Finite Group Databasebeta

← C3×#10591176 #13158355

Subgroups — C3×#10591176 #13158355

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64133C211yesmaximal, characteristic, complemented
2C32×#930C211yesmaximal, characteristic
3C2×C32×#208C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8C3×#5134—31nomaximal
9C3×#5042—31nomaximal
10#10591176C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×#208—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23C3×#3716—22no—
24C2×C32×M16—31no—
25C32×#121—31no—
26#64133C2×C311yescharacteristic, complemented
27C2×C3×#214—31no—
28C3×#930—21no—
29C3×#930D311yescharacteristic
30C3×#930C2×C311yescharacteristic
31C3×#924—31no—
32C3×#924—31no—
33C3×#924—61no—
34C2×C3×#208—21no—
35C2×C3×#208D311yescharacteristic, complemented
36C2×C3×#208C2×C311yescharacteristic, complemented
37C2×C3×#204—31no—
38C2×#3716C2×C311yescharacteristic, complemented
39C3×#832—31no—
40C3×#821—31no—
41C3×#818—31no—
42C3×#816—31no—
43C3×#810—31no—
44C3×#807—31no—
45#63575C2×C311yescharacteristic, complemented
46#63567C2×C311yescharacteristic, complemented
47#63563C2×C311yescharacteristic, complemented
48C23×C33D411yescharacteristic
49C22×C32×D3—21no—
50C2×C33×C4—21no—
51C2×C33×C4—21no—
52C2×C33×C4D411yescharacteristic
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C3×#634—21no—
57C3×#634—21no—
58C32×#67—42no—
59C2×C32×Dic3—21no—
60C32×D12—21no—
61C32×D12—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8D412yescomplemented
64C3×#468—91no—
65#5134—31no—
66#5042—31no—
67C2×C32×D4—31no—
68C2×C32×D4—61no—
69C22×C32×C4—21no—
70C22×C32×C4—21no—
71C22×C32×C4—21no—
72C22×C32×C4—41no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic
77C2×C3×#67—22no—
78C2×C3×#67—21no—
79C2×C3×#67—21no—
80C2×C3×D12—21no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C32×M16—62no—
87C2×C32×C8—31no—
88C2×C32×C8—31no—
89C32×C4⋊C4—31no—
90C32×C22⋊C4—61no—
91#3757—21no—
92C4⋊(C3⋊Dic3)C22×C311yescharacteristic
93#3750—21no—
94C3×#217—62no—
95C3×#217—61no—
96C3×#217—61no—
97C3×#214—62no—
98C3×(C2×D3)⋊C4—22no—
99C3×(C2×D3)⋊C4—21no—
100C3×(C2×D3)⋊C4—21no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—61no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C3×Dic3⋊C4—62no—
108C3×Dic3⋊C4—61no—
109C3×Dic3⋊C4—61no—
110C3×#208—24no—
111C3×#208—22no—
112C3×#208—22no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8—21no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118C2×C3×C3⋊C8C22×C311yescharacteristic
119C3×#204—62no—
120#3716—22no—
121C22×C33—21no—
122C22×C33—21no—
123C22×C33—21no—
124C22×C33C2×D411yescharacteristic
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C33×C4—21no—
129C33×C4C2×D411yescharacteristic
130C33×C4#4111yescharacteristic
131C3×C3⋊Dic3—41no—
132C3×C3⋊Dic3—41no—
133C32×Dic3—41no—
134C2×C3×SD16—91no—
135C2×C3×M16—31no—
136C2×C3×M16—31no—
137C2×C3×M16—61no—
138C3×C22⋊Q8—91no—
139C3×#121—31no—
140C3×#121—31no—
141C3×#121—61no—
142C2×#214—31no—
143#930C3×D311yescharacteristic
144#924—31no—
145#924—31no—
146#924—61no—
147C2×#208C3×D311yescharacteristic, complemented
148C2×#204—31no—
149C3×#107—91no—
150C3×#103—91no—
151C3×#102—91no—
152#832—31no—
153#821—31no—
154#818—31no—
155#816—31no—
156#810—31no—
157#807—31no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—61no—
163C23×C32D1211yescharacteristic
164C23×C32#6711yescharacteristic
165C23×C32C3×D411yescharacteristic
166C22×C3×D3—22no—
167C22×C3×D3—21no—
168C22×C3×D3—21no—
169C32×D4—61no—
170C32×D4—61no—
171C32×D4—122no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—22no—
175C2×C32×C4—22no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—41no—
191C2×C32×C4—41no—
192C2×C32×C4—42no—
193C2×C32×C4—42no—
194C2×C32×C4—61no—
195C2×C32×C4D1211yescharacteristic
196C2×C32×C4#6711yescharacteristic
197C2×C32×C4C3×D411yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C23×C311yescharacteristic
201C2×C3⋊Dic3—21no—
202C2×C3⋊Dic3—21no—
203#634—21no—
204#634—21no—
205C3×#67—44no—
206C3×#67—42no—
207C3×#67—42no—
208C2×C3×Dic3—22no—
209C2×C3×Dic3—21no—
210C2×C3×Dic3—21no—
211C2×C3×Dic3—62no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—61no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—21no—
221C3×D12—41no—
222C3×D12—41no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—61no—
226C3×Dic6—61no—
227C3×Dic6—121no—
228C3×Dic6—121no—
229C32×C8—32no—
230C32×C8—61no—
231C3×C3⋊C8—22no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—21no—
234C3×C3⋊C8—22no—
235C3×C3⋊C8#6712yescomplemented
236C3×C3⋊C8C3×D412yescomplemented
237#468—91noSylow
238C2×C33—41no—
239C2×C33#12111yescharacteristic
240C2×C33AGL(1,ℤ/8)11yescharacteristic
241C2×C33#13711yescharacteristic
242C32×D3—81no—
243C2×C3×Q8—91no—
244C2×C3×D4—31no—
245C2×C3×D4—31no—
246C2×C3×D4—61no—
247C2×C3×D4—62no—
248C2×C3×D4—61no—
249C2×C3×D4—61no—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—21no—
253C22×C3×C4—41no—
254C22×C3×C4D3211yescharacteristic
255C22×C3×C4C2×C3×D311yescharacteristic
256C22×C3×C4C2×C3×D311yescharacteristic
257C2×#67—21no—
258C2×D12C2×C3×D311yescharacteristic
259C2×Dic6—31no—
260C2×Dic6—31no—
261C2×Dic6—61no—
262C3×SD16—182no—
263C3×M16—62no—
264C3×M16—62no—
265C3×M16—64no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—61no—
271C2×C3×C8—61no—
272C3×C4⋊C4—31no—
273C3×C4⋊C4—31no—
274C3×C4⋊C4—61no—
275C3×C4⋊C4—91no—
276C3×C4⋊C4—181no—
277C3×C22⋊C4—62no—
278C3×C22⋊C4—61no—
279C3×C22⋊C4—61no—
280C3×C22⋊C4—181no—
281#217—62no—
282#217—61no—
283#217—61no—
284#214—62no—
285(C2×D3)⋊C4—21no—
286C4⋊Dic3—31no—
287C4⋊Dic3—31no—
288C4⋊Dic3—61no—
289C4⋊Dic3C2×C3×D311yescharacteristic
290Dic3⋊C4—62no—
291Dic3⋊C4—61no—
292Dic3⋊C4—61no—
293#208—22no—
294C2×C3⋊C8C2×C3×D311yescharacteristic
295C2×C3⋊C8C2×C3×D311yescharacteristic
296#204—62no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—22no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—41no—
313C22×C32—41no—
314C22×C32—41no—
315C22×C32—41no—
316C22×C32—41no—
317C22×C32—41no—
318C22×C32—41no—
319C22×C32—42no—
320C22×C32—42no—
321C22×C32—42no—
322C22×C32—121no—
323C22×C32—121no—
324C22×C32—121no—
325C22×C32C2×D1211yescharacteristic
326C22×C32C2×#6711yescharacteristic
327C22×C32C2×C3×D411yescharacteristic
328C2×C3×D3—42no—
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—41no—
336C2×C3×D3—41no—
337C32×C4—21no—
338C32×C4—22no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—41no—
348C32×C4—41no—
349C32×C4—41no—
350C32×C4—41no—
351C32×C4—42no—
352C32×C4—121no—
353C32×C4#23511yescharacteristic
354C32×C4D3×D411yescharacteristic
355C32×C4#23711yescharacteristic
356C32×C4C2×#6711yescharacteristic
357C32×C4C2×C3×D411yescharacteristic
358C32×C4C3×#4111yescharacteristic
359C3⋊Dic3—41no—
360C3⋊Dic3—41no—
361C3×Dic3—42no—
362C3×Dic3—41no—
363C3×Dic3—41no—
364C3×Dic3—122no—
365C3×Dic3—122no—
366C3×Dic3—121no—
367C3×Dic3—121no—
368C3×Dic3—121no—
369C3×Dic3—121no—
370C2×SD16—91no—
371C2×M16—31no—
372C22⋊Q8—91no—
373#121—31no—
374#107—91no—
375#103—91no—
376#102—91no—
377C33#46811yescharacteristic, complemented, Sylow
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—41no—
382C23×C3—62no—
383C23×C3—61no—
384C23×C3—61no—
385C23×C3#62611yescharacteristic
386C23×C3C3×D1211yescharacteristic
387C23×C3C3×#6711yescharacteristic
388C22×D3—21no—
389C3×Q8—181no—
390C3×Q8—181no—
391C3×D4—61no—
392C3×D4—61no—
393C3×D4—61no—
394C3×D4—61no—
395C3×D4—121no—
396C3×D4—121no—
397C3×D4—124no—
398C3×D4—122no—
399C3×D4—122no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—62no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—181no—
426C2×C3×C4—181no—
427C2×C3×C4#62611yescharacteristic
428C2×C3×C4C3×D1211yescharacteristic
429C2×C3×C4C3×#6711yescharacteristic
430C2×C3×C4C2×D3211yescharacteristic
431C2×C3×C4C22×C3×D311yescharacteristic
432C2×C3×C4C22×C3×D311yescharacteristic
433#67—42no—
434C2×Dic3—21no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—61no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441D12—21no—
442D12—21no—
443Dic6—61no—
444Dic6—61no—
445Dic6—61no—
446Dic6—61no—
447Dic6—121no—
448Dic6—121no—
449C3×C8—32no—
450C3×C8—32no—
451C3×C8—62no—
452C3×C8—62no—
453C3×C8—61no—
454C3×C8—61no—
455C3⋊C8—21no—
456C3⋊C8C3×#6712yescomplemented
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—41no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32—42no—
470C2×C32—42no—
471C2×C32—42no—
472C2×C32—44no—
473C2×C32—41no—
474C2×C32—41no—
475C2×C32—41no—
476C2×C32—241no—
477C2×C32#90811yescharacteristic
478C2×C32#90911yescharacteristic
479C2×C32#93011yescharacteristic
480C2×C32#93211yescharacteristic
481C2×C32#93911yescharacteristic
482C2×C32#95111yescharacteristic
483C2×C32C3×#12111yescharacteristic
484C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
485C2×C32C3×#13711yescharacteristic
486C3×D3—82no—
487C3×D3—81no—
488C3×D3—81no—
489C2×Q8—91no—
490C2×D4—31no—
491C2×D4—61no—
492C22×C4C3×D3211yescharacteristic
493SD16—182no—
494M16—62no—
495C2×C8—31no—
496C2×C8—31no—
497C4⋊C4—31no—
498C4⋊C4—91no—
499C4⋊C4—181no—
500C22⋊C4—61no—
501C22⋊C4—181no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—41no—
527C22×C3—122no—
528C22×C3—122no—
529C22×C3—122no—
530C22×C3—121no—
531C22×C3—121no—
532C22×C3—121no—
533C22×C3—121no—
534C22×C3—121no—
535C22×C3—121no—
536C22×C3C2×#62611yescharacteristic
537C22×C3C2×C3×D1211yescharacteristic
538C22×C3C2×C3×#6711yescharacteristic
539C2×D3—41no—
540C2×D3—41no—
541C2×D3—41no—
542C3×C4—22no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—42no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—122no—
558C3×C4—121no—
559C3×C4—121no—
560C3×C4—361no—
561C3×C4—361no—
562C3×C4#380411yescharacteristic
563C3×C4D3×#6711yescharacteristic
564C3×C4C3×#23511yescharacteristic
565C3×C4C3×D3×D411yescharacteristic
566C3×C4C3×#23711yescharacteristic
567C3×C4C2×C3×#6711yescharacteristic
568Dic3—41no—
569Dic3—122no—
570Dic3—122no—
571Dic3—121no—
572Dic3—121no—
573Dic3—121no—
574Dic3—121no—
575C32—21no—
576C32—21no—
577C32—21no—
578C32—41no—
579C32#504211yescharacteristic, complemented
580C32#513411yescharacteristic, complemented
581C32C3×#46811yescharacteristic, complemented
582C23—61no—
583C23C3×#62611yescharacteristic
584Q8—181no—
585Q8—181no—
586D4—61no—
587D4—61no—
588D4—122no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—61no—
592C2×C4—181no—
593C2×C4—181no—
594C2×C4C3×#62611yescharacteristic
595C2×C4C2×C3×D3211yescharacteristic
596C8—32no—
597C8—61no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—42no—
608C2×C3—42no—
609C2×C3—42no—
610C2×C3—44no—
611C2×C3—41no—
612C2×C3—41no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—41no—
617C2×C3—242no—
618C2×C3—241no—
619C2×C3—241no—
620C2×C3#6383011yescharacteristic
621C2×C3#6383111yescharacteristic
622C2×C3#6397811yescharacteristic
623C2×C3C3×#90811yescharacteristic
624C2×C3C3×#90911yescharacteristic
625C2×C3C3×#93011yescharacteristic
626C2×C3C3×#93211yescharacteristic
627C2×C3C3×#93911yescharacteristic
628C2×C3C3×#95111yescharacteristic
629D3—81no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22C2×C3×#62611yescharacteristic
637C4—21no—
638C4—121no—
639C4—361no—
640C4—361no—
641C4C3×#380411yescharacteristic
642C4C3×D3×#6711yescharacteristic
643C3—21no—
644C3—21no—
645C3—21no—
646C3—41no—
647C3#1059117611yescharacteristic, direct product factor
648C3C3×#504211yescharacteristic, complemented
649C3C3×#513411yescharacteristic, complemented
650C2—41no—
651C2—241no—
652C2C3×#6383011yescharacteristic
653C2C3×#6383111yescharacteristic
654C2C3×#6397811yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC32×#930C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64133C2semidirect
3. C2×C32×#208C2semidirect
4. C2×C3×#3716C2semidirect
5. C3×#63575C2semidirect
6. C3×#63567C2semidirect
7. C3×#63563C2semidirect
10. #10591176C3direct (N × H)
26. #64133C2×C3semidirect
35. C2×C3×#208D3semidirect
36. C2×C3×#208C2×C3semidirect
38. C2×#3716C2×C3semidirect
45. #63575C2×C3semidirect
46. #63567C2×C3semidirect
47. #63563C2×C3semidirect
63. C32×C3⋊C8D4semidirect
147. C2×#208C3×D3semidirect
235. C3×C3⋊C8#67semidirect
236. C3×C3⋊C8C3×D4semidirect
377. C33#468semidirect
456. C3⋊C8C3×#67semidirect
579. C32#5042semidirect
580. C32#5134semidirect
581. C32C3×#468semidirect
647. C3#10591176direct (N × H)
648. C3C3×#5042semidirect
649. C3C3×#5134semidirect

Maximal subgroups

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

subgroupindex
C3×#641332
C32×#9302
C2×C32×#2082
C2×C3×#37162
C3×#635752
C3×#635672
C3×#635632
C3×#51343
C3×#50423
#105911763