Finite Group Databasebeta

← C3×#10590766 #13157752

Subgroups — C3×#10590766 #13157752

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#204C211yesmaximal, characteristic
2C3×Dic3×D12C211yesmaximal, characteristic
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#815C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8C3×#5164—31nomaximal
9C3×#4710—31nomaximal
10#10590766C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C2×C33×C8C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C32×#204C414yescomplemented
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×Dic32—21no—
23C2×C32×SD16—31no—
24C3×Q8×Dic3—31no—
25C3×D4×Dic3—31no—
26C2×C3×#204—21no—
27C2×C3×#204D311yescharacteristic
28C2×C3×#204C2×C311yescharacteristic
29C3×C4×D12—31no—
30C3×C4×Dic6—31no—
31Dic3×D12C2×C311yescharacteristic
32Dic3×Dic6C2×C311yescharacteristic, complemented
33C32×#97—31no—
34#63645C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#832—31no—
37C3×#821—31no—
38C3×#818—31no—
39C3×#818—31no—
40C3×#818—61no—
41C3×#816—31no—
42C3×#815—21no—
43C3×#815D311yescharacteristic, complemented
44C3×#815C2×C311yescharacteristic, complemented
45#63566C2×C311yescharacteristic, complemented
46#63563C2×C311yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C2×C32×Dic3D411yescharacteristic
54C32×D12—21no—
55C32×D12C2×C412yes—
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58C3×D3×Dic3—42no—
59C33×C8C2×C412yescomplemented
60C32×C3⋊C8—21no—
61C3×#440—91no—
62#5164—31no—
63#4710—31no—
64C2×C32×Q8—31no—
65C2×C32×D4—31no—
66C22×C3×Dic3—61no—
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C22×C311yescharacteristic
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2×D311yescharacteristic
73C2×C3×Dic6C22×C311yescharacteristic
74C2×D3×Dic3—21no—
75C32×SD16—34no—
76C2×C32×C8—21no—
77C2×C32×C8—21no—
78C2×C32×C8—21no—
79C2×C32×C8—31no—
80C2×C32×C8—41no—
81C2×C32×C8C2×D311yescharacteristic
82C2×C32×C8C2×D311yescharacteristic
83C2×C32×C8C22×C311yescharacteristic
84C32×C42—31no—
85C4⋊(C3⋊Dic3)C22×C311yescharacteristic
86C4⋊(C3⋊Dic3)C22×C311yescharacteristic
87C3×#217—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C22×C311yescharacteristic
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105C3×#204—24no—
106C3×#204Dic314yescomplemented
107C3×#204C3×C414yescomplemented
108Dic3⋊Dic3—21no—
109(C2×D3)⋊Dic3—21no—
110Dic32—21no—
111C22×C33C2×D411yescharacteristic
112C2×C32×D3—22no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4C22×C411yescharacteristic
116C33×C4#4111yescharacteristic
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C32×Dic3—22no—
120C32×Dic3—21no—
121C32×Dic3—21no—
122C2×C3×SD16—31no—
123C2×C3×SD16—31no—
124C2×C3×SD16—61no—
125C3×C4×Q8—91no—
126C3×C4×D4—91no—
127Q8×Dic3—31no—
128D4×Dic3—31no—
129C2×#204C3×D311yescharacteristic
130C4×D12—31no—
131C4×Dic6—31no—
132C3×#107—91no—
133C3×#103—91no—
134C3×#102—91no—
135C3×#97—31no—
136C3×#97—31no—
137C3×#97—61no—
138#835—31no—
139#832—31no—
140#821—31no—
141#818—31no—
142#818—31no—
143#818—61no—
144#816—31no—
145#815C3×D311yescharacteristic, complemented
146C23×C32—61no—
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—41no—
150C32×Q8—32no—
151C32×Q8—61no—
152C32×D4—32no—
153C32×D4—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—31no—
158C2×C32×C4—31no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C23×C311yescharacteristic
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—41no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3—121no—
177C2×C3×Dic3—122no—
178C2×C3×Dic3D1211yescharacteristic
179C2×C3×Dic3D1211yescharacteristic
180C2×C3×Dic3C3×D411yescharacteristic
181C2×C3×Dic3C3×D411yescharacteristic
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—22no—
185C3×D12—41no—
186C3×D12C2×Dic312yes—
187C3×D12C2×C3×C412yes—
188C3×C4×D3—122no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—22no—
192C3×Dic6—41no—
193C3×Dic6C2×Dic312yescomplemented
194C3×Dic6C2×C3×C412yescomplemented
195D3×Dic3—42no—
196C32×C8—22no—
197C32×C8—22no—
198C32×C8—22no—
199C32×C8—42no—
200C32×C8—61no—
201C32×C8C4×D312yescomplemented
202C32×C8C2×Dic312yescomplemented
203C32×C8C2×C3×C412yescomplemented
204C3×C3⋊C8—22no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—21no—
207#440—91noSylow
208C2×C33C4×D411yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C2×C33#13711yescharacteristic
211C32×D3—42no—
212C2×C3×Q8—31no—
213C2×C3×Q8—31no—
214C2×C3×Q8—61no—
215C2×C3×D4—31no—
216C2×C3×D4—31no—
217C2×C3×D4—61no—
218C22×C3×C4—181no—
219C22×Dic3—61no—
220C2×D12C2×C3×D311yescharacteristic
221C2×C4×D3—61no—
222C2×Dic6C2×C3×D311yescharacteristic
223C3×SD16—34no—
224C3×SD16—34no—
225C3×SD16—64no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—41no—
232C2×C3×C8—61no—
233C2×C3×C8D3211yescharacteristic
234C2×C3×C8C2×C3×D311yescharacteristic
235C2×C3×C8C2×C3×D311yescharacteristic
236C3×C4⋊C4—91no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—181no—
239C3×C22⋊C4—181no—
240C3×C42—31no—
241C3×C42—31no—
242C3×C42—61no—
243C3×C42—181no—
244#217—61no—
245(C2×D3)⋊C4—61no—
246C4⋊Dic3—31no—
247C4⋊Dic3—31no—
248C4⋊Dic3—31no—
249C4⋊Dic3—31no—
250C4⋊Dic3—61no—
251C4⋊Dic3—61no—
252C4⋊Dic3—61no—
253Dic3⋊C4—61no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3C2×C3×D311yescharacteristic
257C2×C3⋊C8C2×C3×D311yescharacteristic
258#204C3×Dic314yescomplemented
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32—62no—
264C22×C32—121no—
265C22×C32—121no—
266C22×C32C2×D1211yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×C3×D411yescharacteristic
269C2×C3×D3—22no—
270C2×C3×D3—22no—
271C2×C3×D3—42no—
272C2×C3×D3—41no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—81no—
277C2×C3×D3—81no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—62no—
287C32×C4—61no—
288C32×C4—61no—
289C32×C4C2×C4×D311yescharacteristic
290C32×C4#23511yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4C22×Dic311yescharacteristic
293C32×C4C22×C3×C411yescharacteristic
294C32×C4C3×#4111yescharacteristic
295C3⋊Dic3—41no—
296C3⋊Dic3—41no—
297C3×Dic3—21no—
298C3×Dic3—21no—
299C3×Dic3—22no—
300C3×Dic3—22no—
301C3×Dic3—21no—
302C3×Dic3—21no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—241no—
311C3×Dic3—241no—
312C2×SD16—31no—
313C4×Q8—91no—
314C4×D4—91no—
315#107—91no—
316#103—91no—
317#102—91no—
318#97—31no—
319C33#44011yescharacteristic, complemented, Sylow
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—121no—
323C22×D3—21no—
324C3×Q8—32no—
325C3×Q8—32no—
326C3×Q8—62no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—121no—
330C3×D4—32no—
331C3×D4—32no—
332C3×D4—62no—
333C3×D4—61no—
334C3×D4—61no—
335C3×D4—121no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—41no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—121no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—362no—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C22×C3×D311yescharacteristic
354C2×C3×C4C22×C3×D311yescharacteristic
355C2×Dic3—21no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—121no—
361C2×Dic3—121no—
362C2×Dic3—122no—
363C2×Dic3C3×D1211yescharacteristic
364C2×Dic3C3×D1211yescharacteristic
365D12—21no—
366D12C2×C3×Dic312yes—
367C4×D3—122no—
368Dic6—21no—
369Dic6C2×C3×Dic312yescomplemented
370C3×C8—22no—
371C3×C8—22no—
372C3×C8—22no—
373C3×C8—42no—
374C3×C8—62no—
375C3×C8—61no—
376C3×C8—61no—
377C3×C8D3×Dic312yescomplemented
378C3×C8C3×C4×D312yescomplemented
379C3×C8C2×C3×Dic312yescomplemented
380C3⋊C8—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—122no—
394C2×C32C4×D1211yescharacteristic
395C2×C32#90811yescharacteristic
396C2×C32#90911yescharacteristic
397C2×C32#91411yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32D4×Dic311yescharacteristic
400C2×C32C3×C4×D411yescharacteristic
401C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
402C2×C32C3×#13711yescharacteristic
403C3×D3—42no—
404C3×D3—42no—
405C3×D3—82no—
406C2×Q8—31no—
407C2×D4—31no—
408C22×C4—181no—
409SD16—34no—
410C2×C8—31no—
411C2×C8C3×D3211yescharacteristic
412C4⋊C4—91no—
413C4⋊C4—91no—
414C4⋊C4—181no—
415C22⋊C4—181no—
416C42—31no—
417C42—181no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—122no—
425C22×C3—121no—
426C22×C3—121no—
427C22×C3—121no—
428C22×C3—121no—
429C22×C3—241no—
430C22×C3—241no—
431C22×C3D3×D1211yescharacteristic
432C22×C3C2×C3×D1211yescharacteristic
433C22×C3C3×D3×D411yescharacteristic
434C2×D3—22no—
435C2×D3—41no—
436C2×D3—41no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—122no—
454C3×C4—361no—
455C3×C4—361no—
456C3×C4#379511yescharacteristic
457C3×C4C2×D3×Dic311yescharacteristic
458C3×C4C2×C3×C4×D311yescharacteristic
459C3×C4C3×#23511yescharacteristic
460C3×C4C3×#23711yescharacteristic
461C3×C4C22×C3×Dic311yescharacteristic
462Dic3—21no—
463Dic3—21no—
464Dic3—22no—
465Dic3—121no—
466Dic3—121no—
467Dic3—121no—
468Dic3—121no—
469Dic3—241no—
470Dic3—241no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#471011yescharacteristic, complemented
476C32#516411yescharacteristic, complemented
477C32C3×#44011yescharacteristic, complemented
478C23—61no—
479Q8—32no—
480Q8—61no—
481D4—32no—
482D4—61no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—61no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—362no—
489C2×C4C2×C3×D3211yescharacteristic
490C8—61no—
491C8C3×D3×Dic312yescomplemented
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—122no—
505C2×C3—122no—
506C2×C3—242no—
507C2×C3#6379411yescharacteristic
508C2×C3#6380011yescharacteristic
509C2×C3Dic3×D1211yescharacteristic
510C2×C3C3×C4×D1211yescharacteristic
511C2×C3C3×#90811yescharacteristic
512C2×C3C3×#90911yescharacteristic
513C2×C3C3×#91411yescharacteristic
514C2×C3C3×#91511yescharacteristic
515C2×C3C3×D4×Dic311yescharacteristic
516D3—42no—
517C22—62no—
518C22—121no—
519C22—121no—
520C22C3×D3×D1211yescharacteristic
521C4—61no—
522C4—61no—
523C4—62no—
524C4—361no—
525C4—361no—
526C4C3×#379511yescharacteristic
527C4C2×C3×D3×Dic311yescharacteristic
528C3—21no—
529C3—21no—
530C3—21no—
531C3—41no—
532C3#1059076611yescharacteristic, direct product factor
533C3C3×#471011yescharacteristic, complemented
534C3C3×#516411yescharacteristic, complemented
535C2—122no—
536C2C3×#6379411yescharacteristic
537C2C3×#6380011yescharacteristic
538C2C3×Dic3×D1211yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#440
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×Dic3×Dic6C2semidirect
4. C3×#63645C2semidirect
5. C32×#815C2semidirect
6. C3×#63566C2semidirect
10. #10590766C3direct (N × H)
19. C32×#204C4semidirect
32. Dic3×Dic6C2×C3semidirect
34. #63645C2×C3semidirect
43. C3×#815D3semidirect
44. C3×#815C2×C3semidirect
45. #63566C2×C3semidirect
57. C32×Dic6C2×C4semidirect
59. C33×C8C2×C4semidirect
106. C3×#204Dic3semidirect
107. C3×#204C3×C4semidirect
145. #815C3×D3semidirect
193. C3×Dic6C2×Dic3semidirect
194. C3×Dic6C2×C3×C4semidirect
201. C32×C8C4×D3semidirect
202. C32×C8C2×Dic3semidirect
203. C32×C8C2×C3×C4semidirect
258. #204C3×Dic3semidirect
319. C33#440semidirect
369. Dic6C2×C3×Dic3semidirect
377. C3×C8D3×Dic3semidirect
378. C3×C8C3×C4×D3semidirect
379. C3×C8C2×C3×Dic3semidirect
475. C32#4710semidirect
476. C32#5164semidirect
477. C32C3×#440semidirect
491. C8C3×D3×Dic3semidirect
532. C3#10590766direct (N × H)
533. C3C3×#4710semidirect
534. C3C3×#5164semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
C3×Dic3×D122
C3×Dic3×Dic62
C3×#636452
C32×#8152
C3×#635662
C3×#635632
C3×#51643
C3×#47103
#105907663