169#102—92no—
170C33#46911yescharacteristic, complemented, Sylow
171C23×C3—28no—
172C23×C3—22no—
173C23×C3—48no—
174C23×C3—91no—
175C22×D3—34no—
176C3×D4—216no—
177C3×D4—28no—
178C3×D4—28no—
179C3×D4—22no—
180C3×D4—22no—
181C3×D4—48no—
182C3×D4—364no—
183C2×C3×C4—24no—
184C2×C3×C4—91no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—182no—
188C2×C3×C4C22×C3×D314yes—
189C2×C3×C4C22×C3⋊D311yescharacteristic
190#67—1216no—
191C2×Dic3—34no—
192C2×Dic3—68no—
193C4×D3—64no—
194C4×D3—64no—
195C3×C8—28no—
196C3×C8—181no—
197C3×C8C3×#6718yescomplemented
198C3×C8#63812yescomplemented
199C3⋊C8—64no—
200C2×C32—28no—
201C2×C32—24no—
202C2×C32—816no—
203C2×C32—88no—
204C2×C32—82no—
205C2×C32#91118yes—
206C2×C32#93814yes—
207C2×C32C3×#12111yescharacteristic
208C2×C32C3×AGL(1,ℤ/8)12yes—
209C3⋊D3—41no—
210C3×D3—124no—
211C2×D4—182no—
212C2×D4C2×C3×C3⋊D312yes—
213C22×C4—91no—
214D8—22no—
215M16—182no—
216C2×C8—91no—
217C2×C8C2×C3×C3⋊D311yescharacteristic
218C4⋊C4—92no—
219C22⋊C4—182no—
220C22×C3—24no—
221C22×C3—416no—
222C22×C3—48no—
223C22×C3—48no—
224C22×C3—48no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—88no—
229C22×C3—88no—
230C22×C3—182no—
231C22×C3—181no—
232C22×C3C3×D3×D414yes—
233C22×C3D4×C3⋊D311yescharacteristic
234C2×D3—68no—
235C2×D3—64no—
236C3×C4—24no—
237C3×C4—24no—
238C3×C4—181no—
239C3×C4—362no—
240C3×C4C3×#23714yes—
241C3×C4C2×C3×#6714yes—
242C3×C4#383011yescharacteristic
243C3×C4C2×#63811yescharacteristic
244Dic3—64no—
245Dic3—128no—
246C32—24no—
247C32#515914yescomplemented
248C32C3×#46911yescharacteristic, complemented
249C23—22no—
250C23—91no—
251D4—22no—
252D4—22no—
253D4—364no—
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—182no—
258C2×C4C22×C3×C3⋊D311yescharacteristic
259C8—181no—
260C8C3×#63812yescomplemented
261C2×C3—28no—
262C2×C3—24no—
263C2×C3—816no—
264C2×C3—88no—
265C2×C3—82no—
266C2×C3—361no—
267C2×C3C3×#91118yes—
268C2×C3C3×#93814yes—
269C2×C3#6412012yes—
270C2×C3#6414711yescharacteristic
271D3—124no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—182no—
276C22—181no—
277C22C3×D4×C3⋊D311yescharacteristic
278C4—181no—
279C4—362no—
280C4C3×#383011yescharacteristic
281C4C2×C3×#63811yescharacteristic
282C3—24no—
283C3C3×#515914yescomplemented
284C3#1059285811yescharacteristic, direct product factor
285C2—82no—
286C2—361no—
287C2C3×#6412012yes—
288C2C3×#6414711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×D8C2
socleC22×C33C2×D4
hypercenterC2×C3×D8C3⋊D3
hypocenterC32C3×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D8

Complements

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

NH ≅ G/Ntype
1. C2×C33×D8C2semidirect
2. C3×#64147C2semidirect
3. C2×C3×#3743C2semidirect
4. C3×#63659C2semidirect
5. C3×#63644C2semidirect
6. #10592858C3direct (N × H)
18. C2×C32×D8D3semidirect
19. C2×C32×D8C2×C3semidirect
20. #64147C2×C3semidirect
21. C2×#3743C2×C3semidirect
24. #63659C2×C3semidirect
25. #63644C2×C3semidirect
38. C33×C8D4semidirect
74. C2×C3×D8C3×D3semidirect
75. C2×C3×D8C3⋊D3semidirect
109. C32×C8#67semidirect
110. C32×C8C3×D4semidirect
165. C2×D8C3×C3⋊D3semidirect
170. C33#469semidirect
197. C3×C8C3×#67semidirect
198. C3×C8#638semidirect
247. C32#5159semidirect
248. C32C3×#469semidirect
260. C8C3×#638semidirect
283. C3C3×#5159semidirect
284. C3#10592858direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×D82
C3×#641472
C2×C3×#37432
C3×#636592
C3×#636442
#105928583
C3×#51593