Finite Group Databasebeta

← C3×#10589758 #13152995

Subgroups — C3×#10589758 #13152995

Order 1728.

1068 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C22×C3×C4×D9C211yesmaximal, characteristic, complemented
3C3×C9×#123C211yesmaximal, characteristic, complemented
4C3×#63499C211yesmaximal, characteristic, complemented
5C3×#63491C211yesmaximal, characteristic, complemented
6C3×#63458C212yesmaximal, complemented
7C3×#63455C212yesmaximal, complemented
8C3×#63446C212yesmaximal, complemented
9C3×#63444C211yesmaximal, characteristic, complemented
10C3×#611⋊C4C212yesmaximal, complemented
11C3×#63439C211yesmaximal, characteristic, complemented
12C3×#5656—31nomaximal
13#10589758C311yesmaximal, characteristic, direct product factor
14C23×C3×D9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2212yes—
19C2×C3×#611C2211yescharacteristic, complemented
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic, complemented
22C3×#3699—24no—
23C2×C3×D36C2212yes—
24C2×C3×C4×D9—24no—
25C2×C3×C4×D9C2212yes—
26C2×C3×C4×D9C2212yes—
27C2×C3×C4×D9C2211yescharacteristic, complemented
28C2×C3×Dic18C2211yescharacteristic, complemented
29C3×C9×C4⋊C4C2212yescomplemented
30C3×C9×C22⋊C4C2211yescharacteristic
31C3×C9×C22⋊C4C2212yescomplemented
32C3×#3676C2211yescharacteristic
33C3×(C2×D9)⋊C4C2212yes—
34C3×(C2×D9)⋊C4C2212yes—
35C3×(C2×D9)⋊C4C2212yescomplemented
36C3×Dic9⋊C4C2212yescomplemented
37C3×Dic9⋊C4C2212yescomplemented
38C3×C4×Dic9C2212yescomplemented
39C2×C3×#237—31no—
40C22×C3×C4×D3—31no—
41C32×#123D311yescharacteristic
42C3×#937—31no—
43C3×#929—31no—
44C3×#896—32no—
45C3×#893—32no—
46C3×#884—32no—
47C3×#882—31no—
48C3×#67⋊C4—32no—
49C3×#877—31no—
50C2×#3699C2×C311yescharacteristic, complemented
51C22×C4×D9C2×C311yescharacteristic, complemented
52C9×#123—21no—
53C9×#123C2×C311yescharacteristic, complemented
54#63499C2×C311yescharacteristic, complemented
55#63491C2×C311yescharacteristic, complemented
56#63458C2×C312yescomplemented
57#63455C2×C312yescomplemented
58#63446C2×C312yescomplemented
59#63444C2×C311yescharacteristic, complemented
60#611⋊C4C2×C312yescomplemented
61#63439C2×C311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×C3×D9—22no—
65C22×C3×D9—24no—
66C22×C3×D9C2311yescharacteristic
67C22×C3×D9C2312yes—
68C3×D4×C9—22no—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2312yes—
72C2×C3×C4×C9C2312yes—
73C3×#611—22no—
74C3×#611—24no—
75C3×#611—22no—
76C2×C3×Dic9—22no—
77C2×C3×Dic9D414yescomplemented
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2311yescharacteristic
82C3×D36—24no—
83C3×C4×D9—24no—
84C3×C4×D9—24no—
85C3×C4×D9—22no—
86C3×Dic18—22no—
87C3×#525—91no—
88#5656—31no—
89C23×C3×D3—31no—
90C2×C32×D4C2×D311yescharacteristic
91C22×C32×C4C2×D311yescharacteristic
92C2×C3×#67—32no—
93C2×C3×#67—31no—
94C2×C3×#67—31no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C3×#237—64no—
98C2×C3×D12—32no—
99C2×C3×C4×D3—32no—
100C2×C3×C4×D3—32no—
101C2×C3×C4×D3—31no—
102C2×C3×C4×D3—64no—
103C2×C3×Dic6—31no—
104C23×D9C22×C311yescharacteristic
105C32×C4⋊C4C2×D312yes—
106C32×C22⋊C4C2×D311yescharacteristic
107C32×C22⋊C4C2×D312yes—
108C3×#217—31no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×C4×Dic3—32no—
115C2×D4×C9—21no—
116C2×D4×C9C22×C311yescharacteristic
117C22×C4×C9—21no—
118C22×C4×C9C22×C311yescharacteristic
119C2×#611C22×C311yescharacteristic
120C2×#611C22×C312yes—
121C2×#611C22×C311yescharacteristic, complemented
122C22×Dic9C22×C311yescharacteristic
123C22×Dic9C22×C311yescharacteristic, complemented
124#3699—24no—
125C2×D36C22×C312yes—
126C2×C4×D9—24no—
127C2×C4×D9C22×C312yes—
128C2×C4×D9C22×C312yes—
129C2×C4×D9C22×C311yescharacteristic, complemented
130C2×Dic18C22×C311yescharacteristic, complemented
131C9×C4⋊C4—22no—
132C9×C4⋊C4C22×C312yescomplemented
133C9×C22⋊C4—21no—
134C9×C22⋊C4—22no—
135C9×C22⋊C4C22×C311yescharacteristic
136C9×C22⋊C4C22×C312yescomplemented
137#3676C22×C311yescharacteristic
138(C2×D9)⋊C4C22×C312yes—
139(C2×D9)⋊C4C22×C312yes—
140(C2×D9)⋊C4C22×C312yescomplemented
141Dic9⋊C4C22×C312yescomplemented
142Dic9⋊C4C22×C312yescomplemented
143C4×Dic9C22×C312yescomplemented
144C22×C3×C9—22no—
145C22×C3×C9—21no—
146C22×C3×C9—22no—
147C22×C3×C9C2×D412yes—
148C22×C3×C9C2411yescharacteristic
149C2×C3×D9—28no—
150C2×C3×D9—22no—
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9#4114yescomplemented
155C3×C4×C9—21no—
156C3×C4×C9—22no—
157C3×C4×C9—22no—
158C3×Dic9—21no—
159C3×Dic9—22no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C2×C3×#41—91no—
163C23×C3×C4—91no—
164C2×#237—31no—
165C22×C4×D3—31no—
166C3×#123—21no—
167C3×#123—91no—
168C3×#123D911yescharacteristic, complemented
169C3×#123C3×D311yescharacteristic
170C3×C22⋊Q8—92no—
171C3×#121—92no—
172C3×#120—91no—
173C3×#120—91no—
174C3×C4×D4—92no—
175C3×C4×D4—92no—
176C3×#117—91no—
177#937—31no—
178#929—31no—
179#896—32no—
180#893—32no—
181#884—32no—
182#882—31no—
183#67⋊C4—32no—
184#877—31no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D311yescharacteristic
187C22×C3×D3—32no—
188C22×C3×D3—31no—
189C22×C3×D3—64no—
190C22×C3×D3—62no—
191C32×D4—22no—
192C2×C32×C4—22no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C3×#67—62no—
197C3×#67—64no—
198C3×#67—62no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—62no—
205C3×D12—64no—
206C3×C4×D3—64no—
207C3×C4×D3—64no—
208C3×C4×D3—62no—
209C3×Dic6—62no—
210C23×C9—21no—
211C23×C9—21no—
212C23×C9C23×C311yescharacteristic
213C23×C9C23×C311yescharacteristic
214C22×D9—22no—
215C22×D9—24no—
216C22×D9C23×C311yescharacteristic
217C22×D9C23×C312yes—
218D4×C9—22no—
219D4×C9—42no—
220C2×C4×C9—22no—
221C2×C4×C9—22no—
222C2×C4×C9—22no—
223C2×C4×C9—21no—
224C2×C4×C9—42no—
225C2×C4×C9C23×C312yes—
226C2×C4×C9C23×C312yes—
227C2×C4×C9C23×C311yescharacteristic
228#611—22no—
229#611—24no—
230#611—22no—
231C2×Dic9—22no—
232C2×Dic9C3×D414yescomplemented
233C2×Dic9C23×C312yes—
234C2×Dic9C23×C311yescharacteristic
235C2×Dic9C23×C311yescharacteristic
236C2×Dic9C23×C311yescharacteristic
237D36—24no—
238C4×D9—24no—
239C4×D9—24no—
240C4×D9—22no—
241Dic18—22no—
242#525—91noSylow
243C2×C3×C9—22no—
244C2×C3×C9—41no—
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9C2×#4112yes—
247C3×D9—24no—
248C3×D9—41no—
249C24×C3—91no—
250C23×D3—31no—
251C3×#41—184no—
252C2×C3×Q8—91no—
253C2×C3×D4—21no—
254C2×C3×D4—92no—
255C2×C3×D4—91no—
256C2×C3×D4—91no—
257C2×C3×D4—92no—
258C2×C3×D4C2×D911yescharacteristic
259C2×C3×D4C2×C3×D311yescharacteristic
260C22×C3×C4—21no—
261C22×C3×C4—91no—
262C22×C3×C4—91no—
263C22×C3×C4—92no—
264C22×C3×C4—91no—
265C22×C3×C4—92no—
266C22×C3×C4—184no—
267C22×C3×C4C2×D911yescharacteristic
268C22×C3×C4C2×C3×D311yescharacteristic
269C2×#67—32no—
270C2×#67—31no—
271C2×#67—31no—
272C22×Dic3—31no—
273C22×Dic3—31no—
274#237—64no—
275C2×D12—32no—
276C2×C4×D3—32no—
277C2×C4×D3—32no—
278C2×C4×D3—31no—
279C2×C4×D3—64no—
280C2×Dic6—31no—
281C3×C4⋊C4—22no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4C2×D912yescomplemented
285C3×C4⋊C4C2×C3×D312yes—
286C3×C22⋊C4—21no—
287C3×C22⋊C4—22no—
288C3×C22⋊C4—91no—
289C3×C22⋊C4—92no—
290C3×C22⋊C4—92no—
291C3×C22⋊C4—92no—
292C3×C22⋊C4C2×D911yescharacteristic
293C3×C22⋊C4C2×D912yescomplemented
294C3×C22⋊C4C2×C3×D311yescharacteristic
295C3×C22⋊C4C2×C3×D312yes—
296C3×C42—92no—
297#217—31no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—32no—
300(C2×D3)⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303C4×Dic3—32no—
304C22×C32—22no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32D3×D412yes—
308C22×C32C23×D311yescharacteristic
309C2×C3×D3—34no—
310C2×C3×D3—68no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—61no—
314C2×C3×D3—62no—
315C32×C4—21no—
316C32×C4—22no—
317C32×C4—22no—
318C3×Dic3—32no—
319C3×Dic3—32no—
320C3×Dic3—61no—
321C3×Dic3—62no—
322C22×C9—22no—
323C22×C9—21no—
324C22×C9—22no—
325C22×C9—21no—
326C22×C9—22no—
327C22×C9—42no—
328C22×C9—41no—
329C22×C9—42no—
330C22×C9C2×C3×D412yes—
331C22×C9C24×C311yescharacteristic
332C2×D9—28no—
333C2×D9—22no—
334C2×D9—22no—
335C2×D9—21no—
336C2×D9—22no—
337C2×D9C3×#4114yescomplemented
338C4×C9—21no—
339C4×C9—22no—
340C4×C9—22no—
341C4×C9—42no—
342C4×C9—42no—
343C4×C9—41no—
344Dic9—21no—
345Dic9—22no—
346Dic9C2×C3×D412yes—
347Dic9C2×C3×D412yes—
348C2×#41—91no—
349C23×C4—91no—
350#123—91no—
351#123C3×D911yescharacteristic, complemented
352C22⋊Q8—92no—
353#121—92no—
354#120—91no—
355#120—91no—
356C4×D4—92no—
357C4×D4—92no—
358#117—91no—
359C3×C9#52511yescharacteristic, complemented, Sylow
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—92no—
363C23×C3—91no—
364C23×C3—184no—
365C23×C3—182no—
366C23×C3C22×D911yescharacteristic
367C23×C3C22×D911yescharacteristic
368C23×C3C22×C3×D311yescharacteristic
369C23×C3C22×C3×D311yescharacteristic
370C22×D3—32no—
371C22×D3—31no—
372C22×D3—64no—
373C22×D3—62no—
374C3×Q8—182no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—42no—
378C3×D4—184no—
379C3×D4—182no—
380C3×D4—184no—
381C3×D4—182no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—42no—
388C2×C3×C4—94no—
389C2×C3×C4—91no—
390C2×C3×C4—91no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—184no—
394C2×C3×C4—184no—
395C2×C3×C4—182no—
396C2×C3×C4—182no—
397C2×C3×C4C22×D912yes—
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C22×D912yes—
400C2×C3×C4C22×C3×D312yes—
401C2×C3×C4C22×C3×D312yes—
402C2×C3×C4C22×C3×D311yescharacteristic
403#67—62no—
404#67—64no—
405#67—62no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—34no—
409C2×Dic3—32no—
410C2×Dic3—31no—
411C2×Dic3—62no—
412D12—64no—
413C4×D3—64no—
414C4×D3—64no—
415C4×D3—62no—
416Dic6—62no—
417C2×C32—22no—
418C2×C32—41no—
419C2×C32C2×D3×D411yescharacteristic
420C2×C32D3×#4112yes—
421C3×D3—64no—
422C3×D3—121no—
423C2×C9—22no—
424C2×C9—21no—
425C2×C9—22no—
426C2×C9—42no—
427C2×C9—41no—
428C2×C9—81no—
429C2×C9C22×C3×D411yescharacteristic
430C2×C9C2×C3×#4112yes—
431D9—24no—
432D9—41no—
433C24—91no—
434#41—184no—
435C2×Q8—91no—
436C2×D4—92no—
437C2×D4—91no—
438C2×D4—91no—
439C2×D4—92no—
440C2×D4C2×C3×D911yescharacteristic
441C22×C4—91no—
442C22×C4—91no—
443C22×C4—92no—
444C22×C4—91no—
445C22×C4—92no—
446C22×C4—184no—
447C22×C4C2×C3×D911yescharacteristic
448C4⋊C4—92no—
449C4⋊C4—92no—
450C4⋊C4C2×C3×D912yescomplemented
451C22⋊C4—91no—
452C22⋊C4—92no—
453C22⋊C4—92no—
454C22⋊C4—92no—
455C22⋊C4C2×C3×D911yescharacteristic
456C22⋊C4C2×C3×D912yescomplemented
457C42—92no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—22no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—42no—
467C22×C3—41no—
468C22×C3—42no—
469C22×C3—94no—
470C22×C3—188no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3—181no—
474C22×C3—182no—
475C22×C3D4×D912yes—
476C22×C3C23×D911yescharacteristic
477C22×C3C3×D3×D412yes—
478C22×C3C23×C3×D311yescharacteristic
479C2×D3—34no—
480C2×D3—68no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—61no—
484C2×D3—62no—
485C3×C4—21no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—21no—
490C3×C4—22no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—42no—
494C3×C4—92no—
495C3×C4—92no—
496C3×C4—181no—
497C3×C4—182no—
498Dic3—32no—
499Dic3—32no—
500Dic3—61no—
501Dic3—62no—
502C32#565611yescharacteristic
503C9—21no—
504C9C3×#52511yescharacteristic, complemented
505C23—92no—
506C23—91no—
507C23—184no—
508C23—182no—
509C23C22×C3×D911yescharacteristic
510C23C22×C3×D911yescharacteristic
511Q8—182no—
512D4—22no—
513D4—184no—
514D4—182no—
515D4—184no—
516D4—182no—
517C2×C4—22no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—92no—
522C2×C4—91no—
523C2×C4—184no—
524C2×C4—184no—
525C2×C4—182no—
526C2×C4—182no—
527C2×C4C22×C3×D912yes—
528C2×C4C22×C3×D911yescharacteristic
529C2×C4C22×C3×D912yes—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—42no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—81no—
538C2×C3—184no—
539C2×C3—361no—
540C2×C3C2×D4×D911yescharacteristic
541C2×C3#41×D912yes—
542C2×C3C2×C3×D3×D411yescharacteristic
543C2×C3C3×D3×#4112yes—
544D3—64no—
545D3—121no—
546C22—22no—
547C22—21no—
548C22—22no—
549C22—94no—
550C22—188no—
551C22—182no—
552C22—182no—
553C22—181no—
554C22—182no—
555C22C3×D4×D912yes—
556C22C23×C3×D911yescharacteristic
557C4—22no—
558C4—21no—
559C4—22no—
560C4—92no—
561C4—92no—
562C4—181no—
563C4—182no—
564C3—21no—
565C3#1058975811yescharacteristic, direct product factor
566C3C3×#565611yescharacteristic
567C2—22no—
568C2—41no—
569C2—184no—
570C2—361no—
571C2C2×C3×D4×D911yescharacteristic
572C2C3×#41×D912yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#123C2
socleC22×C32C23×D3
hypercenterC3×#123D9
hypocenterC9C3×#525
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C22×C3×C4×D9C2semidirect
3. C3×C9×#123C2semidirect
4. C3×#63499C2semidirect
5. C3×#63491C2semidirect
6. C3×#63458C2semidirect
7. C3×#63455C2semidirect
8. C3×#63446C2semidirect
9. C3×#63444C2semidirect
10. C3×#611⋊C4C2semidirect
11. C3×#63439C2semidirect
13. #10589758C3direct (N × H)
19. C2×C3×#611C22semidirect
21. C22×C3×Dic9C22semidirect
27. C2×C3×C4×D9C22semidirect
28. C2×C3×Dic18C22semidirect
29. C3×C9×C4⋊C4C22semidirect
31. C3×C9×C22⋊C4C22semidirect
35. C3×(C2×D9)⋊C4C22semidirect
36. C3×Dic9⋊C4C22semidirect
37. C3×Dic9⋊C4C22semidirect
38. C3×C4×Dic9C22semidirect
50. C2×#3699C2×C3semidirect
51. C22×C4×D9C2×C3semidirect
53. C9×#123C2×C3semidirect
54. #63499C2×C3semidirect
55. #63491C2×C3semidirect
56. #63458C2×C3semidirect
57. #63455C2×C3semidirect
58. #63446C2×C3semidirect
59. #63444C2×C3semidirect
60. #611⋊C4C2×C3semidirect
61. #63439C2×C3semidirect
77. C2×C3×Dic9D4semidirect
121. C2×#611C22×C3semidirect
123. C22×Dic9C22×C3semidirect
129. C2×C4×D9C22×C3semidirect
130. C2×Dic18C22×C3semidirect
132. C9×C4⋊C4C22×C3semidirect
136. C9×C22⋊C4C22×C3semidirect
140. (C2×D9)⋊C4C22×C3semidirect
141. Dic9⋊C4C22×C3semidirect
142. Dic9⋊C4C22×C3semidirect
143. C4×Dic9C22×C3semidirect
154. C2×C3×D9#41semidirect
168. C3×#123D9semidirect
232. C2×Dic9C3×D4semidirect
284. C3×C4⋊C4C2×D9semidirect
293. C3×C22⋊C4C2×D9semidirect
337. C2×D9C3×#41semidirect
351. #123C3×D9semidirect
359. C3×C9#525semidirect
450. C4⋊C4C2×C3×D9semidirect
456. C22⋊C4C2×C3×D9semidirect
504. C9C3×#525semidirect
565. C3#10589758direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C22×C3×C4×D92
C3×C9×#1232
C3×#634992
C3×#634912
C3×#634582
C3×#634552
C3×#634462
C3×#634442
C3×#611⋊C42
C3×#634392
C3×#56563
#105897583