Finite Group Databasebeta

← C3×#10591431 #13158439

Subgroups — C3×#10591431 #13158439

Order 1728.

752 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#936C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic
3C3×Dic3×D12C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8C3×#5153—31nomaximal
9C3×#5057—31nomaximal
10#10591431C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×#217—21no—
19C32×#213—22no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3715—22no—
24C2×C32×D8—31no—
25C32×#124—31no—
26C2×C3×#215—31no—
27C3×#936—21no—
28C3×#936D311yescharacteristic, complemented
29C3×#936C2×C311yescharacteristic, complemented
30C3×D4×Dic3—31no—
31C2×C3×#214—31no—
32C2×C3×#213—21no—
33C2×C3×#213D311yescharacteristic
34C2×C3×#213C2×C311yescharacteristic
35C3×C4×D12—31no—
36Dic3×D12C2×C311yescharacteristic, complemented
37C2×#3715C2×C311yescharacteristic, complemented
38#63659C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#832—31no—
41C3×#832—31no—
42C3×#832—61no—
43C3×#816—31no—
44C3×Dic3⋊C8—31no—
45C3×C4⋊(C3⋊C8)—31no—
46Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
47#63566C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3—21no—
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12D412yes—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×D3×Dic3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#5153—31no—
66#5057—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—31no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C22×C3×Dic3—61no—
77C2×C3×D12—21no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C22×C311yescharacteristic
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C22×C311yescharacteristic
84C2×D3×Dic3—21no—
85C32×D8—62no—
86C2×C32×C8—31no—
87C32×C22⋊C4—61no—
88C32×C22⋊C4—61no—
89C32×C42—31no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—22no—
93C3×#217—22no—
94C3×#217—21no—
95C3×#217—21no—
96C3×#217—21no—
97C3×#217—21no—
98C3×#217—61no—
99C3×#215—62no—
100C3×#214—62no—
101C3×#213—24no—
102C3×#213—22no—
103C3×#213—22no—
104C3×(C2×D3)⋊C4—61no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—61no—
108C3×C4×Dic3—21no—
109C3×C4×Dic3C2×D311yescharacteristic
110C3×C4×Dic3C22×C311yescharacteristic
111C2×C3×C3⋊C8—21no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—61no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117(C2×D3)⋊Dic3—21no—
118#3715—22no—
119C22×C33—41no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33C2×D411yescharacteristic
123C2×C32×D3—22no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—41no—
129C32×Dic3—21no—
130C32×Dic3—21no—
131C32×Dic3—41no—
132C2×C3×SD16—91no—
133C2×C3×D8—31no—
134C2×C3×D8—31no—
135C2×C3×D8—61no—
136C3×#124—31no—
137C3×#124—31no—
138C3×#124—61no—
139C3×C4×D4—91no—
140C2×#215—31no—
141#936C3×D311yescharacteristic, complemented
142D4×Dic3—31no—
143C2×#214—31no—
144C2×#213C3×D311yescharacteristic
145C4×D12—31no—
146C3×C4⋊C8—91no—
147C3×#103—91no—
148C3×#102—91no—
149#835—31no—
150#832—31no—
151#832—31no—
152#832—61no—
153#816—31no—
154Dic3⋊C8—31no—
155C4⋊(C3⋊C8)—31no—
156C23×C32—21no—
157C23×C32—22no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—41no—
161C23×C32—41no—
162C23×C32—42no—
163C23×C32—61no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—41no—
167C32×Q8—61no—
168C32×Q8—61no—
169C32×D4—21no—
170C32×D4—21no—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—32no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—42no—
183C32×D4—42no—
184C32×D4—61no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—31no—
189C2×C32×C4—31no—
190C2×C32×C4—41no—
191C2×C32×C4—61no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195C2×C3⋊Dic3—21no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3—122no—
205C2×C3×Dic3#6711yescharacteristic
206C2×C3×Dic3#6711yescharacteristic
207C2×C3×Dic3C3×D411yescharacteristic
208C2×C3×Dic3C3×D411yescharacteristic
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—22no—
212C3×D12—41no—
213C3×D12#6712yes—
214C3×D12C3×D412yes—
215C3×C4×D3—122no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—21no—
219C3×Dic6—21no—
220C3×Dic6—41no—
221C3×Dic6—41no—
222D3×Dic3—42no—
223C32×C8—61no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—22no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—121no—
231#463—91noSylow
232C2×C33—81no—
233C2×C33#12111yescharacteristic
234C2×C33#13511yescharacteristic
235C2×C33AGL(1,ℤ/8)11yescharacteristic
236C32×D3—42no—
237C2×C3×Q8—31no—
238C2×C3×Q8—31no—
239C2×C3×Q8—61no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—31no—
244C2×C3×D4—31no—
245C2×C3×D4—41no—
246C2×C3×D4—61no—
247C2×C3×D4D3211yescharacteristic
248C2×C3×D4C2×C3×D311yescharacteristic
249C2×C3×D4C2×C3×D311yescharacteristic
250C22×C3×C4—181no—
251C22×Dic3—61no—
252C2×D12C2×C3×D311yescharacteristic
253C2×C4×D3—61no—
254C2×Dic6C2×C3×D311yescharacteristic
255C3×SD16—182no—
256C3×D8—62no—
257C3×D8—62no—
258C3×D8—64no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—61no—
262C2×C3×C8—91no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—61no—
265C3×C22⋊C4—61no—
266C3×C22⋊C4—61no—
267C3×C22⋊C4—61no—
268C3×C22⋊C4—62no—
269C3×C22⋊C4—62no—
270C3×C22⋊C4—181no—
271C3×C42—31no—
272C3×C42—31no—
273C3×C42—61no—
274#217—21no—
275#217—21no—
276#217—61no—
277#215—62no—
278#214—62no—
279#213—22no—
280(C2×D3)⋊C4—61no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—61no—
284C4×Dic3C2×C3×D311yescharacteristic
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8C2×C3×D311yescharacteristic
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—42no—
294C22×C32—42no—
295C22×C32—42no—
296C22×C32—44no—
297C22×C32—41no—
298C22×C32—41no—
299C22×C32—41no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—42no—
303C22×C32—42no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—62no—
309C22×C32—81no—
310C22×C32—81no—
311C22×C32—81no—
312C22×C32—81no—
313C22×C32—82no—
314C22×C32—82no—
315C22×C32—121no—
316C22×C32—121no—
317C22×C32D3×D411yescharacteristic
318C22×C32C2×#6711yescharacteristic
319C22×C32C2×C3×D411yescharacteristic
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—42no—
323C2×C3×D3—41no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—81no—
328C2×C3×D3—81no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—41no—
336C32×C4—41no—
337C32×C4—61no—
338C32×C4—61no—
339C32×C4—121no—
340C32×C4D3×D411yescharacteristic
341C32×C4#23711yescharacteristic
342C32×C4#23711yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C32×C4C2×C3×D411yescharacteristic
345C32×C4C3×#4111yescharacteristic
346C3⋊Dic3—41no—
347C3×Dic3—21no—
348C3×Dic3—21no—
349C3×Dic3—21no—
350C3×Dic3—21no—
351C3×Dic3—41no—
352C3×Dic3—41no—
353C3×Dic3—42no—
354C3×Dic3—41no—
355C3×Dic3—41no—
356C3×Dic3—121no—
357C3×Dic3—121no—
358C3×Dic3—241no—
359C2×SD16—91no—
360C2×D8—31no—
361#124—31no—
362C4×D4—91no—
363C4⋊C8—91no—
364#103—91no—
365#102—91no—
366C33#46311yescharacteristic, complemented, Sylow
367C23×C3—22no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—42no—
374C23×C3—61no—
375C23×C3—61no—
376C23×C3—121no—
377C22×D3—21no—
378C3×Q8—61no—
379C3×Q8—61no—
380C3×Q8—61no—
381C3×Q8—61no—
382C3×Q8—121no—
383C3×Q8—121no—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—21no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—32no—
393C3×D4—32no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—62no—
401C3×D4—61no—
402C3×D4—61no—
403C3×D4—121no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—31no—
408C2×C3×C4—31no—
409C2×C3×C4—31no—
410C2×C3×C4—31no—
411C2×C3×C4—41no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—62no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—181no—
418C2×C3×C4—362no—
419C2×C3×C4C2×D3211yescharacteristic
420C2×C3×C4C22×C3×D311yescharacteristic
421C2×C3×C4C22×C3×D311yescharacteristic
422C2×Dic3—21no—
423C2×Dic3—61no—
424C2×Dic3—61no—
425C2×Dic3—121no—
426C2×Dic3—122no—
427C2×Dic3C3×#6711yescharacteristic
428C2×Dic3C3×#6711yescharacteristic
429D12—21no—
430D12C3×#6712yes—
431C4×D3—122no—
432Dic6—21no—
433Dic6—21no—
434C3×C8—62no—
435C3×C8—61no—
436C3×C8—61no—
437C3×C8—181no—
438C3⋊C8—21no—
439C3⋊C8—61no—
440C3⋊C8—61no—
441C3⋊C8—121no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—41no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—82no—
455C2×C32—82no—
456C2×C32—82no—
457C2×C32—84no—
458C2×C32—81no—
459C2×C32—81no—
460C2×C32—81no—
461C2×C32—122no—
462C2×C32#91111yescharacteristic
463C2×C32#91211yescharacteristic
464C2×C32#93811yescharacteristic
465C2×C32#93911yescharacteristic
466C2×C32#94911yescharacteristic
467C2×C32#95011yescharacteristic
468C2×C32C3×#12111yescharacteristic
469C2×C32C3×#13511yescharacteristic
470C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
471C3×D3—42no—
472C3×D3—42no—
473C3×D3—82no—
474C2×Q8—31no—
475C2×D4—31no—
476C2×D4C3×D3211yescharacteristic
477C22×C4—181no—
478SD16—182no—
479D8—62no—
480C2×C8—31no—
481C2×C8—91no—
482C4⋊C4—91no—
483C22⋊C4—61no—
484C22⋊C4—61no—
485C22⋊C4—181no—
486C42—31no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—44no—
496C22×C3—41no—
497C22×C3—41no—
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—62no—
507C22×C3—62no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—82no—
513C22×C3—82no—
514C22×C3—122no—
515C22×C3—121no—
516C22×C3—121no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3—241no—
520C22×C3—241no—
521C22×C3D3×#6711yescharacteristic
522C22×C3C3×D3×D411yescharacteristic
523C22×C3C2×C3×#6711yescharacteristic
524C2×D3—22no—
525C2×D3—41no—
526C2×D3—41no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—41no—
534C3×C4—41no—
535C3×C4—61no—
536C3×C4—61no—
537C3×C4—61no—
538C3×C4—61no—
539C3×C4—121no—
540C3×C4—121no—
541C3×C4—122no—
542C3×C4—121no—
543C3×C4—121no—
544C3×C4—361no—
545C3×C4#380511yescharacteristic
546C3×C4D3×#6711yescharacteristic
547C3×C4C3×D3×D411yescharacteristic
548C3×C4C3×#23711yescharacteristic
549C3×C4C3×#23711yescharacteristic
550C3×C4C2×C3×#6711yescharacteristic
551Dic3—21no—
552Dic3—21no—
553Dic3—41no—
554Dic3—121no—
555Dic3—121no—
556Dic3—241no—
557C32—21no—
558C32—21no—
559C32—21no—
560C32—41no—
561C32#505711yescharacteristic, complemented
562C32#515311yescharacteristic, complemented
563C32C3×#46311yescharacteristic, complemented
564C23—21no—
565C23—61no—
566Q8—61no—
567Q8—61no—
568D4—21no—
569D4—21no—
570D4—32no—
571D4—61no—
572C2×C4—31no—
573C2×C4—31no—
574C2×C4—61no—
575C2×C4—181no—
576C2×C4—362no—
577C2×C4C2×C3×D3211yescharacteristic
578C8—61no—
579C8—181no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—41no—
592C2×C3—82no—
593C2×C3—82no—
594C2×C3—82no—
595C2×C3—84no—
596C2×C3—81no—
597C2×C3—81no—
598C2×C3—81no—
599C2×C3—122no—
600C2×C3—122no—
601C2×C3—242no—
602C2×C3#6392511yescharacteristic
603C2×C3#6393311yescharacteristic
604C2×C3#6397011yescharacteristic
605C2×C3C3×#91111yescharacteristic
606C2×C3C3×#91211yescharacteristic
607C2×C3C3×#93811yescharacteristic
608C2×C3C3×#93911yescharacteristic
609C2×C3C3×#94911yescharacteristic
610C2×C3C3×#95011yescharacteristic
611D3—42no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—62no—
616C22—121no—
617C22—121no—
618C22C3×D3×#6711yescharacteristic
619C4—61no—
620C4—61no—
621C4—121no—
622C4—361no—
623C4C3×#380511yescharacteristic
624C4C3×D3×#6711yescharacteristic
625C3—21no—
626C3—21no—
627C3—21no—
628C3—41no—
629C3#1059143111yescharacteristic, direct product factor
630C3C3×#505711yescharacteristic, complemented
631C3C3×#515311yescharacteristic, complemented
632C2—81no—
633C2—122no—
634C2C3×#6392511yescharacteristic
635C2C3×#6393311yescharacteristic
636C2C3×#6397011yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#213C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#463
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. C32×#936C2semidirect
3. C3×Dic3×D12C2semidirect
4. C2×C3×#3715C2semidirect
5. C3×#63659C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63566C2semidirect
10. #10591431C3direct (N × H)
28. C3×#936D3semidirect
29. C3×#936C2×C3semidirect
36. Dic3×D12C2×C3semidirect
37. C2×#3715C2×C3semidirect
38. #63659C2×C3semidirect
46. Dic3⋊(C3⋊C8)C2×C3semidirect
47. #63566C2×C3semidirect
141. #936C3×D3semidirect
366. C33#463semidirect
561. C32#5057semidirect
562. C32#5153semidirect
563. C32C3×#463semidirect
629. C3#10591431direct (N × H)
630. C3C3×#5057semidirect
631. C3C3×#5153semidirect

Maximal subgroups

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

subgroupindex
C32×#9362
C2×C32×#2132
C3×Dic3×D122
C2×C3×#37152
C3×#636592
C3×Dic3⋊(C3⋊C8)2
C3×#635662
C3×#51533
C3×#50573
#105914313