Finite Group Databasebeta

← C3×#10591444 #13158525

Subgroups — C3×#10591444 #13158525

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64149C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C3×#63843C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5163—31nomaximal
9C3×#4762—31nomaximal
10#10591444C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×#638—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#214—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3717—22no—
24C2×C32×SD16—31no—
25#64149C2×C311yescharacteristic
26C3×Q8×Dic3—31no—
27C2×C3×#215—31no—
28C3×#940—31no—
29C3×#940—31no—
30C3×#940—61no—
31C2×C3×#214—21no—
32C2×C3×#214D311yescharacteristic, complemented
33C2×C3×#214C2×C311yescharacteristic, complemented
34C2×C3×#204—31no—
35C3×#887—31no—
36#63843C2×C311yescharacteristic, complemented
37C2×#3717C2×C311yescharacteristic, complemented
38C32×#102—31no—
39C3×#832—21no—
40C3×#832D311yescharacteristic, complemented
41C3×#832C2×C311yescharacteristic, complemented
42C3×#821—31no—
43C3×Dic3⋊C8—31no—
44C3×Dic3⋊C8—31no—
45C3×#809—31no—
46#63572C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C3×#638—42no—
54C3×#638—42no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C32×Dic6—21no—
62C32×Dic6D412yescomplemented
63C32×C3⋊C8—21no—
64C32×C3⋊C8—21no—
65C3×#460—91no—
66#5163—31no—
67#4762—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C2×#638—21no—
77C2×#638—21no—
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C2×C3×#67—61no—
80C2×C3×#67—61no—
81C2×C3×#67—61no—
82C2×C3×#67—61no—
83C2×C3×#67—62no—
84C2×C3×#67—62no—
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
88C2×C3×Dic6—21no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×C3×Dic6C22×C311yescharacteristic
91C32×SD16—62no—
92C2×C32×C8—31no—
93C2×C32×C8—31no—
94C32×C4⋊C4—31no—
95C4×C3⋊Dic3C22×C311yescharacteristic
96C3×#215—62no—
97C3×#214—24no—
98C3×#214—22no—
99C3×#214—22no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C22×C311yescharacteristic
104C3×Dic3⋊C4—61no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—61no—
108C3×C4×Dic3—61no—
109C3×C4×Dic3—61no—
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×#204—62no—
117Dic3⋊Dic3—21no—
118Dic32—21no—
119#3717—22no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊D3—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3SD1612yescomplemented
131C32×Dic3—22no—
132C32×Dic3—41no—
133C2×C3×SD16—31no—
134C2×C3×SD16—31no—
135C2×C3×SD16—61no—
136C2×C3×SD16—91no—
137C3×#127—91no—
138C3×C4×Q8—91no—
139Q8×Dic3—31no—
140C2×#215—31no—
141#940—31no—
142#940—31no—
143#940—61no—
144C2×#214C3×D311yescharacteristic, complemented
145C2×#204—31no—
146#887—31no—
147C3×C4⋊C8—91no—
148C3×#102—31no—
149C3×#102—31no—
150C3×#102—61no—
151C3×#102—91no—
152#832C3×D311yescharacteristic, complemented
153#821—31no—
154Dic3⋊C8—31no—
155Dic3⋊C8—31no—
156#809—31no—
157C23×C32—21no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C22×C3⋊D3—21no—
165C22×C3×D3—62no—
166C22×C3×D3—61no—
167C22×C3×D3—61no—
168C32×Q8—32no—
169C32×Q8—61no—
170C32×D4—21no—
171C32×D4—21no—
172C32×D4—22no—
173C32×D4—22no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—42no—
183C32×D4—42no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4—61no—
189C2×C32×C4—61no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C23×C311yescharacteristic
193#638—42no—
194#638—42no—
195C2×C3⋊Dic3C3×D411yescharacteristic
196C2×C3⋊Dic3C3×D411yescharacteristic
197C4⋊(C3⋊D3)—21no—
198C4⋊(C3⋊D3)—21no—
199C3×#67—122no—
200C3×#67—122no—
201C3×#67—124no—
202C3×#67—124no—
203C3×#67—122no—
204C3×#67—122no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—21no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—41no—
214C2×C3×Dic3—41no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C3×D12—61no—
218C3×D12—61no—
219C3×D12—61no—
220C3×D12—61no—
221C3×D12—121no—
222C3×D12—121no—
223C3×Dic6—21no—
224C3×Dic6—21no—
225C3×Dic6—22no—
226C3×Dic6—41no—
227C3×Dic6#6712yescomplemented
228C3×Dic6C3×D412yescomplemented
229C32×C8—61no—
230C32×C8—61no—
231C3×C3⋊C8—22no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—21no—
234C3×C3⋊C8—21no—
235C3×C3⋊C8—21no—
236C3×C3⋊C8—41no—
237#460—91noSylow
238C2×C33—81no—
239C2×C33#12111yescharacteristic
240C2×C33C2×SD1611yescharacteristic
241C2×C33AGL(1,ℤ/8)11yescharacteristic
242C3×C3⋊D3—81no—
243C2×C3×Q8—31no—
244C2×C3×Q8—31no—
245C2×C3×Q8—61no—
246C2×C3×D4—21no—
247C2×C3×D4—21no—
248C2×C3×D4—21no—
249C2×C3×D4—41no—
250C2×C3×D4—91no—
251C2×C3×D4—181no—
252C2×C3×D4—181no—
253C2×C3×D4D3211yescharacteristic
254C2×C3×D4C2×C3×D311yescharacteristic
255C2×C3×D4C2×C3×D311yescharacteristic
256C2×#67—61no—
257C2×#67—61no—
258C2×#67—61no—
259C2×#67—61no—
260C2×#67—62no—
261C2×#67—62no—
262C2×D12—31no—
263C2×D12—31no—
264C2×D12—61no—
265C2×Dic6C2×C3×D311yescharacteristic
266C3×SD16—62no—
267C3×SD16—62no—
268C3×SD16—64no—
269C3×SD16—182no—
270C2×C3×C8—31no—
271C2×C3×C8—31no—
272C2×C3×C8—31no—
273C2×C3×C8—31no—
274C2×C3×C8—61no—
275C2×C3×C8—61no—
276C3×C4⋊C4—31no—
277C3×C4⋊C4—31no—
278C3×C4⋊C4—61no—
279C3×C4⋊C4—181no—
280C3×C42—91no—
281C3×C42—181no—
282#215—62no—
283#214—22no—
284C4⋊Dic3—61no—
285C4⋊Dic3C2×C3×D311yescharacteristic
286Dic3⋊C4—61no—
287C4×Dic3—31no—
288C4×Dic3—31no—
289C4×Dic3—61no—
290C4×Dic3—61no—
291C4×Dic3—61no—
292C2×C3⋊C8C2×C3×D311yescharacteristic
293C2×C3⋊C8C2×C3×D311yescharacteristic
294#204—62no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—42no—
302C22×C32—44no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—42no—
309C22×C32—42no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32—41no—
313C22×C32—41no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—81no—
317C22×C32—81no—
318C22×C32—82no—
319C22×C32—82no—
320C22×C32D3×D411yescharacteristic
321C22×C32D3×D411yescharacteristic
322C22×C32C2×C3×D411yescharacteristic
323C2×C3⋊D3—41no—
324C2×C3⋊D3—41no—
325C2×C3⋊D3—41no—
326C2×C3×D3—122no—
327C2×C3×D3—122no—
328C2×C3×D3—122no—
329C2×C3×D3—121no—
330C2×C3×D3—121no—
331C2×C3×D3—121no—
332C2×C3×D3—121no—
333C2×C3×D3—121no—
334C2×C3×D3—121no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—41no—
342C32×C4—41no—
343C32×C4—62no—
344C32×C4—121no—
345C32×C4#23511yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4#23711yescharacteristic
348C32×C4C2×#6711yescharacteristic
349C32×C4C2×C3×D411yescharacteristic
350C32×C4C3×#4111yescharacteristic
351C3⋊Dic3—21no—
352C3⋊Dic3C3×SD1612yescomplemented
353C3×Dic3—22no—
354C3×Dic3—22no—
355C3×Dic3—32no—
356C3×Dic3—32no—
357C3×Dic3—42no—
358C3×Dic3—41no—
359C3×Dic3—41no—
360C3×Dic3—62no—
361C3×Dic3—62no—
362C3×Dic3—61no—
363C3×Dic3—61no—
364C3×Dic3—81no—
365C2×SD16—31no—
366C2×SD16—91no—
367#127—91no—
368C4×Q8—91no—
369C4⋊C8—91no—
370#102—31no—
371#102—91no—
372C33#46011yescharacteristic, complemented, Sylow
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—42no—
380C23×C3—181no—
381C22×D3—62no—
382C22×D3—61no—
383C22×D3—61no—
384C3×Q8—32no—
385C3×Q8—32no—
386C3×Q8—62no—
387C3×Q8—61no—
388C3×Q8—61no—
389C3×Q8—121no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—21no—
393C3×D4—21no—
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—21no—
397C3×D4—21no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—42no—
403C3×D4—42no—
404C3×D4—181no—
405C3×D4—181no—
406C3×D4—362no—
407C3×D4—362no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—41no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—61no—
415C2×C3×C4—61no—
416C2×C3×C4—91no—
417C2×C3×C4—91no—
418C2×C3×C4—121no—
419C2×C3×C4—121no—
420C2×C3×C4C2×D3211yescharacteristic
421C2×C3×C4C22×C3×D311yescharacteristic
422C2×C3×C4C22×C3×D311yescharacteristic
423#67—122no—
424#67—122no—
425#67—124no—
426#67—124no—
427#67—122no—
428#67—122no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—31no—
432C2×Dic3—31no—
433C2×Dic3—31no—
434C2×Dic3—31no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437D12—61no—
438D12—61no—
439D12—61no—
440D12—61no—
441D12—121no—
442D12—121no—
443Dic6—21no—
444Dic6C3×#6712yescomplemented
445C3×C8—62no—
446C3×C8—61no—
447C3×C8—61no—
448C3×C8—61no—
449C3×C8—61no—
450C3×C8—121no—
451C3⋊C8—21no—
452C3⋊C8—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—41no—
463C2×C32—41no—
464C2×C32—41no—
465C2×C32—82no—
466C2×C32—82no—
467C2×C32—82no—
468C2×C32—84no—
469C2×C32—81no—
470C2×C32—81no—
471C2×C32—81no—
472C2×C32#88411yescharacteristic
473C2×C32#91111yescharacteristic
474C2×C32D3×SD1611yescharacteristic
475C2×C32D3×SD1611yescharacteristic
476C2×C32#91411yescharacteristic
477C2×C32#93911yescharacteristic
478C2×C32C3×#12111yescharacteristic
479C2×C32C2×C3×SD1611yescharacteristic
480C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
481C3⋊D3—81no—
482C3×D3—242no—
483C3×D3—241no—
484C3×D3—241no—
485C2×Q8—31no—
486C2×D4—91no—
487C2×D4—181no—
488C2×D4—181no—
489C2×D4C3×D3211yescharacteristic
490SD16—62no—
491SD16—182no—
492C2×C8—31no—
493C2×C8—31no—
494C4⋊C4—31no—
495C4⋊C4—181no—
496C42—91no—
497C42—181no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—44no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—82no—
522C22×C3—82no—
523C22×C3—361no—
524C22×C3—361no—
525C22×C3—361no—
526C22×C3#381111yescharacteristic
527C22×C3C3×D3×D411yescharacteristic
528C22×C3C3×D3×D411yescharacteristic
529C2×D3—122no—
530C2×D3—122no—
531C2×D3—122no—
532C2×D3—121no—
533C2×D3—121no—
534C2×D3—121no—
535C2×D3—121no—
536C2×D3—121no—
537C2×D3—121no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—62no—
547C3×C4—62no—
548C3×C4—92no—
549C3×C4—122no—
550C3×C4—121no—
551C3×C4—121no—
552C3×C4—181no—
553C3×C4—241no—
554C3×C4#380411yescharacteristic
555C3×C4D3×#6711yescharacteristic
556C3×C4C3×#23511yescharacteristic
557C3×C4C3×D3×D411yescharacteristic
558C3×C4C3×#23711yescharacteristic
559C3×C4C2×C3×#6711yescharacteristic
560Dic3—22no—
561Dic3—32no—
562Dic3—32no—
563Dic3—41no—
564Dic3—62no—
565Dic3—62no—
566Dic3—61no—
567Dic3—61no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#476211yescharacteristic, complemented
573C32#516311yescharacteristic, complemented
574C32C3×#46011yescharacteristic, complemented
575C23—21no—
576C23—181no—
577Q8—32no—
578Q8—61no—
579D4—21no—
580D4—21no—
581D4—181no—
582D4—181no—
583D4—362no—
584D4—362no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—91no—
588C2×C4—91no—
589C2×C4C2×C3×D3211yescharacteristic
590C8—61no—
591C8—61no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—41no—
604C2×C3—82no—
605C2×C3—82no—
606C2×C3—82no—
607C2×C3—84no—
608C2×C3—81no—
609C2×C3—81no—
610C2×C3—81no—
611C2×C3—721no—
612C2×C3#6393011yescharacteristic
613C2×C3#6393711yescharacteristic
614C2×C3#6397111yescharacteristic
615C2×C3C3×#88411yescharacteristic
616C2×C3C3×#91111yescharacteristic
617C2×C3C3×D3×SD1611yescharacteristic
618C2×C3C3×D3×SD1611yescharacteristic
619C2×C3C3×#91411yescharacteristic
620C2×C3C3×#93911yescharacteristic
621D3—242no—
622D3—241no—
623D3—241no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—361no—
628C22—361no—
629C22—361no—
630C22C3×#381111yescharacteristic
631C4—62no—
632C4—92no—
633C4—121no—
634C4—181no—
635C4C3×#380411yescharacteristic
636C4C3×D3×#6711yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059144411yescharacteristic, direct product factor
642C3C3×#476211yescharacteristic, complemented
643C3C3×#516311yescharacteristic, complemented
644C2—81no—
645C2—721no—
646C2C3×#6393011yescharacteristic
647C2C3×#6393711yescharacteristic
648C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#460
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
2. C2×C32×#214C2semidirect
3. C3×#63843C2semidirect
4. C2×C3×#3717C2semidirect
5. C32×#832C2semidirect
6. C3×#63572C2semidirect
7. C3×#63564C2semidirect
10. #10591444C3direct (N × H)
32. C2×C3×#214D3semidirect
33. C2×C3×#214C2×C3semidirect
36. #63843C2×C3semidirect
37. C2×#3717C2×C3semidirect
40. C3×#832D3semidirect
41. C3×#832C2×C3semidirect
46. #63572C2×C3semidirect
47. #63564C2×C3semidirect
62. C32×Dic6D4semidirect
130. C3×C3⋊Dic3SD16semidirect
144. C2×#214C3×D3semidirect
152. #832C3×D3semidirect
227. C3×Dic6#67semidirect
228. C3×Dic6C3×D4semidirect
352. C3⋊Dic3C3×SD16semidirect
372. C33#460semidirect
444. Dic6C3×#67semidirect
572. C32#4762semidirect
573. C32#5163semidirect
574. C32C3×#460semidirect
641. C3#10591444direct (N × H)
642. C3C3×#4762semidirect
643. C3C3×#5163semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C32×#2142
C3×#638432
C2×C3×#37172
C32×#8322
C3×#635722
C3×#635642
C3×#51633
C3×#47623
#105914443