Finite Group Databasebeta

← C3×#10590534 #13157510

Subgroups — C3×#10590534 #13157510

Order 1728.

708 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic, complemented
2C3×#63864C211yesmaximal, characteristic, complemented
3C2×C3×#3715C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5016—31nomaximal
9C3×#4819—31nomaximal
10#10590534C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C2×C32×Dic6C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×#3722—21no—
22C3×#3715—22no—
23C3×#3714—22no—
24C32×C4×D4—31no—
25C3×#947—31no—
26C2×C3×#215—31no—
27C2×C3×#214—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C3×#893—21no—
31C3×#893D311yescharacteristic, complemented
32C3×#893C2×C311yescharacteristic, complemented
33C3×#875—31no—
34#63864C2×C311yescharacteristic, complemented
35C2×#3715C2×C311yescharacteristic, complemented
36C2×#3714C2×C311yescharacteristic
37C32×#103—31no—
38#63636C2×C311yescharacteristic, complemented
39C3×Dic3⋊C8—31no—
40C3×#810—21no—
41C3×#810D311yescharacteristic, complemented
42C3×#810C2×C311yescharacteristic, complemented
43C3×#809—31no—
44C3×#809—31no—
45C3×#809—61no—
46C3×C4⋊(C3⋊C8)—31no—
47Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12D412yes—
59C32×C4×D3—42no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#5016—31no—
66#4819—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C22×C32×C4—61no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C22×C311yescharacteristic
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3—21no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C2×C32×C8—31no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4—41no—
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C22×C311yescharacteristic
91C32×C22⋊C4—61no—
92C32×C42—31no—
93C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
94C3×#215—62no—
95C3×#214—62no—
96C3×#213—62no—
97C3×(C2×D3)⋊C4—22no—
98C3×(C2×D3)⋊C4—21no—
99C3×(C2×D3)⋊C4—21no—
100C3×(C2×D3)⋊C4—61no—
101C3×(C2×D3)⋊C4—61no—
102C3×(C2×D3)⋊C4—61no—
103C3×(C2×D3)⋊C4—61no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3C2×D311yescharacteristic
106C3×C4×Dic3C22×C311yescharacteristic
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C3×D24—62no—
114#3722—21no—
115#3722—21no—
116#3715—22no—
117#3714—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C32×D3—22no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C32×Dic3—21no—
129C32×Dic3—21no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×D8—91no—
133C3×#124—91no—
134C3×C4×D4—31no—
135C3×C4×D4—31no—
136C3×C4×D4—61no—
137#947—31no—
138C2×#215—31no—
139C2×#214—31no—
140C2×#213—31no—
141C2×D24—31no—
142#893C3×D311yescharacteristic, complemented
143#875—31no—
144C3×C4⋊C8—91no—
145C3×#103—31no—
146C3×#103—31no—
147C3×#103—61no—
148C3×#102—91no—
149Dic3⋊C8—31no—
150#810C3×D311yescharacteristic, complemented
151#809—31no—
152#809—31no—
153#809—61no—
154C4⋊(C3⋊C8)—31no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3×D3—22no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—121no—
163C32×Q8—61no—
164C32×Q8—61no—
165C32×D4—32no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—31no—
175C2×C32×C4—31no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—42no—
180C2×C32×C4—61no—
181C2×C32×C4—122no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C4⋊(C3⋊D3)—21no—
186C4⋊(C3⋊D3)—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—41no—
192C2×C3×Dic3#6711yescharacteristic
193C2×C3×Dic3#6711yescharacteristic
194C2×C3×Dic3C3×D411yescharacteristic
195C2×C3×Dic3C3×D411yescharacteristic
196C3×D12—22no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—22no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—121no—
206C3×D12D1212yes—
207C3×D12C3×D412yes—
208C3×C4×D3—44no—
209C3×C4×D3—42no—
210C3×C4×D3—42no—
211C3×Dic6—22no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—41no—
217C32×C8—61no—
218C3⋊(C3⋊C8)—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—41no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—121no—
225#463—91noSylow
226C2×C33#12111yescharacteristic
227C2×C33#13511yescharacteristic
228C2×C33AGL(1,ℤ/8)11yescharacteristic
229C3×C3⋊D3—81no—
230C32×D3—42no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—61no—
234C2×C3×D4—31no—
235C2×C3×D4—31no—
236C2×C3×D4—61no—
237C2×C3×D4—91no—
238C22×C3×C4—61no—
239C22×C3×C4—61no—
240C22×C3×C4—62no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—61no—
244C2×D12C2×C3×D311yescharacteristic
245C2×C4×D3—21no—
246C2×Dic6C2×C3×D311yescharacteristic
247C3×SD16—182no—
248C3×D8—182no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—61no—
252C2×C3×C8—91no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4D3211yescharacteristic
258C3×C4⋊C4C2×C3×D311yescharacteristic
259C3×C4⋊C4C2×C3×D311yescharacteristic
260C3×C22⋊C4—61no—
261C3×C22⋊C4—61no—
262C3×C22⋊C4—62no—
263C3×C22⋊C4—181no—
264C3×C22⋊C4—181no—
265C3×C42—31no—
266C3×C42—31no—
267C3×C42—61no—
268#215—62no—
269#214—62no—
270#213—62no—
271(C2×D3)⋊C4—21no—
272(C2×D3)⋊C4—61no—
273(C2×D3)⋊C4—61no—
274(C2×D3)⋊C4—61no—
275(C2×D3)⋊C4—61no—
276C4×Dic3C2×C3×D311yescharacteristic
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8C2×C3×D311yescharacteristic
281D24—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—62no—
287C22×C32—121no—
288C22×C32—121no—
289C22×C32D3×D411yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C22×C32C2×C3×D411yescharacteristic
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3×D3—24no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—122no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—241no—
312C2×C3×D3—241no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4—42no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—61no—
326C32×C4—61no—
327C32×C4—81no—
328C32×C4—81no—
329C32×C4—82no—
330C32×C4—121no—
331C32×C4C2×D1211yescharacteristic
332C32×C4#23511yescharacteristic
333C32×C4#23511yescharacteristic
334C32×C4D3×D411yescharacteristic
335C32×C4C2×C3×D411yescharacteristic
336C32×C4C3×#4111yescharacteristic
337C3×Dic3—22no—
338C3×Dic3—21no—
339C3×Dic3—21no—
340C3×Dic3—21no—
341C3×Dic3—21no—
342C3×Dic3—41no—
343C3×Dic3—42no—
344C3×Dic3—41no—
345C3×Dic3—41no—
346C2×SD16—91no—
347C2×D8—91no—
348#124—91no—
349C4×D4—31no—
350C4⋊C8—91no—
351#103—31no—
352#102—91no—
353C33#46311yescharacteristic, complemented, Sylow
354C23×C3—62no—
355C23×C3—61no—
356C23×C3—61no—
357C23×C3—181no—
358C22×D3—21no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—121no—
362C3×Q8—62no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—61no—
366C3×Q8—61no—
367C3×Q8—121no—
368C3×D4—32no—
369C3×D4—32no—
370C3×D4—62no—
371C3×D4—62no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—181no—
375C3×D4—181no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—121no—
396C2×C3×C4—124no—
397C2×C3×C4—122no—
398C2×C3×C4—122no—
399C2×C3×C4C2×D3211yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×Dic3—21no—
403C2×Dic3C3×#6711yescharacteristic
404C2×Dic3C3×#6711yescharacteristic
405D12—21no—
406D12—62no—
407D12—61no—
408D12—61no—
409D12—61no—
410D12—61no—
411D12—121no—
412D12C3×D1212yes—
413C4×D3—42no—
414Dic6—21no—
415Dic6—21no—
416C3×C8—61no—
417C3×C8—61no—
418C3×C8—121no—
419C3×C8—181no—
420C3⋊C8—21no—
421C3⋊C8—61no—
422C3⋊C8—61no—
423C3⋊C8—121no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—122no—
437C2×C32#88411yescharacteristic
438C2×C32#91411yescharacteristic
439C2×C32#91911yescharacteristic
440C2×C32#93011yescharacteristic
441C2×C32#93211yescharacteristic
442C2×C32#95011yescharacteristic
443C2×C32C3×#12111yescharacteristic
444C2×C32C3×#13511yescharacteristic
445C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
446C3⋊D3—81no—
447C3×D3—44no—
448C3×D3—42no—
449C3×D3—42no—
450C3×D3—242no—
451C3×D3—241no—
452C3×D3—241no—
453C2×Q8—31no—
454C2×D4—31no—
455C2×D4—91no—
456C22×C4—61no—
457SD16—182no—
458D8—182no—
459C2×C8—31no—
460C2×C8—91no—
461C4⋊C4C3×D3211yescharacteristic
462C22⋊C4—61no—
463C22⋊C4—181no—
464C22⋊C4—181no—
465C42—31no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—64no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—361no—
480C22×C3—361no—
481C22×C3—361no—
482C22×C3D3×#6711yescharacteristic
483C22×C3C3×D3×D411yescharacteristic
484C22×C3C2×C3×#6711yescharacteristic
485C2×D3—22no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—122no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—241no—
496C2×D3—241no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—62no—
510C3×C4—61no—
511C3×C4—61no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—82no—
517C3×C4—121no—
518C3×C4—122no—
519C3×C4—121no—
520C3×C4—121no—
521C3×C4#379811yescharacteristic
522C3×C4D3×D1211yescharacteristic
523C3×C4C2×C3×D1211yescharacteristic
524C3×C4C3×#23511yescharacteristic
525C3×C4C3×#23511yescharacteristic
526C3×C4C3×D3×D411yescharacteristic
527Dic3—21no—
528Dic3—21no—
529Dic3—41no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#481911yescharacteristic, complemented
535C32#501611yescharacteristic, complemented
536C32C3×#46311yescharacteristic, complemented
537C23—61no—
538C23—181no—
539Q8—61no—
540Q8—61no—
541D4—32no—
542D4—61no—
543D4—181no—
544D4—181no—
545C2×C4—21no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—61no—
549C2×C4—122no—
550C2×C4C2×C3×D3211yescharacteristic
551C8—61no—
552C8—181no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—124no—
566C2×C3—122no—
567C2×C3—122no—
568C2×C3—721no—
569C2×C3#6390611yescharacteristic
570C2×C3#6393411yescharacteristic
571C2×C3#6394911yescharacteristic
572C2×C3C3×#88411yescharacteristic
573C2×C3C3×#91411yescharacteristic
574C2×C3C3×#91911yescharacteristic
575C2×C3C3×#93011yescharacteristic
576C2×C3C3×#93211yescharacteristic
577C2×C3C3×#95011yescharacteristic
578D3—42no—
579D3—242no—
580D3—241no—
581D3—241no—
582C22—62no—
583C22—121no—
584C22—121no—
585C22—361no—
586C22—361no—
587C22—361no—
588C22C3×D3×#6711yescharacteristic
589C4—41no—
590C4—61no—
591C4—61no—
592C4—121no—
593C4C3×#379811yescharacteristic
594C4C3×D3×D1211yescharacteristic
595C3—21no—
596C3—21no—
597C3—21no—
598C3—41no—
599C3#1059053411yescharacteristic, direct product factor
600C3C3×#481911yescharacteristic, complemented
601C3C3×#501611yescharacteristic, complemented
602C2—122no—
603C2—721no—
604C2C3×#6390611yescharacteristic
605C2C3×#6393411yescharacteristic
606C2C3×#6394911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#893C2semidirect
2. C3×#63864C2semidirect
3. C2×C3×#3715C2semidirect
5. C3×#63636C2semidirect
6. C32×#810C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590534C3direct (N × H)
31. C3×#893D3semidirect
32. C3×#893C2×C3semidirect
34. #63864C2×C3semidirect
35. C2×#3715C2×C3semidirect
38. #63636C2×C3semidirect
41. C3×#810D3semidirect
42. C3×#810C2×C3semidirect
47. Dic3⋊(C3⋊C8)C2×C3semidirect
142. #893C3×D3semidirect
150. #810C3×D3semidirect
353. C33#463semidirect
534. C32#4819semidirect
535. C32#5016semidirect
536. C32C3×#463semidirect
599. C3#10590534direct (N × H)
600. C3C3×#4819semidirect
601. C3C3×#5016semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#638642
C2×C3×#37152
C2×C3×#37142
C3×#636362
C32×#8102
C3×Dic3⋊(C3⋊C8)2
C3×#50163
C3×#48193
#105905343