Finite Group Databasebeta

← C2×#67×C22⋊C9 #13164642

Subgroups — C2×#67×C22⋊C9 #13164642

Order 1728.

1338 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C22⋊C9C211yesmaximal, characteristic, complemented
2C22×D3×C22⋊C9C211yesmaximal, characteristic, complemented
3#67×C22⋊C9C214yesmaximal, direct product factor
4C2×Dic3×C22⋊C9C211yesmaximal, characteristic, complemented
5C23×C3×#67C311yesmaximal, characteristic
6C2×D4×C22⋊C9—31nomaximal
7C22×C3×C22⋊C9—22no—
8C22×C3×C22⋊C9C2211yescharacteristic
9C22×C3×C22⋊C9C2212yescomplemented
10C2×D3×C22⋊C9—22no—
11C2×D3×C22⋊C9C2212yescomplemented
12C2×C9×#67—41nomaximal
13Dic3×C22⋊C9C2212yescomplemented
14C25×C32C2×C311yescharacteristic
15C24×C3×D3C2×C311yescharacteristic
16C22×C3×#67—31no—
17C22×C3×#67—31no—
18C22×C3×#67—31no—
19C22×C3×#67—31no—
20C22×C3×#67—34no—
21C22×C3×#67C2×C314yes—
22C23×C3×Dic3C2×C311yescharacteristic
23C23×C22⋊C9—21no—
24C23×C22⋊C9—31no—
25C23×C22⋊C9D311yescharacteristic, complemented
26D4×C22⋊C9—34no—
27C2×C4×C22⋊C9—31no—
28C23×C3×C9—41no—
29C22×D3×C9—41no—
30C2×C3×C22⋊C9—22no—
31C2×C3×C22⋊C9D412yescomplemented
32C2×C3×C22⋊C9C2311yescharacteristic
33D3×C22⋊C9—22no—
34C9×#67—44no—
35C2×C9×Dic3—41no—
36C23×C3×D4—31no—
37C23×#67C911yescharacteristic, complemented
38C24×C32—22no—
39C24×C32—31no—
40C24×C32—32no—
41C24×C32—31no—
42C24×C32—62no—
43C24×C32C22×C312yes—
44C24×C32C22×C311yescharacteristic
45C23×C3×D3—22no—
46C23×C3×D3—31no—
47C23×C3×D3—31no—
48C23×C3×D3—32no—
49C23×C3×D3—62no—
50C23×C3×D3C22×C312yes—
51C2×C3×#67—34no—
52C2×C3×#67—34no—
53C2×C3×#67—32no—
54C2×C3×#67—34no—
55C2×C3×#67—31no—
56C2×C3×#67—31no—
57C2×C3×#67—34no—
58C2×C3×#67—34no—
59C2×C3×#67—34no—
60C2×C3×#67—34no—
61C2×C3×#67—34no—
62C2×C3×#67—31no—
63C2×C3×#67A411yescharacteristic
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3C22×C312yes—
68C22×C22⋊C9—24no—
69C22×C22⋊C9—21no—
70C22×C22⋊C9—22no—
71C22×C22⋊C9—22no—
72C22×C22⋊C9—32no—
73C22×C22⋊C9—62no—
74C22×C22⋊C9C2×D311yescharacteristic
75C22×C22⋊C9C2×D312yescomplemented
76C2×D4×C9—121no—
77C4×C22⋊C9—32no—
78C22×C3×C9—41no—
79C22×C3×C9—42no—
80C22×C3×C9—82no—
81C2×D3×C9—42no—
82C2×D3×C9—82no—
83C3×C22⋊C9C2×D411yescharacteristic, complemented
84C9×Dic3—42no—
85C25×C3—21no—
86C25×C3—31no—
87C25×C3C2×C911yescharacteristic, complemented
88C25×C3C3×D311yescharacteristic
89C24×D3C2×C911yescharacteristic, complemented
90C22×C3×D4—34no—
91C22×C3×D4—91no—
92C22×C3×D4—91no—
93C22×C3×D4—94no—
94C22×C3×D4—91no—
95C22×C3×D4—91no—
96C23×C3×C4—31no—
97C22×#67—31no—
98C22×#67—31no—
99C22×#67—31no—
100C22×#67—31no—
101C22×#67—34no—
102C22×#67C2×C914yescomplemented
103C23×Dic3C2×C911yescharacteristic, complemented
104C23×C32—22no—
105C23×C32—31no—
106C23×C32—32no—
107C23×C32—31no—
108C23×C32—32no—
109C23×C32—32no—
110C23×C32—32no—
111C23×C32—32no—
112C23×C32—31no—
113C23×C32—62no—
114C23×C32—62no—
115C23×C32—62no—
116C23×C32—64no—
117C23×C32—62no—
118C23×C32—62no—
119C23×C32—64no—
120C23×C32C3×D412yes—
121C23×C32C2×A411yescharacteristic
122C23×C32C23×C311yescharacteristic
123C22×C3×D3—22no—
124C22×C3×D3—32no—
125C22×C3×D3—31no—
126C22×C3×D3—32no—
127C22×C3×D3—32no—
128C22×C3×D3—32no—
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3C2×A411yescharacteristic
137C3×#67—38no—
138C3×#67—34no—
139C3×#67—34no—
140C3×#67—34no—
141C3×#67C2×A414yes—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3C2×A411yescharacteristic
148C23×C9—41no—
149C23×C9—81no—
150C23×C9—121no—
151C2×C22⋊C9—24no—
152C2×C22⋊C9—22no—
153C2×C22⋊C9—22no—
154C2×C22⋊C9—21no—
155C2×C22⋊C9—62no—
156C2×C22⋊C9#6712yesdirect product factor
157C2×C22⋊C9C22×D311yescharacteristic
158D4×C9—124no—
159C2×C4×C9—121no—
160C23×D4—31noSylow
161C2×C3×C9—42no—
162C2×C3×C9—41no—
163C2×C3×C9—82no—
164D3×C9—82no—
165C24×C3—24no—
166C24×C3—22no—
167C24×C3—22no—
168C24×C3—21no—
169C24×C3—22no—
170C24×C3—32no—
171C24×C3—31no—
172C24×C3—32no—
173C24×C3—31no—
174C24×C3—31no—
175C24×C3—31no—
176C24×C3—32no—
177C24×C3—61no—
178C24×C3—62no—
179C24×C3—61no—
180C24×C3—62no—
181C24×C3—62no—
182C24×C3—64no—
183C24×C3—62no—
184C24×C3—91no—
185C24×C3—92no—
186C24×C3—91no—
187C24×C3—182no—
188C24×C3C22×C911yescharacteristic
189C24×C3C22×C912yescomplemented
190C24×C3C2×C3×D311yescharacteristic
191C24×C3C2×C3×D312yes—
192C23×D3—22no—
193C23×D3—31no—
194C23×D3—31no—
195C23×D3—32no—
196C23×D3—62no—
197C23×D3C22×C912yescomplemented
198C2×C3×D4—31no—
199C2×C3×D4—91no—
200C2×C3×D4—94no—
201C2×C3×D4—94no—
202C2×C3×D4—91no—
203C2×C3×D4—94no—
204C2×C3×D4—94no—
205C2×C3×D4—94no—
206C2×C3×D4—92no—
207C2×C3×D4—91no—
208C2×C3×D4—94no—
209C2×C3×D4—94no—
210C2×C3×D4—94no—
211C22×C3×C4—32no—
212C22×C3×C4—91no—
213C22×C3×C4—92no—
214C22×C3×C4—91no—
215C2×#67—34no—
216C2×#67—34no—
217C2×#67—32no—
218C2×#67—34no—
219C2×#67—31no—
220C2×#67—31no—
221C2×#67—34no—
222C2×#67—34no—
223C2×#67—34no—
224C2×#67—34no—
225C2×#67—34no—
226C2×#67—31no—
227C2×#67C22⋊C911yescharacteristic, direct product factor
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—32no—
231C22×Dic3C22×C912yescomplemented
232C22×C32—22no—
233C22×C32—32no—
234C22×C32—31no—
235C22×C32—32no—
236C22×C32—32no—
237C22×C32—32no—
238C22×C32—31no—
239C22×C32—31no—
240C22×C32—32no—
241C22×C32—62no—
242C22×C32—62no—
243C22×C32—64no—
244C22×C32—62no—
245C22×C32—62no—
246C22×C32—64no—
247C22×C32—62no—
248C22×C32C2×C3×D411yescharacteristic
249C22×C32C22×A412yes—
250C22×C32C22×A411yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3—32no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—62no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3C22×A412yes—
261C3×Dic3—32no—
262C3×Dic3C22×A412yes—
263C22×C9—42no—
264C22×C9—41no—
265C22×C9—82no—
266C22×C9—84no—
267C22×C9—82no—
268C22×C9—81no—
269C22×C9—122no—
270C22×C9—242no—
271C22⋊C9—21no—
272C22⋊C9C2×#6711yescharacteristic, direct product factor
273C4×C9—122no—
274C25—31no—
275C25D3×C911yescharacteristic, complemented
276C22×D4—34no—
277C22×D4—91no—
278C22×D4—91no—
279C22×D4—94no—
280C22×D4—91no—
281C22×D4—91no—
282C23×C4—31no—
283C3×C9—41noSylow
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—31no—
291C23×C3—31no—
292C23×C3—32no—
293C23×C3—32no—
294C23×C3—31no—
295C23×C3—31no—
296C23×C3—32no—
297C23×C3—32no—
298C23×C3—32no—
299C23×C3—32no—
300C23×C3—32no—
301C23×C3—32no—
302C23×C3—31no—
303C23×C3—31no—
304C23×C3—31no—
305C23×C3—32no—
306C23×C3—32no—
307C23×C3—61no—
308C23×C3—62no—
309C23×C3—61no—
310C23×C3—62no—
311C23×C3—61no—
312C23×C3—62no—
313C23×C3—62no—
314C23×C3—62no—
315C23×C3—62no—
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—64no—
320C23×C3—64no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—64no—
324C23×C3—64no—
325C23×C3—62no—
326C23×C3—62no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—64no—
330C23×C3—64no—
331C23×C3—64no—
332C23×C3—64no—
333C23×C3—68no—
334C23×C3—64no—
335C23×C3—68no—
336C23×C3—62no—
337C23×C3—92no—
338C23×C3—92no—
339C23×C3—92no—
340C23×C3—92no—
341C23×C3—91no—
342C23×C3—182no—
343C23×C3—182no—
344C23×C3—182no—
345C23×C3—184no—
346C23×C3—182no—
347C23×C3—182no—
348C23×C3—184no—
349C23×C3D4×C912yescomplemented
350C23×C3C2×C22⋊C911yescharacteristic, complemented
351C23×C3C23×C911yescharacteristic
352C23×C3C3×#6712yes—
353C23×C3D3×A411yescharacteristic
354C23×C3C22×C3×D311yescharacteristic
355C22×D3—22no—
356C22×D3—32no—
357C22×D3—31no—
358C22×D3—32no—
359C22×D3—32no—
360C22×D3—32no—
361C22×D3—62no—
362C22×D3—62no—
363C22×D3—64no—
364C22×D3—64no—
365C22×D3—62no—
366C22×D3—62no—
367C22×D3—62no—
368C22×D3C2×C22⋊C911yescharacteristic, complemented
369C3×D4—34no—
370C3×D4—94no—
371C3×D4—94no—
372C3×D4—98no—
373C3×D4—94no—
374C2×C3×C4—31no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—92no—
378C2×C3×C4—91no—
379C2×C3×C4—92no—
380#67—38no—
381#67—34no—
382#67—34no—
383#67—34no—
384#67C2×C22⋊C914yesdirect product factor
385C2×Dic3—32no—
386C2×Dic3—31no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3C2×C22⋊C911yescharacteristic, complemented
391C2×C32—22no—
392C2×C32—32no—
393C2×C32—31no—
394C2×C32—31no—
395C2×C32—62no—
396C2×C32D4×A412yes—
397C2×C32C23×A411yescharacteristic
398C3×D3—22no—
399C3×D3—62no—
400C2×C9—42no—
401C2×C9—41no—
402C2×C9—82no—
403C2×C9—84no—
404C2×C9—82no—
405C2×C9—81no—
406C2×C9—242no—
407C24—22no—
408C24—32no—
409C24—31no—
410C24—31no—
411C24—32no—
412C24—62no—
413C24—62no—
414C24—91no—
415C24—92no—
416C24—91no—
417C24—182no—
418C24C2×D3×C911yescharacteristic
419C24C2×D3×C912yescomplemented
420C2×D4—31no—
421C2×D4—91no—
422C2×D4—94no—
423C2×D4—94no—
424C2×D4—91no—
425C2×D4—94no—
426C2×D4—94no—
427C2×D4—94no—
428C2×D4—92no—
429C2×D4—91no—
430C2×D4—94no—
431C2×D4—94no—
432C2×D4—94no—
433C22×C4—32no—
434C22×C4—91no—
435C22×C4—92no—
436C22×C4—91no—
437C22×C3—24no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—21no—
441C22×C3—22no—
442C22×C3—21no—
443C22×C3—32no—
444C22×C3—31no—
445C22×C3—32no—
446C22×C3—31no—
447C22×C3—32no—
448C22×C3—32no—
449C22×C3—32no—
450C22×C3—32no—
451C22×C3—31no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—32no—
456C22×C3—32no—
457C22×C3—32no—
458C22×C3—32no—
459C22×C3—32no—
460C22×C3—61no—
461C22×C3—62no—
462C22×C3—61no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—62no—
466C22×C3—61no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—64no—
471C22×C3—62no—
472C22×C3—64no—
473C22×C3—62no—
474C22×C3—64no—
475C22×C3—64no—
476C22×C3—68no—
477C22×C3—64no—
478C22×C3—62no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—64no—
483C22×C3—64no—
484C22×C3—64no—
485C22×C3—64no—
486C22×C3—68no—
487C22×C3—62no—
488C22×C3—62no—
489C22×C3—62no—
490C22×C3—92no—
491C22×C3—182no—
492C22×C3—182no—
493C22×C3—184no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—184no—
497C22×C3—182no—
498C22×C3C2×D4×C911yescharacteristic, complemented
499C22×C3C22×C22⋊C911yescharacteristic
500C22×C3C22×C22⋊C912yescomplemented
501C22×C3C2×C3×#6711yescharacteristic
502C22×C3C2×D3×A412yes—
503C22×C3C2×D3×A411yescharacteristic
504C2×D3—22no—
505C2×D3—32no—
506C2×D3—62no—
507C2×D3—64no—
508C2×D3—62no—
509C2×D3—64no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3C22×C22⋊C912yescomplemented
514C3×C4—32no—
515C3×C4—92no—
516Dic3—32no—
517Dic3C22×C22⋊C912yescomplemented
518C32C2×D4×A411yescharacteristic
519C9—41no—
520C9—81no—
521C23—22no—
522C23—31no—
523C23—32no—
524C23—31no—
525C23—32no—
526C23—32no—
527C23—31no—
528C23—31no—
529C23—32no—
530C23—32no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—64no—
535C23—62no—
536C23—62no—
537C23—64no—
538C23—62no—
539C23—92no—
540C23—92no—
541C23—92no—
542C23—92no—
543C23—91no—
544C23—182no—
545C23—182no—
546C23—182no—
547C23—184no—
548C23—182no—
549C23—182no—
550C23—184no—
551C23C9×#6712yescomplemented
552C23D3×C22⋊C911yescharacteristic, complemented
553C23C22×D3×C911yescharacteristic
554D4—34no—
555D4—94no—
556D4—94no—
557D4—98no—
558D4—94no—
559C2×C4—31no—
560C2×C4—92no—
561C2×C4—92no—
562C2×C4—92no—
563C2×C4—91no—
564C2×C4—92no—
565C2×C3—24no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—22no—
570C2×C3—31no—
571C2×C3—32no—
572C2×C3—31no—
573C2×C3—31no—
574C2×C3—32no—
575C2×C3—31no—
576C2×C3—61no—
577C2×C3—62no—
578C2×C3—61no—
579C2×C3—64no—
580C2×C3—62no—
581C2×C3—62no—
582C2×C3—62no—
583C2×C3—182no—
584C2×C3D4×C22⋊C912yescomplemented
585C2×C3C23×C22⋊C911yescharacteristic
586C2×C3A4×#6712yes—
587C2×C3C22×D3×A411yescharacteristic
588D3—22no—
589D3—62no—
590C22—22no—
591C22—32no—
592C22—31no—
593C22—32no—
594C22—31no—
595C22—31no—
596C22—32no—
597C22—32no—
598C22—32no—
599C22—32no—
600C22—62no—
601C22—62no—
602C22—64no—
603C22—62no—
604C22—62no—
605C22—64no—
606C22—62no—
607C22—62no—
608C22—92no—
609C22—182no—
610C22—182no—
611C22—184no—
612C22—182no—
613C22—182no—
614C22—184no—
615C22—182no—
616C22C2×C9×#6711yescharacteristic, complemented
617C22C2×D3×C22⋊C911yescharacteristic
618C22C2×D3×C22⋊C912yescomplemented
619C4—32no—
620C4—92no—
621C3—21no—
622C3C2×D4×C22⋊C911yescharacteristic, complemented
623C3C2×A4×#6711yescharacteristic
624C2—22no—
625C2—31no—
626C2—32no—
627C2—31no—
628C2—62no—
629C2—62no—
630C2—182no—
631C2#67×C22⋊C912yesdirect product factor
632C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutatorC23×C3C23×C9
FrattiniC2×C3C22×D3×A4
FittingC25×C32C2×C3
socleC24×C32C22×C3
prime radicalC23×C3×#67C3
cosocleC23×C32C23×C3
hypercenterC23×C3D3×A4
hypocenterC22×C3C2×D4×C9
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×C22⋊C9C2semidirect
2. C22×D3×C22⋊C9C2semidirect
3. #67×C22⋊C9C2direct (N × H)
4. C2×Dic3×C22⋊C9C2semidirect
9. C22×C3×C22⋊C9C22semidirect
11. C2×D3×C22⋊C9C22semidirect
13. Dic3×C22⋊C9C22semidirect
25. C23×C22⋊C9D3semidirect
31. C2×C3×C22⋊C9D4semidirect
37. C23×#67C9semidirect
75. C22×C22⋊C9C2×D3semidirect
83. C3×C22⋊C9C2×D4semidirect
87. C25×C3C2×C9semidirect
89. C24×D3C2×C9semidirect
102. C22×#67C2×C9semidirect
103. C23×Dic3C2×C9semidirect
156. C2×C22⋊C9#67direct (N × H)
189. C24×C3C22×C9semidirect
197. C23×D3C22×C9semidirect
227. C2×#67C22⋊C9direct (N × H)
231. C22×Dic3C22×C9semidirect
272. C22⋊C9C2×#67direct (N × H)
275. C25D3×C9semidirect
349. C23×C3D4×C9semidirect
350. C23×C3C2×C22⋊C9semidirect
368. C22×D3C2×C22⋊C9semidirect
384. #67C2×C22⋊C9direct (N × H)
390. C2×Dic3C2×C22⋊C9semidirect
419. C24C2×D3×C9semidirect
498. C22×C3C2×D4×C9semidirect
500. C22×C3C22×C22⋊C9semidirect
513. C2×D3C22×C22⋊C9semidirect
517. Dic3C22×C22⋊C9semidirect
551. C23C9×#67semidirect
552. C23D3×C22⋊C9semidirect
584. C2×C3D4×C22⋊C9semidirect
616. C22C2×C9×#67semidirect
618. C22C2×D3×C22⋊C9semidirect
622. C3C2×D4×C22⋊C9semidirect
631. C2#67×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C22×D3×C22⋊C92
#67×C22⋊C92
C2×Dic3×C22⋊C92
C23×C3×#673
C2×D4×C22⋊C93
C2×C9×#674