Finite Group Databasebeta

← C3×#10591169 #13158348

Subgroups — C3×#10591169 #13158348

Order 1728.

1166 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D12C211yesmaximal, characteristic
2C32×#924C211yesmaximal, characteristic, complemented
3C2×C3×#3715C212yesmaximal, complemented
4C3×#63657C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#5218—31nomaximal
7C3×#5046—31nomaximal
8#10591169C311yesmaximal, characteristic, direct product factor
9C23×C32×D3—21no—
10C22×C33×C4C2211yescharacteristic
11C2×C32×D12—21no—
12C2×C32×D12—24no—
13C2×C32×D12C2212yes—
14C2×C32×Dic6C2211yescharacteristic, complemented
15C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
16C32×#217—21no—
17C32×C4⋊Dic3C2211yescharacteristic, complemented
18C32×Dic3⋊C4—21no—
19C3×#3715—24no—
20C22×C32×D4—31no—
21C22×C3×D12—21no—
22C22×C3×D12D311yescharacteristic
23C22×C3×D12C2×C311yescharacteristic
24C32×C22⋊Q8—31no—
25C2×C3×#215—32no—
26C2×C3×#214—32no—
27C3×#924—21no—
28C3×#924D311yescharacteristic, complemented
29C3×#924C2×C311yescharacteristic, complemented
30C2×#3715C2×C312yescomplemented
31#63657C2×C311yescharacteristic, complemented
32C3×#832—32no—
33C3×#830—31no—
34C3×#830—31no—
35C3×#830—61no—
36C3×#809—32no—
37#63562C2×C312yescomplemented
38C23×C33D411yescharacteristic
39C22×C32×D3—24no—
40C22×C32×D3—22no—
41C22×C32×D3—42no—
42C22×C32×D3—42no—
43C2×C33×C4—22no—
44C2×C33×C4D411yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C2×C32×Dic3—21no—
47C2×C32×Dic3—21no—
48C32×D12—24no—
49C32×D12—22no—
50C32×D12D414yes—
51C32×Dic6—22no—
52C3×C3⋊(C3⋊C8)—22no—
53C3×#450—91no—
54#5218—31no—
55#5046—31no—
56C24×C32—61no—
57C23×C3×D3—22no—
58C23×C3×D3—21no—
59C23×C3×D3—21no—
60C2×C32×Q8—31no—
61C2×C32×D4—32no—
62C2×C32×D4—64no—
63C2×C32×D4—61no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—21no—
67C22×C32×C4—41no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C2×C3×D12—22no—
72C2×C3×D12—28no—
73C2×C3×D12—22no—
74C2×C3×D12—21no—
75C2×C3×D12—21no—
76C2×C3×D12—24no—
77C2×C3×D12—24no—
78C2×C3×D12C2×D312yes—
79C2×C3×D12C22×C312yes—
80C2×C3×Dic6—21no—
81C2×C3×Dic6C2×D311yescharacteristic, complemented
82C2×C3×Dic6C22×C311yescharacteristic, complemented
83C32×C4⋊C4—31no—
84C32×C4⋊C4—61no—
85C32×C22⋊C4—61no—
86C2×C3⋊(C3⋊C8)C22×C312yescomplemented
87C3×#217—22no—
88C3×#217—21no—
89C3×#217—21no—
90C3×#215—64no—
91C3×#214—64no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3C2×D311yescharacteristic, complemented
94C3×C4⋊Dic3C22×C311yescharacteristic, complemented
95C3×Dic3⋊C4—22no—
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—21no—
98C2×C3×C3⋊C8—32no—
99C2×C3×C3⋊C8—32no—
100C2×C3×C3⋊C8—62no—
101#3715—24no—
102C22×C33—21no—
103C22×C33—21no—
104C22×C33SD1612yescomplemented
105C22×C33C2×D411yescharacteristic
106C2×C32×D3—24no—
107C2×C32×D3—48no—
108C2×C32×D3—42no—
109C2×C32×D3—42no—
110C33×C4—21no—
111C33×C4C2×D412yes—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C22×C3×D4—31no—
115C22×C3×D4—31no—
116C22×C3×D4—61no—
117C22×D12C3×D311yescharacteristic
118C2×C3×SD16—92no—
119C3×C22⋊Q8—31no—
120C3×C22⋊Q8—31no—
121C3×C22⋊Q8—61no—
122C2×#215—32no—
123C2×#214—32no—
124#924C3×D311yescharacteristic, complemented
125C3×#102—92no—
126C3×C22⋊C8—91no—
127#832—32no—
128#830—31no—
129#830—31no—
130#830—61no—
131#809—32no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—41no—
136C23×C32—64no—
137C23×C32—62no—
138C23×C32—122no—
139C23×C32—122no—
140C23×C32#6711yescharacteristic
141C23×C32#6711yescharacteristic
142C23×C32C3×D411yescharacteristic
143C22×C3×D3—28no—
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—44no—
150C22×C3×D3—42no—
151C22×C3×D3—42no—
152C22×C3×D3—44no—
153C22×C3×D3—42no—
154C22×C3×D3—42no—
155C32×Q8—62no—
156C32×D4—34no—
157C32×D4—64no—
158C32×D4—62no—
159C2×C32×C4—22no—
160C2×C32×C4—24no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—42no—
172C2×C32×C4—42no—
173C2×C32×C4—44no—
174C2×C32×C4—61no—
175C2×C32×C4—61no—
176C2×C32×C4#6711yescharacteristic
177C2×C32×C4#6711yescharacteristic
178C2×C32×C4C3×D411yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C3×Dic3—22no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C3×D12—28no—
189C3×D12—24no—
190C3×D12—24no—
191C3×D12—24no—
192C3×D12—22no—
193C3×D12—22no—
194C3×D12—24no—
195C3×D12#6714yes—
196C3×D12C3×D414yes—
197C3×Dic6—22no—
198C3×Dic6—22no—
199C3×Dic6—42no—
200C3⋊(C3⋊C8)—22no—
201C3×C3⋊C8—62no—
202C3×C3⋊C8—62no—
203C3×C3⋊C8—122no—
204#450—91noSylow
205C2×C33—22no—
206C2×C33#12011yescharacteristic
207C2×C33C2×SD1611yescharacteristic
208C2×C33AGL(1,ℤ/8)11yescharacteristic
209C32×D3—44no—
210C24×C3—62no—
211C24×C3—61no—
212C24×C3—61no—
213C23×D3—21no—
214C2×C3×Q8—31no—
215C2×C3×Q8—31no—
216C2×C3×Q8—61no—
217C2×C3×D4—32no—
218C2×C3×D4—32no—
219C2×C3×D4—62no—
220C2×C3×D4—64no—
221C2×C3×D4—64no—
222C2×C3×D4—68no—
223C2×C3×D4—62no—
224C2×C3×D4—61no—
225C2×C3×D4—61no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—41no—
230C22×C3×C4D3211yescharacteristic
231C22×C3×C4C2×C3×D311yescharacteristic
232C22×C3×C4C2×C3×D311yescharacteristic
233C2×D12—21no—
234C2×D12—24no—
235C2×D12C2×C3×D312yes—
236C2×Dic6C2×C3×D311yescharacteristic, complemented
237C3×SD16—184no—
238C2×C3×C8—92no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—61no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—61no—
245C3×C22⋊C4—62no—
246C3×C22⋊C4—61no—
247C3×C22⋊C4—61no—
248#217—21no—
249#215—64no—
250#214—64no—
251C4⋊Dic3C2×C3×D311yescharacteristic, complemented
252Dic3⋊C4—21no—
253C2×C3⋊C8—32no—
254C2×C3⋊C8—32no—
255C2×C3⋊C8—62no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—42no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32—41no—
276C22×C32—42no—
277C22×C32—42no—
278C22×C32—64no—
279C22×C32—128no—
280C22×C32—122no—
281C22×C32—122no—
282C22×C32#21412yescomplemented
283C22×C32#21512yescomplemented
284C22×C32C3×SD1612yescomplemented
285C22×C32C2×#6711yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C22×C32C2×C3×D411yescharacteristic
288C2×C3×D3—28no—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—416no—
292C2×C3×D3—48no—
293C2×C3×D3—48no—
294C2×C3×D3—44no—
295C2×C3×D3—44no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C32×C4—21no—
301C32×C4—22no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—22no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—42no—
308C32×C4—41no—
309C32×C4—41no—
310C32×C4—42no—
311C32×C4—121no—
312C32×C4—121no—
313C32×C4D3×D412yes—
314C32×C4C2×#6712yes—
315C32×C4C2×C3×D412yes—
316C3×Dic3—42no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C22×D4—31no—
323C2×SD16—92no—
324C22⋊Q8—31no—
325#102—92no—
326C22⋊C8—91no—
327C33#45011yescharacteristic, complemented, Sylow
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—41no—
332C23×C3—68no—
333C23×C3—64no—
334C23×C3—64no—
335C23×C3—64no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—124no—
339C23×C3—122no—
340C23×C3—122no—
341C23×C3—124no—
342C23×C3—122no—
343C23×C3—122no—
344C23×C3#62511yescharacteristic
345C23×C3C3×#6711yescharacteristic
346C23×C3C3×#6711yescharacteristic
347C22×D3—24no—
348C22×D3—22no—
349C22×D3—42no—
350C22×D3—42no—
351C3×Q8—62no—
352C3×Q8—62no—
353C3×Q8—122no—
354C3×D4—34no—
355C3×D4—34no—
356C3×D4—64no—
357C3×D4—68no—
358C3×D4—64no—
359C3×D4—64no—
360C3×D4—64no—
361C3×D4—62no—
362C3×D4—62no—
363C2×C3×C4—24no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—44no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4#62511yescharacteristic
385C2×C3×C4C3×#6711yescharacteristic
386C2×C3×C4C3×#6711yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C22×C3×D311yescharacteristic
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392D12—24no—
393D12—22no—
394D12C3×#6714yes—
395Dic6—22no—
396C3×C8—182no—
397C3⋊C8—62no—
398C3⋊C8—62no—
399C3⋊C8—122no—
400C2×C32—22no—
401C2×C32—24no—
402C2×C32—22no—
403C2×C32—22no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—41no—
416C2×C32—42no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32—124no—
420C2×C32#93211yescharacteristic
421C2×C32C2×#21411yescharacteristic
422C2×C32#93711yescharacteristic
423C2×C32C2×#21511yescharacteristic
424C2×C32#94911yescharacteristic
425C2×C32#95311yescharacteristic
426C2×C32C3×#12011yescharacteristic
427C2×C32C2×C3×SD1611yescharacteristic
428C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
429C3×D3—48no—
430C3×D3—44no—
431C3×D3—44no—
432C24—61no—
433C2×Q8—31no—
434C2×D4—32no—
435C2×D4—64no—
436C2×D4—61no—
437C22×C4C3×D3211yescharacteristic
438SD16—184no—
439C2×C8—92no—
440C4⋊C4—31no—
441C4⋊C4—61no—
442C22⋊C4—61no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—68no—
466C22×C3—64no—
467C22×C3—64no—
468C22×C3—1216no—
469C22×C3—128no—
470C22×C3—128no—
471C22×C3—124no—
472C22×C3—124no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—122no—
476C22×C3—122no—
477C22×C3#371512yescomplemented
478C22×C3C3×#21412yescomplemented
479C22×C3C3×#21512yescomplemented
480C22×C3C2×#62511yescharacteristic
481C22×C3C2×C3×#6711yescharacteristic
482C22×C3C2×C3×#6711yescharacteristic
483C2×D3—24no—
484C2×D3—48no—
485C2×D3—42no—
486C2×D3—42no—
487C3×C4—22no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—122no—
499C3×C4—122no—
500C3×C4—121no—
501C3×C4—121no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4D3×#6712yes—
505C3×C4C3×D3×D412yes—
506C3×C4C2×C3×#6712yes—
507Dic3—41no—
508Dic3—41no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#504611yescharacteristic, complemented
514C32#521811yescharacteristic, complemented
515C32C3×#45011yescharacteristic, complemented
516C23—64no—
517C23—62no—
518C23—122no—
519C23—122no—
520C23C3×#62511yescharacteristic
521Q8—62no—
522D4—34no—
523D4—64no—
524D4—62no—
525C2×C4—22no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4C3×#62511yescharacteristic
529C2×C4C2×C3×D3211yescharacteristic
530C8—182no—
531C2×C3—24no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—42no—
545C2×C3—42no—
546C2×C3—44no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—128no—
551C2×C3—124no—
552C2×C3—124no—
553C2×C3#6382311yescharacteristic
554C2×C3C2×#371511yescharacteristic
555C2×C3#6397611yescharacteristic
556C2×C3C3×#93211yescharacteristic
557C2×C3C2×C3×#21411yescharacteristic
558C2×C3C3×#93711yescharacteristic
559C2×C3C2×C3×#21511yescharacteristic
560C2×C3C3×#94911yescharacteristic
561C2×C3C3×#95311yescharacteristic
562D3—44no—
563C22—21no—
564C22—21no—
565C22—64no—
566C22—128no—
567C22—122no—
568C22—122no—
569C22C3×#371512yescomplemented
570C22C2×C3×#62511yescharacteristic
571C4—21no—
572C4—121no—
573C4—121no—
574C4C3×D3×#6712yes—
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059116911yescharacteristic, direct product factor
580C3C3×#504611yescharacteristic, complemented
581C3C3×#521811yescharacteristic, complemented
582C2—22no—
583C2—124no—
584C2C3×#6382311yescharacteristic
585C2C2×C3×#371511yescharacteristic
586C2C3×#6397611yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D12C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#450
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
2. C32×#924C2semidirect
3. C2×C3×#3715C2semidirect
4. C3×#63657C2semidirect
5. C3×#63562C2semidirect
8. #10591169C3direct (N × H)
14. C2×C32×Dic6C22semidirect
15. C2×C3×C3⋊(C3⋊C8)C22semidirect
17. C32×C4⋊Dic3C22semidirect
28. C3×#924D3semidirect
29. C3×#924C2×C3semidirect
30. C2×#3715C2×C3semidirect
31. #63657C2×C3semidirect
37. #63562C2×C3semidirect
81. C2×C3×Dic6C2×D3semidirect
82. C2×C3×Dic6C22×C3semidirect
86. C2×C3⋊(C3⋊C8)C22×C3semidirect
93. C3×C4⋊Dic3C2×D3semidirect
94. C3×C4⋊Dic3C22×C3semidirect
104. C22×C33SD16semidirect
124. #924C3×D3semidirect
236. C2×Dic6C2×C3×D3semidirect
251. C4⋊Dic3C2×C3×D3semidirect
282. C22×C32#214semidirect
283. C22×C32#215semidirect
284. C22×C32C3×SD16semidirect
327. C33#450semidirect
477. C22×C3#3715semidirect
478. C22×C3C3×#214semidirect
479. C22×C3C3×#215semidirect
513. C32#5046semidirect
514. C32#5218semidirect
515. C32C3×#450semidirect
569. C22C3×#3715semidirect
579. C3#10591169direct (N × H)
580. C3C3×#5046semidirect
581. C3C3×#5218semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D122
C32×#9242
C2×C3×#37152
C3×#636572
C3×#635622
C3×#52183
C3×#50463
#105911693