Finite Group Databasebeta

← C3×#10592515 #13162851

Subgroups — C3×#10592515 #13162851

Order 1728.

828 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3743C211yesmaximal, characteristic
4C3×#64106C211yesmaximal, characteristic
5C33×#103C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592515C311yesmaximal, characteristic, direct product factor
9C3×#4816—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3756—22no—
16C3×#3752—21no—
17C3×#3752—21no—
18C3×C4⋊(C3⋊Dic3)—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×#3750—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#3743—22no—
23#64155C2×C311yescharacteristic
24C2×#3756C2×C311yescharacteristic, complemented
25C2×#3743C2×C311yescharacteristic
26#64106C2×C311yescharacteristic
27C3×#946—34no—
28C2×C3×#216—34no—
29C2×C3×ΓL(1,𝔽25)—34no—
30C3×#897—34no—
31C32×#103—24no—
32C32×#103D314yescomplemented
33C32×#103C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63635C2×C311yescharacteristic, complemented
36C3×#816—34no—
37C3×#808—34no—
38C22×C3×C3⋊D3D411yescharacteristic
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C33×C8—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53C3×#470—91no—
54#4816—34no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C22×C311yescharacteristic
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C2×#634C22×C311yescharacteristic
60C2×C3×C4×D3—34no—
61C2×C3×Dic6—34no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C22×C311yescharacteristic
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C22×C311yescharacteristic
68#3756—22no—
69#3752—21no—
70#3752—21no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)C22×C311yescharacteristic
73#3750—21no—
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75#3743—22no—
76C3×#216—68no—
77C3×(C2×D3)⋊C4—64no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4⋊Dic3—34no—
80C3×C4⋊Dic3—64no—
81C3×Dic3⋊C4—64no—
82C2×C3×C3⋊C8—34no—
83C3×ΓL(1,𝔽25)—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C33×C4—41no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—41no—
94C3×C3⋊Dic3—41no—
95C2×C3×Q16—91no—
96C2×C3×M16—91no—
97C3×C22⋊Q8—91no—
98C3×C22⋊Q8—91no—
99#946—34no—
100C2×#216—34no—
101C2×ΓL(1,𝔽25)—34no—
102#897—34no—
103C3×#106—91no—
104C3×#103—24no—
105C3×#103—91no—
106C3×#103C3×D314yescomplemented
107C3×#103C3⋊D311yescharacteristic, complemented
108#816—34no—
109#808—34no—
110C22×C3⋊D3C3×D411yescharacteristic
111C22×C3×D3—34no—
112C32×Q8—28no—
113C32×Q8—24no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—24no—
117C32×Q8—44no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—44no—
124C2×C32×C4—44no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3C3×D411yescharacteristic
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132#634—21no—
133#634—21no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—64no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139C3×Dic6—64no—
140C3×Dic6—64no—
141C32×C8—21no—
142C32×C8—24no—
143C32×C8—44no—
144C3⋊(C3⋊C8)C3×D412yes—
145C3×C3⋊C8—38no—
146#470—91noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C33#13711yescharacteristic
150C3×C3⋊D3—41no—
151C2×C3×Q8—24no—
152C2×C3×Q8—91no—
153C2×C3×Q8C2×C3×D314yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C22×C3×C4—91no—
156C2×C4×D3—34no—
157C2×Dic6—34no—
158C3×Q16—182no—
159C3×M16—182no—
160C2×C3×C8—24no—
161C2×C3×C8—91no—
162C2×C3×C8C2×C3×D314yes—
163C2×C3×C8C2×C3⋊D311yescharacteristic
164C3×C4⋊C4—24no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4—181no—
168C3×C4⋊C4C2×C3×D314yes—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170C3×C22⋊C4—181no—
171C3×C22⋊C4—181no—
172#216—68no—
173(C2×D3)⋊C4—64no—
174(C2×D3)⋊C4—64no—
175C4⋊Dic3—34no—
176C4⋊Dic3—64no—
177Dic3⋊C4—64no—
178C2×C3⋊C8—34no—
179ΓL(1,𝔽25)—68no—
180C22×C32—24no—
181C22×C32D3×D414yes—
182C22×C32C2×C3×D411yescharacteristic
183C2×C3⋊D3—21no—
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—48no—
192C32×C4—44no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—44no—
196C32×C4—84no—
197C32×C4#23514yes—
198C32×C4D3×D414yes—
199C32×C4C2×C3×D411yescharacteristic
200C32×C4C3×#4111yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3×Dic3—64no—
205C3×Dic3—124no—
206C3×Dic3—124no—
207C2×Q16—91no—
208C2×M16—91no—
209C22⋊Q8—91no—
210C22⋊Q8—91no—
211#106—91no—
212#103—91no—
213#103C3×C3⋊D311yescharacteristic, complemented
214C33#47011yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—34no—
217C3×Q8—28no—
218C3×Q8—24no—
219C3×Q8—24no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—44no—
223C3×Q8—181no—
224C3×Q8—181no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—44no—
231C2×C3×C4—44no—
232C2×C3×C4—91no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4C22×C3×D314yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—34no—
240C2×Dic3—64no—
241C2×Dic3—64no—
242C4×D3—64no—
243C4×D3—64no—
244Dic6—64no—
245Dic6—64no—
246C3×C8—24no—
247C3×C8—21no—
248C3×C8—44no—
249C3×C8—92no—
250C3⋊C8—38no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32#88414yes—
255C2×C32#91514yes—
256C2×C32#91814yes—
257C2×C32C3×#12111yescharacteristic
258C2×C32C3×#13711yescharacteristic
259C2×C32C3×#13711yescharacteristic
260C3⋊D3—41no—
261C3×D3—124no—
262C2×Q8—91no—
263C2×Q8C2×C3×C3⋊D311yescharacteristic
264C22×C4—91no—
265Q16—182no—
266M16—182no—
267C2×C8—91no—
268C2×C8C2×C3×C3⋊D311yescharacteristic
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—181no—
272C4⋊C4C2×C3×C3⋊D311yescharacteristic
273C22⋊C4—181no—
274C22⋊C4—181no—
275C22×C3—24no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3C3×D3×D414yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—64no—
282C2×D3—64no—
283C2×D3—64no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—48no—
287C3×C4—44no—
288C3×C4—44no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—84no—
292C3×C4—181no—
293C3×C4—361no—
294C3×C4—361no—
295C3×C4C3×#23514yes—
296C3×C4C3×D3×D414yes—
297C3×C4#382811yescharacteristic
298C3×C4D4×C3⋊D311yescharacteristic
299Dic3—64no—
300Dic3—124no—
301Dic3—124no—
302C32—24no—
303C32#481614yescomplemented
304C32C3×#47011yescharacteristic, complemented
305C23—91no—
306Q8—21no—
307Q8—21no—
308Q8—181no—
309Q8—181no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4C22×C3×C3⋊D311yescharacteristic
318C8—21no—
319C8—92no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—361no—
324C2×C3C3×#88414yes—
325C2×C3C3×#91514yes—
326C2×C3C3×#91814yes—
327C2×C3#6409311yescharacteristic
328C2×C3#6412411yescharacteristic
329C2×C3#6412711yescharacteristic
330D3—124no—
331C22—181no—
332C22—181no—
333C22—181no—
334C22C3×D4×C3⋊D311yescharacteristic
335C4—41no—
336C4—41no—
337C4—181no—
338C4—361no—
339C4—361no—
340C4C3×#382811yescharacteristic
341C4C3×D4×C3⋊D311yescharacteristic
342C3—24no—
343C3C3×#481614yescomplemented
344C3#1059251511yescharacteristic, direct product factor
345C2—361no—
346C2C3×#6409311yescharacteristic
347C2C3×#6412411yescharacteristic
348C2C3×#6412711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
5. C33×#103C2semidirect
6. C3×#63643C2semidirect
7. C3×#63635C2semidirect
8. #10592515C3direct (N × H)
24. C2×#3756C2×C3semidirect
32. C32×#103D3semidirect
33. C32×#103C2×C3semidirect
34. #63643C2×C3semidirect
35. #63635C2×C3semidirect
106. C3×#103C3×D3semidirect
107. C3×#103C3⋊D3semidirect
213. #103C3×C3⋊D3semidirect
214. C33#470semidirect
303. C32#4816semidirect
304. C32C3×#470semidirect
343. C3C3×#4816semidirect
344. C3#10592515direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641552
C2×C3×#37562
C2×C3×#37432
C3×#641062
C33×#1032
C3×#636432
C3×#636352
#105925153
C3×#48163