Finite Group Databasebeta

← C3×#10591499 #13158468

Subgroups — C3×#10591499 #13158468

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#938C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C3×#63856C211yesmaximal, characteristic, complemented
4C2×C3×D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5171—31nomaximal
9C3×#5041—31nomaximal
10#10591499C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×#214—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3724—21no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×D3×C3⋊C8—22no—
24C2×C32×SD16—31no—
25C32×#121—31no—
26C3×#946—31no—
27C3×#938—21no—
28C3×#938D311yescharacteristic
29C3×#938C2×C311yescharacteristic
30C2×C3×#214—21no—
31C2×C3×#214D311yescharacteristic, complemented
32C2×C3×#214C2×C311yescharacteristic, complemented
33C3×#924—31no—
34C22×C3×C3⋊C8—31no—
35C2×C3×D3×C8—31no—
36#63856C2×C311yescharacteristic, complemented
37C2×D3×C3⋊C8C2×C311yescharacteristic, complemented
38#63659C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#832—31no—
41C3×#832—31no—
42C3×#832—61no—
43C3×#817—31no—
44C3×#810—31no—
45C3×#808—31no—
46#63573C2×C311yescharacteristic, complemented
47#63565C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3D411yescharacteristic
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C32×#67—42no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3—21no—
59C32×C4×D3—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8D412yescomplemented
64C3×#465—91no—
65#5171—31no—
66#5041—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—41no—
72C2×C32×D4—61no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C22×C32×C4—31no—
77C2×C3×#67—22no—
78C2×C3×#67—21no—
79C2×C3×#67—21no—
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3C2×D311yescharacteristic
82C2×C3×C4×D3C22×C311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89C32×C22⋊C4—61no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—22no—
93C3×#217—21no—
94C3×#217—21no—
95C3×#217—61no—
96C3×#214—24no—
97C3×#214—22no—
98C3×#214—22no—
99C3×(C2×D3)⋊C4—61no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×C4⋊Dic3C2×D311yescharacteristic
105C3×C4⋊Dic3C22×C311yescharacteristic
106C3×Dic3⋊C4—61no—
107C3×Dic3⋊C4—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—62no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×D3×C8—62no—
116#3724—21no—
117(C2×D3)⋊Dic3—21no—
118D3×C3⋊C8—22no—
119C22×C33—41no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33C2×D411yescharacteristic
123C2×C32×D3—21no—
124C2×C32×D3—21no—
125C2×C32×D3SD1612yescomplemented
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—41no—
129C32×Dic3—21no—
130C32×Dic3—41no—
131C32×Dic3—41no—
132C2×C3×SD16—31no—
133C2×C3×SD16—31no—
134C2×C3×SD16—61no—
135C22×C3×C8—91no—
136C3×C22⋊Q8—91no—
137C3×#121—31no—
138C3×#121—31no—
139C3×#121—61no—
140#946—31no—
141#938C3×D311yescharacteristic
142C2×#214C3×D311yescharacteristic, complemented
143#924—31no—
144C22×C3⋊C8—31no—
145C2×D3×C8—31no—
146C3×#106—91no—
147C3×#103—91no—
148C3×#102—91no—
149#835—31no—
150#832—31no—
151#832—31no—
152#832—61no—
153#817—31no—
154#810—31no—
155#808—31no—
156C23×C32—21no—
157C23×C32—22no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—31no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C22×C3×D3—21no—
165C22×C3×D3#6711yescharacteristic
166C22×C3×D3C3×D411yescharacteristic
167C32×Q8—61no—
168C32×Q8—61no—
169C32×D4—21no—
170C32×D4—21no—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—42no—
182C32×D4—42no—
183C32×D4—122no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—31no—
188C2×C32×C4—41no—
189C2×C32×C4—61no—
190C2×C32×C4—61no—
191C2×C32×C4—61no—
192C2×C32×C4—61no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196C2×C3⋊Dic3—21no—
197C3×#67—44no—
198C3×#67—42no—
199C3×#67—42no—
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—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—121no—
210C2×C3×Dic3#6711yescharacteristic
211C2×C3×Dic3C3×D411yescharacteristic
212C3×C4×D3—21no—
213C3×C4×D3—21no—
214C3×C4×D3—21no—
215C3×C4×D3—21no—
216C3×C4×D3—41no—
217C3×C4×D3—41no—
218C3×Dic6—22no—
219C3×Dic6—21no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—41no—
224C32×C8—32no—
225C3⋊(C3⋊C8)—21no—
226C3×C3⋊C8—22no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—121no—
230C3×C3⋊C8#6712yescomplemented
231C3×C3⋊C8C3×D412yescomplemented
232#465—91noSylow
233C2×C33—81no—
234C2×C33#12111yescharacteristic
235C2×C33C2×SD1611yescharacteristic
236C2×C33#13511yescharacteristic
237C32×D3—22no—
238C2×C3×Q8—31no—
239C2×C3×Q8—31no—
240C2×C3×Q8—61no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—41no—
245C2×C3×D4—61no—
246C2×C3×D4—61no—
247C2×C3×D4—62no—
248C2×C3×D4D3211yescharacteristic
249C2×C3×D4C2×C3×D311yescharacteristic
250C2×C3×D4C2×C3×D311yescharacteristic
251C22×C3×C4—31no—
252C22×C3×C4—31no—
253C22×C3×C4—61no—
254C2×#67—21no—
255C2×C4×D3C2×C3×D311yescharacteristic
256C2×Dic6C2×C3×D311yescharacteristic
257C3×SD16—62no—
258C3×SD16—62no—
259C3×SD16—64no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—61no—
263C2×C3×C8—91no—
264C2×C3×C8—182no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—91no—
269C3×C4⋊C4—181no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—61no—
272C3×C22⋊C4—62no—
273C3×C22⋊C4—181no—
274#217—21no—
275#217—61no—
276#214—22no—
277(C2×D3)⋊C4—61no—
278C4⋊Dic3—31no—
279C4⋊Dic3—31no—
280C4⋊Dic3—61no—
281C4⋊Dic3C2×C3×D311yescharacteristic
282Dic3⋊C4—61no—
283Dic3⋊C4—61no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8—62no—
288C2×C3⋊C8C2×C3×D311yescharacteristic
289D3×C8—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—32no—
294C22×C32—41no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—42no—
298C22×C32—44no—
299C22×C32—41no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—61no—
311C22×C32—61no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32D3×D411yescharacteristic
319C22×C32C2×#6711yescharacteristic
320C22×C32C2×C3×D411yescharacteristic
321C2×C3×D3—21no—
322C2×C3×D3—21no—
323C2×C3×D3—21no—
324C2×C3×D3—21no—
325C2×C3×D3—22no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C2×C3×D3#21412yescomplemented
329C2×C3×D3C3×SD1612yescomplemented
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4—61no—
339C32×C4—121no—
340C32×C4—121no—
341C32×C4D3×D411yescharacteristic
342C32×C4#23711yescharacteristic
343C32×C4#23711yescharacteristic
344C32×C4C2×#6711yescharacteristic
345C32×C4C2×C3×D411yescharacteristic
346C32×C4C3×#4111yescharacteristic
347C3⋊Dic3—41no—
348C3×Dic3—21no—
349C3×Dic3—21no—
350C3×Dic3—41no—
351C3×Dic3—42no—
352C3×Dic3—42no—
353C3×Dic3—41no—
354C3×Dic3—41no—
355C3×Dic3—41no—
356C3×Dic3—41no—
357C3×Dic3—121no—
358C3×Dic3—121no—
359C3×Dic3—241no—
360C2×SD16—31no—
361C22×C8—91no—
362C22⋊Q8—91no—
363#121—31no—
364#106—91no—
365#103—91no—
366#102—91no—
367C33#46511yescharacteristic, complemented, Sylow
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—31no—
373C23×C3—31no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—61no—
378C22×D3C3×#6711yescharacteristic
379C3×Q8—62no—
380C3×Q8—61no—
381C3×Q8—61no—
382C3×Q8—61no—
383C3×Q8—61no—
384C3×Q8—121no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—21no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—122no—
400C3×D4—122no—
401C3×D4—124no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—31no—
406C2×C3×C4—31no—
407C2×C3×C4—41no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—62no—
414C2×C3×C4—61no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—121no—
419C2×C3×C4—121no—
420C2×C3×C4—121no—
421C2×C3×C4—181no—
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C22×C3×D311yescharacteristic
424C2×C3×C4C22×C3×D311yescharacteristic
425#67—42no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—121no—
431C2×Dic3C3×#6711yescharacteristic
432C4×D3—21no—
433C4×D3—21no—
434Dic6—21no—
435Dic6—21no—
436C3×C8—32no—
437C3×C8—32no—
438C3×C8—62no—
439C3×C8—181no—
440C3⋊C8—61no—
441C3⋊C8—61no—
442C3⋊C8—121no—
443C3⋊C8C3×#6712yescomplemented
444C2×C32—21no—
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—41no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—62no—
457C2×C32—82no—
458C2×C32—82no—
459C2×C32—82no—
460C2×C32—84no—
461C2×C32—81no—
462C2×C32—81no—
463C2×C32—81no—
464C2×C32D3×SD1611yescharacteristic
465C2×C32#91611yescharacteristic
466C2×C32C2×#21411yescharacteristic
467C2×C32#93811yescharacteristic
468C2×C32#93911yescharacteristic
469C2×C32#95011yescharacteristic
470C2×C32C3×#12111yescharacteristic
471C2×C32C2×C3×SD1611yescharacteristic
472C2×C32C3×#13511yescharacteristic
473C3×D3—22no—
474C3×D3—22no—
475C3×D3—42no—
476C2×Q8—31no—
477C2×D4—61no—
478C2×D4C3×D3211yescharacteristic
479C22×C4—31no—
480SD16—62no—
481C2×C8—31no—
482C2×C8—91no—
483C2×C8—182no—
484C4⋊C4—31no—
485C4⋊C4—91no—
486C4⋊C4—181no—
487C22⋊C4—61no—
488C22⋊C4—181no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—32no—
493C22×C3—32no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—44no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—62no—
511C22×C3—61no—
512C22×C3—61no—
513C22×C3—61no—
514C22×C3—61no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—82no—
520C22×C3—82no—
521C22×C3—121no—
522C22×C3—121no—
523C22×C3D3×#6711yescharacteristic
524C22×C3C3×D3×D411yescharacteristic
525C22×C3C2×C3×#6711yescharacteristic
526C2×D3—21no—
527C2×D3—21no—
528C2×D3C3×#21412yescomplemented
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—61no—
538C3×C4—61no—
539C3×C4—121no—
540C3×C4—122no—
541C3×C4—122no—
542C3×C4—121no—
543C3×C4—121no—
544C3×C4—121no—
545C3×C4—121no—
546C3×C4—361no—
547C3×C4#380511yescharacteristic
548C3×C4D3×#6711yescharacteristic
549C3×C4C3×D3×D411yescharacteristic
550C3×C4C3×#23711yescharacteristic
551C3×C4C3×#23711yescharacteristic
552C3×C4C2×C3×#6711yescharacteristic
553Dic3—21no—
554Dic3—41no—
555Dic3—41no—
556Dic3—121no—
557Dic3—121no—
558Dic3—241no—
559C32—21no—
560C32—21no—
561C32—21no—
562C32—41no—
563C32#504111yescharacteristic, complemented
564C32#517111yescharacteristic, complemented
565C32C3×#46511yescharacteristic, complemented
566C23—21no—
567C23—31no—
568Q8—61no—
569Q8—61no—
570D4—21no—
571D4—21no—
572D4—122no—
573C2×C4—31no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—181no—
579C2×C4C2×C3×D3211yescharacteristic
580C8—32no—
581C8—181no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—62no—
595C2×C3—62no—
596C2×C3—82no—
597C2×C3—82no—
598C2×C3—82no—
599C2×C3—84no—
600C2×C3—81no—
601C2×C3—81no—
602C2×C3—81no—
603C2×C3—122no—
604C2×C3D3×#21411yescharacteristic
605C2×C3#6393511yescharacteristic
606C2×C3#6397011yescharacteristic
607C2×C3C3×D3×SD1611yescharacteristic
608C2×C3C3×#91611yescharacteristic
609C2×C3C2×C3×#21411yescharacteristic
610C2×C3C3×#93811yescharacteristic
611C2×C3C3×#93911yescharacteristic
612C2×C3C3×#95011yescharacteristic
613D3—22no—
614C22—32no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—61no—
619C22—61no—
620C22C3×D3×#6711yescharacteristic
621C4—61no—
622C4—121no—
623C4—121no—
624C4—361no—
625C4C3×#380511yescharacteristic
626C4C3×D3×#6711yescharacteristic
627C3—21no—
628C3—21no—
629C3—21no—
630C3—41no—
631C3#1059149911yescharacteristic, direct product factor
632C3C3×#504111yescharacteristic, complemented
633C3C3×#517111yescharacteristic, complemented
634C2—62no—
635C2—81no—
636C2C3×D3×#21411yescharacteristic
637C2C3×#6393511yescharacteristic
638C2C3×#6397011yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#465
LCSG ▷ C2×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
2. C2×C32×#214C2semidirect
3. C3×#63856C2semidirect
4. C2×C3×D3×C3⋊C8C2semidirect
5. C3×#63659C2semidirect
6. C3×#63573C2semidirect
7. C3×#63565C2semidirect
10. #10591499C3direct (N × H)
31. C2×C3×#214D3semidirect
32. C2×C3×#214C2×C3semidirect
36. #63856C2×C3semidirect
37. C2×D3×C3⋊C8C2×C3semidirect
38. #63659C2×C3semidirect
46. #63573C2×C3semidirect
47. #63565C2×C3semidirect
63. C32×C3⋊C8D4semidirect
125. C2×C32×D3SD16semidirect
142. C2×#214C3×D3semidirect
230. C3×C3⋊C8#67semidirect
231. C3×C3⋊C8C3×D4semidirect
328. C2×C3×D3#214semidirect
329. C2×C3×D3C3×SD16semidirect
367. C33#465semidirect
443. C3⋊C8C3×#67semidirect
528. C2×D3C3×#214semidirect
563. C32#5041semidirect
564. C32#5171semidirect
565. C32C3×#465semidirect
631. C3#10591499direct (N × H)
632. C3C3×#5041semidirect
633. C3C3×#5171semidirect

Maximal subgroups

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

subgroupindex
C32×#9382
C2×C32×#2142
C3×#638562
C2×C3×D3×C3⋊C82
C3×#636592
C3×#635732
C3×#635652
C3×#51713
C3×#50413
#105914993