Finite Group Databasebeta

← C3×#10591557 #13158641

Subgroups — C3×#10591557 #13158641

Order 1728.

878 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×Q8C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic
3C2×C3×#3711C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5225—31nomaximal
9C3×#4802—31nomaximal
10#10591557C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C32×D3×Q8—22no—
13C32×D3×Q8C414yescomplemented
14C2×C32×C4×D3—21no—
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6—21no—
17C2×C32×Dic6C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×Dic32—21no—
24C3×#3711—22no—
25C22×C32×Q8—31no—
26C2×C3×D3×Q8—21no—
27C2×C3×D3×Q8D311yescharacteristic
28C2×C3×D3×Q8C2×C311yescharacteristic
29C3×#926—31no—
30C2×C3×#208—31no—
31C2×C3×ΓL(1,𝔽25)—31no—
32C3×#892—31no—
33#63841C2×C311yescharacteristic
34C2×#3711C2×C311yescharacteristic
35C32×#103—31no—
36#63662C2×C311yescharacteristic, complemented
37C3×#835—21no—
38C3×#835—31no—
39C3×#835—31no—
40C3×#835—31no—
41C3×#835—31no—
42C3×#835—61no—
43C3×#835D311yescharacteristic, complemented
44C3×#835C2×C311yescharacteristic, complemented
45C3×#816—31no—
46C3×#810—31no—
47#63566C2×C311yescharacteristic, complemented
48#63565C2×C311yescharacteristic, complemented
49C22×C32×D3D411yescharacteristic
50C33×Q8—21no—
51C33×Q8C2×C412yes—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3—24no—
59C32×C4×D3C2×C412yes—
60C32×C4×D3D412yes—
61C32×Dic6—21no—
62C32×Dic6—24no—
63C32×Dic6C2×C412yes—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66C3×#419—91no—
67#5225—31no—
68#4802—31no—
69C2×C32×Q8—21no—
70C2×C32×Q8—21no—
71C2×C32×Q8—21no—
72C2×C32×Q8—34no—
73C2×C32×Q8—31no—
74C2×C32×Q8—41no—
75C2×C32×Q8—61no—
76C2×C32×Q8—62no—
77C2×C32×Q8C2×D311yescharacteristic
78C2×C32×Q8C2×D311yescharacteristic
79C2×C32×Q8C22×C311yescharacteristic
80C22×C32×C4—31no—
81C22×C32×C4—61no—
82C3×D3×Q8—24no—
83C3×D3×Q8—24no—
84C3×D3×Q8—22no—
85C3×D3×Q8—22no—
86C3×D3×Q8Dic314yescomplemented
87C3×D3×Q8C3×C414yescomplemented
88C2×C3×C4×D3—22no—
89C2×C3×C4×D3—21no—
90C2×C3×C4×D3—21no—
91C2×C3×C4×D3—21no—
92C2×C3×C4×D3C2×D311yescharacteristic
93C2×C3×C4×D3C22×C311yescharacteristic
94C2×C3×Dic6—22no—
95C2×C3×Dic6—21no—
96C2×C3×Dic6—21no—
97C2×C3×Dic6—21no—
98C2×C3×Dic6C2×D311yescharacteristic
99C2×C3×Dic6C22×C311yescharacteristic
100C2×C32×C8—31no—
101C32×C4⋊C4—31no—
102C4⋊(C3⋊Dic3)C22×C311yescharacteristic
103C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
104C3×#217—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×C4⋊Dic3—21no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×C4⋊Dic3C22×C311yescharacteristic
112C3×C4×Dic3—61no—
113C3×C4×Dic3—61no—
114C3×#208—62no—
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—61no—
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C22×C311yescharacteristic
121C3×ΓL(1,𝔽25)—62no—
122(C2×D3)⋊Dic3—21no—
123Dic32—21no—
124#3711—22no—
125C22×C33C2×D411yescharacteristic
126C2×C32×D3—21no—
127C2×C32×D3—21no—
128C2×C32×D3C22⋊C412yes—
129C33×C4—22no—
130C33×C4C22×C411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—41no—
133C32×Dic3—22no—
134C32×Dic3—41no—
135C32×Dic3C22⋊C412yes—
136C22×C3×Q8—31no—
137C22×C3×Q8—31no—
138C22×C3×Q8—61no—
139C2×D3×Q8C3×D311yescharacteristic
140C2×C3×M16—91no—
141C3×#117—91no—
142#926—31no—
143C2×#208—31no—
144C2×ΓL(1,𝔽25)—31no—
145#892—31no—
146C3×#103—31no—
147C3×#103—31no—
148C3×#103—61no—
149C3×#103—91no—
150C3×#103—91no—
151C3×#103—91no—
152#835—31no—
153#835—31no—
154#835—31no—
155#835—31no—
156#835—61no—
157#835C3×D311yescharacteristic, complemented
158#816—31no—
159#810—31no—
160C23×C32—31no—
161C22×C3×D3—21no—
162C22×C3×D3#6711yescharacteristic
163C22×C3×D3C3×D411yescharacteristic
164C32×Q8—21no—
165C32×Q8—22no—
166C32×Q8—21no—
167C32×Q8—21no—
168C32×Q8—22no—
169C32×Q8—22no—
170C32×Q8—22no—
171C32×Q8—32no—
172C32×Q8—42no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—42no—
176C32×Q8—64no—
177C32×Q8—61no—
178C32×Q8C4×D312yes—
179C32×Q8C2×Dic312yes—
180C32×Q8C2×C3×C412yes—
181C2×C32×C4—21no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—31no—
189C2×C32×C4—32no—
190C2×C32×C4—32no—
191C2×C32×C4—41no—
192C2×C32×C4—41no—
193C2×C32×C4—41no—
194C2×C32×C4—42no—
195C2×C32×C4—64no—
196C2×C32×C4—61no—
197C2×C32×C4—61no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C23×C311yescharacteristic
201C2×C3⋊Dic3—21no—
202C2×C3×Dic3—22no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—21no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—41no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—121no—
212C2×C3×Dic3#6711yescharacteristic
213C2×C3×Dic3C3×D411yescharacteristic
214C3×C4×D3—28no—
215C3×C4×D3—24no—
216C3×C4×D3—24no—
217C3×C4×D3—22no—
218C3×C4×D3—22no—
219C3×C4×D3C2×Dic312yes—
220C3×C4×D3#6712yes—
221C3×C4×D3C2×C3×C412yes—
222C3×C4×D3C3×D412yes—
223C3×Dic6—22no—
224C3×Dic6—28no—
225C3×Dic6—21no—
226C3×Dic6—21no—
227C3×Dic6—24no—
228C3×Dic6—24no—
229C3×Dic6—22no—
230C3×Dic6C2×Dic312yes—
231C3×Dic6C2×C3×C412yes—
232C32×C8—61no—
233C3⋊(C3⋊C8)—21no—
234C3×C3⋊C8—21no—
235C3×C3⋊C8—21no—
236C3×C3⋊C8—41no—
237C3×C3⋊C8—61no—
238C3×C3⋊C8—61no—
239C3×C3⋊C8—121no—
240#419—91noSylow
241C2×C33C2×C22⋊C411yescharacteristic
242C2×C33#13711yescharacteristic
243C2×C33#13711yescharacteristic
244C32×D3—22no—
245C2×C3×Q8—21no—
246C2×C3×Q8—21no—
247C2×C3×Q8—21no—
248C2×C3×Q8—34no—
249C2×C3×Q8—34no—
250C2×C3×Q8—31no—
251C2×C3×Q8—31no—
252C2×C3×Q8—41no—
253C2×C3×Q8—61no—
254C2×C3×Q8—64no—
255C2×C3×Q8—61no—
256C2×C3×Q8—61no—
257C2×C3×Q8—62no—
258C2×C3×Q8—62no—
259C2×C3×Q8—62no—
260C2×C3×Q8—64no—
261C2×C3×Q8D3211yescharacteristic
262C2×C3×Q8C2×C3×D311yescharacteristic
263C2×C3×Q8C2×C3×D311yescharacteristic
264C22×C3×C4—31no—
265C22×C3×C4—31no—
266C22×C3×C4—61no—
267C22×C3×C4—61no—
268C22×C3×C4—61no—
269C22×C3×C4—62no—
270D3×Q8—22no—
271D3×Q8C3×Dic314yescomplemented
272C2×C4×D3—21no—
273C2×C4×D3C2×C3×D311yescharacteristic
274C2×Dic6—21no—
275C2×Dic6C2×C3×D311yescharacteristic
276C3×M16—182no—
277C2×C3×C8—31no—
278C2×C3×C8—31no—
279C2×C3×C8—61no—
280C2×C3×C8—91no—
281C3×C4⋊C4—31no—
282C3×C4⋊C4—31no—
283C3×C4⋊C4—61no—
284C3×C4⋊C4—91no—
285C3×C22⋊C4—181no—
286C3×C42—181no—
287#217—61no—
288(C2×D3)⋊C4—61no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—61no—
292C4⋊Dic3C2×C3×D311yescharacteristic
293C4×Dic3—61no—
294C4×Dic3—61no—
295#208—62no—
296C2×C3⋊C8—31no—
297C2×C3⋊C8—31no—
298C2×C3⋊C8—61no—
299C2×C3⋊C8C2×C3×D311yescharacteristic
300ΓL(1,𝔽25)—62no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—32no—
305C22×C32—41no—
306C22×C32—61no—
307C22×C32—61no—
308C22×C32D3×D411yescharacteristic
309C22×C32C2×#6711yescharacteristic
310C22×C32C2×C3×D411yescharacteristic
311C2×C3×D3—22no—
312C2×C3×D3—22no—
313C2×C3×D3—21no—
314C2×C3×D3—21no—
315C2×C3×D3—21no—
316C2×C3×D3—21no—
317C2×C3×D3—22no—
318C2×C3×D3#21712yes—
319C2×C3×D3C3×C22⋊C412yes—
320C32×C4—22no—
321C32×C4—24no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—32no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—42no—
334C32×C4—42no—
335C32×C4—44no—
336C32×C4—62no—
337C32×C4—121no—
338C32×C4C2×C4×D311yescharacteristic
339C32×C4D3×D411yescharacteristic
340C32×C4C22×Dic311yescharacteristic
341C32×C4C2×#6711yescharacteristic
342C32×C4C22×C3×C411yescharacteristic
343C32×C4C2×C3×D411yescharacteristic
344C3⋊Dic3—41no—
345C3×Dic3—24no—
346C3×Dic3—22no—
347C3×Dic3—22no—
348C3×Dic3—22no—
349C3×Dic3—41no—
350C3×Dic3—41no—
351C3×Dic3—81no—
352C3×Dic3—121no—
353C3×Dic3—121no—
354C3×Dic3—241no—
355C3×Dic3#21712yes—
356C3×Dic3C3×C22⋊C412yes—
357C22×Q8—31no—
358C2×M16—91no—
359#117—91no—
360#103—31no—
361#103—91no—
362#103—91no—
363#103—91no—
364C33#41911yescharacteristic, complemented, Sylow
365C23×C3—31no—
366C23×C3—31no—
367C23×C3—61no—
368C22×D3C3×#6711yescharacteristic
369C3×Q8—22no—
370C3×Q8—21no—
371C3×Q8—21no—
372C3×Q8—21no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8—32no—
377C3×Q8—32no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—42no—
381C3×Q8—42no—
382C3×Q8—62no—
383C3×Q8—64no—
384C3×Q8—64no—
385C3×Q8—68no—
386C3×Q8—62no—
387C3×Q8—61no—
388C3×Q8—61no—
389C3×Q8D3×Dic312yes—
390C3×Q8C3×C4×D312yes—
391C3×Q8C2×C3×Dic312yes—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—32no—
402C2×C3×C4—32no—
403C2×C3×C4—32no—
404C2×C3×C4—32no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—42no—
408C2×C3×C4—41no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—61no—
412C2×C3×C4—64no—
413C2×C3×C4—64no—
414C2×C3×C4—68no—
415C2×C3×C4—62no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—61no—
419C2×C3×C4—61no—
420C2×C3×C4—121no—
421C2×C3×C4—181no—
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C22×C3×D311yescharacteristic
424C2×C3×C4C22×C3×D311yescharacteristic
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—61no—
428C2×Dic3—61no—
429C2×Dic3—121no—
430C2×Dic3C3×#6711yescharacteristic
431C4×D3—24no—
432C4×D3C2×C3×Dic312yes—
433C4×D3C3×#6712yes—
434Dic6—21no—
435Dic6—24no—
436Dic6C2×C3×Dic312yes—
437C3×C8—61no—
438C3×C8—61no—
439C3×C8—121no—
440C3×C8—181no—
441C3⋊C8—21no—
442C3⋊C8—61no—
443C3⋊C8—61no—
444C3⋊C8—121no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—62no—
458C2×C32D3×C22⋊C411yescharacteristic
459C2×C32#91511yescharacteristic
460C2×C32#91811yescharacteristic
461C2×C32#94211yescharacteristic
462C2×C32#94211yescharacteristic
463C2×C32C2×#21711yescharacteristic
464C2×C32C2×C3×C22⋊C411yescharacteristic
465C2×C32C3×#13711yescharacteristic
466C2×C32C3×#13711yescharacteristic
467C3×D3—24no—
468C3×D3—22no—
469C3×D3—22no—
470C2×Q8—34no—
471C2×Q8—31no—
472C2×Q8—61no—
473C2×Q8—62no—
474C2×Q8C3×D3211yescharacteristic
475C22×C4—31no—
476C22×C4—61no—
477M16—182no—
478C2×C8—31no—
479C2×C8—91no—
480C4⋊C4—31no—
481C4⋊C4—91no—
482C22⋊C4—181no—
483C42—181no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—32no—
488C22×C3—32no—
489C22×C3—41no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3—61no—
496C22×C3—61no—
497C22×C3D3×#6711yescharacteristic
498C22×C3C3×D3×D411yescharacteristic
499C22×C3C2×C3×#6711yescharacteristic
500C2×D3—21no—
501C2×D3—21no—
502C2×D3C3×#21712yes—
503C3×C4—24no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—32no—
514C3×C4—32no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—44no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—62no—
521C3×C4—64no—
522C3×C4—62no—
523C3×C4—62no—
524C3×C4—121no—
525C3×C4—121no—
526C3×C4—241no—
527C3×C4—361no—
528C3×C4C2×D3×Dic311yescharacteristic
529C3×C4D3×#6711yescharacteristic
530C3×C4C2×C3×C4×D311yescharacteristic
531C3×C4C3×D3×D411yescharacteristic
532C3×C4C22×C3×Dic311yescharacteristic
533C3×C4C2×C3×#6711yescharacteristic
534Dic3—22no—
535Dic3—41no—
536Dic3—121no—
537Dic3—121no—
538Dic3—241no—
539Dic3C3×#21712yes—
540C32—21no—
541C32—21no—
542C32—21no—
543C32—41no—
544C32#480211yescharacteristic, complemented
545C32#522511yescharacteristic, complemented
546C32C3×#41911yescharacteristic, complemented
547C23—31no—
548Q8—21no—
549Q8—32no—
550Q8—64no—
551Q8—61no—
552Q8C3×D3×Dic312yes—
553C2×C4—21no—
554C2×C4—31no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—64no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—181no—
561C2×C4C2×C3×D3211yescharacteristic
562C8—61no—
563C8—181no—
564C2×C3—21no—
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—41no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—64no—
577C2×C3—62no—
578C2×C3—62no—
579C2×C3#6394611yescharacteristic
580C2×C3#6394811yescharacteristic
581C2×C3D3×#21711yescharacteristic
582C2×C3C3×D3×C22⋊C411yescharacteristic
583C2×C3C3×#91511yescharacteristic
584C2×C3C3×#91811yescharacteristic
585C2×C3C3×#94211yescharacteristic
586C2×C3C3×#94211yescharacteristic
587C2×C3C2×C3×#21711yescharacteristic
588D3—22no—
589C22—32no—
590C22—61no—
591C22—61no—
592C22C3×D3×#6711yescharacteristic
593C4—22no—
594C4—32no—
595C4—62no—
596C4—121no—
597C4—361no—
598C4C2×C3×D3×Dic311yescharacteristic
599C4C3×D3×#6711yescharacteristic
600C3—21no—
601C3—21no—
602C3—21no—
603C3—41no—
604C3#1059155711yescharacteristic, direct product factor
605C3C3×#480211yescharacteristic, complemented
606C3C3×#522511yescharacteristic, complemented
607C2—62no—
608C2C3×#6394611yescharacteristic
609C2C3×#6394811yescharacteristic
610C2C3×D3×#21711yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×Q8D32
hypocenterC32C3×#419
LCSG ▷ 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
4. C3×#63662C2semidirect
5. C32×#835C2semidirect
6. C3×#63566C2semidirect
7. C3×#63565C2semidirect
10. #10591557C3direct (N × H)
13. C32×D3×Q8C4semidirect
36. #63662C2×C3semidirect
43. C3×#835D3semidirect
44. C3×#835C2×C3semidirect
47. #63566C2×C3semidirect
48. #63565C2×C3semidirect
86. C3×D3×Q8Dic3semidirect
87. C3×D3×Q8C3×C4semidirect
157. #835C3×D3semidirect
271. D3×Q8C3×Dic3semidirect
364. C33#419semidirect
544. C32#4802semidirect
545. C32#5225semidirect
546. C32C3×#419semidirect
604. C3#10591557direct (N × H)
605. C3C3×#4802semidirect
606. C3C3×#5225semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C3×#638412
C2×C3×#37112
C3×#636622
C32×#8352
C3×#635662
C3×#635652
C3×#52253
C3×#48023
#105915573