Finite Group Databasebeta

← C3×#10591589 #13158593

Subgroups — C3×#10591589 #13158593

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3755C211yesmaximal, characteristic, complemented
2C32×#946C211yesmaximal, characteristic, complemented
3C3×#63909C211yesmaximal, characteristic
4C2×C3×#3711C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5049—31nomaximal
9C3×#4812—31nomaximal
10#10591589C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C3×#626—21no—
15C3×#3755—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4—21no—
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3722—21no—
23C3×#3711—22no—
24C32×C22⋊Q8—31no—
25C2×#3755C2×C311yescharacteristic, complemented
26C3×#946—21no—
27C3×#946D311yescharacteristic, complemented
28C3×#946C2×C311yescharacteristic, complemented
29C2×C3×#215—31no—
30C2×C3×#215—31no—
31C2×C3×#215—61no—
32C3×#930—31no—
33C2×C3×#208—31no—
34C2×C3×ΓL(1,𝔽25)—31no—
35C3×#895—31no—
36#63909C2×C311yescharacteristic
37C2×#3711C2×C311yescharacteristic, complemented
38C32×#103—31no—
39C3×#835—21no—
40C3×#835D311yescharacteristic, complemented
41C3×#835C2×C311yescharacteristic, complemented
42C3×#821—31no—
43C3×#809—31no—
44C3×#807—31no—
45C3×#807—31no—
46#63576C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3D411yescharacteristic
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3D411yescharacteristic
59C32×C4×D3—21no—
60C32×C4×D3—21no—
61C3×#626—42no—
62C3×C3⋊(C3⋊C8)D412yescomplemented
63C32×C3⋊C8—21no—
64C3×#468—91no—
65#5049—31no—
66#4812—31no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C22×C32×C4—31no—
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×C3×#67—61no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×D12—61no—
81C2×C3×C4×D3—21no—
82C2×C3×C4×D3C2×D311yescharacteristic
83C2×C3×C4×D3C22×C311yescharacteristic
84C2×#626—21no—
85C2×C32×C8—31no—
86C32×C4⋊C4—31no—
87C32×C4⋊C4—31no—
88C32×C4⋊C4—61no—
89C32×C22⋊C4—61no—
90#3755—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#215—62no—
93C3×#215—62no—
94C3×#215—64no—
95C3×(C2×D3)⋊C4—22no—
96C3×(C2×D3)⋊C4—21no—
97C3×(C2×D3)⋊C4—21no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—61no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4⋊Dic3C22×C311yescharacteristic
105C3×C4⋊Dic3C22×C311yescharacteristic
106C3×Dic3⋊C4—22no—
107C3×Dic3⋊C4—21no—
108C3×Dic3⋊C4—21no—
109C3×#208—62no—
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—61no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×ΓL(1,𝔽25)—62no—
117#3722—21no—
118#3711—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C32×D3—21no—
124C2×C32×D3—21no—
125C2×C32×D3—21no—
126C33×C4—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C32×Dic3—21no—
130C32×Dic3—41no—
131C32×Dic3—41no—
132C2×C3×SD16—91no—
133C2×C3×M16—91no—
134C3×C22⋊Q8—31no—
135C3×C22⋊Q8—31no—
136C3×C22⋊Q8—61no—
137C3×#121—91no—
138#946C3×D311yescharacteristic, complemented
139C2×#215—31no—
140C2×#215—31no—
141C2×#215—61no—
142#930—31no—
143C2×#208—31no—
144C2×ΓL(1,𝔽25)—31no—
145#895—31no—
146C3×#107—91no—
147C3×#103—31no—
148C3×#103—31no—
149C3×#103—61no—
150C3×#102—91no—
151#835C3×D311yescharacteristic, complemented
152#821—31no—
153#809—31no—
154#807—31no—
155#807—31no—
156C23×C32—31no—
157C22×C3⋊D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C22×C3×D3#6711yescharacteristic
163C22×C3×D3C3×D411yescharacteristic
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—22no—
167C32×Q8—22no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—21no—
171C32×Q8—21no—
172C32×Q8—41no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—41no—
176C32×Q8—42no—
177C32×Q8—42no—
178C2×C32×C4—21no—
179C2×C32×C4—22no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—31no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—42no—
190C2×C32×C4—61no—
191C2×C32×C4—61no—
192C2×C32×C4—61no—
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—122no—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—41no—
207C2×C3×Dic3#6711yescharacteristic
208C2×C3×Dic3C3×D411yescharacteristic
209C3×D12—62no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—121no—
215C3×D12—122no—
216C3×C4×D3—21no—
217C3×C4×D3—21no—
218C3×C4×D3—21no—
219C3×C4×D3—21no—
220C3×C4×D3—41no—
221C3×C4×D3—41no—
222#626—42no—
223C32×C8—61no—
224C3⋊(C3⋊C8)C3×D412yescomplemented
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—32no—
228C3×C3⋊C8—32no—
229C3×C3⋊C8—41no—
230C3×C3⋊C8—62no—
231#468—91noSylow
232C2×C33#12111yescharacteristic
233C2×C33AGL(1,ℤ/8)11yescharacteristic
234C2×C33#13711yescharacteristic
235C3×C3⋊D3—81no—
236C32×D3—41no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—41no—
241C2×C3×Q8D3211yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×Q8C2×C3×D311yescharacteristic
244C2×C3×D4—91no—
245C2×C3×D4—181no—
246C22×C3×C4—31no—
247C22×C3×C4—31no—
248C22×C3×C4—61no—
249C2×#67—61no—
250C2×D12—31no—
251C2×D12—31no—
252C2×D12—61no—
253C2×D12—61no—
254C2×C4×D3C2×C3×D311yescharacteristic
255C3×SD16—182no—
256C3×M16—182no—
257C2×C3×C8—31no—
258C2×C3×C8—31no—
259C2×C3×C8—61no—
260C2×C3×C8—91no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—31no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4—62no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—61no—
272C3×C22⋊C4—62no—
273C3×C22⋊C4—181no—
274#215—62no—
275#215—62no—
276#215—64no—
277(C2×D3)⋊C4—21no—
278(C2×D3)⋊C4—61no—
279(C2×D3)⋊C4—61no—
280C4⋊Dic3C2×C3×D311yescharacteristic
281C4⋊Dic3C2×C3×D311yescharacteristic
282Dic3⋊C4—21no—
283#208—62no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8C2×C3×D311yescharacteristic
288ΓL(1,𝔽25)—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—61no—
294C22×C32—61no—
295C22×C32—61no—
296C22×C32D3×D411yescharacteristic
297C22×C32C2×#6711yescharacteristic
298C22×C32C2×C3×D411yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3×D3—22no—
303C2×C3×D3—21no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—122no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—241no—
319C2×C3×D3—241no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4—42no—
329C32×C4—42no—
330C32×C4—42no—
331C32×C4—44no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4—61no—
336C32×C4—121no—
337C32×C4—121no—
338C32×C4#23511yescharacteristic
339C32×C4D3×D411yescharacteristic
340C32×C4D3×D411yescharacteristic
341C32×C4#23711yescharacteristic
342C32×C4C2×C3×D411yescharacteristic
343C32×C4C3×#4111yescharacteristic
344C3×Dic3—21no—
345C3×Dic3—21no—
346C3×Dic3—41no—
347C3×Dic3—42no—
348C3×Dic3—41no—
349C3×Dic3—41no—
350C3×Dic3—41no—
351C3×Dic3—41no—
352C3×Dic3—81no—
353C2×SD16—91no—
354C2×M16—91no—
355C22⋊Q8—31no—
356#121—91no—
357#107—91no—
358#103—31no—
359#102—91no—
360C33#46811yescharacteristic, complemented, Sylow
361C23×C3—31no—
362C23×C3—31no—
363C23×C3—61no—
364C23×C3—181no—
365C22×D3—61no—
366C22×D3—61no—
367C22×D3—121no—
368C22×D3C3×#6711yescharacteristic
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—21no—
372C3×Q8—21no—
373C3×Q8—21no—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—21no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—41no—
381C3×Q8—42no—
382C3×Q8—42no—
383C3×D4—181no—
384C3×D4—181no—
385C3×D4—362no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—42no—
398C2×C3×C4—41no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—62no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—121no—
410C2×C3×C4—121no—
411C2×C3×C4—121no—
412C2×C3×C4C2×D3211yescharacteristic
413C2×C3×C4C22×C3×D311yescharacteristic
414C2×C3×C4C22×C3×D311yescharacteristic
415#67—122no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3C3×#6711yescharacteristic
419D12—62no—
420D12—61no—
421D12—61no—
422D12—61no—
423D12—61no—
424D12—121no—
425D12—122no—
426C4×D3—21no—
427C4×D3—21no—
428C3×C8—61no—
429C3×C8—61no—
430C3×C8—92no—
431C3×C8—121no—
432C3⋊C8—21no—
433C3⋊C8—32no—
434C3⋊C8—32no—
435C3⋊C8—62no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—121no—
449C2×C32#88411yescharacteristic
450C2×C32#91411yescharacteristic
451C2×C32#91811yescharacteristic
452C2×C32#93911yescharacteristic
453C2×C32#94211yescharacteristic
454C2×C32#94911yescharacteristic
455C2×C32C3×#12111yescharacteristic
456C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
457C2×C32C3×#13711yescharacteristic
458C3⋊D3—81no—
459C3×D3—42no—
460C3×D3—41no—
461C3×D3—41no—
462C3×D3—242no—
463C3×D3—241no—
464C3×D3—241no—
465C2×Q8C3×D3211yescharacteristic
466C2×D4—91no—
467C2×D4—181no—
468C22×C4—31no—
469SD16—182no—
470M16—182no—
471C2×C8—31no—
472C2×C8—91no—
473C4⋊C4—31no—
474C4⋊C4—31no—
475C4⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—181no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—62no—
483C22×C3—61no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—121no—
490C22×C3—121no—
491C22×C3—361no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3D3×#6711yescharacteristic
495C22×C3C3×D3×D411yescharacteristic
496C22×C3C2×C3×#6711yescharacteristic
497C2×D3—21no—
498C2×D3—21no—
499C2×D3—21no—
500C2×D3—122no—
501C2×D3—121no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—241no—
508C2×D3—241no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—61no—
525C3×C4—61no—
526C3×C4—121no—
527C3×C4—122no—
528C3×C4—121no—
529C3×C4—121no—
530C3×C4—121no—
531C3×C4—121no—
532C3×C4—241no—
533C3×C4#380411yescharacteristic
534C3×C4#381111yescharacteristic
535C3×C4C3×#23511yescharacteristic
536C3×C4C3×D3×D411yescharacteristic
537C3×C4C3×D3×D411yescharacteristic
538C3×C4C3×#23711yescharacteristic
539Dic3—21no—
540Dic3—41no—
541Dic3—41no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#481211yescharacteristic, complemented
547C32#504911yescharacteristic, complemented
548C32C3×#46811yescharacteristic, complemented
549C23—31no—
550C23—181no—
551Q8—21no—
552Q8—21no—
553D4—181no—
554D4—181no—
555D4—362no—
556C2×C4—21no—
557C2×C4—31no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4C2×C3×D3211yescharacteristic
563C8—61no—
564C8—92no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—122no—
578C2×C3—121no—
579C2×C3—121no—
580C2×C3—721no—
581C2×C3#6393911yescharacteristic
582C2×C3#6394811yescharacteristic
583C2×C3#6397111yescharacteristic
584C2×C3C3×#88411yescharacteristic
585C2×C3C3×#91411yescharacteristic
586C2×C3C3×#91811yescharacteristic
587C2×C3C3×#93911yescharacteristic
588C2×C3C3×#94211yescharacteristic
589C2×C3C3×#94911yescharacteristic
590D3—41no—
591D3—242no—
592D3—241no—
593D3—241no—
594C22—61no—
595C22—61no—
596C22—61no—
597C22—361no—
598C22—361no—
599C22—361no—
600C22C3×D3×#6711yescharacteristic
601C4—41no—
602C4—61no—
603C4—121no—
604C4—121no—
605C4C3×#380411yescharacteristic
606C4C3×#381111yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059158911yescharacteristic, direct product factor
612C3C3×#481211yescharacteristic, complemented
613C3C3×#504911yescharacteristic, complemented
614C2—121no—
615C2—721no—
616C2C3×#6393911yescharacteristic
617C2C3×#6394811yescharacteristic
618C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC3×#63909C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3755C2semidirect
2. C32×#946C2semidirect
4. C2×C3×#3711C2semidirect
5. C32×#835C2semidirect
6. C3×#63576C2semidirect
7. C3×#63564C2semidirect
10. #10591589C3direct (N × H)
25. C2×#3755C2×C3semidirect
27. C3×#946D3semidirect
28. C3×#946C2×C3semidirect
37. C2×#3711C2×C3semidirect
40. C3×#835D3semidirect
41. C3×#835C2×C3semidirect
46. #63576C2×C3semidirect
47. #63564C2×C3semidirect
62. C3×C3⋊(C3⋊C8)D4semidirect
138. #946C3×D3semidirect
151. #835C3×D3semidirect
224. C3⋊(C3⋊C8)C3×D4semidirect
360. C33#468semidirect
546. C32#4812semidirect
547. C32#5049semidirect
548. C32C3×#468semidirect
611. C3#10591589direct (N × H)
612. C3C3×#4812semidirect
613. C3C3×#5049semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37552
C32×#9462
C3×#639092
C2×C3×#37112
C32×#8352
C3×#635762
C3×#635642
C3×#50493
C3×#48123
#105915893