Finite Group Databasebeta

← C3×#10589779 #13153075

Subgroups — C3×#10589779 #13153075

Order 1728.

946 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×D9C211yesmaximal, characteristic, complemented
2C2×C3×#3699C211yesmaximal, characteristic, complemented
3C3×C9×#124C211yesmaximal, characteristic, complemented
4C3×#63508C211yesmaximal, characteristic, complemented
5C3×#63500C211yesmaximal, characteristic, complemented
6C3×#63447C212yesmaximal, complemented
7C3×#63446C212yesmaximal, complemented
8C3×#611⋊C4C212yesmaximal, complemented
9C3×#63440C212yesmaximal, complemented
10C3×C4×D36C211yesmaximal, characteristic, complemented
11C3×C4×Dic18C211yesmaximal, characteristic, complemented
12C3×#5677—31nomaximal
13#10589779C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×#611C2212yes—
18C22×C3×Dic9C2212yes—
19C3×Q8×D9—24no—
20C3×#3699—24no—
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C3×C4×D9C2212yes—
25C2×C3×Dic18C2212yes—
26C2×C3×Dic18C2211yescharacteristic
27C3×C9×C22⋊C4C2212yes—
28C3×C9×C22⋊C4C2212yes—
29C3×C42×C9C2211yescharacteristic
30C3×#3676C2212yes—
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C3×(C2×D9)⋊C4C2212yes—
34C3×C4⋊Dic9C2211yescharacteristic
35C3×C4⋊Dic9C2211yescharacteristic
36C3×Dic9⋊C4C2212yes—
37C3×Dic9⋊C4C2212yes—
38C3×C4×Dic9C2212yes—
39C2×C3×D3×Q8—31no—
40C2×C3×#237—31no—
41C32×#124D311yescharacteristic
42C3×#946—31no—
43C3×#938—31no—
44C3×#885—32no—
45C3×#884—32no—
46C3×#67⋊C4—32no—
47C3×#878—32no—
48C3×C4×D12—31no—
49C3×C4×Dic6—31no—
50C2×Q8×D9C2×C311yescharacteristic, complemented
51C2×#3699C2×C311yescharacteristic, complemented
52C9×#124—21no—
53C9×#124C2×C311yescharacteristic, complemented
54#63508C2×C311yescharacteristic, complemented
55#63500C2×C311yescharacteristic, complemented
56#63447C2×C312yescomplemented
57#63446C2×C312yescomplemented
58#611⋊C4C2×C312yescomplemented
59#63440C2×C312yescomplemented
60C4×D36C2×C311yescharacteristic, complemented
61C4×Dic18C2×C311yescharacteristic, complemented
62C23×C3×C9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C3×Q8×C9—22no—
66C3×D4×C9—22no—
67C2×C3×C4×C9C2312yes—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2311yescharacteristic
71C3×#611—24no—
72C3×#611—24no—
73C2×C3×Dic9—24no—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2312yes—
78C3×D36—22no—
79C3×C4×D9—22no—
80C3×C4×D9—22no—
81C3×C4×D9—24no—
82C3×Dic18—24no—
83C3×Dic18D414yescomplemented
84C3×#548—91no—
85#5677—31no—
86C2×C32×Q8C2×D311yescharacteristic
87C2×C32×D4C2×D311yescharacteristic
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C22×C3×Dic3—32no—
91C3×D3×Q8—64no—
92C3×#237—64no—
93C2×C3×D12—31no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—31no—
96C2×C3×C4×D3—31no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6—32no—
99C32×C22⋊C4C2×D312yes—
100C32×C22⋊C4C2×D312yes—
101C32×C42C2×D311yescharacteristic
102C3×#217—32no—
103C3×(C2×D3)⋊C4—31no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—32no—
111C2×Q8×C9—21no—
112C2×Q8×C9C22×C311yescharacteristic
113C2×D4×C9—21no—
114C2×D4×C9C22×C311yescharacteristic
115C2×#611C22×C312yes—
116C2×#611C22×C312yes—
117C22×Dic9C22×C312yes—
118Q8×D9—24no—
119#3699—24no—
120C2×D36C22×C311yescharacteristic
121C2×C4×D9C22×C311yescharacteristic
122C2×C4×D9C22×C311yescharacteristic
123C2×C4×D9C22×C312yes—
124C2×Dic18C22×C312yes—
125C2×Dic18C22×C311yescharacteristic
126C9×C22⋊C4—22no—
127C9×C22⋊C4—22no—
128C9×C22⋊C4C22×C312yes—
129C9×C22⋊C4C22×C312yes—
130C42×C9—21no—
131C42×C9C22×C311yescharacteristic
132#3676C22×C312yes—
133(C2×D9)⋊C4C22×C312yes—
134(C2×D9)⋊C4C22×C311yescharacteristic
135(C2×D9)⋊C4C22×C311yescharacteristic
136C4⋊Dic9C22×C311yescharacteristic
137C4⋊Dic9C22×C311yescharacteristic
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140C4×Dic9C22×C312yes—
141C22×C3×C9—22no—
142C22×C3×C9—22no—
143C22×C3×C9—22no—
144C22×C3×C9C2411yescharacteristic
145C2×C3×D9—21no—
146C2×C3×D9—21no—
147C2×C3×D9—21no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9#4112yescomplemented
151C3×C4×C9—22no—
152C3×C4×C9—21no—
153C3×C4×C9—21no—
154C3×C4×C9C2×D412yes—
155C3×Dic9—21no—
156C3×Dic9—21no—
157C3×Dic9—22no—
158C3×Dic9C2×D414yes—
159C2×C3×#41—91no—
160C22×C3×Q8—91no—
161C2×D3×Q8—31no—
162C2×#237—31no—
163C3×#124—21no—
164C3×#124—92no—
165C3×#124D911yescharacteristic, complemented
166C3×#124C3×D311yescharacteristic
167C3×C22⋊Q8—91no—
168C3×C22⋊Q8—92no—
169C3×#121—91no—
170C3×#121—92no—
171C3×C4×Q8—91no—
172C3×C4×D4—92no—
173C3×C4×D4—91no—
174#946—31no—
175#938—31no—
176#885—32no—
177#884—32no—
178#67⋊C4—32no—
179#878—32no—
180C4×D12—31no—
181C4×Dic6—31no—
182C23×C32C22×D312yes—
183C22×C3×D3—31no—
184C22×C3×D3—31no—
185C32×Q8—22no—
186C32×D4—22no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C3×#67—64no—
192C3×#67—64no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—64no—
198C3×D12—62no—
199C3×C4×D3—64no—
200C3×C4×D3—62no—
201C3×C4×D3—62no—
202C3×Dic6—34no—
203C3×Dic6—64no—
204C23×C9—22no—
205C23×C9C23×C312yes—
206C22×D9C23×C311yescharacteristic
207C22×D9C23×C311yescharacteristic
208Q8×C9—22no—
209Q8×C9—42no—
210D4×C9—22no—
211D4×C9—42no—
212C2×C4×C9—21no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—22no—
216C2×C4×C9C23×C311yescharacteristic
217C2×C4×C9C23×C311yescharacteristic
218C2×C4×C9C23×C312yes—
219C2×C4×C9C23×C311yescharacteristic
220#611—24no—
221#611—24no—
222C2×Dic9—24no—
223C2×Dic9C23×C312yes—
224C2×Dic9C23×C311yescharacteristic
225C2×Dic9C23×C311yescharacteristic
226C2×Dic9C23×C312yes—
227D36—22no—
228C4×D9—22no—
229C4×D9—22no—
230C4×D9—24no—
231Dic18—24no—
232Dic18C3×D414yescomplemented
233#548—91noSylow
234C2×C3×C9—42no—
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C2×C3×C92−1+411yescharacteristic
238C3×D9—22no—
239C3×D9—41no—
240C3×#41—184no—
241C2×C3×Q8—21no—
242C2×C3×Q8—91no—
243C2×C3×Q8—92no—
244C2×C3×Q8—184no—
245C2×C3×Q8C2×D911yescharacteristic
246C2×C3×Q8C2×C3×D311yescharacteristic
247C2×C3×D4—21no—
248C2×C3×D4—92no—
249C2×C3×D4—92no—
250C2×C3×D4—91no—
251C2×C3×D4C2×D911yescharacteristic
252C2×C3×D4C2×C3×D311yescharacteristic
253C22×C3×C4—92no—
254C22×C3×C4—92no—
255C22×C3×C4—91no—
256C22×C3×C4—91no—
257C2×#67—32no—
258C2×#67—32no—
259C22×Dic3—32no—
260D3×Q8—64no—
261#237—64no—
262C2×D12—31no—
263C2×C4×D3—31no—
264C2×C4×D3—31no—
265C2×C4×D3—32no—
266C2×Dic6—31no—
267C2×Dic6—32no—
268C3×C4⋊C4—92no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—91no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—91no—
277C3×C22⋊C4—92no—
278C3×C22⋊C4C2×D912yes—
279C3×C22⋊C4C2×D912yes—
280C3×C22⋊C4C2×C3×D312yes—
281C3×C22⋊C4C2×C3×D312yes—
282C3×C42—21no—
283C3×C42—92no—
284C3×C42C2×D911yescharacteristic
285C3×C42C2×C3×D311yescharacteristic
286#217—32no—
287(C2×D3)⋊C4—31no—
288(C2×D3)⋊C4—31no—
289(C2×D3)⋊C4—32no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294C4×Dic3—32no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32C23×D311yescharacteristic
299C2×C3×D3—32no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—22no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4D3×D412yes—
309C3×Dic3—34no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—62no—
313C22×C9—22no—
314C22×C9—22no—
315C22×C9—22no—
316C22×C9—21no—
317C22×C9—42no—
318C22×C9—42no—
319C22×C9—42no—
320C22×C9C24×C311yescharacteristic
321C2×D9—21no—
322C2×D9—21no—
323C2×D9—21no—
324C2×D9—21no—
325C2×D9—21no—
326C2×D9C3×#4112yescomplemented
327C4×C9—22no—
328C4×C9—21no—
329C4×C9—21no—
330C4×C9—22no—
331C4×C9—41no—
332C4×C9—41no—
333C4×C9—42no—
334C4×C9C2×C3×D412yes—
335Dic9—21no—
336Dic9—21no—
337Dic9—22no—
338Dic9C2×C3×D414yes—
339C2×#41—91no—
340C22×Q8—91no—
341#124—92no—
342#124C3×D911yescharacteristic, complemented
343C22⋊Q8—91no—
344C22⋊Q8—92no—
345#121—91no—
346#121—92no—
347C4×Q8—91no—
348C4×D4—92no—
349C4×D4—91no—
350C3×C9#54811yescharacteristic, complemented, Sylow
351C23×C3—22no—
352C23×C3—91no—
353C23×C3—91no—
354C23×C3C22×D912yes—
355C23×C3C22×C3×D312yes—
356C22×D3—31no—
357C22×D3—31no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—42no—
361C3×Q8—94no—
362C3×Q8—184no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—42no—
366C3×D4—182no—
367C3×D4—184no—
368C3×D4—184no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—92no—
374C2×C3×C4—92no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—184no—
378C2×C3×C4—184no—
379C2×C3×C4—182no—
380C2×C3×C4—182no—
381C2×C3×C4C22×D911yescharacteristic
382C2×C3×C4C22×D911yescharacteristic
383C2×C3×C4C22×D912yes—
384C2×C3×C4C22×D911yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×C3×C4C22×C3×D311yescharacteristic
387C2×C3×C4C22×C3×D312yes—
388C2×C3×C4C22×C3×D311yescharacteristic
389#67—64no—
390#67—64no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—64no—
396D12—62no—
397C4×D3—64no—
398C4×D3—62no—
399C4×D3—62no—
400Dic6—34no—
401Dic6—64no—
402C2×C32—42no—
403C2×C32C2×D3×D411yescharacteristic
404C2×C32D3×#4111yescharacteristic
405C2×C32#101011yescharacteristic
406C3×D3—62no—
407C3×D3—121no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—42no—
412C2×C9—82no—
413C2×C9C22×C3×D411yescharacteristic
414C2×C9C2×C3×#4111yescharacteristic
415C2×C9C3×2−1+411yescharacteristic
416D9—22no—
417D9—41no—
418#41—184no—
419C2×Q8—91no—
420C2×Q8—92no—
421C2×Q8—184no—
422C2×Q8C2×C3×D911yescharacteristic
423C2×D4—92no—
424C2×D4—92no—
425C2×D4—91no—
426C2×D4C2×C3×D911yescharacteristic
427C22×C4—92no—
428C22×C4—92no—
429C22×C4—91no—
430C22×C4—91no—
431C4⋊C4—92no—
432C4⋊C4—91no—
433C4⋊C4—92no—
434C4⋊C4—91no—
435C22⋊C4—92no—
436C22⋊C4—91no—
437C22⋊C4—91no—
438C22⋊C4—92no—
439C22⋊C4C2×C3×D912yes—
440C22⋊C4C2×C3×D912yes—
441C42—92no—
442C42C2×C3×D911yescharacteristic
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—92no—
454C22×C3—181no—
455C22×C3—181no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C23×D911yescharacteristic
460C22×C3C23×C3×D311yescharacteristic
461C2×D3—32no—
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C3×C4—22no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—22no—
474C3×C4—42no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—94no—
478C3×C4—181no—
479C3×C4—181no—
480C3×C4—182no—
481C3×C4D4×D912yes—
482C3×C4C3×D3×D412yes—
483Dic3—34no—
484Dic3—61no—
485Dic3—61no—
486Dic3—62no—
487C32#567711yescharacteristic
488C9—21no—
489C9C3×#54811yescharacteristic, complemented
490C23—91no—
491C23—91no—
492C23C22×C3×D912yes—
493Q8—22no—
494Q8—94no—
495Q8—184no—
496D4—22no—
497D4—182no—
498D4—184no—
499D4—184no—
500C2×C4—92no—
501C2×C4—92no—
502C2×C4—91no—
503C2×C4—91no—
504C2×C4—184no—
505C2×C4—184no—
506C2×C4—182no—
507C2×C4—182no—
508C2×C4C22×C3×D911yescharacteristic
509C2×C4C22×C3×D911yescharacteristic
510C2×C4C22×C3×D912yes—
511C2×C4C22×C3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—42no—
517C2×C3—82no—
518C2×C3—182no—
519C2×C3—361no—
520C2×C3C2×D4×D911yescharacteristic
521C2×C3#41×D911yescharacteristic
522C2×C3#6371611yescharacteristic
523C2×C3C2×C3×D3×D411yescharacteristic
524C2×C3C3×D3×#4111yescharacteristic
525C2×C3C3×#101011yescharacteristic
526D3—62no—
527D3—121no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—92no—
532C22—181no—
533C22—181no—
534C22—181no—
535C22—181no—
536C22—181no—
537C22C23×C3×D911yescharacteristic
538C4—22no—
539C4—21no—
540C4—21no—
541C4—94no—
542C4—181no—
543C4—181no—
544C4—182no—
545C4C3×D4×D912yes—
546C3—21no—
547C3#1058977911yescharacteristic, direct product factor
548C3C3×#567711yescharacteristic
549C2—42no—
550C2—182no—
551C2—361no—
552C2C2×C3×D4×D911yescharacteristic
553C2C3×#41×D911yescharacteristic
554C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#124C2
socleC22×C32C23×D3
hypercenterC3×#124D9
hypocenterC9C3×#548
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×D9C2semidirect
2. C2×C3×#3699C2semidirect
3. C3×C9×#124C2semidirect
4. C3×#63508C2semidirect
5. C3×#63500C2semidirect
6. C3×#63447C2semidirect
7. C3×#63446C2semidirect
8. C3×#611⋊C4C2semidirect
9. C3×#63440C2semidirect
10. C3×C4×D36C2semidirect
11. C3×C4×Dic18C2semidirect
13. #10589779C3direct (N × H)
50. C2×Q8×D9C2×C3semidirect
51. C2×#3699C2×C3semidirect
53. C9×#124C2×C3semidirect
54. #63508C2×C3semidirect
55. #63500C2×C3semidirect
56. #63447C2×C3semidirect
57. #63446C2×C3semidirect
58. #611⋊C4C2×C3semidirect
59. #63440C2×C3semidirect
60. C4×D36C2×C3semidirect
61. C4×Dic18C2×C3semidirect
83. C3×Dic18D4semidirect
150. C2×C3×D9#41semidirect
165. C3×#124D9semidirect
232. Dic18C3×D4semidirect
326. C2×D9C3×#41semidirect
342. #124C3×D9semidirect
350. C3×C9#548semidirect
489. C9C3×#548semidirect
547. C3#10589779direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
C2×C3×#36992
C3×C9×#1242
C3×#635082
C3×#635002
C3×#634472
C3×#634462
C3×#611⋊C42
C3×#634402
C3×C4×D362
C3×C4×Dic182
C3×#56773
#105897793