Finite Group Databasebeta

← C3×#10592471 #13162809

Subgroups — C3×#10592471 #13162809

Order 1728.

1148 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#64101C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592471C311yesmaximal, characteristic, direct product factor
9C3×#4772—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830C414yescomplemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C3×#3752—21no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3—21no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×C8×C3⋊D3—22no—
24C2×#3830C2×C311yescharacteristic
25C2×C3×#237—34no—
26C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
27#64101C2×C311yescharacteristic, complemented
28C2×C3×D3×C8—34no—
29C3×#892—34no—
30C32×#102—24no—
31C32×#102D314yescomplemented
32C32×#102C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×#832—34no—
37C3×#816—34no—
38C3×#810—34no—
39C23×C33—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×D4—21no—
42C33×D4C2×C412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C3×#638—24no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—24no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3C2×C412yes—
51C3×C4×C3⋊D3D412yes—
52C3×#634—21no—
53C3×#634C2×C412yes—
54C33×C8—21no—
55C3×C3⋊(C3⋊C8)—21no—
56C3×#416—91no—
57#4772—34no—
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C22×C311yescharacteristic
61C2×#638—21no—
62C22×C3⋊Dic3—21no—
63#3830—22no—
64#3830C3×C414yescomplemented
65C2×C4×C3⋊D3C22×C311yescharacteristic
66C2×#634C22×C311yescharacteristic
67C2×C3×#67—64no—
68C22×C3×Dic3—64no—
69C3×#237—316no—
70C3×#237—68no—
71C2×C3×C4×D3—34no—
72C2×C3×Dic6—34no—
73C2×C32×C8—24no—
74C2×C32×C8C2×D314yes—
75C2×C32×C8C22×C311yescharacteristic
76C32×C4⋊C4—24no—
77C32×C4⋊C4C2×D314yes—
78C32×C4⋊C4C22×C311yescharacteristic
79#3752—21no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4×C3⋊Dic3—21no—
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C8×C3⋊D3—22no—
84C3×(C2×D3)⋊C4—64no—
85C3×C4⋊Dic3—34no—
86C3×C4×Dic3—64no—
87C2×C3×C3⋊C8—34no—
88C3×D3×C8—68no—
89C22×C33—22no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33C2×D411yescharacteristic
93C2×C3×C3⋊D3—21no—
94C2×C3×C3⋊D3—21no—
95C2×C3×C3⋊D3C22⋊C412yes—
96C33×C4—41no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—22no—
100C3×C3⋊Dic3—41no—
101C3×C3⋊Dic3C22⋊C412yes—
102C2×C3×#41—91no—
103C2×#237—34no—
104C22×C3×C8—91no—
105C3×#117—91no—
106C2×D3×C8—34no—
107#892—34no—
108C3×#103—91no—
109C3×#103—91no—
110C3×#102—24no—
111C3×#102—91no—
112C3×#102C3×D314yescomplemented
113C3×#102C3⋊D311yescharacteristic, complemented
114#832—34no—
115#816—34no—
116#810—34no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32—44no—
120C22×C3⋊D3C3×D411yescharacteristic
121C22×C3×D3—34no—
122C32×D4—21no—
123C32×D4—28no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4C4×D318yes—
127C32×D4C2×C3×C412yes—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C23×C311yescharacteristic
134#638—24no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—24no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3C3×D411yescharacteristic
139C4×C3⋊D3C2×C3×C412yes—
140C4×C3⋊D3C3×D412yes—
141#634—21no—
142#634C2×C3×C412yes—
143C3×#67—616no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—616no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C3×C4×D3—38no—
149C3×C4×D3—38no—
150C3×Dic6—38no—
151C3×Dic6—64no—
152C32×C8—21no—
153C32×C8—24no—
154C32×C8—44no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—64no—
157#416—91noSylow
158C2×C33—42no—
159C2×C33C2×C22⋊C411yescharacteristic
160C2×C33#13511yescharacteristic
161C2×C33#13511yescharacteristic
162C3×C3⋊D3—22no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×Q8—91no—
166C2×C3×D4—24no—
167C2×C3×D4—181no—
168C2×C3×D4C2×C3×D314yes—
169C2×C3×D4C2×C3⋊D311yescharacteristic
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172C2×#67—64no—
173C22×Dic3—64no—
174#237—316no—
175#237—68no—
176C2×C4×D3—34no—
177C2×Dic6—34no—
178C2×C3×C8—24no—
179C2×C3×C8—91no—
180C2×C3×C8—182no—
181C2×C3×C8C2×C3×D314yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C4⋊C4—24no—
184C3×C4⋊C4—91no—
185C3×C4⋊C4C2×C3×D314yes—
186C3×C4⋊C4C2×C3⋊D311yescharacteristic
187C3×C22⋊C4—181no—
188C3×C42—181no—
189(C2×D3)⋊C4—64no—
190C4⋊Dic3—34no—
191C4×Dic3—64no—
192C2×C3⋊C8—34no—
193D3×C8—68no—
194C22×C32—22no—
195C22×C32—24no—
196C22×C32—28no—
197C22×C32—48no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32—84no—
203C22×C32—84no—
204C22×C32D3×D414yes—
205C22×C32C2×C3×D411yescharacteristic
206C2×C3⋊D3—21no—
207C2×C3⋊D3—21no—
208C2×C3⋊D3C3×C22⋊C412yes—
209C2×C3×D3—38no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—41no—
215C32×C4—44no—
216C32×C4—84no—
217C32×C4C2×C4×D314yes—
218C32×C4D3×D414yes—
219C32×C4C22×C3×C411yescharacteristic
220C32×C4C2×C3×D411yescharacteristic
221C3⋊Dic3—22no—
222C3⋊Dic3—41no—
223C3⋊Dic3C3×C22⋊C412yes—
224C3×Dic3—38no—
225C3×Dic3—68no—
226C3×Dic3—124no—
227C2×#41—91no—
228C22×C8—91no—
229#117—91no—
230#103—91no—
231#103—91no—
232#102—91no—
233#102C3×C3⋊D311yescharacteristic, complemented
234C33#41611yescharacteristic, complemented, Sylow
235C23×C3—24no—
236C23×C3—21no—
237C23×C3—44no—
238C23×C3—91no—
239C22×D3—34no—
240C3×Q8—92no—
241C3×Q8—181no—
242C3×D4—24no—
243C3×D4—21no—
244C3×D4—28no—
245C3×D4—44no—
246C3×D4—184no—
247C3×D4C3×C4×D318yes—
248C3×D4C4×C3⋊D312yes—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—44no—
253C2×C3×C4—91no—
254C2×C3×C4—92no—
255C2×C3×C4—92no—
256C2×C3×C4—184no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4C22×C3×D314yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261#67—616no—
262C2×Dic3—34no—
263C2×Dic3—616no—
264C2×Dic3—64no—
265C2×Dic3—64no—
266C4×D3—38no—
267C4×D3—38no—
268Dic6—38no—
269Dic6—64no—
270C3×C8—24no—
271C3×C8—21no—
272C3×C8—44no—
273C3×C8—181no—
274C3⋊C8—64no—
275C2×C32—24no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—42no—
279C2×C32—48no—
280C2×C32—88no—
281C2×C32D3×C22⋊C414yes—
282C2×C32#91214yes—
283C2×C32#91614yes—
284C2×C32C2×C3×C22⋊C411yescharacteristic
285C2×C32C3×#13511yescharacteristic
286C2×C32C3×#13511yescharacteristic
287C3⋊D3—22no—
288C3×D3—68no—
289#41—94no—
290#41—182no—
291C2×Q8—91no—
292C2×D4—181no—
293C2×D4C2×C3×C3⋊D311yescharacteristic
294C22×C4—91no—
295C22×C4—181no—
296C2×C8—91no—
297C2×C8—182no—
298C2×C8C2×C3×C3⋊D311yescharacteristic
299C4⋊C4—91no—
300C4⋊C4C2×C3×C3⋊D311yescharacteristic
301C22⋊C4—181no—
302C42—181no—
303C22×C3—28no—
304C22×C3—22no—
305C22×C3—24no—
306C22×C3—48no—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—84no—
312C22×C3—84no—
313C22×C3—92no—
314C22×C3—181no—
315C22×C3—181no—
316C22×C3C3×D3×D414yes—
317C22×C3D4×C3⋊D311yescharacteristic
318C2×D3—38no—
319C2×D3—64no—
320C2×D3—64no—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—44no—
324C3×C4—41no—
325C3×C4—84no—
326C3×C4—92no—
327C3×C4—182no—
328C3×C4—361no—
329C3×C4C2×C3×C4×D314yes—
330C3×C4C3×D3×D414yes—
331C3×C4C2×C4×C3⋊D311yescharacteristic
332C3×C4D4×C3⋊D311yescharacteristic
333Dic3—38no—
334Dic3—68no—
335Dic3—124no—
336C32—24no—
337C32#477214yescomplemented
338C32C3×#41611yescharacteristic, complemented
339C23—21no—
340C23—91no—
341Q8—92no—
342Q8—181no—
343D4—21no—
344D4—184no—
345D4C3×C4×C3⋊D312yes—
346C2×C4—21no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—184no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4C22×C3×C3⋊D311yescharacteristic
354C8—21no—
355C8—181no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—48no—
360C2×C3—42no—
361C2×C3—88no—
362C2×C3—182no—
363C2×C3C3×D3×C22⋊C414yes—
364C2×C3C3×#91214yes—
365C2×C3C3×#91614yes—
366C2×C3C22⋊C4×C3⋊D311yescharacteristic
367C2×C3#6412111yescharacteristic
368C2×C3#6412511yescharacteristic
369D3—68no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22C3×D4×C3⋊D311yescharacteristic
377C4—41no—
378C4—92no—
379C4—182no—
380C4—361no—
381C4C2×C3×C4×C3⋊D311yescharacteristic
382C4C3×D4×C3⋊D311yescharacteristic
383C3—24no—
384C3C3×#477214yescomplemented
385C3#1059247111yescharacteristic, direct product factor
386C2—42no—
387C2—182no—
388C2C3×C22⋊C4×C3⋊D311yescharacteristic
389C2C3×#6412111yescharacteristic
390C2C3×#6412511yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×C3⋊D3C2semidirect
3. C3×#64101C2semidirect
4. C33×#102C2semidirect
5. C3×#63659C2semidirect
6. C3×#63643C2semidirect
7. C3×#63637C2semidirect
8. #10592471C3direct (N × H)
14. C3×#3830C4semidirect
26. C2×C8×C3⋊D3C2×C3semidirect
27. #64101C2×C3semidirect
31. C32×#102D3semidirect
32. C32×#102C2×C3semidirect
33. #63659C2×C3semidirect
34. #63643C2×C3semidirect
35. #63637C2×C3semidirect
64. #3830C3×C4semidirect
112. C3×#102C3×D3semidirect
113. C3×#102C3⋊D3semidirect
233. #102C3×C3⋊D3semidirect
234. C33#416semidirect
337. C32#4772semidirect
338. C32C3×#416semidirect
384. C3C3×#4772semidirect
385. C3#10592471direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C3×C8×C3⋊D32
C3×#641012
C33×#1022
C3×#636592
C3×#636432
C3×#636372
#105924713
C3×#47723