Finite Group Databasebeta

← C3×#10591121 #13158367

Subgroups — C3×#10591121 #13158367

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64133C211yesmaximal, characteristic
2C32×#924C211yesmaximal, characteristic
3C2×C32×#208C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8C3×#5137—31nomaximal
9C3×#5052—31nomaximal
10#10591121C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3757—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×#208—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23C3×#3719—22no—
24C2×C32×M16—31no—
25C32×C22⋊Q8—31no—
26#64133C2×C311yescharacteristic
27C2×C3×#216—31no—
28C3×#924—21no—
29C3×#924—31no—
30C3×#924—31no—
31C3×#924—61no—
32C3×#924D311yescharacteristic
33C3×#924C2×C311yescharacteristic
34C2×C3×#208—21no—
35C2×C3×#208D311yescharacteristic
36C2×C3×#208C2×C311yescharacteristic
37C2×C3×Dic12—31no—
38C2×#3719C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#817—31no—
41C3×#816—31no—
42C3×#816—31no—
43C3×#810—31no—
44C3×#808—31no—
45#63573C2×C311yescharacteristic, complemented
46#63567C2×C311yescharacteristic, complemented
47#63566C2×C311yescharacteristic, complemented
48C23×C33D411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×C3⋊C8—21no—
62C32×C3⋊C8D412yes—
63C3×#470—91no—
64#5137—31no—
65#5052—31no—
66C2×C32×Q8—31no—
67C22×C32×C4—21no—
68C22×C32×C4—21no—
69C22×C32×C4—21no—
70C22×C32×C4—41no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C22×C311yescharacteristic
74C2×#634C22×C311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×M16—62no—
82C2×C32×C8—31no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—61no—
86C32×C22⋊C4—61no—
87#3757—21no—
88C4⋊(C3⋊Dic3)C22×C311yescharacteristic
89#3750—21no—
90C3×#217—22no—
91C3×#217—21no—
92C3×#217—21no—
93C3×#217—62no—
94C3×#217—61no—
95C3×#217—61no—
96C3×#216—62no—
97C3×C4⋊Dic3—21no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C22×C311yescharacteristic
103C3×Dic3⋊C4—22no—
104C3×Dic3⋊C4—21no—
105C3×Dic3⋊C4—21no—
106C3×Dic3⋊C4—62no—
107C3×Dic3⋊C4—61no—
108C3×Dic3⋊C4—61no—
109C3×#208—24no—
110C3×#208—22no—
111C3×#208—22no—
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118C3×Dic12—62no—
119#3719—22no—
120C22×C33—21no—
121C22×C33—21no—
122C22×C33—21no—
123C22×C33C2×D411yescharacteristic
124C33×C4—21no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—41no—
128C3×C3⋊Dic3—41no—
129C32×Dic3—41no—
130C32×Dic3—41no—
131C2×C3×Q16—91no—
132C2×C3×M16—31no—
133C2×C3×M16—31no—
134C2×C3×M16—61no—
135C3×C22⋊Q8—31no—
136C3×C22⋊Q8—31no—
137C3×C22⋊Q8—61no—
138C3×C22⋊Q8—91no—
139C2×#216—31no—
140#924—31no—
141#924—31no—
142#924—61no—
143#924C3×D311yescharacteristic
144C2×#208C3×D311yescharacteristic
145C2×Dic12—31no—
146C3×#106—91no—
147C3×#103—91no—
148C3×#103—91no—
149#835—31no—
150#817—31no—
151#816—31no—
152#816—31no—
153#810—31no—
154#808—31no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—41no—
159C23×C32D1211yescharacteristic
160C23×C32#6711yescharacteristic
161C23×C32C3×D411yescharacteristic
162C32×Q8—61no—
163C32×Q8—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—41no—
183C2×C32×C4—41no—
184C2×C32×C4—42no—
185C2×C32×C4—42no—
186C2×C32×C4—61no—
187C2×C32×C4—61no—
188C2×C32×C4D1211yescharacteristic
189C2×C32×C4#6711yescharacteristic
190C2×C32×C4C3×D411yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C311yescharacteristic
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—21no—
196#634—21no—
197#634—21no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—41no—
215C3×Dic6—41no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—121no—
221C3×Dic6—121no—
222C32×C8—32no—
223C32×C8—61no—
224C3×C3⋊C8—22no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—22no—
228C3×C3⋊C8#6712yes—
229C3×C3⋊C8C3×D412yes—
230#470—91noSylow
231C2×C33—41no—
232C2×C33#12111yescharacteristic
233C2×C33#13711yescharacteristic
234C2×C33#13711yescharacteristic
235C2×C3×Q8—31no—
236C2×C3×Q8—31no—
237C2×C3×Q8—61no—
238C2×C3×Q8—91no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—41no—
243C22×C3×C4D3211yescharacteristic
244C22×C3×C4C2×C3×D311yescharacteristic
245C22×C3×C4C2×C3×D311yescharacteristic
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—61no—
249C2×Dic6C2×C3×D311yescharacteristic
250C3×Q16—182no—
251C3×M16—62no—
252C3×M16—62no—
253C3×M16—64no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—31no—
258C2×C3×C8—61no—
259C2×C3×C8—61no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—62no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4—181no—
268C3×C22⋊C4—62no—
269C3×C22⋊C4—61no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—181no—
272#217—21no—
273#217—62no—
274#217—61no—
275#217—61no—
276#216—62no—
277C4⋊Dic3—31no—
278C4⋊Dic3—31no—
279C4⋊Dic3—61no—
280C4⋊Dic3C2×C3×D311yescharacteristic
281Dic3⋊C4—21no—
282Dic3⋊C4—62no—
283Dic3⋊C4—61no—
284Dic3⋊C4—61no—
285#208—22no—
286C2×C3⋊C8C2×C3×D311yescharacteristic
287C2×C3⋊C8C2×C3×D311yescharacteristic
288Dic12—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—42no—
312C22×C32—42no—
313C22×C32—42no—
314C22×C32C2×D1211yescharacteristic
315C22×C32C2×#6711yescharacteristic
316C22×C32C2×C3×D411yescharacteristic
317C32×C4—21no—
318C32×C4—22no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—41no—
330C32×C4—41no—
331C32×C4—42no—
332C32×C4—121no—
333C32×C4—121no—
334C32×C4#23511yescharacteristic
335C32×C4D3×D411yescharacteristic
336C32×C4#23711yescharacteristic
337C32×C4C2×#6711yescharacteristic
338C32×C4C2×C3×D411yescharacteristic
339C32×C4C3×#4111yescharacteristic
340C3⋊Dic3—41no—
341C3⋊Dic3—41no—
342C3×Dic3—42no—
343C3×Dic3—42no—
344C3×Dic3—41no—
345C3×Dic3—41no—
346C3×Dic3—41no—
347C3×Dic3—41no—
348C3×Dic3—122no—
349C3×Dic3—122no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—121no—
353C3×Dic3—121no—
354C2×Q16—91no—
355C2×M16—31no—
356C22⋊Q8—31no—
357C22⋊Q8—91no—
358#106—91no—
359#103—91no—
360#103—91no—
361C33#47011yescharacteristic, complemented, Sylow
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3#62611yescharacteristic
367C23×C3C3×D1211yescharacteristic
368C23×C3C3×#6711yescharacteristic
369C3×Q8—61no—
370C3×Q8—61no—
371C3×Q8—61no—
372C3×Q8—61no—
373C3×Q8—121no—
374C3×Q8—121no—
375C3×Q8—181no—
376C3×Q8—181no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—181no—
406C2×C3×C4—181no—
407C2×C3×C4#62611yescharacteristic
408C2×C3×C4C3×D1211yescharacteristic
409C2×C3×C4C3×#6711yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C22×C3×D311yescharacteristic
412C2×C3×C4C22×C3×D311yescharacteristic
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—62no—
416C2×Dic3—62no—
417C2×Dic3—61no—
418C2×Dic3—61no—
419C2×Dic3—61no—
420C2×Dic3—61no—
421Dic6—21no—
422Dic6—21no—
423Dic6—61no—
424Dic6—61no—
425Dic6—61no—
426Dic6—61no—
427Dic6—121no—
428Dic6—121no—
429C3×C8—32no—
430C3×C8—32no—
431C3×C8—62no—
432C3×C8—62no—
433C3×C8—61no—
434C3×C8—61no—
435C3⋊C8—21no—
436C3⋊C8C3×#6712yes—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—42no—
450C2×C32—42no—
451C2×C32—42no—
452C2×C32—44no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32#90911yescharacteristic
457C2×C32#90911yescharacteristic
458C2×C32#93011yescharacteristic
459C2×C32#93911yescharacteristic
460C2×C32#94211yescharacteristic
461C2×C32#95111yescharacteristic
462C2×C32C3×#12111yescharacteristic
463C2×C32C3×#13711yescharacteristic
464C2×C32C3×#13711yescharacteristic
465C2×Q8—31no—
466C2×Q8—91no—
467C22×C4C3×D3211yescharacteristic
468Q16—182no—
469M16—62no—
470C2×C8—31no—
471C2×C8—31no—
472C4⋊C4—31no—
473C4⋊C4—61no—
474C4⋊C4—91no—
475C4⋊C4—181no—
476C22⋊C4—61no—
477C22⋊C4—181no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3C2×#62611yescharacteristic
504C22×C3C2×C3×D1211yescharacteristic
505C22×C3C2×C3×#6711yescharacteristic
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—122no—
522C3×C4—122no—
523C3×C4—121no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4—121no—
527C3×C4—361no—
528C3×C4—361no—
529C3×C4#380411yescharacteristic
530C3×C4D3×#6711yescharacteristic
531C3×C4C3×#23511yescharacteristic
532C3×C4C3×D3×D411yescharacteristic
533C3×C4C3×#23711yescharacteristic
534C3×C4C2×C3×#6711yescharacteristic
535Dic3—41no—
536Dic3—41no—
537Dic3—122no—
538Dic3—122no—
539Dic3—121no—
540Dic3—121no—
541Dic3—121no—
542Dic3—121no—
543C32—21no—
544C32—21no—
545C32—21no—
546C32—41no—
547C32#505211yescharacteristic, complemented
548C32#513711yescharacteristic, complemented
549C32C3×#47011yescharacteristic, complemented
550C23C3×#62611yescharacteristic
551Q8—61no—
552Q8—61no—
553Q8—181no—
554Q8—181no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4C3×#62611yescharacteristic
562C2×C4C2×C3×D3211yescharacteristic
563C8—32no—
564C8—61no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—42no—
575C2×C3—42no—
576C2×C3—42no—
577C2×C3—44no—
578C2×C3—41no—
579C2×C3—41no—
580C2×C3—41no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3#6383111yescharacteristic
585C2×C3#6383311yescharacteristic
586C2×C3#6397811yescharacteristic
587C2×C3C3×#90911yescharacteristic
588C2×C3C3×#90911yescharacteristic
589C2×C3C3×#93011yescharacteristic
590C2×C3C3×#93911yescharacteristic
591C2×C3C3×#94211yescharacteristic
592C2×C3C3×#95111yescharacteristic
593C22—21no—
594C22—21no—
595C22—21no—
596C22C2×C3×#62611yescharacteristic
597C4—21no—
598C4—121no—
599C4—121no—
600C4—361no—
601C4—361no—
602C4C3×#380411yescharacteristic
603C4C3×D3×#6711yescharacteristic
604C3—21no—
605C3—21no—
606C3—21no—
607C3—41no—
608C3#1059112111yescharacteristic, direct product factor
609C3C3×#505211yescharacteristic, complemented
610C3C3×#513711yescharacteristic, complemented
611C2—41no—
612C2C3×#6383111yescharacteristic
613C2C3×#6383311yescharacteristic
614C2C3×#6397811yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#470
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
4. C2×C3×#3719C2semidirect
5. C3×#63573C2semidirect
6. C3×#63567C2semidirect
7. C3×#63566C2semidirect
10. #10591121C3direct (N × H)
38. C2×#3719C2×C3semidirect
45. #63573C2×C3semidirect
46. #63567C2×C3semidirect
47. #63566C2×C3semidirect
361. C33#470semidirect
547. C32#5052semidirect
548. C32#5137semidirect
549. C32C3×#470semidirect
608. C3#10591121direct (N × H)
609. C3C3×#5052semidirect
610. C3C3×#5137semidirect

Maximal subgroups

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

subgroupindex
C3×#641332
C32×#9242
C2×C32×#2082
C2×C3×#37192
C3×#635732
C3×#635672
C3×#635662
C3×#51373
C3×#50523
#105911213