Finite Group Databasebeta

← C3×#10591483 #13158496

Subgroups — C3×#10591483 #13158496

Order 1728.

858 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
1C2×C3×#3753C211yesmaximal, characteristic, complemented
2C32×#936C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic, complemented
5C2×C3×#3717C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5154—31nomaximal
9C3×#5058—31nomaximal
10#10591483C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#626—22no—
16C3×#3753—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—22no—
19C32×#213—22no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3717—22no—
23C3×#3715—22no—
24C2×C32×D8—31no—
25C32×#124—31no—
26C2×#3753C2×C311yescharacteristic, complemented
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C3×#940—31no—
30C3×#936—21no—
31C3×#936D311yescharacteristic, complemented
32C3×#936C2×C311yescharacteristic, complemented
33C2×C3×#214—31no—
34C2×C3×#213—21no—
35C2×C3×#213—31no—
36C2×C3×#213—31no—
37C2×C3×#213—61no—
38C2×C3×#213D311yescharacteristic, complemented
39C2×C3×#213C2×C311yescharacteristic, complemented
40C2×C3×#204—31no—
41C3×#874—31no—
42#63911C2×C311yescharacteristic, complemented
43C2×#3717C2×C311yescharacteristic, complemented
44C2×#3715C2×C311yescharacteristic, complemented
45C3×#815—31no—
46C3×#803—31no—
47#63554C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C33×D4—21no—
52C33×D4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×#626—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65C3×#496—91no—
66#5154—31no—
67#5058—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—31no—
73C2×C32×D4—41no—
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C22×C311yescharacteristic
77C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
78C2×C3×#67—62no—
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×D12—62no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×D12C22×C311yescharacteristic
86C2×C3×Dic6—21no—
87C2×C3×Dic6C2×D311yescharacteristic
88C2×C3×Dic6C22×C311yescharacteristic
89C2×#626—22no—
90C32×D8—62no—
91C2×C32×C8—31no—
92C32×C22⋊C4—62no—
93C32×C42—31no—
94#3753—22no—
95C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
96C3×#217—24no—
97C3×#217—22no—
98C3×#217—22no—
99C3×#215—62no—
100C3×#215—62no—
101C3×#214—62no—
102C3×#213—24no—
103C3×#213—22no—
104C3×#213—22no—
105C3×#213—62no—
106C3×#213—62no—
107C3×#213—64no—
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
117C3×#204—62no—
118#3717—22no—
119#3715—22no—
120C22×C33—41no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—42no—
125C2×C32×D3—41no—
126C2×C32×D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C32×Dic3—22no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×SD16—91no—
133C2×C3×D8—31no—
134C2×C3×D8—31no—
135C2×C3×D8—61no—
136C2×C3×D8—91no—
137C3×#127—91no—
138C3×#124—31no—
139C3×#124—31no—
140C3×#124—61no—
141C2×#215—31no—
142C2×#215—31no—
143#940—31no—
144#936C3×D311yescharacteristic, complemented
145C2×#214—31no—
146C2×#213—31no—
147C2×#213—31no—
148C2×#213—61no—
149C2×#213C3×D311yescharacteristic, complemented
150C2×#204—31no—
151#874—31no—
152C3×#97—91no—
153#815—31no—
154#803—31no—
155C23×C32—21no—
156C23×C32—22no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—41no—
160C23×C32—41no—
161C23×C32—42no—
162C23×C32—61no—
163C22×C3⋊D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C22×C3×D3—41no—
167C22×C3×D3—61no—
168C22×C3×D3—61no—
169C22×C3×D3—121no—
170C32×Q8—61no—
171C32×Q8—61no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—21no—
179C32×D4—21no—
180C32×D4—41no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—41no—
184C32×D4—42no—
185C32×D4—42no—
186C32×D4—61no—
187C32×D4—61no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—32no—
192C2×C32×C4—41no—
193C2×C32×C4—61no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C23×C311yescharacteristic
197C4⋊(C3⋊D3)—21no—
198C4⋊(C3⋊D3)—21no—
199C3×#67—124no—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—22no—
204C2×C3×Dic3#6712yes—
205C2×C3×Dic3C3×D412yes—
206C3×D12—22no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—41no—
212C3×D12—62no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—124no—
219C3×Dic6—21no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—41no—
224C3×Dic6—41no—
225#626—44no—
226C32×C8—32no—
227C3⋊(C3⋊C8)C3×D412yescomplemented
228C3×C3⋊C8—22no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—62no—
232C3×C3⋊C8#6712yescomplemented
233C3×C3⋊C8C3×D412yescomplemented
234#496—91noSylow
235C2×C33—81no—
236C2×C33#12711yescharacteristic
237C2×C33AGL(1,ℤ/8)12yes—
238C3×C3⋊D3—81no—
239C32×D3—81no—
240C2×C3×Q8—31no—
241C2×C3×Q8—31no—
242C2×C3×Q8—61no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—31no—
247C2×C3×D4—31no—
248C2×C3×D4—41no—
249C2×C3×D4—61no—
250C2×C3×D4—91no—
251C2×C3×D4—182no—
252C2×C3×D4D3211yescharacteristic
253C2×C3×D4C2×C3×D311yescharacteristic
254C2×C3×D4C2×C3×D311yescharacteristic
255C2×#67—62no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—61no—
259C2×D12—62no—
260C2×D12C2×C3×D311yescharacteristic
261C2×Dic6C2×C3×D311yescharacteristic
262C3×SD16—182no—
263C3×SD16—182no—
264C3×D8—62no—
265C3×D8—62no—
266C3×D8—64no—
267C3×D8—182no—
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—61no—
271C2×C3×C8—91no—
272C3×C22⋊C4—62no—
273C3×C22⋊C4—62no—
274C3×C22⋊C4—64no—
275C3×C42—31no—
276C3×C42—31no—
277C3×C42—61no—
278#217—22no—
279#215—62no—
280#215—62no—
281#214—62no—
282#213—22no—
283#213—62no—
284#213—62no—
285#213—64no—
286C4×Dic3C2×C3×D311yescharacteristic
287C2×C3⋊C8—31no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—61no—
290C2×C3⋊C8C2×C3×D311yescharacteristic
291#204—62no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32—42no—
297C22×C32—42no—
298C22×C32—42no—
299C22×C32—44no—
300C22×C32—42no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32—44no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—82no—
308C22×C32—82no—
309C22×C32—84no—
310C22×C32—121no—
311C22×C32—122no—
312C22×C32D3×D411yescharacteristic
313C22×C32C2×#6711yescharacteristic
314C22×C32C2×C3×D411yescharacteristic
315C2×C3⋊D3—41no—
316C2×C3⋊D3—42no—
317C2×C3×D3—42no—
318C2×C3×D3—42no—
319C2×C3×D3—41no—
320C2×C3×D3—41no—
321C2×C3×D3—42no—
322C2×C3×D3—82no—
323C2×C3×D3—122no—
324C2×C3×D3—122no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C2×C3×D3—122no—
328C2×C3×D3—242no—
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—62no—
338C32×C4—121no—
339C32×C4D3×D411yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4C2×#6711yescharacteristic
343C32×C4C2×C3×D411yescharacteristic
344C32×C4C2×C3×D411yescharacteristic
345C3×Dic3—22no—
346C3×Dic3—22no—
347C3×Dic3—42no—
348C3×Dic3—42no—
349C3×Dic3—41no—
350C3×Dic3—41no—
351C2×SD16—91no—
352C2×SD16—91no—
353C2×D8—31no—
354C2×D8—91no—
355#127—91no—
356#124—31no—
357#97—91no—
358C33#49611yescharacteristic, complemented, Sylow
359C23×C3—22no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—42no—
366C23×C3—61no—
367C23×C3—61no—
368C23×C3—121no—
369C23×C3—181no—
370C22×D3—21no—
371C22×D3—61no—
372C22×D3—61no—
373C22×D3—121no—
374C3×Q8—61no—
375C3×Q8—61no—
376C3×Q8—61no—
377C3×Q8—61no—
378C3×Q8—121no—
379C3×Q8—121no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—21no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—21no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—42no—
393C3×D4—42no—
394C3×D4—62no—
395C3×D4—61no—
396C3×D4—61no—
397C3×D4—61no—
398C3×D4—61no—
399C3×D4—121no—
400C3×D4—181no—
401C3×D4—181no—
402C3×D4—364no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—32no—
407C2×C3×C4—32no—
408C2×C3×C4—41no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4C2×D3211yescharacteristic
414C2×C3×C4C22×C3×D311yescharacteristic
415C2×C3×C4C22×C3×D311yescharacteristic
416#67—124no—
417C2×Dic3—21no—
418C2×Dic3C3×#6712yes—
419D12—21no—
420D12—21no—
421D12—62no—
422D12—61no—
423D12—61no—
424D12—61no—
425D12—61no—
426D12—121no—
427D12—124no—
428Dic6—21no—
429Dic6—21no—
430C3×C8—32no—
431C3×C8—32no—
432C3×C8—62no—
433C3×C8—92no—
434C3⋊C8—32no—
435C3⋊C8—32no—
436C3⋊C8—62no—
437C3⋊C8C3×#6712yescomplemented
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—22no—
442C2×C32—22no—
443C2×C32—21no—
444C2×C32—41no—
445C2×C32—42no—
446C2×C32—82no—
447C2×C32—82no—
448C2×C32—82no—
449C2×C32—84no—
450C2×C32—81no—
451C2×C32—81no—
452C2×C32—81no—
453C2×C32—241no—
454C2×C32#91112yes—
455C2×C32#94011yescharacteristic
456C2×C32#94011yescharacteristic
457C2×C32#94912yes—
458C2×C32C3×#12711yescharacteristic
459C2×C32C3×AGL(1,ℤ/8)12yes—
460C3⋊D3—81no—
461C3×D3—82no—
462C3×D3—81no—
463C3×D3—81no—
464C3×D3—242no—
465C3×D3—241no—
466C3×D3—241no—
467C2×Q8—31no—
468C2×D4—31no—
469C2×D4—91no—
470C2×D4—182no—
471C2×D4C3×D3211yescharacteristic
472SD16—182no—
473SD16—182no—
474D8—62no—
475D8—182no—
476C2×C8—31no—
477C2×C8—91no—
478C22⋊C4—62no—
479C42—31no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—84no—
498C22×C3—122no—
499C22×C3—121no—
500C22×C3—122no—
501C22×C3—122no—
502C22×C3—121no—
503C22×C3—242no—
504C22×C3—362no—
505C22×C3—361no—
506C22×C3D3×#6711yescharacteristic
507C22×C3C3×D3×D411yescharacteristic
508C22×C3C2×C3×#6711yescharacteristic
509C2×D3—42no—
510C2×D3—41no—
511C2×D3—122no—
512C2×D3—122no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—122no—
516C2×D3—242no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—62no—
526C3×C4—62no—
527C3×C4—122no—
528C3×C4—122no—
529C3×C4—121no—
530C3×C4—121no—
531C3×C4D3×#6711yescharacteristic
532C3×C4#381111yescharacteristic
533C3×C4C3×D3×D411yescharacteristic
534C3×C4C3×D3×D411yescharacteristic
535C3×C4C3×D3×D411yescharacteristic
536C3×C4C2×C3×#6711yescharacteristic
537Dic3—22no—
538Dic3—41no—
539C32—21no—
540C32—21no—
541C32—21no—
542C32—41no—
543C32#505811yescharacteristic, complemented
544C32#515411yescharacteristic, complemented
545C32C3×#49611yescharacteristic, complemented
546C23—21no—
547C23—61no—
548C23—181no—
549Q8—61no—
550Q8—61no—
551D4—21no—
552D4—21no—
553D4—61no—
554D4—61no—
555D4—181no—
556D4—181no—
557D4—364no—
558C2×C4—32no—
559C2×C4—61no—
560C2×C4C2×C3×D3211yescharacteristic
561C8—32no—
562C8—92no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—22no—
569C2×C3—41no—
570C2×C3—42no—
571C2×C3—82no—
572C2×C3—82no—
573C2×C3—82no—
574C2×C3—84no—
575C2×C3—81no—
576C2×C3—81no—
577C2×C3—81no—
578C2×C3—242no—
579C2×C3—241no—
580C2×C3—241no—
581C2×C3—721no—
582C2×C3#6392512yes—
583C2×C3#6397911yescharacteristic
584C2×C3C3×#91112yes—
585C2×C3C3×#94011yescharacteristic
586C2×C3C3×#94011yescharacteristic
587C2×C3C3×#94912yes—
588D3—81no—
589D3—242no—
590D3—241no—
591D3—241no—
592C22—41no—
593C22—42no—
594C22—121no—
595C22—122no—
596C22—361no—
597C22—362no—
598C22C3×D3×#6711yescharacteristic
599C4—62no—
600C4—121no—
601C4C3×D3×#6711yescharacteristic
602C4C3×#381111yescharacteristic
603C3—21no—
604C3—21no—
605C3—21no—
606C3—41no—
607C3#1059148311yescharacteristic, direct product factor
608C3C3×#505811yescharacteristic, complemented
609C3C3×#515411yescharacteristic, complemented
610C2—81no—
611C2—241no—
612C2—721no—
613C2C3×#6392512yes—
614C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#496
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×#3753C2semidirect
2. C32×#936C2semidirect
3. C2×C32×#213C2semidirect
4. C3×#63911C2semidirect
5. C2×C3×#3717C2semidirect
6. C2×C3×#3715C2semidirect
7. C3×#63554C2semidirect
10. #10591483C3direct (N × H)
26. C2×#3753C2×C3semidirect
31. C3×#936D3semidirect
32. C3×#936C2×C3semidirect
38. C2×C3×#213D3semidirect
39. C2×C3×#213C2×C3semidirect
42. #63911C2×C3semidirect
43. C2×#3717C2×C3semidirect
44. C2×#3715C2×C3semidirect
47. #63554C2×C3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
144. #936C3×D3semidirect
149. C2×#213C3×D3semidirect
227. C3⋊(C3⋊C8)C3×D4semidirect
232. C3×C3⋊C8#67semidirect
233. C3×C3⋊C8C3×D4semidirect
358. C33#496semidirect
437. C3⋊C8C3×#67semidirect
543. C32#5058semidirect
544. C32#5154semidirect
545. C32C3×#496semidirect
607. C3#10591483direct (N × H)
608. C3C3×#5058semidirect
609. C3C3×#5154semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×#9362
C2×C32×#2132
C3×#639112
C2×C3×#37172
C2×C3×#37152
C3×#635542
C3×#51543
C3×#50583
#105914833