Finite Group Databasebeta

← C3×#10591482 #13158495

Subgroups — C3×#10591482 #13158495

Order 1728.

962 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
1C2×C3×#3753C211yesmaximal, characteristic, complemented
2C32×#940C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic, complemented
5C2×C3×#3714C211yesmaximal, characteristic, complemented
6C2×C3×#3713C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5151—31nomaximal
9C3×#5073—31nomaximal
10#10591482C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×#67—22no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×C3×#626—22no—
17C3×#3753—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#213—22no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3714—22no—
23C3×#3713—22no—
24C2×C32×D8—31no—
25C32×#127—31no—
26C2×#3753C2×C311yescharacteristic, complemented
27C3×#940—21no—
28C3×#940—31no—
29C3×#940D311yescharacteristic, complemented
30C3×#940C2×C311yescharacteristic, complemented
31C2×C3×#213—21no—
32C2×C3×#213—31no—
33C2×C3×#213—31no—
34C2×C3×#213—31no—
35C2×C3×#213—31no—
36C2×C3×#213—31no—
37C2×C3×#213—61no—
38C2×C3×#213D311yescharacteristic, complemented
39C2×C3×#213C2×C311yescharacteristic, complemented
40C2×C3×D24—31no—
41C3×#874—31no—
42#63911C2×C311yescharacteristic, complemented
43C2×#3714C2×C311yescharacteristic, complemented
44C2×#3713C2×C311yescharacteristic, complemented
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C22×C32×D3—21no—
52C33×D4—21no—
53C33×D4—21no—
54C2×C33×C4C2311yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C32×#67—44no—
58C2×C32×Dic3D412yes—
59C32×D12—21no—
60C32×D12—21no—
61C32×D12—21no—
62C32×D12—21no—
63C3×#626—44no—
64C3×C3⋊(C3⋊C8)D412yescomplemented
65C32×C3⋊C8D412yescomplemented
66C3×#493—91no—
67#5151—31no—
68#5073—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—31no—
73C2×C32×D4—31no—
74C2×C32×D4—41no—
75C2×C32×D4—62no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C2×D311yescharacteristic
78C2×C32×D4C22×C311yescharacteristic
79C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
80C2×C3×#67—24no—
81C2×C3×#67—22no—
82C2×C3×#67—22no—
83C2×C3×#67—62no—
84C2×C3×D12—21no—
85C2×C3×D12—21no—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—61no—
89C2×C3×D12—62no—
90C2×C3×D12C2×D311yescharacteristic
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×D12C22×C311yescharacteristic
93C2×C3×D12C22×C311yescharacteristic
94C2×#626—22no—
95C32×D8—62no—
96C2×C32×C8—31no—
97C32×C42—31no—
98#3753—22no—
99C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
100C3×#213—24no—
101C3×#213—22no—
102C3×#213—22no—
103C3×#213—62no—
104C3×#213—62no—
105C3×#213—62no—
106C3×#213—62no—
107C3×#213—64no—
108C3×#213—62no—
109C3×C4×Dic3—21no—
110C3×C4×Dic3C2×D311yescharacteristic
111C3×C4×Dic3C22×C311yescharacteristic
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118C3×D24—62no—
119#3714—22no—
120#3713—22no—
121C22×C33—41no—
122C22×C33—42no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—42no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C2×C32×D3—42no—
129C2×C32×D3—42no—
130C33×C4C2×D411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C32×Dic3D814yescomplemented
133C2×C3×D8—31no—
134C2×C3×D8—31no—
135C2×C3×D8—61no—
136C2×C3×D8—91no—
137C2×C3×D8—91no—
138C2×C3×D8—91no—
139C3×#127—31no—
140C3×#127—31no—
141C3×#127—61no—
142C3×#127—91no—
143#940—31no—
144#940C3×D311yescharacteristic, complemented
145C2×#213—31no—
146C2×#213—31no—
147C2×#213—31no—
148C2×#213—31no—
149C2×#213—31no—
150C2×#213—61no—
151C2×#213C3×D311yescharacteristic, complemented
152C2×D24—31no—
153#874—31no—
154C3×C4×C8—91no—
155C8×Dic3—31no—
156C4×C3⋊C8—31no—
157C23×C32—21no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C23×C32—61no—
165C23×C32—61no—
166C22×C3⋊D3—21no—
167C22×C3×D3—22no—
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—21no—
171C22×C3×D3—21no—
172C22×C3×D3—41no—
173C22×C3×D3—61no—
174C22×C3×D3—61no—
175C22×C3×D3—121no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—22no—
179C32×D4—22no—
180C32×D4—21no—
181C32×D4—21no—
182C32×D4—21no—
183C32×D4—21no—
184C32×D4—41no—
185C32×D4—41no—
186C32×D4—41no—
187C32×D4—41no—
188C32×D4—42no—
189C32×D4—42no—
190C32×D4—61no—
191C32×D4—61no—
192C32×D4—61no—
193C32×D4—61no—
194C32×D4—124no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—21no—
198C2×C32×C4—32no—
199C2×C32×C4—41no—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C23×C311yescharacteristic
203C4⋊(C3⋊D3)—21no—
204C4⋊(C3⋊D3)—21no—
205C3×#67—48no—
206C3×#67—44no—
207C3×#67—44no—
208C3×#67—124no—
209C2×C3×Dic3—22no—
210C2×C3×Dic3#6712yes—
211C2×C3×Dic3C3×D412yes—
212C3×D12—22no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—21no—
216C3×D12—21no—
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—21no—
221C3×D12—41no—
222C3×D12—41no—
223C3×D12—41no—
224C3×D12—62no—
225C3×D12—61no—
226C3×D12—61no—
227C3×D12—61no—
228C3×D12—61no—
229C3×D12—121no—
230C3×D12—124no—
231#626—44no—
232C32×C8—32no—
233C3⋊(C3⋊C8)C3×D412yescomplemented
234C3×C3⋊C8—22no—
235C3×C3⋊C8—32no—
236C3×C3⋊C8—32no—
237C3×C3⋊C8—62no—
238C3×C3⋊C8#6712yescomplemented
239C3×C3⋊C8C3×D412yescomplemented
240#493—91noSylow
241C2×C33—81no—
242C2×C33#12711yescharacteristic
243C2×C33C2×D812yes—
244C3×C3⋊D3—81no—
245C32×D3—81no—
246C32×D3—81no—
247C2×C3×D4—21no—
248C2×C3×D4—21no—
249C2×C3×D4—21no—
250C2×C3×D4—31no—
251C2×C3×D4—31no—
252C2×C3×D4—31no—
253C2×C3×D4—31no—
254C2×C3×D4—41no—
255C2×C3×D4—61no—
256C2×C3×D4—61no—
257C2×C3×D4—62no—
258C2×C3×D4—62no—
259C2×C3×D4—64no—
260C2×C3×D4—91no—
261C2×C3×D4—182no—
262C2×C3×D4D3211yescharacteristic
263C2×C3×D4C2×C3×D311yescharacteristic
264C2×C3×D4C2×C3×D311yescharacteristic
265C2×#67—22no—
266C2×#67—62no—
267C2×D12—31no—
268C2×D12—31no—
269C2×D12—61no—
270C2×D12—62no—
271C2×D12C2×C3×D311yescharacteristic
272C2×D12C2×C3×D311yescharacteristic
273C3×D8—62no—
274C3×D8—62no—
275C3×D8—64no—
276C3×D8—182no—
277C3×D8—182no—
278C3×D8—182no—
279C2×C3×C8—31no—
280C2×C3×C8—31no—
281C2×C3×C8—61no—
282C2×C3×C8—91no—
283C3×C42—31no—
284C3×C42—31no—
285C3×C42—61no—
286#213—22no—
287#213—62no—
288#213—62no—
289#213—62no—
290#213—62no—
291#213—64no—
292#213—62no—
293C4×Dic3C2×C3×D311yescharacteristic
294C2×C3⋊C8—31no—
295C2×C3⋊C8—31no—
296C2×C3⋊C8—61no—
297C2×C3⋊C8C2×C3×D311yescharacteristic
298D24—62no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—41no—
303C22×C32—42no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—44no—
307C22×C32—42no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—44no—
312C22×C32—42no—
313C22×C32—42no—
314C22×C32—82no—
315C22×C32—82no—
316C22×C32—84no—
317C22×C32—122no—
318C22×C32—121no—
319C22×C32—121no—
320C22×C32—122no—
321C22×C32D3×D411yescharacteristic
322C22×C32C2×#6711yescharacteristic
323C22×C32C2×C3×D411yescharacteristic
324C2×C3⋊D3—41no—
325C2×C3⋊D3—42no—
326C2×C3×D3—42no—
327C2×C3×D3—42no—
328C2×C3×D3—44no—
329C2×C3×D3—42no—
330C2×C3×D3—42no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C2×C3×D3—42no—
336C2×C3×D3—42no—
337C2×C3×D3—82no—
338C2×C3×D3—122no—
339C2×C3×D3—122no—
340C2×C3×D3—122no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3—242no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—21no—
350C32×C4—34no—
351C32×C4—41no—
352C32×C4—41no—
353C32×C4D3×D411yescharacteristic
354C32×C4D3×D411yescharacteristic
355C32×C4D3×D411yescharacteristic
356C32×C4C2×#6711yescharacteristic
357C32×C4C2×C3×D411yescharacteristic
358C32×C4C2×C3×D411yescharacteristic
359C3×Dic3—24no—
360C3×Dic3#21314yescomplemented
361C3×Dic3C3×D814yescomplemented
362C2×D8—31no—
363C2×D8—91no—
364C2×D8—91no—
365C2×D8—91no—
366#127—31no—
367#127—91no—
368C4×C8—91no—
369C33#49311yescharacteristic, complemented, Sylow
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—62no—
378C23×C3—61no—
379C23×C3—61no—
380C23×C3—61no—
381C23×C3—61no—
382C23×C3—121no—
383C23×C3—181no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—61no—
387C22×D3—61no—
388C22×D3—121no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—21no—
392C3×D4—21no—
393C3×D4—21no—
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—21no—
397C3×D4—41no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—42no—
402C3×D4—42no—
403C3×D4—62no—
404C3×D4—61no—
405C3×D4—61no—
406C3×D4—61no—
407C3×D4—61no—
408C3×D4—61no—
409C3×D4—61no—
410C3×D4—61no—
411C3×D4—61no—
412C3×D4—121no—
413C3×D4—121no—
414C3×D4—121no—
415C3×D4—124no—
416C3×D4—124no—
417C3×D4—128no—
418C3×D4—181no—
419C3×D4—181no—
420C3×D4—364no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—32no—
425C2×C3×C4—32no—
426C2×C3×C4—41no—
427C2×C3×C4—62no—
428C2×C3×C4C2×D3211yescharacteristic
429C2×C3×C4C22×C3×D311yescharacteristic
430C2×C3×C4C22×C3×D311yescharacteristic
431#67—44no—
432#67—124no—
433C2×Dic3C3×#6712yes—
434D12—21no—
435D12—21no—
436D12—21no—
437D12—21no—
438D12—62no—
439D12—61no—
440D12—61no—
441D12—61no—
442D12—61no—
443D12—121no—
444D12—124no—
445C3×C8—32no—
446C3×C8—32no—
447C3×C8—62no—
448C3×C8—92no—
449C3⋊C8—32no—
450C3⋊C8—32no—
451C3⋊C8—62no—
452C3⋊C8C3×#6712yescomplemented
453C2×C32—22no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—22no—
458C2×C32—21no—
459C2×C32—41no—
460C2×C32—42no—
461C2×C32—82no—
462C2×C32—82no—
463C2×C32—82no—
464C2×C32—84no—
465C2×C32—81no—
466C2×C32—81no—
467C2×C32—81no—
468C2×C32—241no—
469C2×C32—241no—
470C2×C32D3×D812yes—
471C2×C32C2×#21312yes—
472C2×C32#94011yescharacteristic
473C2×C32#94011yescharacteristic
474C2×C32C3×#12711yescharacteristic
475C2×C32C2×C3×D812yes—
476C3⋊D3—81no—
477C3×D3—82no—
478C3×D3—82no—
479C3×D3—81no—
480C3×D3—81no—
481C3×D3—81no—
482C3×D3—81no—
483C3×D3—242no—
484C3×D3—241no—
485C3×D3—241no—
486C2×D4—31no—
487C2×D4—31no—
488C2×D4—62no—
489C2×D4—91no—
490C2×D4—182no—
491C2×D4C3×D3211yescharacteristic
492D8—62no—
493D8—182no—
494D8—182no—
495D8—182no—
496C2×C8—31no—
497C2×C8—91no—
498C42—31no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—42no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—84no—
517C22×C3—122no—
518C22×C3—122no—
519C22×C3—124no—
520C22×C3—122no—
521C22×C3—122no—
522C22×C3—121no—
523C22×C3—121no—
524C22×C3—122no—
525C22×C3—122no—
526C22×C3—121no—
527C22×C3—121no—
528C22×C3—242no—
529C22×C3—362no—
530C22×C3—361no—
531C22×C3D3×#6711yescharacteristic
532C22×C3C3×D3×D411yescharacteristic
533C22×C3C2×C3×#6711yescharacteristic
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—41no—
537C2×D3—41no—
538C2×D3—122no—
539C2×D3—122no—
540C2×D3—122no—
541C2×D3—121no—
542C2×D3—121no—
543C2×D3—242no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—34no—
551C3×C4—34no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—64no—
555C3×C4D3×#6711yescharacteristic
556C3×C4#381111yescharacteristic
557C3×C4C3×D3×D411yescharacteristic
558C3×C4C3×D3×D411yescharacteristic
559C3×C4C3×D3×D411yescharacteristic
560C3×C4C2×C3×#6711yescharacteristic
561Dic3C3×#21314yescomplemented
562C32—21no—
563C32—21no—
564C32—21no—
565C32—41no—
566C32#507311yescharacteristic, complemented
567C32#515111yescharacteristic, complemented
568C32C3×#49311yescharacteristic, complemented
569C23—21no—
570C23—61no—
571C23—61no—
572C23—181no—
573D4—21no—
574D4—21no—
575D4—61no—
576D4—61no—
577D4—61no—
578D4—61no—
579D4—124no—
580D4—181no—
581D4—181no—
582D4—364no—
583C2×C4—32no—
584C2×C4C2×C3×D3211yescharacteristic
585C8—32no—
586C8—92no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3—41no—
594C2×C3—42no—
595C2×C3—82no—
596C2×C3—82no—
597C2×C3—82no—
598C2×C3—84no—
599C2×C3—81no—
600C2×C3—81no—
601C2×C3—81no—
602C2×C3—242no—
603C2×C3—242no—
604C2×C3—241no—
605C2×C3—241no—
606C2×C3—241no—
607C2×C3—241no—
608C2×C3—721no—
609C2×C3D3×#21312yes—
610C2×C3#6397911yescharacteristic
611C2×C3C3×D3×D812yes—
612C2×C3C2×C3×#21312yes—
613C2×C3C3×#94011yescharacteristic
614C2×C3C3×#94011yescharacteristic
615D3—81no—
616D3—81no—
617D3—242no—
618D3—241no—
619D3—241no—
620C22—41no—
621C22—42no—
622C22—121no—
623C22—121no—
624C22—122no—
625C22—122no—
626C22—361no—
627C22—362no—
628C22C3×D3×#6711yescharacteristic
629C4—34no—
630C4C3×D3×#6711yescharacteristic
631C4C3×#381111yescharacteristic
632C3—21no—
633C3—21no—
634C3—21no—
635C3—41no—
636C3#1059148211yescharacteristic, direct product factor
637C3C3×#507311yescharacteristic, complemented
638C3C3×#515111yescharacteristic, complemented
639C2—81no—
640C2—241no—
641C2—241no—
642C2—721no—
643C2C3×D3×#21312yes—
644C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#493
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
1. C2×C3×#3753C2semidirect
2. C32×#940C2semidirect
3. C2×C32×#213C2semidirect
4. C3×#63911C2semidirect
5. C2×C3×#3714C2semidirect
6. C2×C3×#3713C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591482C3direct (N × H)
26. C2×#3753C2×C3semidirect
29. C3×#940D3semidirect
30. C3×#940C2×C3semidirect
38. C2×C3×#213D3semidirect
39. C2×C3×#213C2×C3semidirect
42. #63911C2×C3semidirect
43. C2×#3714C2×C3semidirect
44. C2×#3713C2×C3semidirect
47. Dic3×C3⋊C8C2×C3semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
65. C32×C3⋊C8D4semidirect
132. C32×Dic3D8semidirect
144. #940C3×D3semidirect
151. C2×#213C3×D3semidirect
233. C3⋊(C3⋊C8)C3×D4semidirect
238. C3×C3⋊C8#67semidirect
239. C3×C3⋊C8C3×D4semidirect
360. C3×Dic3#213semidirect
361. C3×Dic3C3×D8semidirect
369. C33#493semidirect
452. C3⋊C8C3×#67semidirect
561. Dic3C3×#213semidirect
566. C32#5073semidirect
567. C32#5151semidirect
568. C32C3×#493semidirect
636. C3#10591482direct (N × H)
637. C3C3×#5073semidirect
638. C3C3×#5151semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×#9402
C2×C32×#2132
C3×#639112
C2×C3×#37142
C2×C3×#37132
C3×Dic3×C3⋊C82
C3×#51513
C3×#50733
#105914823