Finite Group Databasebeta

← C3×#10591186 #13158365

Subgroups — C3×#10591186 #13158365

Order 1728.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C2×C3×#3719C212yesmaximal, complemented
4C32×#830C211yesmaximal, characteristic
5C3×#63566C212yesmaximal, complemented
6C3×#5228—31nomaximal
7C3×#4739—31nomaximal
8#10591186C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C22×C32×Dic3—21no—
12C2×C32×Dic6—21no—
13C2×C32×Dic6—24no—
14C2×C32×Dic6C2212yes—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C2×C32×C3⋊C8C2212yes—
19C3×#3719—24no—
20C22×C32×Q8—31no—
21C22×C3×Dic6—21no—
22C22×C3×Dic6D311yescharacteristic
23C22×C3×Dic6C2×C311yescharacteristic
24#64133C2×C311yescharacteristic
25C2×C3×#216—32no—
26C3×#924—31no—
27C3×#924—31no—
28C3×#924—61no—
29C2×C3×Dic12—32no—
30C2×#3719C2×C312yescomplemented
31C32×C22⋊C8—31no—
32C3×#835—32no—
33C3×#830—21no—
34C3×#830D311yescharacteristic
35C3×#830C2×C311yescharacteristic
36C3×#816—32no—
37#63566C2×C312yescomplemented
38C23×C33D411yescharacteristic
39C2×C33×C4—22no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—22no—
45C2×C32×Dic3—24no—
46C2×C32×Dic3—22no—
47C32×Dic6—24no—
48C32×Dic6—22no—
49C32×Dic6D414yes—
50C32×C3⋊C8—22no—
51C3×#451—91no—
52#5228—31no—
53#4739—31no—
54C2×C32×Q8—32no—
55C2×C32×Q8—64no—
56C2×C32×Q8—61no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—41no—
61C22×C32×C4—61no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C2×#634C22×C311yescharacteristic
66C22×C3×Dic3—22no—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—21no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6—28no—
71C2×C3×Dic6—22no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—24no—
75C2×C3×Dic6—24no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D312yes—
80C2×C3×Dic6C22×C312yes—
81C2×C32×C8—32no—
82#3757—21no—
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84#3750—21no—
85C3×#217—62no—
86C3×#217—61no—
87C3×#217—61no—
88C3×#216—64no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×Dic3⋊C4—62no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C2×C3×C3⋊C8—22no—
96C2×C3×C3⋊C8C2×D312yes—
97C2×C3×C3⋊C8C22×C312yes—
98C3×Dic12—64no—
99#3719—24no—
100C22×C33—21no—
101C22×C33—21no—
102C22×C33Q1612yescomplemented
103C22×C33C2×D411yescharacteristic
104C33×C4—21no—
105C33×C4C2×D412yes—
106C3×C3⋊Dic3—41no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—24no—
109C22×C3×Q8—31no—
110C22×C3×Q8—31no—
111C22×C3×Q8—61no—
112C22×Dic6C3×D311yescharacteristic
113C2×C3×Q16—92no—
114C3×C22⋊Q8—91no—
115C2×#216—32no—
116#924—31no—
117#924—31no—
118#924—61no—
119C2×Dic12—32no—
120C3×#103—92no—
121C3×C22⋊C8—31no—
122C3×C22⋊C8—31no—
123C3×C22⋊C8—61no—
124#835—32no—
125#830C3×D311yescharacteristic
126#816—32no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32—41no—
131C23×C32D1211yescharacteristic
132C23×C32#6711yescharacteristic
133C23×C32C3×D411yescharacteristic
134C32×Q8—34no—
135C32×Q8—64no—
136C32×Q8—62no—
137C2×C32×C4—22no—
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—42no—
150C2×C32×C4—42no—
151C2×C32×C4—44no—
152C2×C32×C4—64no—
153C2×C32×C4—62no—
154C2×C32×C4D1211yescharacteristic
155C2×C32×C4#6711yescharacteristic
156C2×C32×C4C3×D411yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162#634—22no—
163C2×C3×Dic3—28no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C3×Dic6—28no—
176C3×Dic6—24no—
177C3×Dic6—24no—
178C3×Dic6—24no—
179C3×Dic6—22no—
180C3×Dic6—22no—
181C3×Dic6—24no—
182C3×Dic6—62no—
183C3×Dic6—62no—
184C3×Dic6—122no—
185C3×Dic6#6714yes—
186C3×Dic6C3×D414yes—
187C32×C8—62no—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—42no—
191#451—91noSylow
192C2×C33—22no—
193C2×C33#12011yescharacteristic
194C2×C33C2×Q1611yescharacteristic
195C2×C33#13711yescharacteristic
196C2×C3×Q8—32no—
197C2×C3×Q8—32no—
198C2×C3×Q8—62no—
199C2×C3×Q8—64no—
200C2×C3×Q8—64no—
201C2×C3×Q8—68no—
202C2×C3×Q8—62no—
203C2×C3×Q8—61no—
204C2×C3×Q8—61no—
205C2×C3×Q8—91no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—41no—
210C22×C3×C4—62no—
211C22×C3×C4—61no—
212C22×C3×C4—61no—
213C22×C3×C4D3211yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×Dic3—21no—
217C2×Dic6—21no—
218C2×Dic6—24no—
219C2×Dic6—31no—
220C2×Dic6—31no—
221C2×Dic6—61no—
222C2×Dic6C2×C3×D312yes—
223C3×Q16—184no—
224C2×C3×C8—32no—
225C2×C3×C8—32no—
226C2×C3×C8—62no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—181no—
229C3×C22⋊C4—181no—
230#217—62no—
231#217—61no—
232#217—61no—
233#216—64no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—61no—
237Dic3⋊C4—62no—
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240C2×C3⋊C8C2×C3×D312yes—
241Dic12—64no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—42no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32—41no—
260C22×C32—41no—
261C22×C32—41no—
262C22×C32—42no—
263C22×C32—42no—
264C22×C32Dic1212yescomplemented
265C22×C32#21612yescomplemented
266C22×C32C3×Q1612yescomplemented
267C22×C32C2×D1211yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C22×C32C2×C3×D411yescharacteristic
270C32×C4—21no—
271C32×C4—22no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—42no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—42no—
281C32×C4—64no—
282C32×C4D3×D412yes—
283C32×C4C2×#6712yes—
284C32×C4C2×C3×D412yes—
285C3⋊Dic3—41no—
286C3⋊Dic3—41no—
287C3×Dic3—28no—
288C3×Dic3—24no—
289C3×Dic3—24no—
290C3×Dic3—122no—
291C3×Dic3—122no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C22×Q8—31no—
297C2×Q16—92no—
298C22⋊Q8—91no—
299#103—92no—
300C22⋊C8—31no—
301C33#45111yescharacteristic, complemented, Sylow
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—41no—
306C23×C3#62611yescharacteristic
307C23×C3C3×D1211yescharacteristic
308C23×C3C3×#6711yescharacteristic
309C3×Q8—34no—
310C3×Q8—34no—
311C3×Q8—64no—
312C3×Q8—68no—
313C3×Q8—64no—
314C3×Q8—64no—
315C3×Q8—64no—
316C3×Q8—62no—
317C3×Q8—62no—
318C3×Q8—182no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—44no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—68no—
335C2×C3×C4—64no—
336C2×C3×C4—64no—
337C2×C3×C4—64no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4#62611yescharacteristic
343C2×C3×C4C3×D1211yescharacteristic
344C2×C3×C4C3×#6711yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C22×C3×D311yescharacteristic
347C2×C3×C4C22×C3×D311yescharacteristic
348C2×Dic3—24no—
349C2×Dic3—22no—
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356Dic6—24no—
357Dic6—22no—
358Dic6—62no—
359Dic6—62no—
360Dic6—122no—
361Dic6C3×#6714yes—
362C3×C8—62no—
363C3×C8—62no—
364C3×C8—122no—
365C3⋊C8—22no—
366C2×C32—22no—
367C2×C32—24no—
368C2×C32—22no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32—42no—
383C2×C32—42no—
384C2×C32—44no—
385C2×C32#88211yescharacteristic
386C2×C32C2×Dic1211yescharacteristic
387C2×C32#90911yescharacteristic
388C2×C32#94211yescharacteristic
389C2×C32C2×#21611yescharacteristic
390C2×C32#95311yescharacteristic
391C2×C32C3×#12011yescharacteristic
392C2×C32C2×C3×Q1611yescharacteristic
393C2×C32C3×#13711yescharacteristic
394C2×Q8—32no—
395C2×Q8—64no—
396C2×Q8—61no—
397C2×Q8—91no—
398C22×C4—61no—
399C22×C4C3×D3211yescharacteristic
400Q16—184no—
401C2×C8—32no—
402C4⋊C4—91no—
403C4⋊C4—181no—
404C22⋊C4—181no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3#371912yescomplemented
428C22×C3C3×Dic1212yescomplemented
429C22×C3C3×#21612yescomplemented
430C22×C3C2×#62611yescharacteristic
431C22×C3C2×C3×D1211yescharacteristic
432C22×C3C2×C3×#6711yescharacteristic
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—68no—
445C3×C4—64no—
446C3×C4—64no—
447C3×C4—361no—
448C3×C4—361no—
449C3×C4D3×#6712yes—
450C3×C4C3×D3×D412yes—
451C3×C4C2×C3×#6712yes—
452Dic3—24no—
453Dic3—122no—
454Dic3—122no—
455Dic3—121no—
456Dic3—121no—
457Dic3—121no—
458Dic3—121no—
459C32—21no—
460C32—21no—
461C32—21no—
462C32—41no—
463C32#473911yescharacteristic, complemented
464C32#522811yescharacteristic, complemented
465C32C3×#45111yescharacteristic, complemented
466C23C3×#62611yescharacteristic
467Q8—34no—
468Q8—64no—
469Q8—62no—
470Q8—182no—
471C2×C4—22no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4C3×#62611yescharacteristic
477C2×C4C2×C3×D3211yescharacteristic
478C8—62no—
479C2×C3—24no—
480C2×C3—22no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—42no—
493C2×C3—42no—
494C2×C3—44no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3#6383311yescharacteristic
499C2×C3C2×#371911yescharacteristic
500C2×C3#6397711yescharacteristic
501C2×C3C3×#88211yescharacteristic
502C2×C3C2×C3×Dic1211yescharacteristic
503C2×C3C3×#90911yescharacteristic
504C2×C3C3×#94211yescharacteristic
505C2×C3C2×C3×#21611yescharacteristic
506C2×C3C3×#95311yescharacteristic
507C22—21no—
508C22—21no—
509C22C3×#371912yescomplemented
510C22C2×C3×#62611yescharacteristic
511C4—21no—
512C4—64no—
513C4—361no—
514C4—361no—
515C4C3×D3×#6712yes—
516C3—21no—
517C3—21no—
518C3—21no—
519C3—41no—
520C3#1059118611yescharacteristic, direct product factor
521C3C3×#473911yescharacteristic, complemented
522C3C3×#522811yescharacteristic, complemented
523C2—22no—
524C2C3×#6383311yescharacteristic
525C2C2×C3×#371911yescharacteristic
526C2C3×#6397711yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3719C2semidirect
5. C3×#63566C2semidirect
8. #10591186C3direct (N × H)
30. C2×#3719C2×C3semidirect
37. #63566C2×C3semidirect
102. C22×C33Q16semidirect
264. C22×C32Dic12semidirect
265. C22×C32#216semidirect
266. C22×C32C3×Q16semidirect
301. C33#451semidirect
427. C22×C3#3719semidirect
428. C22×C3C3×Dic12semidirect
429. C22×C3C3×#216semidirect
463. C32#4739semidirect
464. C32#5228semidirect
465. C32C3×#451semidirect
509. C22C3×#3719semidirect
520. C3#10591186direct (N × H)
521. C3C3×#4739semidirect
522. C3C3×#5228semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C3×#641332
C2×C3×#37192
C32×#8302
C3×#635662
C3×#52283
C3×#47393
#105911863