Finite Group Databasebeta

← C2×#11721050 #13150529

Subgroups — C2×#11721050 #13150529

Order 1728.

2118 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×D9C211yesmaximal, characteristic, complemented
2C22×C9×D12C211yesmaximal, characteristic, complemented
3C22×#6195C211yesmaximal, characteristic, complemented
4C22×#6194C211yesmaximal, characteristic, complemented
5C2×#88536C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
8#11721050C218yesmaximal, direct product factor
9C2×#63922—31nomaximal
10C2×#63499—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C23×D3×C9C2211yescharacteristic, complemented
14C23×C3×D9C2211yescharacteristic
15C22×D3×D9—24no—
16C22×D3×D9—24no—
17C22×D3×D9—24no—
18C22×D3×D9C2214yescomplemented
19C22×C3×C4×C9C2211yescharacteristic, complemented
20C22×C3⋊Dic9C2211yescharacteristic, complemented
21C22×C3×Dic9C2211yescharacteristic, complemented
22C2×C9×D12—24no—
23C2×C9×D12C2214yescomplemented
24C2×#6195—24no—
25C2×#6195C2214yescomplemented
26C2×#6194—24no—
27C2×#6194C2214yescomplemented
28#88536C2214yescomplemented
29C3×(C2×D9)⋊C4C2214yescomplemented
30(C2×D3)⋊Dic9C2214yescomplemented
31C23×D32—31no—
32C22×C3×D12D311yescharacteristic
33C22×#626—31no—
34C22×#625—31no—
35C24×D9—31no—
36C2×#3752—31no—
37C2×C3×(C2×D3)⋊C4—31no—
38C2×(C2×D3)⋊Dic3—31no—
39#63922—38no—
40C22×D4×C9—31no—
41C22×#611—31no—
42C22×#611—31no—
43C2×#3676—31no—
44#63499—38no—
45C2×(C2×D9)⋊C4—31no—
46C2×(C2×D9)⋊C4—61no—
47C2×(C2×D9)⋊C4D311yescharacteristic, complemented
48C23×C3×C9C2311yescharacteristic
49C22×C3⋊D9—22no—
50C22×C3⋊D9—22no—
51C22×C3⋊D9D414yescomplemented
52C22×C3⋊D9C2312yes—
53C22×D3×C9—22no—
54C22×D3×C9—22no—
55C22×D3×C9—22no—
56C22×D3×C9—22no—
57C22×D3×C9—22no—
58C22×D3×C9D414yescomplemented
59C22×D3×C9C2312yes—
60C22×D3×C9C2312yescomplemented
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
71C2×C3⋊Dic9—22no—
72C2×C3⋊Dic9C2312yescomplemented
73C2×C3×Dic9—22no—
74C2×C3×Dic9C2312yescomplemented
75C9×D12—28no—
76#6195—28no—
77#6194—28no—
78C2×#937—91no—
79C2×#882—91no—
80C23×C3⋊D3—31no—
81C23×C3×D3—31no—
82C23×C3×D3C2×D311yescharacteristic
83C23×C3×D3C2×D311yescharacteristic
84C22×D32—34no—
85C22×D32—64no—
86C22×D32—64no—
87C22×D32—64no—
88C22×C32×C4C2×D311yescharacteristic
89C22×C3⋊Dic3—31no—
90C22×C3×Dic3—31no—
91C2×C3×D12—24no—
92C2×C3×D12C2×D314yes—
93C2×#626—34no—
94C2×#626—64no—
95C2×#625—34no—
96C2×#625—64no—
97C24×C9—31no—
98C24×C9—31no—
99C23×D9—31no—
100C23×D9—34no—
101C23×D9—64no—
102C23×D9—64no—
103C23×D9—64no—
104C23×D9—61no—
105C23×D9C2×D311yescharacteristic
106#3752—34no—
107C3×(C2×D3)⋊C4—34no—
108(C2×D3)⋊Dic3—34no—
109C2×D4×C9—34no—
110C2×D4×C9—64no—
111C22×C4×C9—21no—
112C22×C4×C9C2×D311yescharacteristic, complemented
113C2×#611—34no—
114C2×#611—34no—
115C2×#611—64no—
116C2×#611—64no—
117C22×Dic9—31no—
118C22×Dic9—61no—
119C22×Dic9C2×D311yescharacteristic, complemented
120#3676—34no—
121(C2×D9)⋊C4—34no—
122(C2×D9)⋊C4—64no—
123(C2×D9)⋊C4C2×D314yescomplemented
124C22×C3×C9C2×D412yes—
125C22×C3×C9C2×D412yes—
126C22×C3×C9C2×D412yes—
127C22×C3×C9C2411yescharacteristic
128C2×C3⋊D9—28no—
129C2×C3⋊D9—22no—
130C2×C3⋊D9—22no—
131C2×C3⋊D9C2×D414yescomplemented
132C2×D3×C9—28no—
133C2×D3×C9—22no—
134C2×D3×C9—22no—
135C2×D3×C9—22no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—44no—
139C2×D3×C9C2×D414yescomplemented
140C2×C3×D9—28no—
141C2×C3×D9—22no—
142C2×C3×D9—22no—
143C2×C3×D9C2×D414yescomplemented
144D3×D9—232no—
145C3×C4×C9—22no—
146C3⋊Dic9—22no—
147C3×Dic9—22no—
148C24×D3—91no—
149C24×D3—91no—
150C22×C3×D4—31no—
151C22×#67—91no—
152C22×#67—91no—
153C22×#67—91no—
154C22×D12—91no—
155C22×D12D911yescharacteristic, complemented
156C2×C3×C22⋊C4—91no—
157C2×#217—91no—
158#937—98no—
159C2×(C2×D3)⋊C4—31no—
160C2×(C2×D3)⋊C4—91no—
161C2×(C2×D3)⋊C4—91no—
162C2×(C2×D3)⋊C4—91no—
163C2×(C2×D3)⋊C4—181no—
164#882—98no—
165C23×C32C22×D311yescharacteristic
166C22×C3⋊D3—34no—
167C22×C3⋊D3—32no—
168C22×C3⋊D3—62no—
169C22×C3⋊D3—62no—
170C22×C3×D3—22no—
171C22×C3×D3—22no—
172C22×C3×D3—22no—
173C22×C3×D3—22no—
174C22×C3×D3—22no—
175C22×C3×D3—34no—
176C22×C3×D3—32no—
177C22×C3×D3—62no—
178C22×C3×D3—62no—
179C22×C3×D3#6714yes—
180C22×C3×D3C22×D312yes—
181C22×C3×D3C22×D312yes—
182C2×D32—68no—
183C2×D32—68no—
184C2×D32—632no—
185C2×D32—68no—
186C2×C32×C4—22no—
187C2×C32×C4C22×D312yes—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—62no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—62no—
192C3×D12—28no—
193#626—68no—
194#625—68no—
195C23×C9—21no—
196C23×C9—34no—
197C23×C9—32no—
198C23×C9—32no—
199C23×C9—62no—
200C23×C9—62no—
201C23×C9—62no—
202C23×C9—62no—
203C23×C9—62no—
204C23×C9C22×D311yescharacteristic
205C22×D9—22no—
206C22×D9—22no—
207C22×D9—34no—
208C22×D9—32no—
209C22×D9—62no—
210C22×D9—64no—
211C22×D9—68no—
212C22×D9—68no—
213C22×D9—632no—
214C22×D9—62no—
215C22×D9—62no—
216C22×D9—68no—
217C22×D9—122no—
218C22×D9—122no—
219C22×D9D1214yescomplemented
220C22×D9C22×D312yes—
221D4×C9—68no—
222C2×C4×C9—22no—
223C2×C4×C9—22no—
224C2×C4×C9—42no—
225C2×C4×C9C22×D312yescomplemented
226#611—68no—
227#611—68no—
228C2×Dic9—22no—
229C2×Dic9—32no—
230C2×Dic9—62no—
231C2×Dic9—62no—
232C2×Dic9—122no—
233C2×Dic9C22×D312yescomplemented
234C2×#120—271noSylow
235C2×C3×C9#12014yescomplemented
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C9C22×D411yescharacteristic
238C2×C3×C9C22×D411yescharacteristic
239C3⋊D9—24no—
240D3×C9—24no—
241D3×C9—42no—
242C3×D9—24no—
243C24×C3—31no—
244C24×C3—31no—
245C24×C3—91no—
246C23×D3—31no—
247C23×D3—94no—
248C23×D3—91no—
249C23×D3—91no—
250C23×D3—94no—
251C23×D3—181no—
252C23×D3—184no—
253C23×D3—184no—
254C23×D3—184no—
255C23×D3—184no—
256C23×D3—184no—
257C23×D3—184no—
258C23×D3C2×D911yescharacteristic
259C23×D3C2×D911yescharacteristic, complemented
260C2×C3×D4—34no—
261C2×C3×D4—64no—
262C22×C3×C4—21no—
263C22×C3×C4—91no—
264C22×C3×C4C2×D911yescharacteristic, complemented
265C22×C3×C4D3211yescharacteristic
266C2×#67—94no—
267C2×#67—94no—
268C2×#67—94no—
269C2×#67—184no—
270C2×#67—184no—
271C2×#67—184no—
272C22×Dic3—31no—
273C22×Dic3—91no—
274C22×Dic3—91no—
275C22×Dic3—181no—
276C2×D12—24no—
277C2×D12—94no—
278C2×D12—184no—
279C2×D12C2×D914yescomplemented
280C3×C22⋊C4—94no—
281#217—94no—
282(C2×D3)⋊C4—34no—
283(C2×D3)⋊C4—94no—
284(C2×D3)⋊C4—94no—
285(C2×D3)⋊C4—94no—
286(C2×D3)⋊C4—184no—
287C22×C32D3×D412yes—
288C22×C32D3×D412yes—
289C22×C32C2×#6712yes—
290C22×C32C23×D311yescharacteristic
291C2×C3⋊D3—34no—
292C2×C3⋊D3—68no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3×D3—28no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—22no—
299C2×C3×D3—22no—
300C2×C3×D3—22no—
301C2×C3×D3—34no—
302C2×C3×D3—44no—
303C2×C3×D3—68no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3C2×#6714yes—
307D32—632no—
308C32×C4—22no—
309C3⋊Dic3—62no—
310C3×Dic3—62no—
311C22×C9—21no—
312C22×C9—22no—
313C22×C9—22no—
314C22×C9—22no—
315C22×C9—34no—
316C22×C9—68no—
317C22×C9—62no—
318C22×C9—62no—
319C22×C9—62no—
320C22×C9—62no—
321C22×C9—62no—
322C22×C9—124no—
323C22×C9C2×D1212yes—
324C22×C9D3×D412yes—
325C22×C9D3×D412yes—
326C22×C9C23×D311yescharacteristic
327C2×D9—28no—
328C2×D9—22no—
329C2×D9—22no—
330C2×D9—34no—
331C2×D9—64no—
332C2×D9—632no—
333C2×D9—68no—
334C2×D9—62no—
335C2×D9—62no—
336C2×D9—122no—
337C2×D9—122no—
338C2×D9—128no—
339C2×D9C2×D1214yescomplemented
340C4×C9—22no—
341C4×C9—42no—
342Dic9—22no—
343Dic9—62no—
344Dic9—122no—
345C25—271no—
346C22×D4—31no—
347C22×D4—271no—
348C22×D4—271no—
349#120—278no—
350C2×C22⋊C4—91no—
351C2×C22⋊C4—271no—
352C2×C22⋊C4—271no—
353C3×C9C2×#12011yescharacteristic, complemented, Sylow
354C23×C3—21no—
355C23×C3—32no—
356C23×C3—34no—
357C23×C3—32no—
358C23×C3—62no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—62no—
362C23×C3—62no—
363C23×C3—94no—
364C23×C3—92no—
365C23×C3—182no—
366C23×C3—182no—
367C23×C3C22×D911yescharacteristic
368C23×C3C2×D3211yescharacteristic
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—22no—
372C22×D3—22no—
373C22×D3—22no—
374C22×D3—32no—
375C22×D3—34no—
376C22×D3—62no—
377C22×D3—62no—
378C22×D3—92no—
379C22×D3—94no—
380C22×D3—92no—
381C22×D3—94no—
382C22×D3—182no—
383C22×D3—184no—
384C22×D3—188no—
385C22×D3—188no—
386C22×D3—1832no—
387C22×D3—188no—
388C22×D3—188no—
389C22×D3—188no—
390C22×D3—1832no—
391C22×D3—182no—
392C22×D3—182no—
393C22×D3—182no—
394C22×D3—182no—
395C22×D3—188no—
396C22×D3—362no—
397C22×D3—362no—
398C22×D3#61114yescomplemented
399C22×D3C22×D912yes—
400C22×D3C22×D912yescomplemented
401C3×D4—68no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—42no—
406C2×C3×C4—92no—
407C2×C3×C4—182no—
408C2×C3×C4C22×D912yescomplemented
409C2×C3×C4C2×D3212yes—
410#67—188no—
411#67—188no—
412#67—188no—
413C2×Dic3—32no—
414C2×Dic3—62no—
415C2×Dic3—92no—
416C2×Dic3—92no—
417C2×Dic3—182no—
418C2×Dic3—182no—
419C2×Dic3—182no—
420C2×Dic3—362no—
421D12—28no—
422D12—188no—
423C2×C32#93714yes—
424C2×C32C2×D3×D411yescharacteristic
425C2×C32C2×D3×D411yescharacteristic
426C2×C32C22×#6711yescharacteristic
427C3⋊D3—64no—
428C3×D3—24no—
429C3×D3—42no—
430C3×D3—64no—
431C2×C9—21no—
432C2×C9—21no—
433C2×C9—21no—
434C2×C9—24no—
435C2×C9—64no—
436C2×C9—122no—
437C2×C9#88214yescomplemented
438C2×C9C22×D1211yescharacteristic
439C2×C9C2×D3×D411yescharacteristic
440C2×C9C2×D3×D411yescharacteristic
441D9—24no—
442D9—64no—
443D9—124no—
444C24—31no—
445C24—31no—
446C24—91no—
447C24—274no—
448C24—271no—
449C24—544no—
450C24—544no—
451C24—544no—
452C2×D4—34no—
453C2×D4—64no—
454C2×D4—274no—
455C2×D4—274no—
456C2×D4—544no—
457C2×D4—544no—
458C22×C4—91no—
459C22×C4—271no—
460C22×C4D3×D911yescharacteristic, complemented
461C22⋊C4—94no—
462C22⋊C4—274no—
463C22⋊C4—274no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—34no—
469C22×C3—62no—
470C22×C3—62no—
471C22×C3—68no—
472C22×C3—62no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—94no—
476C22×C3—124no—
477C22×C3—188no—
478C22×C3—182no—
479C22×C3—182no—
480C22×C3D4×D912yes—
481C22×C3D4×D912yes—
482C22×C3C2×#61112yes—
483C22×C3C23×D911yescharacteristic
484C22×C3D3×D1212yes—
485C22×C3#380212yes—
486C22×C3D3×#6712yes—
487C22×C3C22×D3211yescharacteristic
488C2×D3—28no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—34no—
495C2×D3—44no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—68no—
499C2×D3—94no—
500C2×D3—94no—
501C2×D3—184no—
502C2×D3—1832no—
503C2×D3—1832no—
504C2×D3—188no—
505C2×D3—182no—
506C2×D3—182no—
507C2×D3—188no—
508C2×D3—182no—
509C2×D3—182no—
510C2×D3—362no—
511C2×D3—368no—
512C2×D3—362no—
513C2×D3C2×#61114yescomplemented
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—182no—
518Dic3—62no—
519Dic3—182no—
520Dic3—182no—
521Dic3—362no—
522C32C2×#93711yescharacteristic
523C9—21no—
524C9C2×#88211yescharacteristic, complemented
525C23—32no—
526C23—34no—
527C23—32no—
528C23—62no—
529C23—62no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—92no—
534C23—94no—
535C23—182no—
536C23—182no—
537C23—272no—
538C23—274no—
539C23—548no—
540C23—548no—
541C23—548no—
542C23—5432no—
543C23—542no—
544C23—542no—
545C23C2×D3×D911yescharacteristic
546D4—68no—
547D4—548no—
548D4—548no—
549C2×C4—22no—
550C2×C4—92no—
551C2×C4—182no—
552C2×C4—272no—
553C2×C4—542no—
554C2×C4C2×D3×D912yescomplemented
555C2×C3—21no—
556C2×C3—24no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—64no—
560C2×C3—122no—
561C2×C3—184no—
562C2×C3#6349914yescomplemented
563C2×C3C2×D4×D911yescharacteristic
564C2×C3C2×D4×D911yescharacteristic
565C2×C3C22×#61111yescharacteristic
566C2×C3#6392214yes—
567C2×C3C2×D3×D1211yescharacteristic
568C2×C3C2×#380211yescharacteristic
569C2×C3C2×D3×#6711yescharacteristic
570D3—24no—
571D3—42no—
572D3—64no—
573D3—184no—
574D3—184no—
575D3—364no—
576C22—34no—
577C22—68no—
578C22—62no—
579C22—62no—
580C22—62no—
581C22—62no—
582C22—62no—
583C22—94no—
584C22—124no—
585C22—182no—
586C22—188no—
587C22—182no—
588C22—274no—
589C22—5432no—
590C22—542no—
591C22—548no—
592C22—542no—
593C22D9×D1212yes—
594C22#8864612yes—
595C22D3×#61112yes—
596C22C22×D3×D911yescharacteristic
597C4—22no—
598C4—182no—
599C4—542no—
600C3—21no—
601C3C2×#6349911yescharacteristic, complemented
602C3C2×#6392211yescharacteristic
603C2—64no—
604C2—122no—
605C2—184no—
606C2—544no—
607C2#1172105014yesdirect product factor
608C2C2×D9×D1211yescharacteristic
609C2C2×#8864611yescharacteristic
610C2C2×D3×#61111yescharacteristic
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×C9×D12C2semidirect
3. C22×#6195C2semidirect
4. C22×#6194C2semidirect
5. C2×#88536C2semidirect
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C2×(C2×D3)⋊Dic9C2semidirect
8. #11721050C2direct (N × H)
13. C23×D3×C9C22semidirect
18. C22×D3×D9C22semidirect
19. C22×C3×C4×C9C22semidirect
20. C22×C3⋊Dic9C22semidirect
21. C22×C3×Dic9C22semidirect
23. C2×C9×D12C22semidirect
25. C2×#6195C22semidirect
27. C2×#6194C22semidirect
28. #88536C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. (C2×D3)⋊Dic9C22semidirect
47. C2×(C2×D9)⋊C4D3semidirect
51. C22×C3⋊D9D4semidirect
58. C22×D3×C9D4semidirect
60. C22×D3×C9C23semidirect
63. C22×C3×D9D4semidirect
70. C2×C3×C4×C9C23semidirect
72. C2×C3⋊Dic9C23semidirect
74. C2×C3×Dic9C23semidirect
112. C22×C4×C9C2×D3semidirect
119. C22×Dic9C2×D3semidirect
123. (C2×D9)⋊C4C2×D3semidirect
131. C2×C3⋊D9C2×D4semidirect
139. C2×D3×C9C2×D4semidirect
143. C2×C3×D9C2×D4semidirect
155. C22×D12D9semidirect
219. C22×D9D12semidirect
225. C2×C4×C9C22×D3semidirect
233. C2×Dic9C22×D3semidirect
235. C2×C3×C9#120semidirect
259. C23×D3C2×D9semidirect
264. C22×C3×C4C2×D9semidirect
279. C2×D12C2×D9semidirect
339. C2×D9C2×D12semidirect
353. C3×C9C2×#120semidirect
398. C22×D3#611semidirect
400. C22×D3C22×D9semidirect
408. C2×C3×C4C22×D9semidirect
437. C2×C9#882semidirect
460. C22×C4D3×D9semidirect
513. C2×D3C2×#611semidirect
524. C9C2×#882semidirect
554. C2×C4C2×D3×D9semidirect
562. C2×C3#63499semidirect
601. C3C2×#63499semidirect
607. C2#11721050direct (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×C9×D122
C22×#61952
C22×#61942
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×(C2×D3)⋊Dic92
#117210502
C2×#639223
C2×#634993