Finite Group Databasebeta

← C3×#10589701 #13152790

Subgroups — C3×#10589701 #13152790

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C3×C9×#121C211yesmaximal, characteristic, complemented
4C3×#63502C211yesmaximal, characteristic, complemented
5C3×#63502C212yesmaximal, complemented
6C3×#63501C212yesmaximal, complemented
7C3×C4×#611C211yesmaximal, characteristic, complemented
8C3×#63457C211yesmaximal, characteristic, complemented
9C3×#63449C211yesmaximal, characteristic, complemented
10C3×#63446C212yesmaximal, complemented
11C3×#611⋊C4C212yesmaximal, complemented
12C3×#5599—31nomaximal
13#10589701C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2211yescharacteristic
15C2×C3×D4×C9C2212yes—
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C3×#611C2211yescharacteristic
19C2×C3×#611C2212yes—
20C2×C3×#611C2212yes—
21C2×C3×#611C2212yes—
22C22×C3×Dic9C2212yes—
23C3×#3699—24no—
24C3×#3697—24no—
25C2×C3×D36C2211yescharacteristic
26C2×C3×D36C2212yes—
27C2×C3×C4×D9C2212yes—
28C2×C3×C4×D9C2211yescharacteristic
29C2×C3×Dic18C2211yescharacteristic
30C3×C9×C4⋊C4C2211yescharacteristic
31C3×C9×C22⋊C4C2212yes—
32C3×#3676C2211yescharacteristic
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C3×(C2×D9)⋊C4C2212yes—
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2212yes—
37C3×C4×Dic9C2211yescharacteristic
38C3×C4×Dic9C2212yes—
39C2×C3×#237—31no—
40C2×C3×#235—31no—
41C32×#121D311yescharacteristic
42C3×#940—32no—
43C3×#940—31no—
44C3×#939—32no—
45C3×C4×#67—31no—
46C3×#895—31no—
47C3×#887—31no—
48C3×#884—32no—
49C3×#67⋊C4—32no—
50C2×#3699C2×C311yescharacteristic, complemented
51C2×#3697C2×C311yescharacteristic, complemented
52C9×#121—21no—
53C9×#121C2×C311yescharacteristic, complemented
54#63502C2×C311yescharacteristic, complemented
55#63502C2×C312yescomplemented
56#63501C2×C312yescomplemented
57C4×#611C2×C311yescharacteristic, complemented
58#63457C2×C311yescharacteristic, complemented
59#63449C2×C311yescharacteristic, complemented
60#63446C2×C312yescomplemented
61#611⋊C4C2×C312yescomplemented
62C23×C3×C9C2312yes—
63C23×C3×C9C2311yescharacteristic
64C22×C3×D9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C3×D4×C9—22no—
67C3×D4×C9—42no—
68C2×C3×C4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3×C4×C9C2312yes—
71C2×C3×C4×C9C2311yescharacteristic
72C3×#611—24no—
73C3×#611—24no—
74C3×#611—22no—
75C3×#611—24no—
76C3×#611—42no—
77C2×C3×Dic9—24no—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2312yes—
81C2×C3×Dic9C2311yescharacteristic
82C3×D36—22no—
83C3×D36—42no—
84C3×C4×D9—22no—
85C3×C4×D9—24no—
86C3×Dic18D414yescomplemented
87C3×#550—91no—
88#5599—31no—
89C2×C32×D4C2×D311yescharacteristic
90C2×C32×D4C2×D312yes—
91C22×C32×C4C2×D311yescharacteristic
92C2×C3×#67—32no—
93C2×C3×#67—32no—
94C2×C3×#67—31no—
95C2×C3×#67—32no—
96C2×C3×#67—32no—
97C22×C3×Dic3—32no—
98C3×#237—64no—
99C3×#235—64no—
100C2×C3×D12—31no—
101C2×C3×D12—32no—
102C2×C3×C4×D3—32no—
103C2×C3×C4×D3—31no—
104C2×C3×Dic6—31no—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C22⋊C4C2×D312yes—
107C3×#217—31no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—31no—
110C3×Dic3⋊C4—31no—
111C3×Dic3⋊C4—32no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3—31no—
114C2×D4×C9—21no—
115C2×D4×C9—22no—
116C2×D4×C9C22×C311yescharacteristic
117C2×D4×C9C22×C312yes—
118C22×C4×C9—21no—
119C22×C4×C9C22×C311yescharacteristic
120C2×#611C22×C311yescharacteristic
121C2×#611C22×C312yes—
122C2×#611C22×C312yes—
123C2×#611C22×C312yes—
124C2×#611C22×C312yes—
125C22×Dic9C22×C312yes—
126#3699—24no—
127#3697—24no—
128C2×D36C22×C311yescharacteristic
129C2×D36C22×C312yes—
130C2×C4×D9C22×C311yescharacteristic
131C2×C4×D9C22×C312yes—
132C2×Dic18C22×C311yescharacteristic
133C9×C4⋊C4—21no—
134C9×C4⋊C4C22×C311yescharacteristic
135C9×C22⋊C4—22no—
136C9×C22⋊C4C22×C312yes—
137#3676C22×C311yescharacteristic
138(C2×D9)⋊C4C22×C311yescharacteristic
139(C2×D9)⋊C4C22×C312yes—
140Dic9⋊C4C22×C311yescharacteristic
141Dic9⋊C4C22×C312yes—
142C4×Dic9C22×C311yescharacteristic
143C4×Dic9C22×C312yes—
144C22×C3×C9—22no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9—22no—
148C22×C3×C9—21no—
149C22×C3×C9—22no—
150C22×C3×C9C2411yescharacteristic
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9—22no—
157C3×C4×C9—21no—
158C3×C4×C9—22no—
159C3×C4×C9C2×D412yes—
160C3×Dic9—21no—
161C3×Dic9—21no—
162C3×Dic9C2×D414yes—
163C3×Dic9#4112yescomplemented
164C2×C3×#41—91no—
165C2×C3×#41—91no—
166C2×#237—31no—
167C2×#235—31no—
168C3×#127—92no—
169C3×#127—91no—
170C3×#121—21no—
171C3×#121—92no—
172C3×#121—91no—
173C3×#121—92no—
174C3×#121D911yescharacteristic, complemented
175C3×#121C3×D311yescharacteristic
176C3×C4×Q8—91no—
177C3×C4×D4—91no—
178C3×C4×D4—92no—
179#940—32no—
180#940—31no—
181#939—32no—
182C4×#67—31no—
183#895—31no—
184#887—31no—
185#884—32no—
186#67⋊C4—32no—
187C23×C32C22×D311yescharacteristic
188C23×C32C22×D312yes—
189C22×C3×D3—32no—
190C22×C3×D3—31no—
191C32×D4—22no—
192C32×D4—42no—
193C2×C32×C4—22no—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C3×#67—64no—
198C3×#67—64no—
199C3×#67—64no—
200C3×#67—62no—
201C3×#67—122no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—64no—
207C3×D12—62no—
208C3×D12—122no—
209C3×C4×D3—64no—
210C3×C4×D3—62no—
211C3×Dic6—34no—
212C23×C9—21no—
213C23×C9—22no—
214C23×C9C23×C311yescharacteristic
215C23×C9C23×C312yes—
216C22×D9C23×C312yes—
217C22×D9C23×C311yescharacteristic
218D4×C9—22no—
219D4×C9—42no—
220D4×C9—42no—
221D4×C9—44no—
222C2×C4×C9—22no—
223C2×C4×C9—21no—
224C2×C4×C9—21no—
225C2×C4×C9—22no—
226C2×C4×C9—42no—
227C2×C4×C9C23×C312yes—
228C2×C4×C9C23×C311yescharacteristic
229C2×C4×C9C23×C311yescharacteristic
230#611—24no—
231#611—24no—
232#611—22no—
233#611—24no—
234#611—42no—
235C2×Dic9—24no—
236C2×Dic9C23×C311yescharacteristic
237C2×Dic9C23×C311yescharacteristic
238C2×Dic9C23×C312yes—
239C2×Dic9C23×C311yescharacteristic
240D36—22no—
241D36—42no—
242C4×D9—22no—
243C4×D9—24no—
244Dic18C3×D414yescomplemented
245#550—91noSylow
246C2×C3×C9—41no—
247C2×C3×C9—42no—
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4111yescharacteristic
250C2×C3×C9H(5,𝔽2)11yescharacteristic
251C3×D9—42no—
252C3×D9—41no—
253C3×#41—184no—
254C3×#41—184no—
255C2×C3×Q8—91no—
256C2×C3×D4—21no—
257C2×C3×D4—22no—
258C2×C3×D4—91no—
259C2×C3×D4—92no—
260C2×C3×D4—92no—
261C2×C3×D4—92no—
262C2×C3×D4—92no—
263C2×C3×D4—91no—
264C2×C3×D4—92no—
265C2×C3×D4C2×D911yescharacteristic
266C2×C3×D4C2×D912yes—
267C2×C3×D4C2×C3×D311yescharacteristic
268C2×C3×D4C2×C3×D312yes—
269C22×C3×C4—21no—
270C22×C3×C4—92no—
271C22×C3×C4—92no—
272C22×C3×C4—91no—
273C22×C3×C4C2×D911yescharacteristic
274C22×C3×C4C2×C3×D311yescharacteristic
275C2×#67—32no—
276C2×#67—32no—
277C2×#67—31no—
278C2×#67—32no—
279C2×#67—32no—
280C22×Dic3—32no—
281#237—64no—
282#235—64no—
283C2×D12—31no—
284C2×D12—32no—
285C2×C4×D3—32no—
286C2×C4×D3—31no—
287C2×Dic6—31no—
288C3×C4⋊C4—21no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—92no—
291C3×C4⋊C4C2×D911yescharacteristic
292C3×C4⋊C4C2×C3×D311yescharacteristic
293C3×C22⋊C4—22no—
294C3×C22⋊C4—91no—
295C3×C22⋊C4—91no—
296C3×C22⋊C4—92no—
297C3×C22⋊C4C2×D912yes—
298C3×C22⋊C4C2×C3×D312yes—
299C3×C42—92no—
300C3×C42—91no—
301#217—31no—
302(C2×D3)⋊C4—32no—
303(C2×D3)⋊C4—31no—
304Dic3⋊C4—31no—
305Dic3⋊C4—32no—
306C4×Dic3—32no—
307C4×Dic3—31no—
308C22×C32—21no—
309C22×C32—22no—
310C22×C32—21no—
311C22×C32—22no—
312C22×C32—21no—
313C22×C32—22no—
314C22×C32C23×D311yescharacteristic
315C2×C3×D3—62no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—61no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4D3×D412yes—
324C3×Dic3—34no—
325C3×Dic3—32no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C22×C9—22no—
329C22×C9—22no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—22no—
333C22×C9—22no—
334C22×C9—21no—
335C22×C9—21no—
336C22×C9—41no—
337C22×C9—41no—
338C22×C9—42no—
339C22×C9—42no—
340C22×C9—42no—
341C22×C9C24×C311yescharacteristic
342C2×D9—22no—
343C2×D9—21no—
344C2×D9—22no—
345C2×D9—21no—
346C2×D9—21no—
347C2×D9—22no—
348C4×C9—21no—
349C4×C9—22no—
350C4×C9—22no—
351C4×C9—41no—
352C4×C9—42no—
353C4×C9C2×C3×D412yes—
354Dic9—21no—
355Dic9—21no—
356Dic9C2×C3×D414yes—
357Dic9C3×#4112yescomplemented
358C2×#41—91no—
359C2×#41—91no—
360#127—92no—
361#127—91no—
362#121—92no—
363#121—91no—
364#121—92no—
365#121C3×D911yescharacteristic, complemented
366C4×Q8—91no—
367C4×D4—91no—
368C4×D4—92no—
369C3×C9#55011yescharacteristic, complemented, Sylow
370C23×C3—21no—
371C23×C3—22no—
372C23×C3—91no—
373C23×C3—92no—
374C23×C3C22×D911yescharacteristic
375C23×C3C22×D912yes—
376C23×C3C22×C3×D311yescharacteristic
377C23×C3C22×C3×D312yes—
378C22×D3—32no—
379C22×D3—31no—
380C3×Q8—94no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—42no—
384C3×D4—42no—
385C3×D4—42no—
386C3×D4—44no—
387C3×D4—182no—
388C3×D4—184no—
389C3×D4—184no—
390C3×D4—184no—
391C3×D4—182no—
392C3×D4—362no—
393C3×D4—362no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—42no—
400C2×C3×C4—91no—
401C2×C3×C4—92no—
402C2×C3×C4—91no—
403C2×C3×C4—91no—
404C2×C3×C4—184no—
405C2×C3×C4—184no—
406C2×C3×C4—182no—
407C2×C3×C4C22×D912yes—
408C2×C3×C4C22×D911yescharacteristic
409C2×C3×C4C22×D911yescharacteristic
410C2×C3×C4C22×C3×D312yes—
411C2×C3×C4C22×C3×D311yescharacteristic
412C2×C3×C4C22×C3×D311yescharacteristic
413#67—64no—
414#67—64no—
415#67—64no—
416#67—62no—
417#67—122no—
418C2×Dic3—31no—
419C2×Dic3—32no—
420C2×Dic3—31no—
421C2×Dic3—31no—
422C2×Dic3—64no—
423D12—62no—
424D12—122no—
425C4×D3—64no—
426C4×D3—62no—
427Dic6—34no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32C2×D3×D411yescharacteristic
431C2×C32#100411yescharacteristic
432C2×C32D3×#4111yescharacteristic
433C3×D3—122no—
434C3×D3—121no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—42no—
439C2×C9—41no—
440C2×C9—42no—
441C2×C9—82no—
442C2×C9C22×C3×D411yescharacteristic
443C2×C9C2×C3×#4111yescharacteristic
444C2×C9C3×H(5,𝔽2)11yescharacteristic
445D9—42no—
446D9—41no—
447#41—184no—
448#41—184no—
449C2×Q8—91no—
450C2×D4—91no—
451C2×D4—92no—
452C2×D4—92no—
453C2×D4—92no—
454C2×D4—92no—
455C2×D4—91no—
456C2×D4—92no—
457C2×D4C2×C3×D911yescharacteristic
458C2×D4C2×C3×D912yes—
459C22×C4—92no—
460C22×C4—92no—
461C22×C4—91no—
462C22×C4C2×C3×D911yescharacteristic
463C4⋊C4—91no—
464C4⋊C4—92no—
465C4⋊C4C2×C3×D911yescharacteristic
466C22⋊C4—91no—
467C22⋊C4—91no—
468C22⋊C4—92no—
469C22⋊C4C2×C3×D912yes—
470C42—92no—
471C42—91no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—22no—
483C22×C3—21no—
484C22×C3—22no—
485C22×C3—21no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—182no—
492C22×C3—181no—
493C22×C3—181no—
494C22×C3—182no—
495C22×C3—181no—
496C22×C3—182no—
497C22×C3C23×D911yescharacteristic
498C22×C3C23×C3×D311yescharacteristic
499C2×D3—62no—
500C2×D3—61no—
501C2×D3—61no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—61no—
505C3×C4—21no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—94no—
513C3×C4—92no—
514C3×C4—181no—
515C3×C4—181no—
516C3×C4D4×D912yes—
517C3×C4C3×D3×D412yes—
518Dic3—34no—
519Dic3—32no—
520Dic3—61no—
521Dic3—61no—
522C32#559911yescharacteristic
523C9—21no—
524C9C3×#55011yescharacteristic, complemented
525C23—91no—
526C23—92no—
527C23C22×C3×D911yescharacteristic
528C23C22×C3×D912yes—
529Q8—94no—
530D4—22no—
531D4—42no—
532D4—184no—
533D4—184no—
534D4—182no—
535D4—184no—
536D4—182no—
537D4—362no—
538D4—362no—
539C2×C4—22no—
540C2×C4—91no—
541C2×C4—92no—
542C2×C4—91no—
543C2×C4—91no—
544C2×C4—184no—
545C2×C4—184no—
546C2×C4—182no—
547C2×C4C22×C3×D912yes—
548C2×C4C22×C3×D911yescharacteristic
549C2×C4C22×C3×D911yescharacteristic
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3—41no—
555C2×C3—42no—
556C2×C3—41no—
557C2×C3—42no—
558C2×C3—82no—
559C2×C3—362no—
560C2×C3—361no—
561C2×C3C2×D4×D911yescharacteristic
562C2×C3#6371011yescharacteristic
563C2×C3#41×D911yescharacteristic
564C2×C3C2×C3×D3×D411yescharacteristic
565C2×C3C3×#100411yescharacteristic
566C2×C3C3×D3×#4111yescharacteristic
567D3—122no—
568D3—121no—
569C22—21no—
570C22—22no—
571C22—21no—
572C22—22no—
573C22—21no—
574C22—22no—
575C22—182no—
576C22—181no—
577C22—181no—
578C22—182no—
579C22—181no—
580C22—182no—
581C22C23×C3×D911yescharacteristic
582C4—21no—
583C4—22no—
584C4—94no—
585C4—92no—
586C4—181no—
587C4—181no—
588C4C3×D4×D912yes—
589C3—21no—
590C3#1058970111yescharacteristic, direct product factor
591C3C3×#559911yescharacteristic
592C2—41no—
593C2—42no—
594C2—362no—
595C2—361no—
596C2C2×C3×D4×D911yescharacteristic
597C2C3×#6371011yescharacteristic
598C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#550
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C2×C3×#3697C2semidirect
3. C3×C9×#121C2semidirect
4. C3×#63502C2semidirect
5. C3×#63502C2semidirect
6. C3×#63501C2semidirect
7. C3×C4×#611C2semidirect
8. C3×#63457C2semidirect
9. C3×#63449C2semidirect
10. C3×#63446C2semidirect
11. C3×#611⋊C4C2semidirect
13. #10589701C3direct (N × H)
50. C2×#3699C2×C3semidirect
51. C2×#3697C2×C3semidirect
53. C9×#121C2×C3semidirect
54. #63502C2×C3semidirect
55. #63502C2×C3semidirect
56. #63501C2×C3semidirect
57. C4×#611C2×C3semidirect
58. #63457C2×C3semidirect
59. #63449C2×C3semidirect
60. #63446C2×C3semidirect
61. #611⋊C4C2×C3semidirect
86. C3×Dic18D4semidirect
163. C3×Dic9#41semidirect
174. C3×#121D9semidirect
244. Dic18C3×D4semidirect
357. Dic9C3×#41semidirect
365. #121C3×D9semidirect
369. C3×C9#550semidirect
524. C9C3×#550semidirect
590. C3#10589701direct (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
C2×C3×#36972
C3×C9×#1212
C3×#635022
C3×#635012
C3×C4×#6112
C3×#634572
C3×#634492
C3×#634462
C3×#611⋊C42
C3×#55993
#105897013