Finite Group Databasebeta

← C3×#10592480 #13162818

Subgroups — C3×#10592480 #13162818

Order 1728.

1172 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C2×C3×#3744C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C33×#102C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592480C311yesmaximal, characteristic, direct product factor
9C3×#4781—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830—22no—
15C2×C3×C4⋊(C3⋊D3)—21no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×C33×C8C2211yescharacteristic
20C33×C4⋊C4C2211yescharacteristic
21C3×#3753—22no—
22C3×#3752—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#3744—22no—
25C2×#3830C2×C311yescharacteristic, complemented
26C2×C3×#237—34no—
27C2×#3753C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic, complemented
30C2×C3×#213—34no—
31C2×C3×#204—34no—
32C3×#895—34no—
33C32×#102—24no—
34C32×#102D314yescomplemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#810—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×D4—21no—
44C33×D4D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C3×#638—24no—
48C2×C3×C3⋊Dic3—24no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—42no—
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#449—91no—
61#4781—34no—
62C2×C32×D4—24no—
63C2×C32×D4C2×D314yes—
64C2×C32×D4C22×C311yescharacteristic
65C2×#638—21no—
66C22×C3⋊Dic3—21no—
67#3830—22no—
68#3830—22no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3C22×C311yescharacteristic
72C2×#634C22×C311yescharacteristic
73C2×C3×#67—64no—
74C22×C3×Dic3—64no—
75C3×#237—68no—
76C3×#237—68no—
77C2×C3×D12—34no—
78C2×C3×D12—64no—
79C2×C3×C4×D3—34no—
80C2×C3×Dic6—34no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yes—
83C2×C32×C8C22×C311yescharacteristic
84C32×C4⋊C4—24no—
85C32×C4⋊C4C2×D314yes—
86C32×C4⋊C4C22×C311yescharacteristic
87#3753—22no—
88#3752—21no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90#3744—22no—
91C3×#213—68no—
92C3×(C2×D3)⋊C4—64no—
93C2×C3×C3⋊C8—34no—
94C3×#204—68no—
95C22×C33—22no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C33×C4—41no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C3×C3⋊Dic3—21no—
110C2×C3×#41—91no—
111C2×#237—34no—
112C2×C3×SD16—91no—
113C2×C3×D8—91no—
114C3×#121—91no—
115C2×#213—34no—
116C2×#204—34no—
117#895—34no—
118C3×#103—91no—
119C3×#102—24no—
120C3×#102C3×D314yescomplemented
121C3×#102C3⋊D311yescharacteristic, complemented
122C3×C22⋊C8—91no—
123(C2×D3)⋊C8—34no—
124#810—34no—
125C23×C32—21no—
126C23×C32—24no—
127C23×C32—44no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3C3×D411yescharacteristic
130C22×C3×D3—34no—
131C22×C3×D3—64no—
132C32×D4—21no—
133C32×D4—28no—
134C32×D4—24no—
135C32×D4—44no—
136C32×D4D1218yescomplemented
137C32×D4C3×D412yescomplemented
138C2×C32×C4—21no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—44no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C23×C311yescharacteristic
144#638—24no—
145C2×C3⋊Dic3—24no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3C3×D411yescharacteristic
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—42no—
151C4×C3⋊D3—21no—
152C4×C3⋊D3—21no—
153#634—21no—
154#634C3×D412yescomplemented
155C3×#67—616no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—616no—
158C2×C3×Dic3—64no—
159C3×D12—64no—
160C3×D12—64no—
161C3×D12—128no—
162C3×C4×D3—64no—
163C3×C4×D3—64no—
164C3×Dic6—38no—
165C3×Dic6—64no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—44no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—64no—
171#449—91noSylow
172C2×C33—42no—
173C2×C33#12011yescharacteristic
174C2×C33#13511yescharacteristic
175C2×C33AGL(1,ℤ/8)11yescharacteristic
176C3×C3⋊D3—41no—
177C3×C3⋊D3—81no—
178C3×#41—182no—
179C3×#41—182no—
180C2×C3×Q8—91no—
181C2×C3×D4—24no—
182C2×C3×D4—91no—
183C2×C3×D4—181no—
184C2×C3×D4—181no—
185C2×C3×D4C2×C3×D314yes—
186C2×C3×D4C2×C3⋊D311yescharacteristic
187C22×C3×C4—91no—
188C22×C3×C4—181no—
189C2×#67—64no—
190C22×Dic3—64no—
191#237—68no—
192#237—68no—
193C2×D12—34no—
194C2×D12—64no—
195C2×C4×D3—34no—
196C2×Dic6—34no—
197C3×SD16—182no—
198C3×D8—182no—
199C2×C3×C8—24no—
200C2×C3×C8—91no—
201C2×C3×C8C2×C3×D314yes—
202C2×C3×C8C2×C3⋊D311yescharacteristic
203C3×C4⋊C4—24no—
204C3×C4⋊C4C2×C3×D314yes—
205C3×C4⋊C4C2×C3⋊D311yescharacteristic
206C3×C22⋊C4—181no—
207#213—68no—
208(C2×D3)⋊C4—64no—
209C2×C3⋊C8—34no—
210#204—68no—
211C22×C32—22no—
212C22×C32—24no—
213C22×C32—28no—
214C22×C32—48no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—44no—
218C22×C32—44no—
219C22×C32—84no—
220C22×C32—84no—
221C22×C32D3×D414yes—
222C22×C32C2×C3×D411yescharacteristic
223C2×C3⋊D3—21no—
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—64no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—41no—
238C32×C4—44no—
239C32×C4—84no—
240C32×C4C2×D1214yes—
241C32×C4D3×D414yes—
242C32×C4C2×C3×D411yescharacteristic
243C32×C4C2×C3×D411yescharacteristic
244C3⋊Dic3—22no—
245C3⋊Dic3—21no—
246C3×Dic3—68no—
247C3×Dic3—64no—
248C2×#41—91no—
249C2×SD16—91no—
250C2×D8—91no—
251#121—91no—
252#103—91no—
253#102C3×C3⋊D311yescharacteristic, complemented
254C22⋊C8—91no—
255C33#44911yescharacteristic, complemented, Sylow
256C23×C3—24no—
257C23×C3—21no—
258C23×C3—44no—
259C23×C3—91no—
260C23×C3—181no—
261C22×D3—34no—
262C22×D3—64no—
263C3×Q8—92no—
264C3×Q8—181no—
265C3×D4—24no—
266C3×D4—21no—
267C3×D4—28no—
268C3×D4—44no—
269C3×D4—184no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—362no—
273C3×D4C3×D1218yescomplemented
274C3×D4C4⋊(C3⋊D3)12yescomplemented
275C2×C3×C4—24no—
276C2×C3×C4—21no—
277C2×C3×C4—24no—
278C2×C3×C4—44no—
279C2×C3×C4—91no—
280C2×C3×C4—184no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4C22×C3×D314yes—
285C2×C3×C4C22×C3⋊D311yescharacteristic
286#67—616no—
287C2×Dic3—34no—
288C2×Dic3—616no—
289C2×Dic3—64no—
290D12—64no—
291D12—64no—
292D12—128no—
293C4×D3—64no—
294C4×D3—64no—
295Dic6—38no—
296Dic6—64no—
297C3×C8—24no—
298C3×C8—21no—
299C3×C8—44no—
300C3×C8—181no—
301C3⋊C8—64no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32—42no—
306C2×C32—48no—
307C2×C32—88no—
308C2×C32#88214yes—
309C2×C32#91114yes—
310C2×C32#91614yes—
311C2×C32C3×#12011yescharacteristic
312C2×C32C3×#13511yescharacteristic
313C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
314C3⋊D3—41no—
315C3⋊D3—81no—
316C3×D3—124no—
317C3×D3—244no—
318#41—182no—
319#41—182no—
320C2×Q8—91no—
321C2×D4—91no—
322C2×D4—181no—
323C2×D4—181no—
324C2×D4C2×C3×C3⋊D311yescharacteristic
325C22×C4—91no—
326C22×C4—181no—
327SD16—182no—
328D8—182no—
329C2×C8—91no—
330C2×C8C2×C3×C3⋊D311yescharacteristic
331C4⋊C4C2×C3×C3⋊D311yescharacteristic
332C22⋊C4—181no—
333C22×C3—28no—
334C22×C3—22no—
335C22×C3—24no—
336C22×C3—48no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—84no—
342C22×C3—84no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—181no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3C3×D3×D414yes—
350C22×C3D4×C3⋊D311yescharacteristic
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—124no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—44no—
360C3×C4—41no—
361C3×C4—84no—
362C3×C4—181no—
363C3×C4—182no—
364C3×C4C2×C3×D1214yes—
365C3×C4C3×D3×D414yes—
366C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
367C3×C4D4×C3⋊D311yescharacteristic
368Dic3—68no—
369Dic3—64no—
370C32—24no—
371C32#478114yescomplemented
372C32C3×#44911yescharacteristic, complemented
373C23—21no—
374C23—91no—
375C23—181no—
376Q8—92no—
377Q8—181no—
378D4—21no—
379D4—184no—
380D4—181no—
381D4—181no—
382D4—362no—
383D4C3×C4⋊(C3⋊D3)12yescomplemented
384C2×C4—21no—
385C2×C4—91no—
386C2×C4—184no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4C22×C3×C3⋊D311yescharacteristic
391C8—21no—
392C8—181no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—48no—
397C2×C3—42no—
398C2×C3—88no—
399C2×C3—361no—
400C2×C3—721no—
401C2×C3C3×#88214yes—
402C2×C3C3×#91114yes—
403C2×C3C3×#91614yes—
404C2×C3#6409111yescharacteristic
405C2×C3#6412011yescharacteristic
406C2×C3#6412511yescharacteristic
407D3—124no—
408D3—244no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—181no—
413C22—181no—
414C22—181no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22C3×D4×C3⋊D311yescharacteristic
419C4—41no—
420C4—181no—
421C4—182no—
422C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
423C4C3×D4×C3⋊D311yescharacteristic
424C3—24no—
425C3C3×#478114yescomplemented
426C3#1059248011yescharacteristic, direct product factor
427C2—42no—
428C2—361no—
429C2—721no—
430C2C3×#6409111yescharacteristic
431C2C3×#6412011yescharacteristic
432C2C3×#6412511yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#449
LCSG ▷ C2×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
1. C2×C3×#3830C2semidirect
2. C2×C3×#3753C2semidirect
3. C2×C3×#3744C2semidirect
4. C3×#64104C2semidirect
5. C33×#102C2semidirect
6. C3×#63647C2semidirect
7. C3×#63637C2semidirect
8. #10592480C3direct (N × H)
25. C2×#3830C2×C3semidirect
27. C2×#3753C2×C3semidirect
28. C2×#3744C2×C3semidirect
29. #64104C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63637C2×C3semidirect
44. C33×D4D4semidirect
57. C3×#634D4semidirect
120. C3×#102C3×D3semidirect
121. C3×#102C3⋊D3semidirect
136. C32×D4D12semidirect
137. C32×D4C3×D4semidirect
154. #634C3×D4semidirect
253. #102C3×C3⋊D3semidirect
255. C33#449semidirect
273. C3×D4C3×D12semidirect
274. C3×D4C4⋊(C3⋊D3)semidirect
371. C32#4781semidirect
372. C32C3×#449semidirect
383. D4C3×C4⋊(C3⋊D3)semidirect
425. C3C3×#4781semidirect
426. C3#10592480direct (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×#37532
C2×C3×#37442
C3×#641042
C33×#1022
C3×#636472
C3×#636372
#105924803
C3×#47813