Finite Group Databasebeta

← C3×#10592856 #13163195

Subgroups — C3×#10592856 #13163195

Order 1728.

1108 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 198 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×C8×C3⋊D3C211yesmaximal, characteristic, complemented
4C3×#63659C212yesmaximal, complemented
5C3×#63645C211yesmaximal, characteristic, complemented
6#10592856C311yesmaximal, characteristic, direct product factor
7C3×#5157—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×C8×C3⋊D3—22no—
17C2×C32×D8—24no—
18C2×C32×D8D314yescomplemented
19C2×C32×D8C2×C311yescharacteristic, complemented
20#64147C2×C312yescomplemented
21C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
22C3×#938—38no—
23C2×C3×D3×C8—34no—
24#63659C2×C312yescomplemented
25#63645C2×C311yescharacteristic, complemented
26C3×#832—38no—
27C3×#818—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×#466—91no—
41#5157—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
58C8×C3⋊D3—22no—
59C3×#217—68no—
60C3×C4⋊Dic3—38no—
61C2×C3×C3⋊C8—34no—
62C3×D3×C8—68no—
63C22×C33—42no—
64C22×C33—42no—
65C22×C33—42no—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3D812yescomplemented
70C33×C4C2×D411yescharacteristic
71C33×C4#4111yescharacteristic
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3—42no—
74C2×C3×D8—24no—
75C2×C3×D8C3×D314yescomplemented
76C2×C3×D8C3⋊D311yescharacteristic, complemented
77C22×C3×C8—91no—
78C3×#121—92no—
79#938—38no—
80C2×D3×C8—34no—
81C3×#107—91no—
82C3×#102—92no—
83#832—38no—
84#818—34no—
85C23×C32—22no—
86C23×C32—28no—
87C23×C32—48no—
88C22×C3⋊D3C3×D411yescharacteristic
89C22×C3×D3—34no—
90C32×D4—216no—
91C32×D4—28no—
92C32×D4—22no—
93C32×D4—22no—
94C32×D4—28no—
95C32×D4—48no—
96C2×C32×C4—24no—
97C2×C32×C4C22×D314yes—
98C2×C32×C4C23×C311yescharacteristic
99#638—44no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3C3×D411yescharacteristic
102C4×C3⋊D3—21no—
103C4×C3⋊D3—21no—
104C3×#67—1216no—
105C2×C3×Dic3—34no—
106C2×C3×Dic3—68no—
107C3×C4×D3—64no—
108C3×C4×D3—64no—
109C32×C8—28no—
110C32×C8#6718yescomplemented
111C32×C8C3×D412yescomplemented
112C3⋊(C3⋊C8)—21no—
113C3×C3⋊C8—64no—
114#466—91noSylow
115C2×C33—82no—
116C2×C33#12111yescharacteristic
117C2×C33C2×D811yescharacteristic
118C2×C33#13511yescharacteristic
119C3×C3⋊D3—22no—
120C2×C3×D4—28no—
121C2×C3×D4—182no—
122C2×C3×D4C2×C3×D318yes—
123C2×C3×D4C2×C3⋊D312yes—
124C22×C3×C4—91no—
125C2×#67—68no—
126C2×C4×D3—34no—
127C3×D8—216no—
128C3×D8—28no—
129C3×D8—22no—
130C2×C3×C8—24no—
131C2×C3×C8—91no—
132C2×C3×C8—182no—
133C2×C3×C8C2×C3×D314yes—
134C2×C3×C8C2×C3⋊D311yescharacteristic
135C3×C4⋊C4—92no—
136C3×C22⋊C4—182no—
137#217—68no—
138C4⋊Dic3—38no—
139C2×C3⋊C8—34no—
140D3×C8—68no—
141C22×C32—24no—
142C22×C32—416no—
143C22×C32—48no—
144C22×C32—42no—
145C22×C32—42no—
146C22×C32—42no—
147C22×C32—48no—
148C22×C32—48no—
149C22×C32—88no—
150C22×C32—88no—
151C22×C32D3×D414yes—
152C22×C32C2×C3×D411yescharacteristic
153C2×C3⋊D3—21no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3C3×D812yescomplemented
156C2×C3×D3—38no—
157C2×C3×D3—64no—
158C2×C3×D3—64no—
159C32×C4—24no—
160C32×C4—24no—
161C32×C4#23714yes—
162C32×C4C2×#6714yes—
163C32×C4C2×C3×D411yescharacteristic
164C32×C4C3×#4111yescharacteristic
165C3⋊Dic3—21no—
166C3⋊Dic3—42no—
167C3×Dic3—64no—
168C3×Dic3—128no—
169C2×D8C3×C3⋊D311yescharacteristic, complemented
170C22×C8—91no—
171#121—92no—
172#107—91no—
173#102—92no—
174C33#46611yescharacteristic, complemented, Sylow
175C23×C3—28no—
176C23×C3—22no—
177C23×C3—48no—
178C23×C3—91no—
179C22×D3—34no—
180C3×D4—216no—
181C3×D4—28no—
182C3×D4—28no—
183C3×D4—22no—
184C3×D4—22no—
185C3×D4—48no—
186C3×D4—364no—
187C2×C3×C4—24no—
188C2×C3×C4—91no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—182no—
192C2×C3×C4C22×C3×D314yes—
193C2×C3×C4C22×C3⋊D311yescharacteristic
194#67—1216no—
195C2×Dic3—34no—
196C2×Dic3—68no—
197C4×D3—64no—
198C4×D3—64no—
199C3×C8—28no—
200C3×C8—181no—
201C3×C8C3×#6718yescomplemented
202C3×C8#63812yescomplemented
203C3⋊C8—64no—
204C2×C32—24no—
205C2×C32—24no—
206C2×C32—24no—
207C2×C32—816no—
208C2×C32—88no—
209C2×C32—82no—
210C2×C32D3×D814yes—
211C2×C32#91214yes—
212C2×C32#93814yes—
213C2×C32C3×#12111yescharacteristic
214C2×C32C2×C3×D811yescharacteristic
215C2×C32C3×#13511yescharacteristic
216C3⋊D3—22no—
217C3×D3—68no—
218C2×D4—182no—
219C2×D4C2×C3×C3⋊D312yes—
220C22×C4—91no—
221D8—22no—
222C2×C8—91no—
223C2×C8—182no—
224C2×C8C2×C3×C3⋊D311yescharacteristic
225C4⋊C4—92no—
226C22⋊C4—182no—
227C22×C3—24no—
228C22×C3—416no—
229C22×C3—48no—
230C22×C3—48no—
231C22×C3—48no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—88no—
236C22×C3—88no—
237C22×C3—92no—
238C22×C3—181no—
239C22×C3—181no—
240C22×C3C3×D3×D414yes—
241C22×C3D4×C3⋊D311yescharacteristic
242C2×D3—38no—
243C2×D3—64no—
244C2×D3—64no—
245C3×C4—24no—
246C3×C4—24no—
247C3×C4—181no—
248C3×C4—362no—
249C3×C4C3×#23714yes—
250C3×C4C2×C3×#6714yes—
251C3×C4#383011yescharacteristic
252C3×C4C2×#63811yescharacteristic
253Dic3—64no—
254Dic3—128no—
255C32—24no—
256C32#515714yescomplemented
257C32C3×#46611yescharacteristic, complemented
258C23—22no—
259C23—91no—
260D4—22no—
261D4—22no—
262D4—364no—
263C2×C4—91no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—182no—
267C2×C4C22×C3×C3⋊D311yescharacteristic
268C8—181no—
269C8C3×#63812yescomplemented
270C2×C3—24no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—816no—
274C2×C3—82no—
275C2×C3—88no—
276C2×C3—182no—
277C2×C3C3×D3×D814yes—
278C2×C3C3×#91214yes—
279C2×C3C3×#93814yes—
280C2×C3D8×C3⋊D311yescharacteristic
281C2×C3#6412111yescharacteristic
282C2×C3#6414711yescharacteristic
283D3—68no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—92no—
288C22—181no—
289C22—181no—
290C22C3×D4×C3⋊D311yescharacteristic
291C4—181no—
292C4—362no—
293C4C3×#383011yescharacteristic
294C4C2×C3×#63811yescharacteristic
295C3—24no—
296C3C3×#515714yescomplemented
297C3#1059285611yescharacteristic, direct product factor
298C2—82no—
299C2—182no—
300C2C3×D8×C3⋊D311yescharacteristic
301C2C3×#6412111yescharacteristic
302C2C3×#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×#466
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×C8×C3⋊D3C2semidirect
4. C3×#63659C2semidirect
5. C3×#63645C2semidirect
6. #10592856C3direct (N × H)
18. C2×C32×D8D3semidirect
19. C2×C32×D8C2×C3semidirect
20. #64147C2×C3semidirect
21. C2×C8×C3⋊D3C2×C3semidirect
24. #63659C2×C3semidirect
25. #63645C2×C3semidirect
38. C33×C8D4semidirect
69. C2×C3×C3⋊D3D8semidirect
75. C2×C3×D8C3×D3semidirect
76. C2×C3×D8C3⋊D3semidirect
110. C32×C8#67semidirect
111. C32×C8C3×D4semidirect
155. C2×C3⋊D3C3×D8semidirect
169. C2×D8C3×C3⋊D3semidirect
174. C33#466semidirect
201. C3×C8C3×#67semidirect
202. C3×C8#638semidirect
256. C32#5157semidirect
257. C32C3×#466semidirect
269. C8C3×#638semidirect
296. C3C3×#5157semidirect
297. C3#10592856direct (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×C8×C3⋊D32
C3×#636592
C3×#636452
#105928563
C3×#51573