Finite Group Databasebeta

← C3×#10591462 #13158474

Subgroups — C3×#10591462 #13158474

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3754C211yesmaximal, characteristic, complemented
2C32×#938C211yesmaximal, characteristic
3C3×#63857C211yesmaximal, characteristic, complemented
4C2×C3×#3711C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8C3×#5042—31nomaximal
9C3×#4780—31nomaximal
10#10591462C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×#67—21no—
14C2×C32×C4×D3C2211yescharacteristic
15C3×#3754—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3711—22no—
24C32×#121—31no—
25C2×#3754C2×C311yescharacteristic, complemented
26C3×#938—21no—
27C3×#938D311yescharacteristic
28C3×#938C2×C311yescharacteristic
29C2×C3×#214—31no—
30C2×C3×#214—31no—
31C2×C3×#214—61no—
32C3×#924—31no—
33C2×C3×#208—31no—
34C2×C3×ΓL(1,𝔽25)—31no—
35C3×#897—31no—
36#63857C2×C311yescharacteristic, complemented
37C2×#3711C2×C311yescharacteristic, complemented
38C32×#102—31no—
39C3×#832—21no—
40C3×#832D311yescharacteristic, complemented
41C3×#832C2×C311yescharacteristic, complemented
42C3×#816—31no—
43C3×#810—31no—
44C3×#807—31no—
45C3×#807—31no—
46#63576C2×C311yescharacteristic, complemented
47#63567C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3D411yescharacteristic
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C32×#67—42no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D411yescharacteristic
60C32×C4×D3—21no—
61C32×C4×D3—21no—
62C3×C3⋊(C3⋊C8)D412yescomplemented
63C32×C3⋊C8—21no—
64C3×#468—91no—
65#5042—31no—
66#4780—31no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—41no—
71C2×C32×D4—61no—
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C22×C311yescharacteristic
75C22×C32×C4—31no—
76C2×#634C22×C311yescharacteristic
77C2×C3×#67—22no—
78C2×C3×#67—21no—
79C2×C3×#67—21no—
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3C2×D311yescharacteristic
82C2×C3×C4×D3C22×C311yescharacteristic
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×C32×C8—31no—
87C32×C4⋊C4—31no—
88C32×C4⋊C4—31no—
89C32×C22⋊C4—61no—
90#3754—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—22no—
93C3×#217—21no—
94C3×#217—21no—
95C3×#217—61no—
96C3×#214—62no—
97C3×#214—62no—
98C3×#214—64no—
99C3×(C2×D3)⋊C4—61no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—61no—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4⋊Dic3C2×D311yescharacteristic
105C3×C4⋊Dic3C22×C311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C3×Dic3⋊C4—61no—
108C3×#208—62no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×ΓL(1,𝔽25)—62no—
116Dic3⋊Dic3—21no—
117(C2×D3)⋊Dic3—21no—
118#3711—22no—
119C22×C33—41no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33C2×D411yescharacteristic
123C2×C32×D3—21no—
124C2×C32×D3—21no—
125C2×C32×D3—21no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—41no—
129C32×Dic3—21no—
130C32×Dic3—41no—
131C32×Dic3—41no—
132C2×C3×SD16—91no—
133C2×C3×M16—91no—
134C3×C22⋊Q8—91no—
135C3×#121—31no—
136C3×#121—31no—
137C3×#121—61no—
138#938C3×D311yescharacteristic
139C2×#214—31no—
140C2×#214—31no—
141C2×#214—61no—
142#924—31no—
143C2×#208—31no—
144C2×ΓL(1,𝔽25)—31no—
145#897—31no—
146C3×#107—91no—
147C3×#103—91no—
148C3×#102—31no—
149C3×#102—31no—
150C3×#102—61no—
151#832C3×D311yescharacteristic, complemented
152#816—31no—
153#810—31no—
154#807—31no—
155#807—31no—
156C23×C32—21no—
157C23×C32—22no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—31no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C22×C3×D3—21no—
165C22×C3×D3#6711yescharacteristic
166C22×C3×D3C3×D411yescharacteristic
167C32×D4—21no—
168C32×D4—21no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—21no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—41no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—42no—
180C32×D4—42no—
181C32×D4—122no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—31no—
186C2×C32×C4—41no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4—61no—
190C2×C32×C4—61no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C311yescharacteristic
194C2×C3⋊Dic3—21no—
195#634—21no—
196#634—21no—
197C3×#67—44no—
198C3×#67—42no—
199C3×#67—42no—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—41no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—121no—
210C2×C3×Dic3#6711yescharacteristic
211C2×C3×Dic3C3×D411yescharacteristic
212C3×C4×D3—21no—
213C3×C4×D3—21no—
214C3×C4×D3—21no—
215C3×C4×D3—21no—
216C3×C4×D3—41no—
217C3×C4×D3—41no—
218C3×Dic6—62no—
219C3×Dic6—61no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—61no—
223C3×Dic6—121no—
224C32×C8—61no—
225C3⋊(C3⋊C8)C3×D412yescomplemented
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—32no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—41no—
231C3×C3⋊C8—62no—
232#468—91noSylow
233C2×C33—81no—
234C2×C33#12111yescharacteristic
235C2×C33AGL(1,ℤ/8)11yescharacteristic
236C2×C33#13711yescharacteristic
237C32×D3—41no—
238C2×C3×Q8—91no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—41no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—62no—
246C2×C3×D4D3211yescharacteristic
247C2×C3×D4C2×C3×D311yescharacteristic
248C2×C3×D4C2×C3×D311yescharacteristic
249C22×C3×C4—31no—
250C22×C3×C4—31no—
251C22×C3×C4—61no—
252C2×#67—21no—
253C2×C4×D3C2×C3×D311yescharacteristic
254C2×Dic6—31no—
255C2×Dic6—31no—
256C2×Dic6—61no—
257C3×SD16—182no—
258C3×M16—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—61no—
262C2×C3×C8—91no—
263C3×C4⋊C4—31no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4—181no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—61no—
272C3×C22⋊C4—62no—
273C3×C22⋊C4—181no—
274#217—21no—
275#217—61no—
276#214—62no—
277#214—62no—
278#214—64no—
279(C2×D3)⋊C4—61no—
280C4⋊Dic3—61no—
281C4⋊Dic3C2×C3×D311yescharacteristic
282C4⋊Dic3C2×C3×D311yescharacteristic
283Dic3⋊C4—61no—
284#208—62no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8C2×C3×D311yescharacteristic
289ΓL(1,𝔽25)—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—42no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—44no—
298C22×C32—41no—
299C22×C32—41no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—42no—
304C22×C32—42no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—61no—
310C22×C32—61no—
311C22×C32—61no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32D3×D411yescharacteristic
319C22×C32C2×#6711yescharacteristic
320C22×C32C2×C3×D411yescharacteristic
321C2×C3×D3—22no—
322C2×C3×D3—21no—
323C2×C3×D3—21no—
324C2×C3×D3—21no—
325C2×C3×D3—21no—
326C2×C3×D3—21no—
327C2×C3×D3—21no—
328C2×C3×D3—41no—
329C2×C3×D3—41no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4—61no—
339C32×C4—121no—
340C32×C4—121no—
341C32×C4#23511yescharacteristic
342C32×C4D3×D411yescharacteristic
343C32×C4D3×D411yescharacteristic
344C32×C4#23711yescharacteristic
345C32×C4C2×C3×D411yescharacteristic
346C32×C4C3×#4111yescharacteristic
347C3⋊Dic3—41no—
348C3×Dic3—21no—
349C3×Dic3—21no—
350C3×Dic3—41no—
351C3×Dic3—42no—
352C3×Dic3—41no—
353C3×Dic3—41no—
354C3×Dic3—41no—
355C3×Dic3—41no—
356C3×Dic3—81no—
357C3×Dic3—122no—
358C3×Dic3—121no—
359C3×Dic3—121no—
360C2×SD16—91no—
361C2×M16—91no—
362C22⋊Q8—91no—
363#121—31no—
364#107—91no—
365#103—91no—
366#102—31no—
367C33#46811yescharacteristic, complemented, Sylow
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—31no—
373C23×C3—31no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—61no—
378C22×D3C3×#6711yescharacteristic
379C3×Q8—181no—
380C3×Q8—181no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—21no—
384C3×D4—21no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—124no—
396C3×D4—122no—
397C3×D4—122no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—31no—
402C2×C3×C4—31no—
403C2×C3×C4—41no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—62no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—121no—
415C2×C3×C4—121no—
416C2×C3×C4—121no—
417C2×C3×C4—181no—
418C2×C3×C4C2×D3211yescharacteristic
419C2×C3×C4C22×C3×D311yescharacteristic
420C2×C3×C4C22×C3×D311yescharacteristic
421#67—42no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—61no—
425C2×Dic3—61no—
426C2×Dic3—121no—
427C2×Dic3C3×#6711yescharacteristic
428C4×D3—21no—
429C4×D3—21no—
430Dic6—62no—
431Dic6—61no—
432Dic6—61no—
433Dic6—61no—
434Dic6—61no—
435Dic6—121no—
436C3×C8—61no—
437C3×C8—61no—
438C3×C8—92no—
439C3×C8—121no—
440C3⋊C8—21no—
441C3⋊C8—32no—
442C3⋊C8—32no—
443C3⋊C8—62no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—82no—
457C2×C32—82no—
458C2×C32—82no—
459C2×C32—84no—
460C2×C32—81no—
461C2×C32—81no—
462C2×C32—81no—
463C2×C32—121no—
464C2×C32#88411yescharacteristic
465C2×C32#91111yescharacteristic
466C2×C32#91511yescharacteristic
467C2×C32#93211yescharacteristic
468C2×C32#93911yescharacteristic
469C2×C32#95111yescharacteristic
470C2×C32C3×#12111yescharacteristic
471C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
472C2×C32C3×#13711yescharacteristic
473C3×D3—42no—
474C3×D3—41no—
475C3×D3—41no—
476C2×Q8—91no—
477C2×D4—61no—
478C2×D4C3×D3211yescharacteristic
479C22×C4—31no—
480SD16—182no—
481M16—182no—
482C2×C8—31no—
483C2×C8—91no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—181no—
487C22⋊C4—61no—
488C22⋊C4—181no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—44no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—62no—
509C22×C3—61no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—61no—
513C22×C3—61no—
514C22×C3—61no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—82no—
520C22×C3—82no—
521C22×C3—121no—
522C22×C3—121no—
523C22×C3D3×#6711yescharacteristic
524C22×C3C3×D3×D411yescharacteristic
525C22×C3C2×C3×#6711yescharacteristic
526C2×D3—21no—
527C2×D3—21no—
528C2×D3—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—61no—
538C3×C4—61no—
539C3×C4—121no—
540C3×C4—122no—
541C3×C4—121no—
542C3×C4—121no—
543C3×C4—121no—
544C3×C4—121no—
545C3×C4—241no—
546C3×C4—361no—
547C3×C4#380411yescharacteristic
548C3×C4#381111yescharacteristic
549C3×C4C3×#23511yescharacteristic
550C3×C4C3×D3×D411yescharacteristic
551C3×C4C3×D3×D411yescharacteristic
552C3×C4C3×#23711yescharacteristic
553Dic3—21no—
554Dic3—41no—
555Dic3—41no—
556Dic3—122no—
557Dic3—121no—
558Dic3—121no—
559C32—21no—
560C32—21no—
561C32—21no—
562C32—41no—
563C32#478011yescharacteristic, complemented
564C32#504211yescharacteristic, complemented
565C32C3×#46811yescharacteristic, complemented
566C23—21no—
567C23—31no—
568Q8—181no—
569Q8—181no—
570D4—21no—
571D4—21no—
572D4—122no—
573C2×C4—31no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—181no—
579C2×C4C2×C3×D3211yescharacteristic
580C8—61no—
581C8—92no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—82no—
595C2×C3—82no—
596C2×C3—82no—
597C2×C3—84no—
598C2×C3—81no—
599C2×C3—81no—
600C2×C3—81no—
601C2×C3—122no—
602C2×C3—121no—
603C2×C3—121no—
604C2×C3#6392911yescharacteristic
605C2×C3#6393211yescharacteristic
606C2×C3#6397111yescharacteristic
607C2×C3C3×#88411yescharacteristic
608C2×C3C3×#91111yescharacteristic
609C2×C3C3×#91511yescharacteristic
610C2×C3C3×#93211yescharacteristic
611C2×C3C3×#93911yescharacteristic
612C2×C3C3×#95111yescharacteristic
613D3—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—61no—
618C22—61no—
619C22—61no—
620C22C3×D3×#6711yescharacteristic
621C4—61no—
622C4—121no—
623C4—121no—
624C4—361no—
625C4C3×#380411yescharacteristic
626C4C3×#381111yescharacteristic
627C3—21no—
628C3—21no—
629C3—21no—
630C3—41no—
631C3#1059146211yescharacteristic, direct product factor
632C3C3×#478011yescharacteristic, complemented
633C3C3×#504211yescharacteristic, complemented
634C2—81no—
635C2—121no—
636C2C3×#6392911yescharacteristic
637C2C3×#6393211yescharacteristic
638C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3754C2semidirect
3. C3×#63857C2semidirect
4. C2×C3×#3711C2semidirect
5. C32×#832C2semidirect
6. C3×#63576C2semidirect
7. C3×#63567C2semidirect
10. #10591462C3direct (N × H)
25. C2×#3754C2×C3semidirect
36. #63857C2×C3semidirect
37. C2×#3711C2×C3semidirect
40. C3×#832D3semidirect
41. C3×#832C2×C3semidirect
46. #63576C2×C3semidirect
47. #63567C2×C3semidirect
62. C3×C3⋊(C3⋊C8)D4semidirect
151. #832C3×D3semidirect
225. C3⋊(C3⋊C8)C3×D4semidirect
367. C33#468semidirect
563. C32#4780semidirect
564. C32#5042semidirect
565. C32C3×#468semidirect
631. C3#10591462direct (N × H)
632. C3C3×#4780semidirect
633. C3C3×#5042semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9382
C3×#638572
C2×C3×#37112
C32×#8322
C3×#635762
C3×#635672
C3×#50423
C3×#47803
#105914623