Finite Group Databasebeta

← C3×#10591610 #13158649

Subgroups — C3×#10591610 #13158649

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3717C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×#5191—31nomaximal
9C3×#5168—31nomaximal
10#10591610C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3752—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#216—22no—
18C32×#215—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3717—22no—
23C3×#3714—22no—
24C2×C32×Q16—31no—
25C2×C32×SD16—31no—
26#64156C2×C311yescharacteristic, complemented
27C3×#947—31no—
28C3×#947—31no—
29C3×#947—31no—
30C3×#947—61no—
31C2×C3×#216—21no—
32C2×C3×#216D311yescharacteristic, complemented
33C2×C3×#216C2×C311yescharacteristic, complemented
34C2×C3×#215—21no—
35C2×C3×#215—31no—
36C2×C3×#215D311yescharacteristic, complemented
37C2×C3×#215C2×C311yescharacteristic, complemented
38C3×#936—31no—
39C2×C3×#213—31no—
40C2×C3×D24—31no—
41C2×C3×#204—31no—
42#63863C2×C311yescharacteristic, complemented
43C2×#3717C2×C311yescharacteristic, complemented
44C2×#3714C2×C311yescharacteristic
45C3×C8×Dic3—31no—
46C3×C8×Dic3—31no—
47#63553C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3D412yes—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C2×C32×Dic3—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C32×C3⋊C8D412yes—
63C32×C3⋊C8D412yes—
64C3×#495—91no—
65#5191—31no—
66#5168—31no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—31no—
71C2×C32×Q8—41no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C22×C311yescharacteristic
75C2×C32×D4—31no—
76C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
77C2×C3×D12—21no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×Q16—62no—
87C32×SD16—62no—
88C2×C32×C8—31no—
89C2×C32×C8—31no—
90#3752—22no—
91C4×C3⋊Dic3C22×C311yescharacteristic
92C3×#217—62no—
93C3×#216—24no—
94C3×#216—22no—
95C3×#216—22no—
96C3×#215—24no—
97C3×#215—22no—
98C3×#215—22no—
99C3×#215—62no—
100C3×#213—62no—
101C3×(C2×D3)⋊C4—62no—
102C3×(C2×D3)⋊C4—62no—
103C3×(C2×D3)⋊C4—62no—
104C3×(C2×D3)⋊C4—64no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×D24—62no—
115C3×#204—62no—
116(C2×D3)⋊Dic3—22no—
117#3717—22no—
118#3714—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C33×C4—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2×D411yescharacteristic
127C3×C3⋊Dic3—22no—
128C32×Dic3—41no—
129C2×C3×Q16—31no—
130C2×C3×Q16—31no—
131C2×C3×Q16—61no—
132C2×C3×SD16—31no—
133C2×C3×SD16—31no—
134C2×C3×SD16—61no—
135C2×C3×SD16—91no—
136C2×C3×D8—91no—
137C3×#124—91no—
138C3×#124—91no—
139#947—31no—
140#947—31no—
141#947—31no—
142#947—61no—
143C2×#216C3×D311yescharacteristic, complemented
144C2×#215—31no—
145C2×#215C3×D311yescharacteristic, complemented
146#936—31no—
147C2×#213—31no—
148C2×D24—31no—
149C2×#204—31no—
150C3×C4×C8—91no—
151C8×Dic3—31no—
152C8×Dic3—31no—
153C23×C32—61no—
154C22×C3⋊D3—21no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—41no—
158C22×C3×D3—62no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—22no—
164C32×Q8—22no—
165C32×Q8—21no—
166C32×Q8—21no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—41no—
170C32×Q8—41no—
171C32×Q8—41no—
172C32×Q8—41no—
173C32×Q8—42no—
174C32×Q8—42no—
175C32×Q8—61no—
176C32×Q8—61no—
177C32×D4—61no—
178C32×D4—61no—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—42no—
190C2×C32×C4—61no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C311yescharacteristic
194C2×C3⋊Dic3C3×D412yes—
195C4⋊(C3⋊D3)—21no—
196C4⋊(C3⋊D3)—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—41no—
202C2×C3×Dic3—62no—
203C3×D12—22no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—41no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—121no—
214C3×D12—121no—
215C3×Dic6—22no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—21no—
219C3×Dic6—21no—
220C3×Dic6—41no—
221C32×C8—32no—
222C32×C8—32no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—22no—
225C3×C3⋊C8#6712yes—
226C3×C3⋊C8#6712yes—
227C3×C3⋊C8C3×D412yes—
228C3×C3⋊C8C3×D412yes—
229#495—91noSylow
230C2×C33#12711yescharacteristic
231C2×C33#13512yes—
232C3×C3⋊D3—81no—
233C32×D3—81no—
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—31no—
238C2×C3×Q8—31no—
239C2×C3×Q8—41no—
240C2×C3×Q8—61no—
241C2×C3×Q8D3211yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×Q8C2×C3×D311yescharacteristic
244C2×C3×D4—31no—
245C2×C3×D4—31no—
246C2×C3×D4—61no—
247C2×C3×D4—91no—
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—61no—
251C2×D12C2×C3×D311yescharacteristic
252C2×Dic6C2×C3×D311yescharacteristic
253C3×Q16—62no—
254C3×Q16—62no—
255C3×Q16—64no—
256C3×SD16—62no—
257C3×SD16—62no—
258C3×SD16—64no—
259C3×SD16—182no—
260C3×D8—182no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—61no—
267C3×C22⋊C4—182no—
268C3×C22⋊C4—182no—
269C3×C42—91no—
270#217—62no—
271#216—22no—
272#215—22no—
273#215—62no—
274#213—62no—
275(C2×D3)⋊C4—62no—
276(C2×D3)⋊C4—62no—
277(C2×D3)⋊C4—62no—
278(C2×D3)⋊C4—64no—
279C4×Dic3—31no—
280C4×Dic3—31no—
281C4×Dic3—61no—
282C2×C3⋊C8C2×C3×D311yescharacteristic
283C2×C3⋊C8C2×C3×D311yescharacteristic
284D24—62no—
285#204—62no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—121no—
291C22×C32—122no—
292C22×C32D3×D411yescharacteristic
293C22×C32D3×D411yescharacteristic
294C22×C32C2×C3×D411yescharacteristic
295C2×C3⋊D3—41no—
296C2×C3⋊D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—42no—
302C2×C3×D3—82no—
303C2×C3×D3—122no—
304C2×C3×D3—124no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4—42no—
319C32×C4—42no—
320C32×C4—44no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—121no—
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C32×C4C2×C3×D411yescharacteristic
330C32×C4C2×C3×D411yescharacteristic
331C3⋊Dic3—22no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—122no—
338C2×Q16—31no—
339C2×SD16—31no—
340C2×SD16—91no—
341C2×D8—91no—
342#124—91no—
343#124—91no—
344C4×C8—91no—
345C33#49511yescharacteristic, complemented, Sylow
346C23×C3—61no—
347C23×C3—61no—
348C23×C3—121no—
349C23×C3—181no—
350C22×D3—21no—
351C22×D3—62no—
352C22×D3—61no—
353C22×D3—61no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—42no—
367C3×Q8—42no—
368C3×Q8—62no—
369C3×Q8—61no—
370C3×Q8—61no—
371C3×Q8—61no—
372C3×Q8—61no—
373C3×Q8—121no—
374C3×D4—62no—
375C3×D4—61no—
376C3×D4—61no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—121no—
380C3×D4—181no—
381C3×D4—181no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—92no—
396C2×C3×C4—121no—
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—62no—
404D12—21no—
405D12—21no—
406D12—61no—
407D12—61no—
408D12—61no—
409D12—61no—
410D12—121no—
411D12—121no—
412Dic6—21no—
413Dic6—21no—
414C3×C8—32no—
415C3×C8—32no—
416C3×C8—32no—
417C3×C8—32no—
418C3×C8—62no—
419C3×C8—62no—
420C3⋊C8C3×#6712yes—
421C3⋊C8C3×#6712yes—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—241no—
431C2×C32#91612yes—
432C2×C32#91912yes—
433C2×C32#94011yescharacteristic
434C2×C32#94011yescharacteristic
435C2×C32C3×#12711yescharacteristic
436C2×C32C3×#13512yes—
437C3⋊D3—81no—
438C3×D3—82no—
439C3×D3—81no—
440C3×D3—81no—
441C3×D3—242no—
442C3×D3—241no—
443C3×D3—241no—
444C2×Q8—31no—
445C2×Q8C3×D3211yescharacteristic
446C2×D4—31no—
447C2×D4—91no—
448Q16—62no—
449SD16—62no—
450SD16—182no—
451D8—182no—
452C2×C8—31no—
453C2×C8—31no—
454C22⋊C4—182no—
455C22⋊C4—182no—
456C42—91no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—121no—
464C22×C3—122no—
465C22×C3—121no—
466C22×C3—242no—
467C22×C3—362no—
468C22×C3—361no—
469C22×C3#381111yescharacteristic
470C22×C3C3×D3×D411yescharacteristic
471C22×C3C3×D3×D411yescharacteristic
472C2×D3—42no—
473C2×D3—41no—
474C2×D3—122no—
475C2×D3—124no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—121no—
479C2×D3—121no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—122no—
496C3×C4—121no—
497C3×C4—121no—
498C3×C4—182no—
499C3×C4D3×#6711yescharacteristic
500C3×C4D3×#6711yescharacteristic
501C3×C4C3×D3×D411yescharacteristic
502C3×C4C3×D3×D411yescharacteristic
503C3×C4C2×C3×#6711yescharacteristic
504C3×C4C2×C3×#6711yescharacteristic
505Dic3—41no—
506Dic3—62no—
507Dic3—62no—
508Dic3—122no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#516811yescharacteristic, complemented
514C32#519111yescharacteristic, complemented
515C32C3×#49511yescharacteristic, complemented
516C23—61no—
517C23—181no—
518Q8—21no—
519Q8—21no—
520Q8—61no—
521Q8—61no—
522D4—61no—
523D4—61no—
524D4—181no—
525D4—181no—
526C2×C4—21no—
527C2×C4—61no—
528C2×C4—92no—
529C2×C4C2×C3×D3211yescharacteristic
530C8—32no—
531C8—32no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—42no—
539C2×C3—41no—
540C2×C3—242no—
541C2×C3—241no—
542C2×C3—241no—
543C2×C3—721no—
544C2×C3#6395012yes—
545C2×C3#6397911yescharacteristic
546C2×C3C3×#91612yes—
547C2×C3C3×#91912yes—
548C2×C3C3×#94011yescharacteristic
549C2×C3C3×#94011yescharacteristic
550D3—81no—
551D3—242no—
552D3—241no—
553D3—241no—
554C22—122no—
555C22—121no—
556C22—362no—
557C22—361no—
558C22C3×#381111yescharacteristic
559C4—41no—
560C4—121no—
561C4—182no—
562C4C3×D3×#6711yescharacteristic
563C4C3×D3×#6711yescharacteristic
564C3—21no—
565C3—21no—
566C3—21no—
567C3—41no—
568C3#1059161011yescharacteristic, direct product factor
569C3C3×#516811yescharacteristic, complemented
570C3C3×#519111yescharacteristic, complemented
571C2—241no—
572C2—721no—
573C2C3×#6395012yes—
574C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C2×C32×#216C2semidirect
3. C2×C32×#215C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3717C2semidirect
7. C3×#63553C2semidirect
10. #10591610C3direct (N × H)
26. #64156C2×C3semidirect
32. C2×C3×#216D3semidirect
33. C2×C3×#216C2×C3semidirect
36. C2×C3×#215D3semidirect
37. C2×C3×#215C2×C3semidirect
42. #63863C2×C3semidirect
43. C2×#3717C2×C3semidirect
47. #63553C2×C3semidirect
143. C2×#216C3×D3semidirect
145. C2×#215C3×D3semidirect
345. C33#495semidirect
513. C32#5168semidirect
514. C32#5191semidirect
515. C32C3×#495semidirect
568. C3#10591610direct (N × H)
569. C3C3×#5168semidirect
570. C3C3×#5191semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C32×#2162
C2×C32×#2152
C3×#638632
C2×C3×#37172
C2×C3×#37142
C3×#635532
C3×#51913
C3×#51683
#105916103