Finite Group Databasebeta

← C3×#10590370 #13157365

Subgroups — C3×#10590370 #13157365

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64084C211yesmaximal, characteristic, complemented
2C32×#875C211yesmaximal, characteristic, complemented
3C2×C3×#3719C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5C2×C3×#3716C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic
7C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5059—31nomaximal
9C3×#4696—31nomaximal
10#10590370C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C33×C42C2211yescharacteristic
16C3×#3752—22no—
17C32×(C2×D3)⋊C4—22no—
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3719—22no—
21C3×#3717—22no—
22C3×#3716—22no—
23C3×#3714—22no—
24C32×#124—31no—
25#64084C2×C311yescharacteristic, complemented
26C2×C3×#216—31no—
27C2×C3×#215—31no—
28C2×C3×#214—31no—
29C2×C3×#213—31no—
30C2×C3×Dic12—31no—
31C2×C3×D24—31no—
32C2×C3×#204—31no—
33C2×C3×#204—31no—
34C3×#875—21no—
35C3×#875—31no—
36C3×#875—31no—
37C3×#875—61no—
38C3×#875D311yescharacteristic, complemented
39C3×#875C2×C311yescharacteristic, complemented
40C2×#3719C2×C311yescharacteristic, complemented
41C2×#3717C2×C311yescharacteristic, complemented
42C2×#3716C2×C311yescharacteristic, complemented
43C2×#3714C2×C311yescharacteristic
44C32×C4×C8—31no—
45C3×C4×C3⋊C8—21no—
46C3×C4×C3⋊C8D311yescharacteristic, complemented
47C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4D412yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×#634—21no—
56C3×#634—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#5059—31no—
66#4696—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C22×C311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C2×C32×C8—31no—
84C2×C32×C8—31no—
85C32×C22⋊C4—62no—
86C32×C42—21no—
87C32×C42—21no—
88C32×C42—21no—
89C32×C42—41no—
90C32×C42C2×D311yescharacteristic
91C32×C42C2×D311yescharacteristic
92C32×C42C22×C311yescharacteristic
93#3752—22no—
94C3×#216—62no—
95C3×#215—62no—
96C3×#214—62no—
97C3×#213—62no—
98C3×(C2×D3)⋊C4—24no—
99C3×(C2×D3)⋊C4—22no—
100C3×(C2×D3)⋊C4—22no—
101C3×(C2×D3)⋊C4—64no—
102C3×(C2×D3)⋊C4—62no—
103C3×(C2×D3)⋊C4—62no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C3×Dic12—62no—
111C3×D24—62no—
112C3×#204—62no—
113C3×#204—62no—
114#3719—22no—
115#3717—22no—
116#3716—22no—
117#3714—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—42no—
121C2×C32×D3—41no—
122C2×C32×D3—42no—
123C33×C4—22no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×SD16—91no—
130C2×C3×SD16—91no—
131C2×C3×D8—91no—
132C3×#124—31no—
133C3×#124—31no—
134C3×#124—61no—
135C3×#124—91no—
136C2×#216—31no—
137C2×#215—31no—
138C2×#214—31no—
139C2×#213—31no—
140C2×Dic12—31no—
141C2×D24—31no—
142C2×#204—31no—
143C2×#204—31no—
144#875—31no—
145#875—31no—
146#875—61no—
147#875C3×D311yescharacteristic, complemented
148C3×C4×C8—31no—
149C3×C4×C8—31no—
150C3×C4×C8—61no—
151C4×C3⋊C8C3×D311yescharacteristic, complemented
152C23×C32—61no—
153C22×C3⋊D3—21no—
154C22×C3×D3—22no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—62no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C32×Q8—61no—
161C32×Q8—61no—
162C32×D4—61no—
163C32×D4—61no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—42no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4D1212yes—
174C2×C32×C4#6712yes—
175C2×C32×C4C3×D412yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C23×C311yescharacteristic
179C2×C3⋊Dic3—21no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182#634—21no—
183#634—21no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—41no—
195C3×D12—41no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×D12—121no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—41no—
207C3×Dic6—41no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—121no—
213C3×Dic6—121no—
214C32×C8—32no—
215C32×C8—32no—
216C3×C3⋊C8—22no—
217C3×C3⋊C8—22no—
218C3×C3⋊C8D1212yes—
219C3×C3⋊C8D1212yes—
220C3×C3⋊C8C3×D412yes—
221C3×C3⋊C8C3×D412yes—
222#495—91noSylow
223C2×C33#12711yescharacteristic
224C2×C33#13512yes—
225C3×C3⋊D3—81no—
226C32×D3—81no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—91no—
231C2×C3×D4—31no—
232C2×C3×D4—31no—
233C2×C3×D4—61no—
234C2×C3×D4—91no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12C2×C3×D311yescharacteristic
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6C2×C3×D311yescharacteristic
243C3×Q16—182no—
244C3×SD16—182no—
245C3×SD16—182no—
246C3×D8—182no—
247C2×C3×C8—31no—
248C2×C3×C8—31no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—61no—
252C2×C3×C8—61no—
253C3×C22⋊C4—64no—
254C3×C22⋊C4—62no—
255C3×C22⋊C4—62no—
256C3×C22⋊C4—182no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—41no—
261C3×C42D3211yescharacteristic
262C3×C42C2×C3×D311yescharacteristic
263C3×C42C2×C3×D311yescharacteristic
264#216—62no—
265#215—62no—
266#214—62no—
267#213—62no—
268(C2×D3)⋊C4—22no—
269(C2×D3)⋊C4—64no—
270(C2×D3)⋊C4—62no—
271(C2×D3)⋊C4—62no—
272C2×C3⋊C8C2×C3×D311yescharacteristic
273C2×C3⋊C8C2×C3×D311yescharacteristic
274Dic12—62no—
275D24—62no—
276#204—62no—
277#204—62no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32—121no—
283C22×C32—122no—
284C22×C32C2×D1211yescharacteristic
285C22×C32C2×#6711yescharacteristic
286C22×C32C2×C3×D411yescharacteristic
287C2×C3⋊D3—41no—
288C2×C3⋊D3—42no—
289C2×C3×D3—42no—
290C2×C3×D3—44no—
291C2×C3×D3—41no—
292C2×C3×D3—42no—
293C2×C3×D3—41no—
294C2×C3×D3—42no—
295C2×C3×D3—122no—
296C2×C3×D3—124no—
297C2×C3×D3—121no—
298C2×C3×D3—122no—
299C2×C3×D3—121no—
300C2×C3×D3—122no—
301C32×C4—22no—
302C32×C4—24no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—42no—
314C32×C4—42no—
315C32×C4—44no—
316C32×C4—121no—
317C32×C4C2×D1211yescharacteristic
318C32×C4C2×D1211yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4C2×C3×D411yescharacteristic
322C32×C4C2×C3×D411yescharacteristic
323C3⋊Dic3—41no—
324C3×Dic3—42no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—122no—
328C3×Dic3—121no—
329C3×Dic3—121no—
330C2×Q16—91no—
331C2×SD16—91no—
332C2×SD16—91no—
333C2×D8—91no—
334#124—31no—
335#124—91no—
336C4×C8—31no—
337C33#49511yescharacteristic, complemented, Sylow
338C23×C3—62no—
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—181no—
342C22×D3—21no—
343C22×D3—62no—
344C22×D3—61no—
345C22×D3—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—121no—
351C3×Q8—121no—
352C3×Q8—181no—
353C3×Q8—181no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—61no—
357C3×D4—61no—
358C3×D4—121no—
359C3×D4—121no—
360C3×D4—181no—
361C3×D4—181no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—181no—
374C2×C3×C4#62612yes—
375C2×C3×C4C3×D1212yes—
376C2×C3×C4C3×#6712yes—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C22×C3×D311yescharacteristic
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384D12—21no—
385D12—21no—
386D12—61no—
387D12—61no—
388D12—61no—
389D12—61no—
390D12—121no—
391D12—121no—
392Dic6—21no—
393Dic6—21no—
394Dic6—61no—
395Dic6—61no—
396Dic6—61no—
397Dic6—61no—
398Dic6—121no—
399Dic6—121no—
400C3×C8—32no—
401C3×C8—32no—
402C3×C8—32no—
403C3×C8—32no—
404C3×C8—62no—
405C3×C8—62no—
406C3⋊C8C3×D1212yes—
407C3⋊C8C3×D1212yes—
408C2×C32—22no—
409C2×C32—21no—
410C2×C32—22no—
411C2×C32—22no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—41no—
415C2×C32—42no—
416C2×C32—241no—
417C2×C32#87411yescharacteristic
418C2×C32#90412yes—
419C2×C32#94011yescharacteristic
420C2×C32#95012yes—
421C2×C32C3×#12711yescharacteristic
422C2×C32C3×#13512yes—
423C3⋊D3—81no—
424C3×D3—82no—
425C3×D3—81no—
426C3×D3—81no—
427C3×D3—242no—
428C3×D3—241no—
429C3×D3—241no—
430C2×Q8—31no—
431C2×Q8—91no—
432C2×D4—31no—
433C2×D4—91no—
434Q16—182no—
435SD16—182no—
436SD16—182no—
437D8—182no—
438C2×C8—31no—
439C2×C8—31no—
440C22⋊C4—62no—
441C22⋊C4—182no—
442C42C3×D3211yescharacteristic
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—41no—
447C22×C3—122no—
448C22×C3—124no—
449C22×C3—122no—
450C22×C3—122no—
451C22×C3—121no—
452C22×C3—121no—
453C22×C3—362no—
454C22×C3—361no—
455C22×C3C2×#62611yescharacteristic
456C22×C3C2×C3×D1211yescharacteristic
457C22×C3C2×C3×#6711yescharacteristic
458C2×D3—41no—
459C2×D3—42no—
460C2×D3—122no—
461C2×D3—124no—
462C2×D3—122no—
463C2×D3—121no—
464C2×D3—122no—
465C2×D3—121no—
466C3×C4—24no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—44no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—361no—
485C3×C4D3×D1211yescharacteristic
486C3×C4D3×D1211yescharacteristic
487C3×C4C2×C3×D1211yescharacteristic
488C3×C4C2×C3×D1211yescharacteristic
489C3×C4C3×D3×D411yescharacteristic
490C3×C4C3×D3×D411yescharacteristic
491Dic3—41no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#469611yescharacteristic, complemented
500C32#505911yescharacteristic, complemented
501C32C3×#49511yescharacteristic, complemented
502C23—61no—
503C23—181no—
504Q8—61no—
505Q8—61no—
506Q8—181no—
507Q8—181no—
508D4—61no—
509D4—61no—
510D4—181no—
511D4—181no—
512C2×C4—61no—
513C2×C4—181no—
514C2×C4C3×#62612yes—
515C2×C4C2×C3×D3211yescharacteristic
516C8—32no—
517C8—32no—
518C2×C3—22no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—41no—
526C2×C3—242no—
527C2×C3—241no—
528C2×C3—241no—
529C2×C3—721no—
530C2×C3#6382912yes—
531C2×C3#6391111yescharacteristic
532C2×C3C3×#87411yescharacteristic
533C2×C3C3×#90412yes—
534C2×C3C3×#94011yescharacteristic
535C2×C3C3×#95012yes—
536D3—81no—
537D3—242no—
538D3—241no—
539D3—241no—
540C22—122no—
541C22—121no—
542C22—362no—
543C22—361no—
544C22C2×C3×#62611yescharacteristic
545C4—22no—
546C4—121no—
547C4—361no—
548C4C3×D3×D1211yescharacteristic
549C4C3×D3×D1211yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059037011yescharacteristic, direct product factor
555C3C3×#469611yescharacteristic, complemented
556C3C3×#505911yescharacteristic, complemented
557C2—241no—
558C2—721no—
559C2C3×#6382912yes—
560C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypercenterC3×C42D32
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64084C2semidirect
2. C32×#875C2semidirect
3. C2×C3×#3719C2semidirect
4. C2×C3×#3717C2semidirect
5. C2×C3×#3716C2semidirect
7. C32×C4×C3⋊C8C2semidirect
10. #10590370C3direct (N × H)
25. #64084C2×C3semidirect
38. C3×#875D3semidirect
39. C3×#875C2×C3semidirect
40. C2×#3719C2×C3semidirect
41. C2×#3717C2×C3semidirect
42. C2×#3716C2×C3semidirect
46. C3×C4×C3⋊C8D3semidirect
47. C3×C4×C3⋊C8C2×C3semidirect
147. #875C3×D3semidirect
151. C4×C3⋊C8C3×D3semidirect
337. C33#495semidirect
499. C32#4696semidirect
500. C32#5059semidirect
501. C32C3×#495semidirect
554. C3#10590370direct (N × H)
555. C3C3×#4696semidirect
556. C3C3×#5059semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×#8752
C2×C3×#37192
C2×C3×#37172
C2×C3×#37162
C2×C3×#37142
C32×C4×C3⋊C82
C3×#50593
C3×#46963
#105903703