Finite Group Databasebeta

← C2×#11721051 #13150552

Subgroups — C2×#11721051 #13150552

Order 1728.

2222 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×D9C211yesmaximal, characteristic, complemented
2C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C22×#6195C211yesmaximal, characteristic, complemented
4C22×#6192C211yesmaximal, characteristic, complemented
5C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C2×#88445C211yesmaximal, characteristic, complemented
8#11721051C218yesmaximal, direct product factor
9C2×#63923—31nomaximal
10C2×#63444—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×C3⋊D9C2211yescharacteristic, complemented
13C23×D3×C9C2211yescharacteristic
14C23×C3×D9C2211yescharacteristic
15C22×D3×D9—24no—
16C22×D3×D9—24no—
17C22×D3×D9—24no—
18C22×D3×D9C2214yescomplemented
19C22×C3×C4×C9C2211yescharacteristic, complemented
20C2×C4⋊(C3⋊D9)—24no—
21C2×C4⋊(C3⋊D9)C2214yescomplemented
22C22×C9×Dic3C2211yescharacteristic, complemented
23C22×C3×Dic9C2211yescharacteristic, complemented
24C2×#6195—24no—
25C2×#6195C2214yescomplemented
26C2×#6192—24no—
27C2×#6192C2214yescomplemented
28C9×(C2×D3)⋊C4C2214yescomplemented
29C3×(C2×D9)⋊C4C2214yescomplemented
30#88445C2214yescomplemented
31C23×D32—31no—
32C22×C4⋊(C3⋊D3)—31no—
33C22×#626—31no—
34C22×#626—31no—
35C24×D9—31no—
36C2×C3×(C2×D3)⋊C4—31no—
37C2×C3×(C2×D3)⋊C4D311yescharacteristic
38C2×#3722—31no—
39#63923—38no—
40C22×#611—31no—
41C22×D36—31no—
42C22×D36—31no—
43C22×D36—61no—
44C2×C9×C22⋊C4—31no—
45C2×(C2×D9)⋊C4—31no—
46C2×(C2×D9)⋊C4D311yescharacteristic, complemented
47#63444—38no—
48C23×C3×C9C2311yescharacteristic
49C22×C3⋊D9—22no—
50C22×C3⋊D9—22no—
51C22×C3⋊D9—22no—
52C22×C3⋊D9—22no—
53C22×C3⋊D9—22no—
54C22×C3⋊D9D414yescomplemented
55C22×C3⋊D9C2312yes—
56C22×C3⋊D9C2312yescomplemented
57C22×D3×C9—22no—
58C22×D3×C9—22no—
59C22×D3×C9D414yescomplemented
60C22×D3×C9C2312yes—
61C22×C3×D9—22no—
62C22×C3×D9—22no—
63C22×C3×D9D414yescomplemented
64C22×C3×D9C2312yes—
65C2×D3×D9—232no—
66C2×D3×D9—28no—
67C2×D3×D9—28no—
68C2×D3×D9—28no—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2312yescomplemented
71C4⋊(C3⋊D9)—28no—
72C2×C9×Dic3—22no—
73C2×C9×Dic3C2312yescomplemented
74C2×C3×Dic9—22no—
75C2×C3×Dic9C2312yescomplemented
76#6195—28no—
77#6192—28no—
78C2×#882—91no—
79C2×#882—91no—
80C23×C3⋊D3—31no—
81C23×C3⋊D3—31no—
82C23×C3×D3—31no—
83C23×C3×D3C2×D311yescharacteristic
84C22×D32—34no—
85C22×D32—64no—
86C22×D32—64no—
87C22×D32—64no—
88C22×C32×C4C2×D311yescharacteristic
89C2×C4⋊(C3⋊D3)—34no—
90C2×C4⋊(C3⋊D3)—64no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3C2×D311yescharacteristic
93C2×#626—34no—
94C2×#626—34no—
95C2×#626—64no—
96C2×#626—64no—
97C24×C9—31no—
98C23×D9—31no—
99C23×D9—31no—
100C23×D9—34no—
101C23×D9—64no—
102C23×D9—64no—
103C23×D9—64no—
104C23×D9—61no—
105C23×D9—61no—
106C23×D9C2×D311yescharacteristic
107C3×(C2×D3)⋊C4—34no—
108C3×(C2×D3)⋊C4C2×D314yes—
109#3722—34no—
110C22×C4×C9—21no—
111C22×C4×C9—31no—
112C22×C4×C9C2×D311yescharacteristic, complemented
113C2×#611—34no—
114C2×#611—64no—
115C22×Dic9C2×D311yescharacteristic, complemented
116C2×D36—34no—
117C2×D36—34no—
118C2×D36—64no—
119C2×D36—64no—
120C2×D36—68no—
121C2×D36—64no—
122C9×C22⋊C4—34no—
123(C2×D9)⋊C4—34no—
124(C2×D9)⋊C4C2×D314yescomplemented
125C22×C3×C9C2×D412yes—
126C22×C3×C9C2×D412yes—
127C22×C3×C9C2×D412yes—
128C22×C3×C9C2411yescharacteristic
129C2×C3⋊D9—28no—
130C2×C3⋊D9—22no—
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—44no—
136C2×C3⋊D9C2×D414yescomplemented
137C2×D3×C9—28no—
138C2×D3×C9—22no—
139C2×D3×C9—22no—
140C2×D3×C9C2×D414yescomplemented
141C2×C3×D9—28no—
142C2×C3×D9—22no—
143C2×C3×D9—22no—
144C2×C3×D9C2×D414yescomplemented
145D3×D9—232no—
146C3×C4×C9—22no—
147C9×Dic3—22no—
148C3×Dic9—22no—
149C24×D3—91no—
150C24×D3—91no—
151C22×#67—91no—
152C22×#67—91no—
153C22×D12—91no—
154C22×D12—91no—
155C22×D12—91no—
156C22×D12—91no—
157C22×D12—181no—
158C2×C3×C22⋊C4—31no—
159C2×C3×C22⋊C4—91no—
160C2×(C2×D3)⋊C4—31no—
161C2×(C2×D3)⋊C4—91no—
162C2×(C2×D3)⋊C4—91no—
163C2×(C2×D3)⋊C4D911yescharacteristic, complemented
164#882—98no—
165#882—98no—
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—34no—
168C22×C3⋊D3—32no—
169C22×C3⋊D3—32no—
170C22×C3⋊D3—62no—
171C22×C3⋊D3—62no—
172C22×C3⋊D3—62no—
173C22×C3⋊D3—62no—
174C22×C3⋊D3—62no—
175C22×C3×D3—22no—
176C22×C3×D3—22no—
177C22×C3×D3—34no—
178C22×C3×D3—32no—
179C22×C3×D3—62no—
180C22×C3×D3—62no—
181C22×C3×D3D1214yes—
182C22×C3×D3C22×D312yes—
183C2×D32—68no—
184C2×D32—68no—
185C2×D32—632no—
186C2×D32—68no—
187C2×C32×C4—22no—
188C2×C32×C4C22×D312yes—
189C4⋊(C3⋊D3)—68no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3C22×D312yes—
194#626—68no—
195#626—68no—
196C23×C9—21no—
197C23×C9—34no—
198C23×C9—32no—
199C23×C9—62no—
200C23×C9—62no—
201C23×C9C22×D311yescharacteristic
202C22×D9—22no—
203C22×D9—22no—
204C22×D9—34no—
205C22×D9—32no—
206C22×D9—32no—
207C22×D9—64no—
208C22×D9—62no—
209C22×D9—62no—
210C22×D9—68no—
211C22×D9—68no—
212C22×D9—632no—
213C22×D9—62no—
214C22×D9—62no—
215C22×D9—68no—
216C22×D9—64no—
217C22×D9—62no—
218C22×D9—62no—
219C22×D9—62no—
220C22×D9—122no—
221C22×D9—122no—
222C22×D9—122no—
223C22×D9—122no—
224C22×D9D1214yescomplemented
225C22×D9C22×D312yes—
226C2×C4×C9—24no—
227C2×C4×C9—22no—
228C2×C4×C9—22no—
229C2×C4×C9—32no—
230C2×C4×C9—62no—
231C2×C4×C9C22×D312yescomplemented
232#611—68no—
233C2×Dic9—22no—
234C2×Dic9C22×D312yescomplemented
235D36—68no—
236D36—616no—
237D36—68no—
238C2×#120—271noSylow
239C2×C3×C9#12014yescomplemented
240C2×C3×C9C22×D411yescharacteristic
241C2×C3×C9C22×D411yescharacteristic
242C2×C3×C9C22×D411yescharacteristic
243C3⋊D9—24no—
244C3⋊D9—42no—
245D3×C9—24no—
246C3×D9—24no—
247C24×C3—31no—
248C24×C3—91no—
249C23×D3—31no—
250C23×D3—94no—
251C23×D3—91no—
252C23×D3—91no—
253C23×D3—91no—
254C23×D3—91no—
255C23×D3—94no—
256C23×D3—181no—
257C23×D3—181no—
258C23×D3—184no—
259C23×D3—184no—
260C23×D3—184no—
261C23×D3—184no—
262C23×D3—184no—
263C23×D3—184no—
264C23×D3C2×D911yescharacteristic
265C22×C3×C4—21no—
266C22×C3×C4—31no—
267C22×C3×C4—91no—
268C22×C3×C4C2×D911yescharacteristic, complemented
269C22×C3×C4D3211yescharacteristic
270C2×#67—94no—
271C2×#67—94no—
272C2×#67—184no—
273C2×#67—184no—
274C22×Dic3—31no—
275C22×Dic3C2×D911yescharacteristic, complemented
276C2×D12—94no—
277C2×D12—94no—
278C2×D12—94no—
279C2×D12—94no—
280C2×D12—184no—
281C2×D12—184no—
282C2×D12—184no—
283C2×D12—188no—
284C2×D12—184no—
285C2×D12—184no—
286C3×C22⋊C4—34no—
287C3×C22⋊C4—94no—
288(C2×D3)⋊C4—34no—
289(C2×D3)⋊C4—94no—
290(C2×D3)⋊C4—94no—
291(C2×D3)⋊C4C2×D914yescomplemented
292C22×C32C2×D1212yes—
293C22×C32D3×D412yes—
294C22×C32D3×D412yes—
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—34no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—68no—
299C2×C3⋊D3—62no—
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3⋊D3—62no—
303C2×C3⋊D3—124no—
304C2×C3×D3—28no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3—34no—
308C2×C3×D3—68no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3C2×D1214yes—
312D32—632no—
313C32×C4—22no—
314C3×Dic3—22no—
315C3×Dic3—62no—
316C22×C9—21no—
317C22×C9—22no—
318C22×C9—22no—
319C22×C9—22no—
320C22×C9—34no—
321C22×C9—68no—
322C22×C9—62no—
323C22×C9—62no—
324C22×C9C2×D1212yes—
325C22×C9D3×D412yes—
326C22×C9D3×D412yes—
327C22×C9C23×D311yescharacteristic
328C2×D9—28no—
329C2×D9—22no—
330C2×D9—22no—
331C2×D9—34no—
332C2×D9—64no—
333C2×D9—632no—
334C2×D9—68no—
335C2×D9—62no—
336C2×D9—62no—
337C2×D9—64no—
338C2×D9—62no—
339C2×D9—62no—
340C2×D9—62no—
341C2×D9—122no—
342C2×D9—122no—
343C2×D9—128no—
344C2×D9—122no—
345C2×D9—122no—
346C2×D9—128no—
347C2×D9—124no—
348C2×D9C2×D1214yescomplemented
349C4×C9—24no—
350C4×C9—22no—
351C4×C9—62no—
352Dic9—22no—
353C25—271no—
354C22×D4—271no—
355C22×D4—271no—
356C22×D4—271no—
357#120—278no—
358C2×C22⋊C4—31no—
359C2×C22⋊C4—91no—
360C2×C22⋊C4—271no—
361C3×C9C2×#12011yescharacteristic, complemented, Sylow
362C23×C3—21no—
363C23×C3—32no—
364C23×C3—34no—
365C23×C3—62no—
366C23×C3—62no—
367C23×C3—94no—
368C23×C3—92no—
369C23×C3—182no—
370C23×C3—182no—
371C23×C3C22×D911yescharacteristic
372C23×C3C2×D3211yescharacteristic
373C22×D3—22no—
374C22×D3—22no—
375C22×D3—32no—
376C22×D3—34no—
377C22×D3—62no—
378C22×D3—62no—
379C22×D3—92no—
380C22×D3—94no—
381C22×D3—92no—
382C22×D3—94no—
383C22×D3—92no—
384C22×D3—92no—
385C22×D3—182no—
386C22×D3—184no—
387C22×D3—182no—
388C22×D3—188no—
389C22×D3—188no—
390C22×D3—1832no—
391C22×D3—188no—
392C22×D3—188no—
393C22×D3—188no—
394C22×D3—1832no—
395C22×D3—182no—
396C22×D3—182no—
397C22×D3—182no—
398C22×D3—182no—
399C22×D3—188no—
400C22×D3—184no—
401C22×D3—182no—
402C22×D3—182no—
403C22×D3—182no—
404C22×D3—182no—
405C22×D3—182no—
406C22×D3—182no—
407C22×D3—362no—
408C22×D3—362no—
409C22×D3—362no—
410C22×D3—362no—
411C22×D3D3614yescomplemented
412C22×D3C22×D912yes—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—32no—
418C2×C3×C4—62no—
419C2×C3×C4—92no—
420C2×C3×C4—182no—
421C2×C3×C4C22×D912yescomplemented
422C2×C3×C4C2×D3212yes—
423#67—188no—
424#67—188no—
425C2×Dic3—22no—
426C2×Dic3—32no—
427C2×Dic3—62no—
428C2×Dic3C22×D912yescomplemented
429D12—188no—
430D12—188no—
431D12—1816no—
432D12—188no—
433D12—188no—
434C2×C32#88214yes—
435C2×C32C22×D1211yescharacteristic
436C2×C32C2×D3×D411yescharacteristic
437C2×C32C2×D3×D411yescharacteristic
438C3⋊D3—64no—
439C3⋊D3—122no—
440C3×D3—24no—
441C3×D3—64no—
442C2×C9—21no—
443C2×C9—21no—
444C2×C9—21no—
445C2×C9—24no—
446C2×C9—64no—
447C2×C9#88214yescomplemented
448C2×C9C22×D1211yescharacteristic
449C2×C9C2×D3×D411yescharacteristic
450C2×C9C2×D3×D411yescharacteristic
451D9—24no—
452D9—64no—
453D9—124no—
454D9—124no—
455D9—122no—
456C24—31no—
457C24—91no—
458C24—274no—
459C24—271no—
460C24—271no—
461C24—544no—
462C24—544no—
463C24—544no—
464C2×D4—274no—
465C2×D4—274no—
466C2×D4—274no—
467C2×D4—544no—
468C2×D4—544no—
469C2×D4—544no—
470C22×C4—31no—
471C22×C4—91no—
472C22×C4D3×D911yescharacteristic, complemented
473C22⋊C4—34no—
474C22⋊C4—94no—
475C22⋊C4—274no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—34no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—68no—
484C22×C3—94no—
485C22×C3—188no—
486C22×C3—182no—
487C22×C3—182no—
488C22×C3C2×D3612yes—
489C22×C3D4×D912yes—
490C22×C3D4×D912yes—
491C22×C3C23×D911yescharacteristic
492C22×C3D3×D1212yes—
493C22×C3D3×D1212yes—
494C22×C3#381112yes—
495C22×C3C22×D3211yescharacteristic
496C2×D3—28no—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—34no—
500C2×D3—62no—
501C2×D3—62no—
502C2×D3—68no—
503C2×D3—94no—
504C2×D3—94no—
505C2×D3—184no—
506C2×D3—1832no—
507C2×D3—1832no—
508C2×D3—182no—
509C2×D3—182no—
510C2×D3—188no—
511C2×D3—188no—
512C2×D3—182no—
513C2×D3—182no—
514C2×D3—184no—
515C2×D3—182no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—182no—
521C2×D3—368no—
522C2×D3—362no—
523C2×D3—362no—
524C2×D3—362no—
525C2×D3—362no—
526C2×D3—368no—
527C2×D3—364no—
528C2×D3—364no—
529C2×D3C2×D3614yescomplemented
530C3×C4—24no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—62no—
534C3×C4—182no—
535Dic3—22no—
536Dic3—62no—
537C32C2×#88211yescharacteristic
538C9—21no—
539C9C2×#88211yescharacteristic, complemented
540C23—32no—
541C23—34no—
542C23—62no—
543C23—62no—
544C23—92no—
545C23—94no—
546C23—182no—
547C23—182no—
548C23—272no—
549C23—274no—
550C23—272no—
551C23—548no—
552C23—548no—
553C23—548no—
554C23—5432no—
555C23—542no—
556C23—542no—
557C23—542no—
558C23—542no—
559C23—542no—
560C23C2×D3×D911yescharacteristic
561D4—548no—
562D4—548no—
563D4—548no—
564C2×C4—22no—
565C2×C4—32no—
566C2×C4—62no—
567C2×C4—92no—
568C2×C4—182no—
569C2×C4C2×D3×D912yescomplemented
570C2×C3—21no—
571C2×C3—24no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—64no—
575C2×C3—184no—
576C2×C3#6344414yescomplemented
577C2×C3C22×D3611yescharacteristic
578C2×C3C2×D4×D911yescharacteristic
579C2×C3C2×D4×D911yescharacteristic
580C2×C3#6392314yes—
581C2×C3C2×D3×D1211yescharacteristic
582C2×C3C2×D3×D1211yescharacteristic
583C2×C3C2×#381111yescharacteristic
584D3—24no—
585D3—64no—
586D3—184no—
587D3—184no—
588D3—364no—
589D3—364no—
590D3—362no—
591D3—362no—
592C22—34no—
593C22—62no—
594C22—68no—
595C22—62no—
596C22—94no—
597C22—182no—
598C22—188no—
599C22—182no—
600C22—274no—
601C22—5432no—
602C22—548no—
603C22—542no—
604C22—542no—
605C22—542no—
606C22—542no—
607C22—542no—
608C22—1084no—
609C22D3×D3612yes—
610C22D9×D1212yes—
611C22#8866812yes—
612C22C22×D3×D911yescharacteristic
613C4—22no—
614C4—62no—
615C4—182no—
616C3—21no—
617C3C2×#6344411yescharacteristic, complemented
618C3C2×#6392311yescharacteristic
619C2—64no—
620C2—184no—
621C2—544no—
622C2—1082no—
623C2#1172105114yesdirect product factor
624C2C2×D3×D3611yescharacteristic
625C2C2×D9×D1211yescharacteristic
626C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#120
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C23×D3×D9C2semidirect
2. C22×C4⋊(C3⋊D9)C2semidirect
3. C22×#6195C2semidirect
4. C22×#6192C2semidirect
5. C2×C9×(C2×D3)⋊C4C2semidirect
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C2×#88445C2semidirect
8. #11721051C2direct (N × H)
12. C23×C3⋊D9C22semidirect
18. C22×D3×D9C22semidirect
19. C22×C3×C4×C9C22semidirect
21. C2×C4⋊(C3⋊D9)C22semidirect
22. C22×C9×Dic3C22semidirect
23. C22×C3×Dic9C22semidirect
25. C2×#6195C22semidirect
27. C2×#6192C22semidirect
28. C9×(C2×D3)⋊C4C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. #88445C22semidirect
46. C2×(C2×D9)⋊C4D3semidirect
54. C22×C3⋊D9D4semidirect
56. C22×C3⋊D9C23semidirect
59. C22×D3×C9D4semidirect
63. C22×C3×D9D4semidirect
70. C2×C3×C4×C9C23semidirect
73. C2×C9×Dic3C23semidirect
75. C2×C3×Dic9C23semidirect
112. C22×C4×C9C2×D3semidirect
115. C22×Dic9C2×D3semidirect
124. (C2×D9)⋊C4C2×D3semidirect
136. C2×C3⋊D9C2×D4semidirect
140. C2×D3×C9C2×D4semidirect
144. C2×C3×D9C2×D4semidirect
163. C2×(C2×D3)⋊C4D9semidirect
224. C22×D9D12semidirect
231. C2×C4×C9C22×D3semidirect
234. C2×Dic9C22×D3semidirect
239. C2×C3×C9#120semidirect
268. C22×C3×C4C2×D9semidirect
275. C22×Dic3C2×D9semidirect
291. (C2×D3)⋊C4C2×D9semidirect
348. C2×D9C2×D12semidirect
361. C3×C9C2×#120semidirect
411. C22×D3D36semidirect
421. C2×C3×C4C22×D9semidirect
428. C2×Dic3C22×D9semidirect
447. C2×C9#882semidirect
472. C22×C4D3×D9semidirect
529. C2×D3C2×D36semidirect
539. C9C2×#882semidirect
569. C2×C4C2×D3×D9semidirect
576. C2×C3#63444semidirect
617. C3C2×#63444semidirect
623. C2#11721051direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×D92
C22×C4⋊(C3⋊D9)2
C22×#61952
C22×#61922
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×#884452
#117210512
C2×#639233
C2×#634443