Finite Group Databasebeta

← C3×#10591085 #13158113

Subgroups — C3×#10591085 #13158113

Order 1728.

1194 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D12C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C32×#926C211yesmaximal, characteristic, complemented
4C3×#63562C214yesmaximal, complemented
5C3×#5215—31nomaximal
6C3×#5001—31nomaximal
7#10591085C311yesmaximal, characteristic, direct product factor
8C23×C32×D3—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C32×D12—21no—
11C2×C32×D12—22no—
12C2×C32×D12C414yescomplemented
13C2×C32×D12C2212yes—
14C3×#3748—22no—
15C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
16C32×#217—21no—
17C32×C4⋊Dic3C2212yescomplemented
18C32×C4×Dic3—21no—
19C22×C32×D4—31no—
20C22×C3×D12—21no—
21C22×C3×D12D311yescharacteristic
22C22×C3×D12C2×C311yescharacteristic
23C32×#117—31no—
24C2×#3748C2×C311yescharacteristic, complemented
25C3×#926—21no—
26C3×#926D311yescharacteristic, complemented
27C3×#926C2×C311yescharacteristic, complemented
28C2×C3×#208—31no—
29C2×C3×#208—31no—
30C2×C3×#208—61no—
31C3×#832—34no—
32C3×#809—34no—
33#63562C2×C314yescomplemented
34C23×C33D411yescharacteristic
35C22×C32×D3—24no—
36C22×C32×D3—22no—
37C22×C32×D3—44no—
38C2×C33×C4C2×C412yes—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D412yes—
41C2×C33×C4C2311yescharacteristic
42C2×C32×Dic3—22no—
43C32×D12—24no—
44C32×D12—22no—
45C32×D12C2×C414yes—
46C3×C3⋊(C3⋊C8)—22no—
47C3×#418—91no—
48#5215—31no—
49#5001—31no—
50C24×C32—61no—
51C23×C3×D3—22no—
52C23×C3×D3—21no—
53C23×C3×D3—21no—
54C2×C32×D4—34no—
55C2×C32×D4—32no—
56C2×C32×D4—62no—
57C2×C32×D4—61no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—41no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C2×C3×D12—22no—
66C2×C3×D12—24no—
67C2×C3×D12—24no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12—22no—
71C2×C3×D12—21no—
72C2×C3×D12—22no—
73C2×C3×D12Dic314yescomplemented
74C2×C3×D12C3×C414yescomplemented
75C2×C3×D12C2×D312yes—
76C2×C3×D12C22×C312yes—
77C32×C4⋊C4—32no—
78C32×C22⋊C4—61no—
79C32×C42—61no—
80#3748—22no—
81C2×C3⋊(C3⋊C8)C22×C312yescomplemented
82C3×#217—22no—
83C3×#217—21no—
84C3×#217—21no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3C2×D312yescomplemented
87C3×C4⋊Dic3C22×C312yescomplemented
88C3×C4×Dic3—22no—
89C3×C4×Dic3—21no—
90C3×C4×Dic3—21no—
91C3×#208—64no—
92C3×#208—62no—
93C3×#208—62no—
94C2×C3×C3⋊C8—32no—
95C2×C3×C3⋊C8—32no—
96C2×C3×C3⋊C8—62no—
97C22×C33—22no—
98C22×C33C22⋊C412yes—
99C22×C33C2×D411yescharacteristic
100C2×C32×D3—24no—
101C2×C32×D3—48no—
102C2×C32×D3—44no—
103C33×C4C22⋊C412yes—
104C33×C4C22×C411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C32×Dic3—42no—
107C22×C3×D4—31no—
108C22×C3×D4—31no—
109C22×C3×D4—61no—
110C22×D12C3×D311yescharacteristic
111C2×C3×M16—91no—
112C3×#117—31no—
113C3×#117—31no—
114C3×#117—61no—
115#926C3×D311yescharacteristic, complemented
116C2×#208—31no—
117C2×#208—31no—
118C2×#208—61no—
119C3×#102—94no—
120#832—34no—
121#809—34no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—41no—
126C23×C32—64no—
127C23×C32—62no—
128C23×C32—124no—
129C23×C32#6711yescharacteristic
130C23×C32#6711yescharacteristic
131C23×C32C3×D411yescharacteristic
132C22×C3×D3—28no—
133C22×C3×D3—24no—
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3—48no—
139C22×C3×D3—44no—
140C22×C3×D3—44no—
141C32×D4—34no—
142C32×D4—64no—
143C32×D4—62no—
144C2×C32×C4—22no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—42no—
157C2×C32×C4—41no—
158C2×C32×C4—42no—
159C2×C32×C4—41no—
160C2×C32×C4—62no—
161C2×C32×C4C4×D312yes—
162C2×C32×C4D1212yes—
163C2×C32×C4C2×Dic312yes—
164C2×C32×C4#6711yescharacteristic
165C2×C32×C4#6711yescharacteristic
166C2×C32×C4#6712yes—
167C2×C32×C4C2×C3×C412yes—
168C2×C32×C4C3×D412yes—
169C2×C32×C4C3×D411yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C23×C311yescharacteristic
173C2×C3×Dic3—24no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C3×D12—28no—
177C3×D12—24no—
178C3×D12—24no—
179C3×D12—24no—
180C3×D12—22no—
181C3×D12—22no—
182C3×D12—24no—
183C3×D12C2×Dic314yes—
184C3×D12C2×C3×C414yes—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—64no—
187C3×C3⋊C8—62no—
188C3×C3⋊C8—62no—
189#418—91noSylow
190C2×C33—22no—
191C2×C33C2×C22⋊C411yescharacteristic
192C2×C33AGL(1,ℤ/8)12yes—
193C32×D3—44no—
194C24×C3—62no—
195C24×C3—61no—
196C24×C3—61no—
197C23×D3—21no—
198C2×C3×D4—34no—
199C2×C3×D4—34no—
200C2×C3×D4—32no—
201C2×C3×D4—32no—
202C2×C3×D4—64no—
203C2×C3×D4—62no—
204C2×C3×D4—62no—
205C2×C3×D4—64no—
206C2×C3×D4—61no—
207C2×C3×D4—62no—
208C2×C3×D4—62no—
209C2×C3×D4—61no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—41no—
214C22×C3×C4D3211yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×C3×C4C2×C3×D311yescharacteristic
217C2×D12—21no—
218C2×D12—22no—
219C2×D12C3×Dic314yescomplemented
220C2×D12C2×C3×D312yes—
221C3×M16—182no—
222C2×C3×C8—92no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—62no—
226C3×C22⋊C4—62no—
227C3×C22⋊C4—61no—
228C3×C22⋊C4—61no—
229C3×C42—62no—
230C3×C42—61no—
231C3×C42—61no—
232#217—21no—
233C4⋊Dic3C2×C3×D312yescomplemented
234C4×Dic3—21no—
235#208—64no—
236#208—62no—
237#208—62no—
238C2×C3⋊C8—32no—
239C2×C3⋊C8—32no—
240C2×C3⋊C8—62no—
241C22×C32—22no—
242C22×C32—24no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—42no—
252C22×C32—41no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—44no—
256C22×C32—64no—
257C22×C32—128no—
258C22×C32—124no—
259C22×C32(C2×D3)⋊C412yes—
260C22×C32#21712yes—
261C22×C32C3×C22⋊C412yes—
262C22×C32C2×#6711yescharacteristic
263C22×C32C2×#6711yescharacteristic
264C22×C32C2×C3×D411yescharacteristic
265C2×C3×D3—28no—
266C2×C3×D3—24no—
267C2×C3×D3—24no—
268C2×C3×D3—416no—
269C2×C3×D3—48no—
270C2×C3×D3—48no—
271C2×C3×D3—48no—
272C2×C3×D3—44no—
273C2×C3×D3—44no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—122no—
287C32×C4(C2×D3)⋊C412yes—
288C32×C4#21712yes—
289C32×C4C3×C22⋊C412yes—
290C32×C4C2×C4×D311yescharacteristic
291C32×C4C2×D1211yescharacteristic
292C32×C4C22×Dic311yescharacteristic
293C32×C4C2×#6711yescharacteristic
294C32×C4C22×C3×C411yescharacteristic
295C32×C4C2×C3×D411yescharacteristic
296C3×Dic3—44no—
297C3×Dic3—42no—
298C3×Dic3—42no—
299C22×D4—31no—
300C2×M16—91no—
301#117—31no—
302#102—94no—
303C33#41811yescharacteristic, complemented, Sylow
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—64no—
309C23×C3—68no—
310C23×C3—64no—
311C23×C3—64no—
312C23×C3—62no—
313C23×C3—62no—
314C23×C3—128no—
315C23×C3—124no—
316C23×C3—124no—
317C23×C3#62511yescharacteristic
318C23×C3C3×#6711yescharacteristic
319C23×C3C3×#6711yescharacteristic
320C22×D3—24no—
321C22×D3—22no—
322C22×D3—44no—
323C3×D4—34no—
324C3×D4—34no—
325C3×D4—64no—
326C3×D4—64no—
327C3×D4—68no—
328C3×D4—64no—
329C3×D4—64no—
330C3×D4—62no—
331C3×D4—62no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—64no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4D3×Dic312yes—
352C2×C3×C4#62511yescharacteristic
353C2×C3×C4#62612yes—
354C2×C3×C4C3×C4×D312yes—
355C2×C3×C4C3×D1212yes—
356C2×C3×C4C2×C3×Dic312yes—
357C2×C3×C4C3×#6711yescharacteristic
358C2×C3×C4C3×#6711yescharacteristic
359C2×C3×C4C3×#6712yes—
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C22×C3×D311yescharacteristic
362C2×C3×C4C22×C3×D311yescharacteristic
363C2×Dic3—22no—
364D12—24no—
365D12—22no—
366D12C2×C3×Dic314yes—
367C3×C8—182no—
368C3⋊C8—64no—
369C3⋊C8—62no—
370C3⋊C8—62no—
371C2×C32—22no—
372C2×C32—24no—
373C2×C32—22no—
374C2×C32—22no—
375C2×C32—21no—
376C2×C32—22no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—42no—
383C2×C32—42no—
384C2×C32—42no—
385C2×C32—44no—
386C2×C32—124no—
387C2×C32C2×(C2×D3)⋊C411yescharacteristic
388C2×C32#93212yes—
389C2×C32#94912yes—
390C2×C32C2×#21711yescharacteristic
391C2×C32C2×C3×C22⋊C411yescharacteristic
392C2×C32C3×AGL(1,ℤ/8)12yes—
393C3×D3—48no—
394C3×D3—44no—
395C3×D3—44no—
396C24—61no—
397C2×D4—34no—
398C2×D4—32no—
399C2×D4—61no—
400C2×D4—62no—
401C22×C4C3×D3211yescharacteristic
402M16—182no—
403C2×C8—92no—
404C4⋊C4—32no—
405C22⋊C4—61no—
406C42—61no—
407C22×C3—24no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—44no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—68no—
423C22×C3—64no—
424C22×C3—64no—
425C22×C3—1216no—
426C22×C3—128no—
427C22×C3—128no—
428C22×C3—128no—
429C22×C3—124no—
430C22×C3—124no—
431C22×C3(C2×D3)⋊Dic312yes—
432C22×C3C3×(C2×D3)⋊C412yes—
433C22×C3C3×#21712yes—
434C22×C3C2×#62511yescharacteristic
435C22×C3C2×C3×#6711yescharacteristic
436C22×C3C2×C3×#6711yescharacteristic
437C2×D3—24no—
438C2×D3—48no—
439C2×D3—44no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—124no—
453C3×C4—122no—
454C3×C4—122no—
455C3×C4(C2×D3)⋊Dic312yes—
456C3×C4C3×(C2×D3)⋊C412yes—
457C3×C4C3×#21712yes—
458C3×C4C2×D3×Dic311yescharacteristic
459C3×C4C2×#62611yescharacteristic
460C3×C4C2×C3×C4×D311yescharacteristic
461C3×C4C2×C3×D1211yescharacteristic
462C3×C4C22×C3×Dic311yescharacteristic
463C3×C4C2×C3×#6711yescharacteristic
464Dic3—42no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32#500111yescharacteristic, complemented
470C32#521511yescharacteristic, complemented
471C32C3×#41811yescharacteristic, complemented
472C23—62no—
473C23—64no—
474C23—124no—
475C23C3×#62511yescharacteristic
476D4—34no—
477D4—62no—
478D4—64no—
479C2×C4—62no—
480C2×C4C3×D3×Dic312yes—
481C2×C4C3×#62511yescharacteristic
482C2×C4C3×#62612yes—
483C2×C4C2×C3×D3211yescharacteristic
484C8—182no—
485C2×C3—24no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—22no—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—42no—
496C2×C3—42no—
497C2×C3—44no—
498C2×C3—41no—
499C2×C3—42no—
500C2×C3—128no—
501C2×C3—124no—
502C2×C3—124no—
503C2×C3#6382312yes—
504C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
505C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
506C2×C3C3×#93212yes—
507C2×C3C3×#94912yes—
508C2×C3C2×C3×#21711yescharacteristic
509D3—44no—
510C22—22no—
511C22—64no—
512C22—128no—
513C22—124no—
514C22C3×(C2×D3)⋊Dic312yes—
515C22C2×C3×#62511yescharacteristic
516C4—122no—
517C4C3×(C2×D3)⋊Dic312yes—
518C4C2×C3×D3×Dic311yescharacteristic
519C4C2×C3×#62611yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059108511yescharacteristic, direct product factor
525C3C3×#500111yescharacteristic, complemented
526C3C3×#521511yescharacteristic, complemented
527C2—22no—
528C2—124no—
529C2C3×#6382312yes—
530C2C2×C3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D12C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#418
LCSG ▷ 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
2. C2×C3×#3748C2semidirect
3. C32×#926C2semidirect
4. C3×#63562C2semidirect
7. #10591085C3direct (N × H)
12. C2×C32×D12C4semidirect
15. C2×C3×C3⋊(C3⋊C8)C22semidirect
17. C32×C4⋊Dic3C22semidirect
24. C2×#3748C2×C3semidirect
26. C3×#926D3semidirect
27. C3×#926C2×C3semidirect
33. #63562C2×C3semidirect
73. C2×C3×D12Dic3semidirect
74. C2×C3×D12C3×C4semidirect
81. C2×C3⋊(C3⋊C8)C22×C3semidirect
86. C3×C4⋊Dic3C2×D3semidirect
87. C3×C4⋊Dic3C22×C3semidirect
115. #926C3×D3semidirect
219. C2×D12C3×Dic3semidirect
233. C4⋊Dic3C2×C3×D3semidirect
303. C33#418semidirect
469. C32#5001semidirect
470. C32#5215semidirect
471. C32C3×#418semidirect
524. C3#10591085direct (N × H)
525. C3C3×#5001semidirect
526. C3C3×#5215semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D122
C2×C3×#37482
C32×#9262
C3×#635622
C3×#52153
C3×#50013
#105910853