Finite Group Databasebeta

← C3×#10590508 #13157581

Subgroups — C3×#10590508 #13157581

Order 1728.

726 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64104C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic, complemented
4C2×C3×#3712C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6C32×#808C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#4865—31nomaximal
9C3×#4782—31nomaximal
10#10590508C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C3×#626—21no—
16C33×C4⋊C4C2211yescharacteristic
17C3×#3752—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3714—22no—
23C3×#3712—22no—
24#64104C2×C311yescharacteristic, complemented
25C3×#938—31no—
26C2×C3×#213—31no—
27C2×C3×D24—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C3×#895—31no—
31C3×#895—31no—
32C3×#895—31no—
33C3×#895—61no—
34#63916C2×C311yescharacteristic, complemented
35C2×#3714C2×C311yescharacteristic, complemented
36C2×#3712C2×C311yescharacteristic, complemented
37C32×#106—31no—
38C32×#102—31no—
39C3×#821—31no—
40C3×#809—21no—
41C3×#809—31no—
42C3×#809D311yescharacteristic, complemented
43C3×#809C2×C311yescharacteristic, complemented
44C3×#808—21no—
45C3×#808D311yescharacteristic, complemented
46C3×#808C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3D411yescharacteristic
50C22×C32×D3—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—42no—
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59C2×C32×Dic3—21no—
60C32×D12—21no—
61C32×D12—21no—
62C3×#626—42no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8D412yescomplemented
65C3×#469—91no—
66#4865—31no—
67#4782—31no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3C22×C311yescharacteristic
72C2×C3×#67—61no—
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—61no—
78C2×C3×D12—61no—
79C2×C3×D12—62no—
80C2×C3×D12—61no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—61no—
86C2×#626—21no—
87C2×C32×C8—31no—
88C2×C32×C8—31no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4—21no—
92C32×C4⋊C4—31no—
93C32×C4⋊C4—41no—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C22×C311yescharacteristic
97#3752—21no—
98C3×#217—61no—
99C3×#213—62no—
100C3×(C2×D3)⋊C4—61no—
101C3×(C2×D3)⋊C4—61no—
102C3×(C2×D3)⋊C4—61no—
103C3×(C2×D3)⋊C4—62no—
104C3×C4⋊Dic3—21no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C3×D24—62no—
114C3×ΓL(1,𝔽25)—62no—
115C3×ΓL(1,𝔽25)—62no—
116(C2×D3)⋊Dic3—21no—
117#3714—22no—
118#3712—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—21no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C33×C4—41no—
130C33×C4C2×D411yescharacteristic
131C33×C4#4111yescharacteristic
132C3×C3⋊Dic3—21no—
133C32×Dic3—41no—
134C2×C3×D8—91no—
135C2×C3×M16—91no—
136C3×#121—91no—
137C3×#121—91no—
138#938—31no—
139C2×#213—31no—
140C2×D24—31no—
141C2×ΓL(1,𝔽25)—31no—
142C2×ΓL(1,𝔽25)—31no—
143#895—31no—
144#895—31no—
145#895—31no—
146#895—61no—
147C3×#106—31no—
148C3×#106—31no—
149C3×#106—61no—
150C3×#102—31no—
151C3×#102—31no—
152C3×#102—61no—
153C3×#102—91no—
154#821—31no—
155#809—31no—
156#809C3×D311yescharacteristic, complemented
157#808C3×D311yescharacteristic, complemented
158C23×C32—61no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3C3×D411yescharacteristic
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—31no—
164C22×C3×D3—31no—
165C22×C3×D3—41no—
166C22×C3×D3—61no—
167C22×C3×D3—62no—
168C22×C3×D3—61no—
169C22×C3×D3—61no—
170C32×D4—61no—
171C32×D4—61no—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—41no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—42no—
183C2×C32×C4—61no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C23×C311yescharacteristic
187C2×C3⋊Dic3C3×D411yescharacteristic
188C4⋊(C3⋊D3)—21no—
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—42no—
191C4×C3⋊D3—21no—
192C4×C3⋊D3—21no—
193C3×#67—122no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—41no—
199C2×C3×Dic3—61no—
200C3×D12—22no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—41no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—121no—
211C3×D12—121no—
212C3×D12—124no—
213C3×D12—122no—
214C3×D12—122no—
215C3×D12—122no—
216C3×C4×D3—61no—
217C3×C4×D3—61no—
218C3×C4×D3—61no—
219C3×C4×D3—61no—
220C3×C4×D3—121no—
221C3×C4×D3—121no—
222#626—42no—
223C32×C8—32no—
224C32×C8—61no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—22no—
228C3×C3⋊C8—41no—
229C3×C3⋊C8D1212yescomplemented
230C3×C3⋊C8C3×D412yescomplemented
231#469—91noSylow
232C2×C33#12111yescharacteristic
233C2×C33AGL(1,ℤ/8)11yescharacteristic
234C2×C33AGL(1,ℤ/8)11yescharacteristic
235C3×C3⋊D3—41no—
236C3×C3⋊D3—81no—
237C32×D3—81no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—61no—
241C2×C3×D4—91no—
242C2×C3×D4—181no—
243C2×C3×D4—181no—
244C22×C3×C4—91no—
245C2×#67—61no—
246C2×D12—31no—
247C2×D12—31no—
248C2×D12—61no—
249C2×D12—61no—
250C2×D12—61no—
251C2×D12—62no—
252C2×D12—61no—
253C2×D12C2×C3×D311yescharacteristic
254C2×C4×D3—31no—
255C2×C4×D3—31no—
256C2×C4×D3—61no—
257C3×D8—182no—
258C3×M16—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—61no—
264C2×C3×C8—61no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—31no—
270C3×C4⋊C4—41no—
271C3×C4⋊C4—61no—
272C3×C4⋊C4D3211yescharacteristic
273C3×C4⋊C4C2×C3×D311yescharacteristic
274C3×C4⋊C4C2×C3×D311yescharacteristic
275C3×C22⋊C4—181no—
276C3×C22⋊C4—181no—
277#217—61no—
278#213—62no—
279(C2×D3)⋊C4—61no—
280(C2×D3)⋊C4—61no—
281(C2×D3)⋊C4—61no—
282(C2×D3)⋊C4—62no—
283C4⋊Dic3C2×C3×D311yescharacteristic
284C2×C3⋊C8C2×C3×D311yescharacteristic
285C2×C3⋊C8C2×C3×D311yescharacteristic
286D24—62no—
287ΓL(1,𝔽25)—62no—
288ΓL(1,𝔽25)—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—121no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32D3×D411yescharacteristic
297C22×C32D3×D411yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3⋊D3—21no—
300C2×C3⋊D3—21no—
301C2×C3⋊D3—21no—
302C2×C3⋊D3—41no—
303C2×C3⋊D3—41no—
304C2×C3⋊D3—41no—
305C2×C3×D3—42no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—41no—
312C2×C3×D3—62no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—81no—
320C2×C3×D3—81no—
321C2×C3×D3—121no—
322C2×C3×D3—121no—
323C2×C3×D3—122no—
324C2×C3×D3—122no—
325C2×C3×D3—122no—
326C2×C3×D3—121no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C2×C3×D3—121no—
331C2×C3×D3—121no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—41no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4—42no—
342C32×C4—41no—
343C32×C4—41no—
344C32×C4—81no—
345C32×C4—81no—
346C32×C4—82no—
347C32×C4—121no—
348C32×C4C2×D1211yescharacteristic
349C32×C4#23511yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4#23911yescharacteristic
352C32×C4C2×C3×D411yescharacteristic
353C32×C4C3×#4111yescharacteristic
354C3⋊Dic3—21no—
355C3×Dic3—41no—
356C3×Dic3—41no—
357C3×Dic3—61no—
358C3×Dic3—61no—
359C3×Dic3—81no—
360C3×Dic3—121no—
361C2×D8—91no—
362C2×M16—91no—
363#121—91no—
364#121—91no—
365#106—31no—
366#102—31no—
367#102—91no—
368C33#46911yescharacteristic, complemented, Sylow
369C23×C3—61no—
370C23×C3—61no—
371C23×C3—91no—
372C23×C3—121no—
373C23×C3—181no—
374C22×D3—21no—
375C22×D3—31no—
376C22×D3—31no—
377C22×D3—61no—
378C22×D3—62no—
379C22×D3—61no—
380C22×D3—61no—
381C3×D4—62no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—121no—
387C3×D4—181no—
388C3×D4—181no—
389C3×D4—362no—
390C3×D4—362no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—91no—
405C2×C3×C4—121no—
406C2×C3×C4—181no—
407C2×C3×C4—181no—
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C22×C3×D311yescharacteristic
410C2×C3×C4C22×C3×D311yescharacteristic
411#67—122no—
412C2×Dic3—21no—
413C2×Dic3—31no—
414C2×Dic3—31no—
415C2×Dic3—61no—
416D12—21no—
417D12—21no—
418D12—61no—
419D12—61no—
420D12—61no—
421D12—61no—
422D12—121no—
423D12—121no—
424D12—124no—
425D12—122no—
426D12—122no—
427D12—122no—
428C4×D3—61no—
429C4×D3—61no—
430C4×D3—61no—
431C4×D3—61no—
432C4×D3—121no—
433C4×D3—121no—
434C3×C8—32no—
435C3×C8—32no—
436C3×C8—62no—
437C3×C8—61no—
438C3×C8—61no—
439C3×C8—121no—
440C3⋊C8—21no—
441C3⋊C8C3×D1212yescomplemented
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—241no—
455C2×C32#88411yescharacteristic
456C2×C32#89511yescharacteristic
457C2×C32#91111yescharacteristic
458C2×C32#91411yescharacteristic
459C2×C32#91411yescharacteristic
460C2×C32#91411yescharacteristic
461C2×C32C3×#12111yescharacteristic
462C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
463C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
464C3⋊D3—41no—
465C3⋊D3—81no—
466C3×D3—82no—
467C3×D3—81no—
468C3×D3—81no—
469C3×D3—122no—
470C3×D3—121no—
471C3×D3—121no—
472C3×D3—242no—
473C3×D3—241no—
474C3×D3—241no—
475C2×D4—31no—
476C2×D4—91no—
477C2×D4—181no—
478C2×D4—181no—
479C22×C4—91no—
480D8—182no—
481M16—182no—
482C2×C8—31no—
483C2×C8—31no—
484C4⋊C4—31no—
485C4⋊C4C3×D3211yescharacteristic
486C22⋊C4—181no—
487C22⋊C4—181no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—122no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—181no—
500C22×C3—181no—
501C22×C3—181no—
502C22×C3—241no—
503C22×C3—241no—
504C22×C3—361no—
505C22×C3—361no—
506C22×C3—361no—
507C22×C3#381111yescharacteristic
508C22×C3C3×D3×D411yescharacteristic
509C22×C3C3×D3×D411yescharacteristic
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—62no—
514C2×D3—61no—
515C2×D3—61no—
516C2×D3—61no—
517C2×D3—61no—
518C2×D3—61no—
519C2×D3—61no—
520C2×D3—121no—
521C2×D3—121no—
522C2×D3—122no—
523C2×D3—122no—
524C2×D3—122no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—121no—
530C2×D3—121no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—42no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—81no—
544C3×C4—81no—
545C3×C4—82no—
546C3×C4—121no—
547C3×C4—121no—
548C3×C4—181no—
549C3×C4—241no—
550C3×C4#379911yescharacteristic
551C3×C4D3×D1211yescharacteristic
552C3×C4C2×C3×D1211yescharacteristic
553C3×C4C3×#23511yescharacteristic
554C3×C4C3×D3×D411yescharacteristic
555C3×C4C3×#23911yescharacteristic
556Dic3—41no—
557Dic3—61no—
558Dic3—61no—
559Dic3—121no—
560C32—21no—
561C32—21no—
562C32—21no—
563C32—41no—
564C32#478211yescharacteristic, complemented
565C32#486511yescharacteristic, complemented
566C32C3×#46911yescharacteristic, complemented
567C23—61no—
568C23—91no—
569C23—181no—
570D4—61no—
571D4—61no—
572D4—181no—
573D4—181no—
574D4—362no—
575D4—362no—
576C2×C4—21no—
577C2×C4—61no—
578C2×C4—91no—
579C2×C4—181no—
580C2×C4—181no—
581C2×C4C2×C3×D3211yescharacteristic
582C8—32no—
583C8—61no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—41no—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—242no—
597C2×C3—241no—
598C2×C3—241no—
599C2×C3—361no—
600C2×C3—721no—
601C2×C3#6391011yescharacteristic
602C2×C3#6393711yescharacteristic
603C2×C3#6394111yescharacteristic
604C2×C3C3×#88411yescharacteristic
605C2×C3C3×#89511yescharacteristic
606C2×C3C3×#91111yescharacteristic
607C2×C3C3×#91411yescharacteristic
608C2×C3C3×#91411yescharacteristic
609C2×C3C3×#91411yescharacteristic
610D3—81no—
611D3—122no—
612D3—121no—
613D3—121no—
614D3—242no—
615D3—241no—
616D3—241no—
617C22—121no—
618C22—121no—
619C22—121no—
620C22—181no—
621C22—181no—
622C22—181no—
623C22—361no—
624C22—361no—
625C22—361no—
626C22C3×#381111yescharacteristic
627C4—41no—
628C4—121no—
629C4—181no—
630C4C3×#379911yescharacteristic
631C4C3×D3×D1211yescharacteristic
632C3—21no—
633C3—21no—
634C3—21no—
635C3—41no—
636C3#1059050811yescharacteristic, direct product factor
637C3C3×#478211yescharacteristic, complemented
638C3C3×#486511yescharacteristic, complemented
639C2—241no—
640C2—361no—
641C2—721no—
642C2C3×#6391011yescharacteristic
643C2C3×#6393711yescharacteristic
644C2C3×#6394111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#469
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. C3×#64104C2semidirect
2. C3×#63916C2semidirect
3. C2×C3×#3714C2semidirect
4. C2×C3×#3712C2semidirect
5. C32×#809C2semidirect
6. C32×#808C2semidirect
7. C3×#63564C2semidirect
10. #10590508C3direct (N × H)
24. #64104C2×C3semidirect
34. #63916C2×C3semidirect
35. C2×#3714C2×C3semidirect
36. C2×#3712C2×C3semidirect
42. C3×#809D3semidirect
43. C3×#809C2×C3semidirect
45. C3×#808D3semidirect
46. C3×#808C2×C3semidirect
47. #63564C2×C3semidirect
64. C32×C3⋊C8D4semidirect
156. #809C3×D3semidirect
157. #808C3×D3semidirect
229. C3×C3⋊C8D12semidirect
230. C3×C3⋊C8C3×D4semidirect
368. C33#469semidirect
441. C3⋊C8C3×D12semidirect
564. C32#4782semidirect
565. C32#4865semidirect
566. C32C3×#469semidirect
636. C3#10590508direct (N × H)
637. C3C3×#4782semidirect
638. C3C3×#4865semidirect

Maximal subgroups

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

subgroupindex
C3×#641042
C3×#639162
C2×C3×#37142
C2×C3×#37122
C32×#8092
C32×#8082
C3×#635642
C3×#48653
C3×#47823
#105905083