Finite Group Databasebeta

← C3×#10591171 #13158350

Subgroups — C3×#10591171 #13158350

Order 1728.

986 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic, complemented
2C32×#930C211yesmaximal, characteristic
3C2×C3×#3715C212yesmaximal, complemented
4C3×#63657C211yesmaximal, characteristic, complemented
5C3×#63565C212yesmaximal, complemented
6C3×#5227—31nomaximal
7C3×#5040—31nomaximal
8#10591171C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C32×#67—21no—
11C22×C32×Dic3—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6—21no—
14C2×C32×Dic6—24no—
15C2×C32×Dic6C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C3×#3715—24no—
20C22×C32×Q8—31no—
21C22×C3×Dic6—21no—
22C22×C3×Dic6D311yescharacteristic, complemented
23C22×C3×Dic6C2×C311yescharacteristic, complemented
24C32×#121—31no—
25C2×C3×#215—32no—
26C2×C3×#214—32no—
27C3×#930—21no—
28C3×#930D311yescharacteristic
29C3×#930C2×C311yescharacteristic
30C2×#3715C2×C312yescomplemented
31#63657C2×C311yescharacteristic, complemented
32C3×#835—32no—
33C3×#830—31no—
34C3×#830—31no—
35C3×#830—61no—
36C3×#810—32no—
37#63565C2×C312yescomplemented
38C23×C33D411yescharacteristic
39C22×C32×D3—21no—
40C2×C33×C4—22no—
41C2×C33×C4D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C32×#67—42no—
44C2×C32×Dic3—24no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—21no—
47C32×D12—22no—
48C32×Dic6—24no—
49C32×Dic6—22no—
50C32×Dic6D414yescomplemented
51C3×C3⋊(C3⋊C8)—22no—
52C3×#448—91no—
53#5227—31no—
54#5040—31no—
55C2×C32×Q8—32no—
56C2×C32×Q8—64no—
57C2×C32×Q8—61no—
58C2×C32×D4—31no—
59C2×C32×D4—61no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4—41no—
64C22×C32×C4—61no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C2×C3×#67—22no—
69C2×C3×#67—21no—
70C2×C3×#67—21no—
71C22×C3×Dic3—22no—
72C22×C3×Dic3—21no—
73C22×C3×Dic3—21no—
74C2×C3×D12—21no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C22×C311yescharacteristic
77C2×C3×Dic6—22no—
78C2×C3×Dic6—28no—
79C2×C3×Dic6—22no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6—24no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6C2×D312yes—
85C2×C3×Dic6C22×C312yes—
86C32×C4⋊C4—31no—
87C32×C22⋊C4—61no—
88C2×C3⋊(C3⋊C8)C22×C312yes—
89C3×#215—64no—
90C3×#214—64no—
91C3×(C2×D3)⋊C4—22no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C2×C3×C3⋊C8—32no—
98C2×C3×C3⋊C8—32no—
99C2×C3×C3⋊C8—62no—
100#3715—24no—
101C22×C33—21no—
102C22×C33—21no—
103C22×C33SD1612yescomplemented
104C22×C33C2×D411yescharacteristic
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C33×C4—21no—
109C33×C4C2×D412yes—
110C32×Dic3—24no—
111C32×Dic3—41no—
112C22×C3×Q8—31no—
113C22×C3×Q8—31no—
114C22×C3×Q8—61no—
115C22×Dic6C3×D311yescharacteristic, complemented
116C2×C3×SD16—92no—
117C3×#121—31no—
118C3×#121—31no—
119C3×#121—61no—
120C2×#215—32no—
121C2×#214—32no—
122#930C3×D311yescharacteristic
123C3×#103—92no—
124C3×C22⋊C8—91no—
125#835—32no—
126#830—31no—
127#830—31no—
128#830—61no—
129#810—32no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—41no—
134C23×C32—61no—
135C23×C32#6711yescharacteristic
136C23×C32#6711yescharacteristic
137C23×C32C3×D411yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3—21no—
140C22×C3×D3—21no—
141C32×Q8—34no—
142C32×Q8—64no—
143C32×Q8—62no—
144C32×D4—62no—
145C32×D4—122no—
146C2×C32×C4—22no—
147C2×C32×C4—24no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C32×C4—42no—
159C2×C32×C4—42no—
160C2×C32×C4—44no—
161C2×C32×C4—64no—
162C2×C32×C4—62no—
163C2×C32×C4—61no—
164C2×C32×C4#6711yescharacteristic
165C2×C32×C4#6711yescharacteristic
166C2×C32×C4C3×D411yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C23×C311yescharacteristic
170C3×#67—44no—
171C3×#67—42no—
172C3×#67—42no—
173C2×C3×Dic3—28no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—24no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C3×D12—22no—
183C3×D12—22no—
184C3×D12—42no—
185C3×Dic6—28no—
186C3×Dic6—24no—
187C3×Dic6—24no—
188C3×Dic6—24no—
189C3×Dic6—22no—
190C3×Dic6—22no—
191C3×Dic6—24no—
192C3×Dic6#6714yescomplemented
193C3×Dic6C3×D414yescomplemented
194C3⋊(C3⋊C8)—22no—
195C3×C3⋊C8—62no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—122no—
198#448—91noSylow
199C2×C33—22no—
200C2×C33#12011yescharacteristic
201C2×C33C2×SD1611yescharacteristic
202C2×C33#13711yescharacteristic
203C32×D3—81no—
204C2×C3×Q8—32no—
205C2×C3×Q8—32no—
206C2×C3×Q8—62no—
207C2×C3×Q8—64no—
208C2×C3×Q8—64no—
209C2×C3×Q8—68no—
210C2×C3×Q8—62no—
211C2×C3×Q8—61no—
212C2×C3×Q8—61no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C2×C3×D4—62no—
217C2×C3×D4—61no—
218C2×C3×D4—61no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4—62no—
224C22×C3×C4—61no—
225C22×C3×C4—61no—
226C22×C3×C4D3211yescharacteristic
227C22×C3×C4C2×C3×D311yescharacteristic
228C22×C3×C4C2×C3×D311yescharacteristic
229C2×#67—21no—
230C22×Dic3—21no—
231C2×D12C2×C3×D311yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—24no—
234C2×Dic6C2×C3×D312yes—
235C3×SD16—184no—
236C2×C3×C8—92no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—61no—
240C3×C22⋊C4—62no—
241C3×C22⋊C4—61no—
242C3×C22⋊C4—61no—
243#215—64no—
244#214—64no—
245(C2×D3)⋊C4—21no—
246C4⋊Dic3C2×C3×D311yescharacteristic
247C2×C3⋊C8—32no—
248C2×C3⋊C8—32no—
249C2×C3⋊C8—62no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—42no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—42no—
271C22×C32—42no—
272C22×C32—121no—
273C22×C32—121no—
274C22×C32—121no—
275C22×C32#21412yescomplemented
276C22×C32#21512yescomplemented
277C22×C32C3×SD1612yescomplemented
278C22×C32C2×#6711yescharacteristic
279C22×C32C2×#6711yescharacteristic
280C22×C32C2×C3×D411yescharacteristic
281C2×C3×D3—42no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C32×C4—21no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—22no—
295C32×C4—22no—
296C32×C4—22no—
297C32×C4—42no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—42no—
301C32×C4—64no—
302C32×C4—121no—
303C32×C4D3×D412yes—
304C32×C4C2×#6712yes—
305C32×C4C2×C3×D412yes—
306C3×Dic3—28no—
307C3×Dic3—24no—
308C3×Dic3—24no—
309C3×Dic3—42no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C22×Q8—31no—
313C2×SD16—92no—
314#121—31no—
315#103—92no—
316C22⋊C8—91no—
317C33#44811yescharacteristic, complemented, Sylow
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—41no—
322C23×C3—62no—
323C23×C3—61no—
324C23×C3—61no—
325C23×C3#62511yescharacteristic
326C23×C3C3×#6711yescharacteristic
327C23×C3C3×#6711yescharacteristic
328C22×D3—21no—
329C3×Q8—34no—
330C3×Q8—34no—
331C3×Q8—64no—
332C3×Q8—68no—
333C3×Q8—64no—
334C3×Q8—64no—
335C3×Q8—64no—
336C3×Q8—62no—
337C3×Q8—62no—
338C3×D4—62no—
339C3×D4—62no—
340C3×D4—122no—
341C3×D4—124no—
342C3×D4—122no—
343C3×D4—122no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—68no—
360C2×C3×C4—64no—
361C2×C3×C4—64no—
362C2×C3×C4—64no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4#62511yescharacteristic
369C2×C3×C4C3×#6711yescharacteristic
370C2×C3×C4C3×#6711yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×C3×C4C22×C3×D311yescharacteristic
374#67—42no—
375C2×Dic3—24no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378D12—22no—
379Dic6—24no—
380Dic6—22no—
381Dic6C3×#6714yescomplemented
382C3×C8—182no—
383C3⋊C8—62no—
384C3⋊C8—62no—
385C3⋊C8—122no—
386C2×C32—22no—
387C2×C32—24no—
388C2×C32—22no—
389C2×C32—22no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—44no—
405C2×C32—241no—
406C2×C32C2×#21411yescharacteristic
407C2×C32#93711yescharacteristic
408C2×C32C2×#21511yescharacteristic
409C2×C32#94211yescharacteristic
410C2×C32#95111yescharacteristic
411C2×C32#95311yescharacteristic
412C2×C32C3×#12011yescharacteristic
413C2×C32C2×C3×SD1611yescharacteristic
414C2×C32C3×#13711yescharacteristic
415C3×D3—82no—
416C3×D3—81no—
417C3×D3—81no—
418C2×Q8—32no—
419C2×Q8—64no—
420C2×Q8—61no—
421C2×D4—31no—
422C2×D4—61no—
423C22×C4—61no—
424C22×C4C3×D3211yescharacteristic
425SD16—184no—
426C2×C8—92no—
427C4⋊C4—31no—
428C22⋊C4—61no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—41no—
450C22×C3—42no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—122no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—121no—
457C22×C3—121no—
458C22×C3—121no—
459C22×C3—121no—
460C22×C3#371512yescomplemented
461C22×C3C3×#21412yescomplemented
462C22×C3C3×#21512yescomplemented
463C22×C3C2×#62511yescharacteristic
464C22×C3C2×C3×#6711yescharacteristic
465C22×C3C2×C3×#6711yescharacteristic
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—22no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—68no—
481C3×C4—64no—
482C3×C4—64no—
483C3×C4—122no—
484C3×C4—121no—
485C3×C4—121no—
486C3×C4D3×#6712yes—
487C3×C4C3×D3×D412yes—
488C3×C4C2×C3×#6712yes—
489Dic3—24no—
490Dic3—41no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#504011yescharacteristic, complemented
496C32#522711yescharacteristic, complemented
497C32C3×#44811yescharacteristic, complemented
498C23—61no—
499C23C3×#62511yescharacteristic
500Q8—34no—
501Q8—64no—
502Q8—62no—
503D4—62no—
504D4—122no—
505C2×C4—22no—
506C2×C4—64no—
507C2×C4—62no—
508C2×C4—61no—
509C2×C4C3×#62511yescharacteristic
510C2×C4C2×C3×D3211yescharacteristic
511C8—182no—
512C2×C3—24no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—22no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—42no—
526C2×C3—42no—
527C2×C3—44no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—242no—
532C2×C3—241no—
533C2×C3—241no—
534C2×C3C2×#371511yescharacteristic
535C2×C3#6382711yescharacteristic
536C2×C3#6397611yescharacteristic
537C2×C3C2×C3×#21411yescharacteristic
538C2×C3C3×#93711yescharacteristic
539C2×C3C2×C3×#21511yescharacteristic
540C2×C3C3×#94211yescharacteristic
541C2×C3C3×#95111yescharacteristic
542C2×C3C3×#95311yescharacteristic
543D3—81no—
544C22—21no—
545C22—21no—
546C22—121no—
547C22—121no—
548C22—121no—
549C22C3×#371512yescomplemented
550C22C2×C3×#62511yescharacteristic
551C4—21no—
552C4—64no—
553C4—121no—
554C4C3×D3×#6712yes—
555C3—21no—
556C3—21no—
557C3—21no—
558C3—41no—
559C3#1059117111yescharacteristic, direct product factor
560C3C3×#504011yescharacteristic, complemented
561C3C3×#522711yescharacteristic, complemented
562C2—22no—
563C2—241no—
564C2C2×C3×#371511yescharacteristic
565C2C3×#6382711yescharacteristic
566C2C3×#6397611yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC32×#930C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#448
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
1. C22×C32×Dic6C2semidirect
3. C2×C3×#3715C2semidirect
4. C3×#63657C2semidirect
5. C3×#63565C2semidirect
8. #10591171C3direct (N × H)
22. C22×C3×Dic6D3semidirect
23. C22×C3×Dic6C2×C3semidirect
30. C2×#3715C2×C3semidirect
31. #63657C2×C3semidirect
37. #63565C2×C3semidirect
50. C32×Dic6D4semidirect
103. C22×C33SD16semidirect
115. C22×Dic6C3×D3semidirect
192. C3×Dic6#67semidirect
193. C3×Dic6C3×D4semidirect
275. C22×C32#214semidirect
276. C22×C32#215semidirect
277. C22×C32C3×SD16semidirect
317. C33#448semidirect
381. Dic6C3×#67semidirect
460. C22×C3#3715semidirect
461. C22×C3C3×#214semidirect
462. C22×C3C3×#215semidirect
495. C32#5040semidirect
496. C32#5227semidirect
497. C32C3×#448semidirect
549. C22C3×#3715semidirect
559. C3#10591171direct (N × H)
560. C3C3×#5040semidirect
561. C3C3×#5227semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C32×#9302
C2×C3×#37152
C3×#636572
C3×#635652
C3×#52273
C3×#50403
#105911713