Finite Group Databasebeta

← C3×#10591435 #13158442

Subgroups — C3×#10591435 #13158442

Order 1728.

810 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
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic
3C3×#63898C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63553C211yesmaximal, characteristic, complemented
8C3×Dic3×D8—31nomaximal
9C3×#4756—31nomaximal
10#10591435C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C32×D12C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#213C414yescomplemented
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C2×C32×D8—31no—
24D4×C3⋊Dic3C2×C311yescharacteristic, complemented
25C3×D4×Dic3—31no—
26C3×D4×Dic3—31no—
27C3×D4×Dic3—31no—
28C3×D4×Dic3—61no—
29C2×C3×#213—21no—
30C2×C3×#213D311yescharacteristic
31C2×C3×#213C2×C311yescharacteristic
32C3×#893—31no—
33#63898C2×C311yescharacteristic, complemented
34C32×#102—31no—
35C3×#832—21no—
36C3×#832—31no—
37C3×#832D311yescharacteristic, complemented
38C3×#832C2×C311yescharacteristic, complemented
39C3×#821—31no—
40C3×#818—31no—
41C3×C8×Dic3—31no—
42C3×C8×Dic3—31no—
43C3×#807—31no—
44#63575C2×C311yescharacteristic, complemented
45#63563C2×C311yescharacteristic, complemented
46#63553C2×C311yescharacteristic, complemented
47C23×C33—21no—
48C22×C32×D3—21no—
49C33×D4—21no—
50C33×D4C2×C412yescomplemented
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—42no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C32×Dic3—21no—
57C32×D12—21no—
58C32×D12C2×C412yescomplemented
59C3×D3×Dic3—42no—
60C32×C3⋊C8—21no—
61C32×C3⋊C8C2×C412yescomplemented
62C3×C4×D8—91no—
63Dic3×D8—31no—
64#4756—31no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—21no—
68C2×C32×D4—31no—
69C2×C32×D4—41no—
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C2×D311yescharacteristic
72C2×C32×D4C22×C311yescharacteristic
73C22×C3⋊Dic3—21no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3—61no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3—62no—
78C2×C3×D12—21no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C22×C311yescharacteristic
81C2×C3×C4×D3—61no—
82C2×D3×Dic3—21no—
83C32×D8—34no—
84C2×C32×C8—31no—
85C2×C32×C8—31no—
86C32×C4⋊C4—31no—
87#3757—21no—
88C4⋊(C3⋊Dic3)C22×C311yescharacteristic
89C4×C3⋊Dic3C22×C311yescharacteristic
90C3×#217—61no—
91C3×#217—61no—
92C3×#217—61no—
93C3×#217—62no—
94C3×#213—24no—
95C3×#213Dic314yescomplemented
96C3×#213C3×C414yescomplemented
97C3×(C2×D3)⋊C4—61no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C22×C311yescharacteristic
104C3×C4×Dic3—31no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—61no—
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
113(C2×D3)⋊Dic3—21no—
114C22×C33—22no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33C2×D411yescharacteristic
118C2×C32×D3—22no—
119C2×C32×D3—41no—
120C2×C32×D3—41no—
121C33×C4C22×C411yescharacteristic
122C33×C4#4111yescharacteristic
123C3×C3⋊Dic3—21no—
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3D812yescomplemented
126C32×Dic3—41no—
127C2×C3×D8—31no—
128C2×C3×D8—31no—
129C2×C3×D8—61no—
130C3×C4×D4—91no—
131C3×C4×D4—91no—
132D4×Dic3—31no—
133D4×Dic3—31no—
134D4×Dic3—31no—
135D4×Dic3—61no—
136C2×#213C3×D311yescharacteristic
137#893—31no—
138C3×#107—91no—
139C3×#102—31no—
140C3×#102—31no—
141C3×#102—61no—
142C3×#102—91no—
143C3×C4×C8—91no—
144#832—31no—
145#832C3×D311yescharacteristic, complemented
146#821—31no—
147#818—31no—
148C8×Dic3—31no—
149C8×Dic3—31no—
150#807—31no—
151C23×C32—21no—
152C23×C32—22no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—41no—
156C23×C32—41no—
157C23×C32—42no—
158C23×C32—61no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—41no—
162C32×D4—21no—
163C32×D4—22no—
164C32×D4—21no—
165C32×D4—21no—
166C32×D4—22no—
167C32×D4—22no—
168C32×D4—22no—
169C32×D4—32no—
170C32×D4—42no—
171C32×D4—41no—
172C32×D4—41no—
173C32×D4—42no—
174C32×D4—61no—
175C32×D4C4×D312yescomplemented
176C32×D4C2×Dic312yescomplemented
177C32×D4C2×C3×C412yescomplemented
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—41no—
182C2×C32×C4—61no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C23×C311yescharacteristic
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—42no—
188C2×C3⋊Dic3C3×D411yescharacteristic
189C2×C3⋊Dic3C3×D411yescharacteristic
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—41no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—122no—
203C2×C3×Dic3—122no—
204C2×C3×Dic3—122no—
205C2×C3×Dic3—124no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—22no—
209C3×D12—41no—
210C3×D12C2×Dic312yescomplemented
211C3×D12C2×C3×C412yescomplemented
212C3×C4×D3—122no—
213D3×Dic3—42no—
214C32×C8—32no—
215C32×C8—61no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—22no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8C2×Dic312yescomplemented
221C3×C3⋊C8C2×C3×C412yescomplemented
222C4×D8—91noSylow
223C2×C33—42no—
224C2×C33C4×D411yescharacteristic
225C2×C33C2×D811yescharacteristic
226C2×C33#13511yescharacteristic
227C32×D3—42no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—31no—
232C2×C3×D4—31no—
233C2×C3×D4—41no—
234C2×C3×D4—61no—
235C2×C3×D4D3211yescharacteristic
236C2×C3×D4C2×C3×D311yescharacteristic
237C2×C3×D4C2×C3×D311yescharacteristic
238C22×C3×C4—181no—
239C22×C3×C4—181no—
240C22×Dic3—61no—
241C22×Dic3—61no—
242C22×Dic3—61no—
243C22×Dic3—62no—
244C2×D12C2×C3×D311yescharacteristic
245C2×C4×D3—61no—
246C3×D8—34no—
247C3×D8—34no—
248C3×D8—64no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—91no—
259C3×C22⋊C4—181no—
260C3×C22⋊C4—181no—
261C3×C42—91no—
262#217—61no—
263#217—61no—
264#217—61no—
265#217—62no—
266#213C3×Dic314yescomplemented
267(C2×D3)⋊C4—61no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—61no—
271C4⋊Dic3C2×C3×D311yescharacteristic
272C4×Dic3—31no—
273C4×Dic3—31no—
274C4×Dic3—61no—
275C2×C3⋊C8C2×C3×D311yescharacteristic
276C2×C3⋊C8C2×C3×D311yescharacteristic
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—42no—
287C22×C32—44no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—41no—
296C22×C32—62no—
297C22×C32—81no—
298C22×C32—81no—
299C22×C32—81no—
300C22×C32—81no—
301C22×C32—82no—
302C22×C32—82no—
303C22×C32—121no—
304C22×C32—121no—
305C22×C32D3×D411yescharacteristic
306C22×C32D3×D411yescharacteristic
307C22×C32C2×C3×D411yescharacteristic
308C2×C3×D3—22no—
309C2×C3×D3—22no—
310C2×C3×D3—42no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C2×C3×D3—81no—
316C2×C3×D3—81no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—121no—
326C32×C4C2×C4×D311yescharacteristic
327C32×C4#23711yescharacteristic
328C32×C4#23711yescharacteristic
329C32×C4C22×Dic311yescharacteristic
330C32×C4C22×C3×C411yescharacteristic
331C32×C4C3×#4111yescharacteristic
332C3⋊Dic3—21no—
333C3⋊Dic3—41no—
334C3⋊Dic3C3×D812yescomplemented
335C3×Dic3—32no—
336C3×Dic3—32no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—62no—
340C3×Dic3—61no—
341C3×Dic3—61no—
342C3×Dic3—81no—
343C3×Dic3—121no—
344C3×Dic3—122no—
345C3×Dic3—121no—
346C3×Dic3—121no—
347C2×D8—31no—
348C4×D4—91no—
349C4×D4—91no—
350#107—91no—
351#102—31no—
352#102—91no—
353C4×C8—91no—
354C33C4×D811yescharacteristic, complemented, Sylow
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—42no—
362C23×C3—61no—
363C23×C3—61no—
364C23×C3—121no—
365C22×D3—21no—
366C3×D4—22no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—32no—
374C3×D4—32no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—62no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—121no—
383C3×D4D3×Dic312yescomplemented
384C3×D4C3×C4×D312yescomplemented
385C3×D4C2×C3×Dic312yescomplemented
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—121no—
395C2×C3×C4—181no—
396C2×C3×C4—362no—
397C2×C3×C4—362no—
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401C2×Dic3—21no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—62no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—122no—
412C2×Dic3—122no—
413C2×Dic3—122no—
414C2×Dic3—124no—
415D12—21no—
416D12C2×C3×Dic312yescomplemented
417C4×D3—122no—
418C3×C8—32no—
419C3×C8—32no—
420C3×C8—62no—
421C3×C8—61no—
422C3×C8—61no—
423C3×C8—121no—
424C3⋊C8—21no—
425C3⋊C8C2×C3×Dic312yescomplemented
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32—42no—
439C2×C32—44no—
440C2×C32—42no—
441C2×C32—42no—
442C2×C32—82no—
443C2×C32—82no—
444C2×C32—84no—
445C2×C32—122no—
446C2×C32#67⋊C411yescharacteristic
447C2×C32D3×D811yescharacteristic
448C2×C32D3×D811yescharacteristic
449C2×C32#91211yescharacteristic
450C2×C32#91611yescharacteristic
451C2×C32D4×Dic311yescharacteristic
452C2×C32C3×C4×D411yescharacteristic
453C2×C32C2×C3×D811yescharacteristic
454C2×C32C3×#13511yescharacteristic
455C3×D3—42no—
456C3×D3—42no—
457C3×D3—82no—
458C2×D4—31no—
459C2×D4C3×D3211yescharacteristic
460C22×C4—181no—
461C22×C4—181no—
462D8—34no—
463C2×C8—31no—
464C2×C8—31no—
465C4⋊C4—31no—
466C4⋊C4—91no—
467C22⋊C4—181no—
468C22⋊C4—181no—
469C42—91no—
470C22×C3—24no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—62no—
490C22×C3—62no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—122no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3—121no—
502C22×C3—241no—
503C22×C3—241no—
504C22×C3#381111yescharacteristic
505C22×C3C3×D3×D411yescharacteristic
506C22×C3C3×D3×D411yescharacteristic
507C2×D3—22no—
508C2×D3—41no—
509C2×D3—41no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—92no—
519C3×C4—121no—
520C3×C4—121no—
521C3×C4—181no—
522C3×C4—241no—
523C3×C4—361no—
524C3×C4C2×D3×Dic311yescharacteristic
525C3×C4#380511yescharacteristic
526C3×C4C2×C3×C4×D311yescharacteristic
527C3×C4C3×#23711yescharacteristic
528C3×C4C3×#23711yescharacteristic
529C3×C4C22×C3×Dic311yescharacteristic
530Dic3—32no—
531Dic3—32no—
532Dic3—41no—
533Dic3—62no—
534Dic3—61no—
535Dic3—61no—
536Dic3—121no—
537Dic3—122no—
538Dic3—121no—
539Dic3—121no—
540C32—21no—
541C32—21no—
542C32—21no—
543C32—41no—
544C32#475611yescharacteristic, complemented
545C32Dic3×D811yescharacteristic, complemented
546C32C3×C4×D811yescharacteristic, complemented
547C23—21no—
548C23—61no—
549D4—21no—
550D4—32no—
551D4—61no—
552D4C3×D3×Dic312yescomplemented
553C2×C4—61no—
554C2×C4—91no—
555C2×C4—91no—
556C2×C4—181no—
557C2×C4—362no—
558C2×C4—362no—
559C2×C4C2×C3×D3211yescharacteristic
560C8—32no—
561C8—61no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—44no—
572C2×C3—42no—
573C2×C3—42no—
574C2×C3—42no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—41no—
578C2×C3—82no—
579C2×C3—82no—
580C2×C3—84no—
581C2×C3—122no—
582C2×C3—122no—
583C2×C3—242no—
584C2×C3#6392611yescharacteristic
585C2×C3#6393611yescharacteristic
586C2×C3#67⋊Dic311yescharacteristic
587C2×C3C3×#67⋊C411yescharacteristic
588C2×C3C3×D3×D811yescharacteristic
589C2×C3C3×D3×D811yescharacteristic
590C2×C3C3×#91211yescharacteristic
591C2×C3C3×#91611yescharacteristic
592C2×C3C3×D4×Dic311yescharacteristic
593D3—42no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—62no—
598C22—121no—
599C22—121no—
600C22C3×#381111yescharacteristic
601C4—92no—
602C4—121no—
603C4—181no—
604C4—361no—
605C4C2×C3×D3×Dic311yescharacteristic
606C4C3×#380511yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059143511yescharacteristic, direct product factor
612C3C3×#475611yescharacteristic, complemented
613C3C3×Dic3×D811yescharacteristic, complemented
614C2—42no—
615C2—122no—
616C2C3×#6392611yescharacteristic
617C2C3×#6393611yescharacteristic
618C2C3×#67⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#213C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×C4×D8
LCSG ▷ 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. C3×D4×C3⋊Dic3C2semidirect
3. C3×#63898C2semidirect
4. C32×#832C2semidirect
5. C3×#63575C2semidirect
6. C3×#63563C2semidirect
7. C3×#63553C2semidirect
10. #10591435C3direct (N × H)
18. C32×#213C4semidirect
24. D4×C3⋊Dic3C2×C3semidirect
33. #63898C2×C3semidirect
37. C3×#832D3semidirect
38. C3×#832C2×C3semidirect
44. #63575C2×C3semidirect
45. #63563C2×C3semidirect
46. #63553C2×C3semidirect
50. C33×D4C2×C4semidirect
58. C32×D12C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
95. C3×#213Dic3semidirect
96. C3×#213C3×C4semidirect
125. C3×C3⋊Dic3D8semidirect
145. #832C3×D3semidirect
175. C32×D4C4×D3semidirect
176. C32×D4C2×Dic3semidirect
177. C32×D4C2×C3×C4semidirect
210. C3×D12C2×Dic3semidirect
211. C3×D12C2×C3×C4semidirect
220. C3×C3⋊C8C2×Dic3semidirect
221. C3×C3⋊C8C2×C3×C4semidirect
266. #213C3×Dic3semidirect
334. C3⋊Dic3C3×D8semidirect
354. C33C4×D8semidirect
383. C3×D4D3×Dic3semidirect
384. C3×D4C3×C4×D3semidirect
385. C3×D4C2×C3×Dic3semidirect
416. D12C2×C3×Dic3semidirect
425. C3⋊C8C2×C3×Dic3semidirect
544. C32#4756semidirect
545. C32Dic3×D8semidirect
546. C32C3×C4×D8semidirect
552. D4C3×D3×Dic3semidirect
611. C3#10591435direct (N × H)
612. C3C3×#4756semidirect
613. C3C3×Dic3×D8semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C32×#2132
C3×#638982
C32×#8322
C3×#635752
C3×#635632
C3×#635532
C3×Dic3×D83
C3×#47563
#105914353