Finite Group Databasebeta

← C3×#10595171 #13174734

Subgroups — C3×#10595171 #13174734

Order 1728.

1450 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C32×#936C212yesmaximal, complemented
4C3×#63982C214yesmaximal, complemented
5C3×#63975C214yesmaximal, complemented
6C3×#63874C211yesmaximal, characteristic, complemented
7C3×#63866C212yesmaximal, complemented
8C3×#5675—32nomaximal
9#10595171C311yesmaximal, characteristic, direct product factor
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638C2212yes—
12C22×C3×C3⋊Dic3C2212yes—
13C3×#3830—24no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×Dic6C2212yes—
17C2×C3×#627C2212yes—
18C2×C3×#624C2212yes—
19C3×#3797—24no—
20C32×#217C2218yes—
21C32×C4×Dic3C2212yes—
22C3×#3724C2212yes—
23C3×Dic3⋊Dic3C2218yes—
24C3×#3722C2212yes—
25C2×#3830C2×C311yescharacteristic
26C2×C3×D3×Q8—32no—
27C2×C3×#237—32no—
28C2×C3×#237—61no—
29C2×#3797C2×C311yescharacteristic, complemented
30C32×#124—32no—
31C3×#944—32no—
32C3×#936—22no—
33C3×#936D312yescomplemented
34C3×#936C2×C312yescomplemented
35C3×#883—38no—
36C3×#878—38no—
37C3×#872—32no—
38C3×#869—32no—
39#63982C2×C314yescomplemented
40#63975C2×C314yescomplemented
41#63874C2×C311yescharacteristic, complemented
42#63866C2×C312yescomplemented
43C23×C33C2312yes—
44C22×C3×C3⋊D3C2311yescharacteristic
45C33×D4—22no—
46C2×C33×C4C2311yescharacteristic
47C3×#638—24no—
48C2×C3×C3⋊Dic3—24no—
49C2×C3×C3⋊Dic3C2312yes—
50C2×C3×C3⋊Dic3C2311yescharacteristic
51C3×C4×C3⋊D3D414yes—
52C3×#634—22no—
53C2×C32×Dic3C2314yes—
54C2×C32×Dic3C2314yes—
55C32×Dic6—24no—
56C3×#627—24no—
57C3×#624—24no—
58C3×#536—91no—
59#5675—32no—
60C2×C32×Q8—32no—
61C2×C32×D4—21no—
62C2×C32×D4—22no—
63C2×C32×D4—41no—
64C2×C32×D4C2×D312yes—
65C2×C32×D4C22×C311yescharacteristic
66C2×#638C22×C312yes—
67C22×C3⋊Dic3C22×C312yes—
68#3830—24no—
69C2×C4×C3⋊D3C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×#67—34no—
72C2×C3×#67—62no—
73C22×C3×Dic3—34no—
74C22×C3×Dic3—62no—
75C3×D3×Q8—68no—
76C3×#237—68no—
77C3×#237—68no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—34no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—22no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—34no—
84C2×C3×Dic6—61no—
85C2×C3×Dic6C2×D312yes—
86C2×C3×Dic6C22×C312yes—
87C2×#627C22×C312yes—
88C2×#624C22×C312yes—
89#3797—24no—
90C32×C22⋊C4—38no—
91C32×C42—32no—
92C3×#217—28no—
93C3×#217C2×D318yes—
94C3×#217C22×C318yes—
95C3×(C2×D3)⋊C4—34no—
96C3×C4⋊Dic3—38no—
97C3×Dic3⋊C4—34no—
98C3×Dic3⋊C4—38no—
99C3×C4×Dic3—22no—
100C3×C4×Dic3C2×D312yes—
101C3×C4×Dic3C22×C312yes—
102#3724C22×C312yes—
103Dic3⋊Dic3C22×C318yes—
104#3722C22×C312yes—
105C22×C33—22no—
106C22×C33—24no—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3C2×D412yes—
110C33×C4C2×D412yes—
111C3×C3⋊Dic3—22no—
112C3×C3⋊Dic3C2×D412yes—
113C32×Dic3—24no—
114C32×Dic3—24no—
115C2×C3×#41—91no—
116C22×C3×Q8—91no—
117C2×D3×Q8—32no—
118C2×#237—32no—
119C2×#237—61no—
120C3×C4⋊Q8—92no—
121C3×#124—32no—
122C3×#124—32no—
123C3×#124—62no—
124C3×#123—94no—
125C3×C22⋊Q8—94no—
126C3×#117—91no—
127#944—32no—
128#936C3×D312yescomplemented
129#883—38no—
130#878—38no—
131#872—32no—
132#869—32no—
133C23×C32—22no—
134C23×C32—24no—
135C23×C32—42no—
136C23×C32C22×D314yes—
137C23×C32C23×C312yes—
138C22×C3⋊D3C23×C311yescharacteristic
139C22×C3×D3—32no—
140C22×C3×D3—61no—
141C32×Q8—64no—
142C32×D4—22no—
143C32×D4—24no—
144C32×D4—24no—
145C32×D4—44no—
146C32×D4—44no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—34no—
150C2×C32×C4—34no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C23×C311yescharacteristic
154#638—24no—
155C2×C3⋊Dic3—24no—
156C2×C3⋊Dic3C23×C312yes—
157C2×C3⋊Dic3C23×C311yescharacteristic
158C4×C3⋊D3C3×D414yes—
159#634—22no—
160C3×#67—68no—
161C3×#67—68no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—68no—
169C2×C3×Dic3—68no—
170C2×C3×Dic3C22×D314yes—
171C2×C3×Dic3C22×D314yes—
172C2×C3×Dic3C23×C314yes—
173C2×C3×Dic3C23×C314yes—
174C3×C4×D3—38no—
175C3×C4×D3—64no—
176C3×C4×D3—68no—
177C3×Dic6—24no—
178C3×Dic6—24no—
179C3×Dic6—44no—
180C3×Dic6—68no—
181C3×Dic6—64no—
182C3×Dic6—64no—
183#627—24no—
184#624—24no—
185#536—91noSylow
186C2×C33—42no—
187C2×C33C22×D411yescharacteristic
188C2×C332−1+412yes—
189C3×C3⋊D3—22no—
190C3×#41—184no—
191C2×C3×Q8—32no—
192C2×C3×Q8—32no—
193C2×C3×Q8—62no—
194C2×C3×Q8—92no—
195C2×C3×Q8—91no—
196C2×C3×Q8—184no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C2×C3×D4—92no—
201C2×C3×D4D3211yescharacteristic
202C2×C3×D4C2×C3×D312yes—
203C22×C3×C4—92no—
204C22×C3×C4—91no—
205C22×C3×C4—92no—
206C2×#67—34no—
207C2×#67—62no—
208C22×Dic3—34no—
209C22×Dic3—62no—
210D3×Q8—68no—
211#237—68no—
212#237—68no—
213C2×C4×D3—32no—
214C2×C4×D3—34no—
215C2×C4×D3—61no—
216C2×Dic6—32no—
217C2×Dic6—34no—
218C2×Dic6—61no—
219C2×Dic6C2×C3×D312yes—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—98no—
222C3×C22⋊C4—38no—
223C3×C22⋊C4—38no—
224C3×C22⋊C4—68no—
225C3×C22⋊C4—92no—
226C3×C42—32no—
227C3×C42—32no—
228C3×C42—62no—
229#217C2×C3×D318yes—
230(C2×D3)⋊C4—34no—
231C4⋊Dic3—38no—
232Dic3⋊C4—34no—
233Dic3⋊C4—38no—
234C4×Dic3C2×C3×D312yes—
235C22×C32—22no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—28no—
239C22×C32—24no—
240C22×C32—21no—
241C22×C32—22no—
242C22×C32—41no—
243C22×C32—44no—
244C22×C32—44no—
245C22×C32—44no—
246C22×C32—48no—
247C22×C32—84no—
248C22×C32C23×D312yes—
249C22×C32C24×C311yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3C2×C3×D412yes—
252C2×C3×D3—34no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—122no—
256C32×C4—22no—
257C32×C4—24no—
258C32×C4—42no—
259C32×C4—64no—
260C32×C4—64no—
261C32×C4D3×D414yes—
262C32×C4C2×C3×D412yes—
263C3⋊Dic3—22no—
264C3⋊Dic3C2×C3×D412yes—
265C3×Dic3—24no—
266C3×Dic3—24no—
267C3×Dic3—24no—
268C3×Dic3—24no—
269C3×Dic3—34no—
270C3×Dic3—44no—
271C3×Dic3—44no—
272C3×Dic3—62no—
273C3×Dic3—64no—
274C3×Dic3—64no—
275C2×#41—91no—
276C22×Q8—91no—
277C4⋊Q8—92no—
278#124—32no—
279#123—94no—
280C22⋊Q8—94no—
281#117—91no—
282C33#53611yescharacteristic, complemented, Sylow
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—91no—
287C23×C3C2×D3212yes—
288C23×C3C22×C3×D314yes—
289C22×D3—32no—
290C22×D3—61no—
291C3×Q8—64no—
292C3×Q8—64no—
293C3×Q8—124no—
294C3×Q8—184no—
295C3×Q8—182no—
296C3×D4—24no—
297C3×D4—22no—
298C3×D4—24no—
299C3×D4—44no—
300C3×D4—44no—
301C3×D4—184no—
302C2×C3×C4—22no—
303C2×C3×C4—21no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4—34no—
307C2×C3×C4—34no—
308C2×C3×C4—41no—
309C2×C3×C4—64no—
310C2×C3×C4—64no—
311C2×C3×C4—91no—
312C2×C3×C4—94no—
313C2×C3×C4—92no—
314C2×C3×C4—184no—
315C2×C3×C4—184no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C22×C3×D312yes—
318#67—68no—
319#67—68no—
320C2×Dic3—32no—
321C2×Dic3—34no—
322C2×Dic3—62no—
323C2×Dic3—61no—
324C2×Dic3—68no—
325C2×Dic3—68no—
326C2×Dic3C22×C3×D314yes—
327C2×Dic3C22×C3×D314yes—
328C4×D3—38no—
329C4×D3—64no—
330C4×D3—68no—
331Dic6—24no—
332Dic6—68no—
333Dic6—64no—
334Dic6—64no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—22no—
338C2×C32—24no—
339C2×C32—41no—
340C2×C32—42no—
341C2×C32—42no—
342C2×C32—44no—
343C2×C32—44no—
344C2×C32—84no—
345C2×C32—84no—
346C2×C32C2×D3×D412yes—
347C2×C32#101014yes—
348C2×C32C22×C3×D411yescharacteristic
349C2×C32C3×2−1+412yes—
350C3⋊D3—22no—
351C3×D3—64no—
352C3×D3—122no—
353#41—184no—
354C2×Q8—32no—
355C2×Q8—92no—
356C2×Q8—91no—
357C2×Q8—184no—
358C2×D4—92no—
359C2×D4C3×D3211yescharacteristic
360C22×C4—92no—
361C22×C4—91no—
362C22×C4—92no—
363C4⋊C4—92no—
364C4⋊C4—98no—
365C22⋊C4—38no—
366C22⋊C4—92no—
367C42—32no—
368C22×C3—24no—
369C22×C3—24no—
370C22×C3—22no—
371C22×C3—24no—
372C22×C3—28no—
373C22×C3—21no—
374C22×C3—22no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—48no—
378C22×C3—44no—
379C22×C3—41no—
380C22×C3—84no—
381C22×C3—92no—
382C22×C3—182no—
383C22×C3C22×D3211yescharacteristic
384C22×C3C23×C3×D312yes—
385C2×D3—34no—
386C2×D3—62no—
387C2×D3—64no—
388C2×D3—122no—
389C3×C4—22no—
390C3×C4—24no—
391C3×C4—42no—
392C3×C4—64no—
393C3×C4—64no—
394C3×C4—64no—
395C3×C4—64no—
396C3×C4—92no—
397C3×C4—124no—
398C3×C4—124no—
399C3×C4—182no—
400C3×C4#381112yes—
401C3×C4C3×D3×D414yes—
402Dic3—24no—
403Dic3—24no—
404Dic3—34no—
405Dic3—62no—
406Dic3—64no—
407Dic3—64no—
408C32—21no—
409C32—22no—
410C32—41no—
411C32#567512yescomplemented
412C32C3×#53611yescharacteristic, complemented
413C23—91no—
414C23C2×C3×D3212yes—
415Q8—64no—
416Q8—184no—
417Q8—182no—
418D4—22no—
419D4—184no—
420C2×C4—34no—
421C2×C4—34no—
422C2×C4—91no—
423C2×C4—94no—
424C2×C4—92no—
425C2×C4—184no—
426C2×C4—184no—
427C2×C4C2×C3×D3211yescharacteristic
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—24no—
431C2×C3—22no—
432C2×C3—44no—
433C2×C3—42no—
434C2×C3—44no—
435C2×C3—41no—
436C2×C3—42no—
437C2×C3—84no—
438C2×C3—84no—
439C2×C3—182no—
440C2×C3#6430912yes—
441C2×C3C2×#381111yescharacteristic
442C2×C3C2×C3×D3×D412yes—
443C2×C3C3×#101014yes—
444D3—64no—
445D3—122no—
446C22—22no—
447C22—24no—
448C22—92no—
449C22—182no—
450C22C22×C3×D3211yescharacteristic
451C4—64no—
452C4—64no—
453C4—92no—
454C4—182no—
455C4C3×#381112yes—
456C3—21no—
457C3—22no—
458C3—41no—
459C3#1059517111yescharacteristic, direct product factor
460C3C3×#567512yescomplemented
461C2—42no—
462C2—182no—
463C2C3×#6430912yes—
464C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
prime radicalC2×C3×#3830C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C32×#936C2semidirect
4. C3×#63982C2semidirect
5. C3×#63975C2semidirect
6. C3×#63874C2semidirect
7. C3×#63866C2semidirect
9. #10595171C3direct (N × H)
29. C2×#3797C2×C3semidirect
33. C3×#936D3semidirect
34. C3×#936C2×C3semidirect
39. #63982C2×C3semidirect
40. #63975C2×C3semidirect
41. #63874C2×C3semidirect
42. #63866C2×C3semidirect
128. #936C3×D3semidirect
282. C33#536semidirect
411. C32#5675semidirect
412. C32C3×#536semidirect
459. C3#10595171direct (N × H)
460. C3C3×#5675semidirect

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×#37972
C32×#9362
C3×#639822
C3×#639752
C3×#638742
C3×#638662
C3×#56753
#105951713