Finite Group Databasebeta

← C3×#10592857 #13163196

Subgroups — C3×#10592857 #13163196

Order 1728.

1276 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C33×D8C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592857C311yesmaximal, characteristic, direct product factor
9C3×#5158—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×#638—21no—
14C22×C3×C3⋊Dic3—21no—
15C3×#3830—22no—
16C3×#3830—22no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C33×D8—22no—
20C2×C33×C8C2211yescharacteristic
21C3×#3757—21no—
22C3×#3754—22no—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×#3830C2×C311yescharacteristic, complemented
26C2×C3×#237—34no—
27C2×C32×D8—24no—
28C2×C32×D8D314yescomplemented
29C2×C32×D8C2×C311yescharacteristic, complemented
30#64147C2×C311yescharacteristic, complemented
31C2×#3754C2×C311yescharacteristic, complemented
32C3×#938—34no—
33C2×C3×#214—34no—
34#63659C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63643C2×C311yescharacteristic, complemented
37C3×#832—34no—
38C3×(C2×D3)⋊C8—34no—
39C3×#816—34no—
40C23×C33—21no—
41C23×C33—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×D4—21no—
44C33×D4—21no—
45C33×D4—21no—
46C33×D4D412yescomplemented
47C2×C33×C4C2311yescharacteristic
48C3×#638—24no—
49C3×#638—42no—
50C2×C3×C3⋊Dic3—24no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
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#5158—34no—
62C2×C32×D4—24no—
63C2×C32×D4—24no—
64C2×C32×D4C2×D314yes—
65C2×C32×D4C2×D314yes—
66C2×C32×D4C22×C311yescharacteristic
67C2×C32×D4C22×C311yescharacteristic
68C2×#638—21no—
69C2×#638—21no—
70C22×C3⋊Dic3—21no—
71#3830—22no—
72#3830—22no—
73C2×C4×C3⋊D3C22×C311yescharacteristic
74C2×#634C22×C311yescharacteristic
75C2×C3×#67—64no—
76C2×C3×#67—64no—
77C22×C3×Dic3—64no—
78C3×#237—68no—
79C3×#237—68no—
80C2×C3×C4×D3—34no—
81C2×C3×Dic6—34no—
82C32×D8—216no—
83C32×D8—28no—
84C32×D8—22no—
85C2×C32×C8—24no—
86C2×C32×C8C2×D314yes—
87C2×C32×C8C22×C311yescharacteristic
88#3757—21no—
89#3754—22no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—64no—
93C3×#214—68no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8—34no—
96C22×C33—22no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33—41no—
100C22×C33—41no—
101C22×C33—41no—
102C22×C33C2×D411yescharacteristic
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—21no—
105C2×C3×C3⋊D3—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C3×C3⋊Dic3—21no—
110C3×C3⋊Dic3—41no—
111C2×C3×#41—91no—
112C2×#237—34no—
113C2×C3×SD16—91no—
114C2×C3×D8—24no—
115C2×C3×D8C3×D314yescomplemented
116C2×C3×D8C3⋊D311yescharacteristic, complemented
117C3×#121—91no—
118#938—34no—
119C2×#214—34no—
120C3×#103—91no—
121C3×#102—91no—
122C3×C22⋊C8—91no—
123#832—34no—
124(C2×D3)⋊C8—34no—
125#816—34no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—24no—
129C23×C32—24no—
130C23×C32—44no—
131C23×C32—44no—
132C22×C3⋊D3C3×D411yescharacteristic
133C22×C3×D3—34no—
134C32×D4—28no—
135C32×D4—24no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—28no—
140C32×D4—24no—
141C32×D4—24no—
142C32×D4—44no—
143C32×D4—44no—
144C32×D4#6718yescomplemented
145C32×D4C3×D412yescomplemented
146C2×C32×C4—24no—
147C2×C32×C4C22×D314yes—
148C2×C32×C4C23×C311yescharacteristic
149#638—24no—
150#638—42no—
151C2×C3⋊Dic3—24no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3C3×D411yescharacteristic
155C4×C3⋊D3—21no—
156C4×C3⋊D3—21no—
157#634—21no—
158#634C3×D412yescomplemented
159C3×#67—616no—
160C3×#67—128no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—616no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3—64no—
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167C3×Dic6—38no—
168C3×Dic6—64no—
169C32×C8—28no—
170C32×C8—24no—
171C32×C8—21no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—64no—
174#449—91noSylow
175C2×C33—42no—
176C2×C33—81no—
177C2×C33#12011yescharacteristic
178C2×C33#13511yescharacteristic
179C2×C33AGL(1,ℤ/8)11yescharacteristic
180C3×C3⋊D3—41no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—91no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—181no—
187C2×C3×D4—181no—
188C2×C3×D4C2×C3×D314yes—
189C2×C3×D4C2×C3×D314yes—
190C2×C3×D4C2×C3⋊D311yescharacteristic
191C2×C3×D4C2×C3⋊D311yescharacteristic
192C22×C3×C4—91no—
193C22×C3×C4—181no—
194C2×#67—64no—
195C2×#67—64no—
196C22×Dic3—64no—
197#237—68no—
198#237—68no—
199C2×C4×D3—34no—
200C2×Dic6—34no—
201C3×SD16—182no—
202C3×D8—216no—
203C3×D8—28no—
204C3×D8—22no—
205C2×C3×C8—24no—
206C2×C3×C8—91no—
207C2×C3×C8C2×C3×D314yes—
208C2×C3×C8C2×C3⋊D311yescharacteristic
209C3×C4⋊C4—91no—
210C3×C22⋊C4—181no—
211#217—64no—
212#214—68no—
213C4⋊Dic3—34no—
214C2×C3⋊C8—34no—
215C22×C32—22no—
216C22×C32—24no—
217C22×C32—28no—
218C22×C32—48no—
219C22×C32—48no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—44no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—44no—
227C22×C32—44no—
228C22×C32—44no—
229C22×C32—44no—
230C22×C32—84no—
231C22×C32—84no—
232C22×C32—84no—
233C22×C32—84no—
234C22×C32D3×D414yes—
235C22×C32C2×C3×D411yescharacteristic
236C2×C3⋊D3—21no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—21no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—64no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4D3×D414yes—
245C32×C4C2×#6714yes—
246C32×C4C2×C3×D411yescharacteristic
247C32×C4C2×C3×D411yescharacteristic
248C3⋊Dic3—22no—
249C3⋊Dic3—21no—
250C3⋊Dic3—41no—
251C3×Dic3—68no—
252C3×Dic3—64no—
253C3×Dic3—124no—
254C2×#41—91no—
255C2×SD16—91no—
256C2×D8C3×C3⋊D311yescharacteristic, complemented
257#121—91no—
258#103—91no—
259#102—91no—
260C22⋊C8—91no—
261C33#44911yescharacteristic, complemented, Sylow
262C23×C3—24no—
263C23×C3—21no—
264C23×C3—24no—
265C23×C3—21no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—91no—
269C22×D3—34no—
270C3×Q8—92no—
271C3×Q8—181no—
272C3×D4—28no—
273C3×D4—24no—
274C3×D4—24no—
275C3×D4—24no—
276C3×D4—21no—
277C3×D4—21no—
278C3×D4—21no—
279C3×D4—28no—
280C3×D4—44no—
281C3×D4—44no—
282C3×D4—184no—
283C3×D4—362no—
284C3×D4C3×#6718yescomplemented
285C3×D4#63812yescomplemented
286C2×C3×C4—24no—
287C2×C3×C4—91no—
288C2×C3×C4—184no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4—181no—
293C2×C3×C4C22×C3×D314yes—
294C2×C3×C4C22×C3⋊D311yescharacteristic
295#67—616no—
296#67—128no—
297C2×Dic3—34no—
298C2×Dic3—616no—
299C2×Dic3—64no—
300C2×Dic3—64no—
301C4×D3—64no—
302C4×D3—64no—
303Dic6—38no—
304Dic6—64no—
305C3×C8—28no—
306C3×C8—24no—
307C3×C8—21no—
308C3×C8—181no—
309C3⋊C8—64no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—42no—
314C2×C32—48no—
315C2×C32—88no—
316C2×C32—88no—
317C2×C32—84no—
318C2×C32—81no—
319C2×C32#91114yes—
320C2×C32#91214yes—
321C2×C32#93714yes—
322C2×C32C3×#12011yescharacteristic
323C2×C32C3×#13511yescharacteristic
324C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
325C3⋊D3—41no—
326C3×D3—124no—
327#41—182no—
328#41—182no—
329C2×Q8—91no—
330C2×D4—181no—
331C2×D4—181no—
332C2×D4C2×C3×C3⋊D311yescharacteristic
333C2×D4C2×C3×C3⋊D311yescharacteristic
334C22×C4—91no—
335C22×C4—181no—
336SD16—182no—
337D8—22no—
338C2×C8—91no—
339C2×C8C2×C3×C3⋊D311yescharacteristic
340C4⋊C4—91no—
341C22⋊C4—181no—
342C22×C3—28no—
343C22×C3—22no—
344C22×C3—24no—
345C22×C3—48no—
346C22×C3—48no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—84no—
358C22×C3—84no—
359C22×C3—84no—
360C22×C3—84no—
361C22×C3—181no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3C3×D3×D414yes—
365C22×C3D4×C3⋊D311yescharacteristic
366C2×D3—64no—
367C2×D3—64no—
368C2×D3—64no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—181no—
372C3×C4—182no—
373C3×C4—361no—
374C3×C4C3×D3×D414yes—
375C3×C4C2×C3×#6714yes—
376C3×C4D4×C3⋊D311yescharacteristic
377C3×C4C2×#63811yescharacteristic
378Dic3—68no—
379Dic3—64no—
380Dic3—124no—
381C32—24no—
382C32#515814yescomplemented
383C32C3×#44911yescharacteristic, complemented
384C23—21no—
385C23—21no—
386C23—91no—
387Q8—92no—
388Q8—181no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—184no—
393D4—362no—
394D4C3×#63812yescomplemented
395C2×C4—91no—
396C2×C4—184no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4C22×C3×C3⋊D311yescharacteristic
402C8—21no—
403C8—181no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—42no—
408C2×C3—48no—
409C2×C3—88no—
410C2×C3—88no—
411C2×C3—81no—
412C2×C3—84no—
413C2×C3—361no—
414C2×C3C3×#91114yes—
415C2×C3C3×#91214yes—
416C2×C3C3×#93714yes—
417C2×C3#6412011yescharacteristic
418C2×C3#6412111yescharacteristic
419C2×C3#6414611yescharacteristic
420D3—124no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—181no—
428C22—181no—
429C22—181no—
430C22C3×D4×C3⋊D311yescharacteristic
431C4—181no—
432C4—182no—
433C4—361no—
434C4C3×D4×C3⋊D311yescharacteristic
435C4C2×C3×#63811yescharacteristic
436C3—24no—
437C3C3×#515814yescomplemented
438C3#1059285711yescharacteristic, direct product factor
439C2—42no—
440C2—81no—
441C2—361no—
442C2C3×#6412011yescharacteristic
443C2C3×#6412111yescharacteristic
444C2C3×#6414611yescharacteristic
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×#449
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×C3×#3830C2semidirect
2. C2×C33×D8C2semidirect
3. C3×#64147C2semidirect
4. C2×C3×#3754C2semidirect
5. C3×#63659C2semidirect
6. C3×#63647C2semidirect
7. C3×#63643C2semidirect
8. #10592857C3direct (N × H)
25. C2×#3830C2×C3semidirect
28. C2×C32×D8D3semidirect
29. C2×C32×D8C2×C3semidirect
30. #64147C2×C3semidirect
31. C2×#3754C2×C3semidirect
34. #63659C2×C3semidirect
35. #63647C2×C3semidirect
36. #63643C2×C3semidirect
46. C33×D4D4semidirect
57. C3×#634D4semidirect
115. C2×C3×D8C3×D3semidirect
116. C2×C3×D8C3⋊D3semidirect
144. C32×D4#67semidirect
145. C32×D4C3×D4semidirect
158. #634C3×D4semidirect
256. C2×D8C3×C3⋊D3semidirect
261. C33#449semidirect
284. C3×D4C3×#67semidirect
285. C3×D4#638semidirect
382. C32#5158semidirect
383. C32C3×#449semidirect
394. D4C3×#638semidirect
437. C3C3×#5158semidirect
438. C3#10592857direct (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×C33×D82
C3×#641472
C2×C3×#37542
C3×#636592
C3×#636472
C3×#636432
#105928573
C3×#51583