Finite Group Databasebeta

← C22×#73105 #72269824

Subgroups — C22×#73105 #72269824

Order 1536.

3643 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#5409C212yesmaximal
2C2×#73105C2112yesmaximal, direct product factor
3C23×C8×A4C211yesmaximal, characteristic
4C22×#1810—31nomaximal, Sylow
5C23×C4×A4C2211yescharacteristic
6C22×C8×A4C416yes—
7C22×C8×A4C412yes—
8C22×C8×A4C2216yes—
9C2×#5409—26no—
10C2×#5409C22112yes—
11C22×A4⋊C8—22no—
12C22×#819—41nomaximal
13#73105C22116yesdirect product factor
14C25×C8D311yescharacteristic
15C23×#108—31no—
16C23×#108—31no—
17C22×#407—32no—
18C2×#1810—312no—
19C2×#1810—312no—
20C22×#341—32no—
21C24×A4Q811yescharacteristic
22C22×C4×A4C2×C413yes—
23C22×C4×A4C2×C411yescharacteristic
24C22×C4×A4D411yescharacteristic
25C22×C4×A4D413yes—
26C22×C4×A4Q813yes—
27C22×C4×A4C2313yes—
28C23×C3×C8—41no—
29C22×#208—42no—
30C2×C8×A4C2×C4112yes—
31C2×C8×A4C2×C4112yes—
32C2×C8×A4C2314yes—
33#5409—212no—
34#5409C2318yes—
35C2×A4⋊C8—212no—
36C2×#819—412no—
37C25×C4C2×D311yescharacteristic
38C23×M16—32no—
39C23×M16—32no—
40C24×C8—31no—
41C24×C8—32no—
42C24×C8—36no—
43C24×C8—31no—
44C24×C8—36no—
45C24×C8Dic316yes—
46C24×C8Dic312yes—
47C24×C8C2×D316yes—
48C22×#108—312no—
49C22×#108—312no—
50C22×#108—61no—
51C22×#108—62no—
52C22×#108—66no—
53C22×#108—66no—
54C22×#108—61no—
55C2×#407—312no—
56C2×#407—312no—
57C2×#407—66no—
58C2×#407—66no—
59C22×C22⋊C8—62no—
60C22×C22⋊C8—62no—
61#1810—348no—
62#1810—316no—
63C2×#341—324no—
64C23×A4—24no—
65C23×A4C4⋊C413yes—
66C23×A4C4⋊C411yescharacteristic
67C23×A4C2×Q813yes—
68C23×C3×C4—41no—
69C2×C4×A4C4⋊C416yes—
70C2×C4×A4C4⋊C413yes—
71C2×C4×A4C4⋊C413yes—
72C2×C4×A4C22×C413yes—
73C2×C4×A4C22×C413yes—
74C2×C4×A4C2×D413yes—
75C2×C4×A4C2×D413yes—
76C2×C4×A4C2×Q813yes—
77C2×C4×A4C2411yescharacteristic
78C22×C3×C8—42no—
79C22×C3×C8—46no—
80C22×C3×C8—46no—
81C2×#208—412no—
82C2×#208—86no—
83C22×C3⋊C8—82no—
84C8×A4C22×C418yes—
85A4⋊C8—28no—
86#819—416no—
87C26Dic611yescharacteristic
88C24×C4—31no—
89C24×C4—31no—
90C24×C4—33no—
91C24×C4—31no—
92C24×C4—33no—
93C24×C4—33no—
94C24×C4—31no—
95C24×C4—33no—
96C24×C4Dic613yes—
97C24×C4D1211yescharacteristic
98C24×C4D1213yes—
99C24×C4C2×Dic313yes—
100C24×C4C2×Dic311yescharacteristic
101C24×C4C22×D313yes—
102C22×M16—312no—
103C22×M16—312no—
104C22×M16—66no—
105C22×M16—66no—
106C22×M16—62no—
107C22×M16—66no—
108C22×M16—66no—
109C22×M16—62no—
110C22×M16—66no—
111C22×M16—66no—
112C22×M16—62no—
113C22×M16—62no—
114C23×C8—31no—
115C23×C8—32no—
116C23×C8—36no—
117C23×C8—32no—
118C23×C8—36no—
119C23×C8—36no—
120C23×C8—36no—
121C23×C8—312no—
122C23×C8—36no—
123C23×C8—32no—
124C23×C8—36no—
125C23×C8—312no—
126C23×C8—312no—
127C23×C8—324no—
128C23×C8—312no—
129C23×C8—312no—
130C23×C8—612no—
131C23×C8—612no—
132C23×C8—64no—
133C23×C8—61no—
134C23×C8—63no—
135C23×C8—63no—
136C23×C8—62no—
137C23×C8—62no—
138C23×C8C2×Dic3112yes—
139C23×C8C2×Dic3112yes—
140C23×C8S411yescharacteristic
141C23×C8C22×D314yes—
142C2×#108—316no—
143C2×#108—316no—
144C2×#108—612no—
145C2×#108—624no—
146C2×#108—624no—
147C2×#108—624no—
148C2×#108—612no—
149#407—324no—
150#407—38no—
151#407—624no—
152#407—612no—
153#407—612no—
154C2×C22⋊C8—612no—
155C2×C22⋊C8—612no—
156C2×C22⋊C8—612no—
157C2×C22⋊C8—612no—
158#341—332no—
159C24×C3—41no—
160C22×A4—26no—
161C22×A4—26no—
162C22×A4#10814yescomplemented
163C22×A4C2×C4⋊C413yes—
164C22×A4C2×C4⋊C413yes—
165C22×A4C22×Q811yescharacteristic
166C22×C3×C4—43no—
167C22×C3×C4—43no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—43no—
171C22×C3×C4—43no—
172C4×A4C2×C4⋊C413yes—
173C4×A4C2×C4⋊C413yes—
174C4×A4C23×C411yescharacteristic
175C4×A4C22×D411yescharacteristic
176C2×C3×C8—412no—
177C2×C3×C8—412no—
178C2×C3×C8—44no—
179#208—48no—
180#208—812no—
181C2×C3⋊C8—812no—
182C25—24no—
183C25—31no—
184C25—31no—
185C25—33no—
186C25—33no—
187C25—64no—
188C25C4⋊Dic313yes—
189C25C4⋊Dic311yescharacteristic
190C25C2×Dic613yes—
191C23×C4—31no—
192C23×C4—33no—
193C23×C4—33no—
194C23×C4—33no—
195C23×C4—33no—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4—31no—
199C23×C4—33no—
200C23×C4—31no—
201C23×C4—33no—
202C23×C4—33no—
203C23×C4—33no—
204C23×C4—33no—
205C23×C4—33no—
206C23×C4—36no—
207C23×C4—33no—
208C23×C4—33no—
209C23×C4—33no—
210C23×C4—33no—
211C23×C4—36no—
212C23×C4—33no—
213C23×C4—33no—
214C23×C4—36no—
215C23×C4—31no—
216C23×C4—33no—
217C23×C4—33no—
218C23×C4—33no—
219C23×C4—36no—
220C23×C4—33no—
221C23×C4—33no—
222C23×C4—36no—
223C23×C4—31no—
224C23×C4—33no—
225C23×C4—33no—
226C23×C4—33no—
227C23×C4—36no—
228C23×C4—33no—
229C23×C4—33no—
230C23×C4—33no—
231C23×C4—33no—
232C23×C4—63no—
233C23×C4—63no—
234C23×C4—61no—
235C23×C4—63no—
236C23×C4—63no—
237C23×C4—63no—
238C23×C4—63no—
239C23×C4—63no—
240C23×C4—66no—
241C23×C4—63no—
242C23×C4—61no—
243C23×C4—63no—
244C23×C4C4⋊Dic316yes—
245C23×C4C4⋊Dic313yes—
246C23×C4C4⋊Dic313yes—
247C23×C4C2×Dic613yes—
248C23×C4C2×D1213yes—
249C23×C4C2×D1213yes—
250C23×C4C22×Dic313yes—
251C23×C4C22×Dic313yes—
252C23×C4C2×S411yescharacteristic
253C23×C4C23×D311yescharacteristic
254C2×M16—38no—
255C2×M16—38no—
256C2×M16—612no—
257C2×M16—612no—
258C2×M16—612no—
259C2×M16—612no—
260C2×M16—612no—
261C2×M16—612no—
262C2×M16—612no—
263C2×M16—612no—
264C2×M16—612no—
265C2×M16—612no—
266C2×M16—126no—
267C2×M16—126no—
268C2×M16—1212no—
269C2×M16—1212no—
270C2×M16—126no—
271C2×M16—126no—
272C22×C8—32no—
273C22×C8—36no—
274C22×C8—36no—
275C22×C8—312no—
276C22×C8—312no—
277C22×C8—312no—
278C22×C8—324no—
279C22×C8—312no—
280C22×C8—312no—
281C22×C8—312no—
282C22×C8—34no—
283C22×C8—312no—
284C22×C8—312no—
285C22×C8—312no—
286C22×C8—324no—
287C22×C8—324no—
288C22×C8—38no—
289C22×C8—34no—
290C22×C8—624no—
291C22×C8—68no—
292C22×C8—624no—
293C22×C8—66no—
294C22×C8—66no—
295C22×C8—66no—
296C22×C8—612no—
297C22×C8—66no—
298C22×C8—66no—
299C22×C8—612no—
300C22×C8—612no—
301C22×C8—122no—
302C22×C8—126no—
303C22×C8—126no—
304C22×C8—122no—
305C22×C8SL(2,ℤ/4)12yes—
306C22×C8SL(2,ℤ/4)16yes—
307C22×C8C22×Dic318yes—
308C22×C8C2×S416yes—
309#108—616no—
310#108—632no—
311#108—616no—
312C22⋊C8—624no—
313C22⋊C8—624no—
314C22⋊C8—68no—
315C22⋊C8—68no—
316C23×C3—43no—
317C23×C3—41no—
318C23×C3—43no—
319C23×C3—84no—
320C2×A4—23no—
321C2×A4—21no—
322C2×A4C2×#10816yescomplemented
323C2×A4C22×C4⋊C411yescharacteristic
324C2×C3×C4—43no—
325C2×C3×C4—43no—
326C2×C3×C4—43no—
327C2×C3×C4—43no—
328C2×C3×C4—46no—
329C2×C3×C4—43no—
330C2×C3×C4—43no—
331C2×C3×C4—43no—
332C2×C3×C4—41no—
333C3×C8—48no—
334C3⋊C8—88no—
335C24—26no—
336C24—26no—
337C24—31no—
338C24—33no—
339C24—33no—
340C24—33no—
341C24—34no—
342C24—38no—
343C24—33no—
344C24—33no—
345C24—33no—
346C24—33no—
347C24—31no—
348C24—36no—
349C24—33no—
350C24—33no—
351C24—64no—
352C24—64no—
353C24—66no—
354C24—66no—
355C24—63no—
356C24—61no—
357C24—63no—
358C24—64no—
359C24—612no—
360C24—612no—
361C24—612no—
362C24—612no—
363C24#81914yescomplemented
364C24C2×C4⋊Dic313yes—
365C24C2×C4⋊Dic313yes—
366C24A4⋊Q811yescharacteristic
367C24C22×Dic611yescharacteristic
368C22×C4—33no—
369C22×C4—33no—
370C22×C4—31no—
371C22×C4—33no—
372C22×C4—33no—
373C22×C4—31no—
374C22×C4—33no—
375C22×C4—33no—
376C22×C4—36no—
377C22×C4—33no—
378C22×C4—33no—
379C22×C4—33no—
380C22×C4—33no—
381C22×C4—33no—
382C22×C4—36no—
383C22×C4—33no—
384C22×C4—33no—
385C22×C4—33no—
386C22×C4—36no—
387C22×C4—33no—
388C22×C4—36no—
389C22×C4—33no—
390C22×C4—33no—
391C22×C4—36no—
392C22×C4—36no—
393C22×C4—33no—
394C22×C4—33no—
395C22×C4—33no—
396C22×C4—33no—
397C22×C4—33no—
398C22×C4—33no—
399C22×C4—33no—
400C22×C4—33no—
401C22×C4—36no—
402C22×C4—33no—
403C22×C4—31no—
404C22×C4—33no—
405C22×C4—33no—
406C22×C4—33no—
407C22×C4—31no—
408C22×C4—33no—
409C22×C4—33no—
410C22×C4—31no—
411C22×C4—33no—
412C22×C4—33no—
413C22×C4—36no—
414C22×C4—36no—
415C22×C4—36no—
416C22×C4—36no—
417C22×C4—36no—
418C22×C4—33no—
419C22×C4—33no—
420C22×C4—33no—
421C22×C4—33no—
422C22×C4—33no—
423C22×C4—33no—
424C22×C4—33no—
425C22×C4—33no—
426C22×C4—33no—
427C22×C4—33no—
428C22×C4—33no—
429C22×C4—31no—
430C22×C4—63no—
431C22×C4—61no—
432C22×C4—63no—
433C22×C4—61no—
434C22×C4—63no—
435C22×C4—63no—
436C22×C4—63no—
437C22×C4—63no—
438C22×C4—63no—
439C22×C4—63no—
440C22×C4—66no—
441C22×C4—63no—
442C22×C4—63no—
443C22×C4—66no—
444C22×C4—66no—
445C22×C4—66no—
446C22×C4—63no—
447C22×C4—63no—
448C22×C4—63no—
449C22×C4—63no—
450C22×C4—63no—
451C22×C4—63no—
452C22×C4—66no—
453C22×C4—63no—
454C22×C4—66no—
455C22×C4—63no—
456C22×C4—63no—
457C22×C4—63no—
458C22×C4C2×C4⋊Dic313yes—
459C22×C4C2×C4⋊Dic313yes—
460C22×C4A4⋊Q813yes—
461C22×C4C4⋊S413yes—
462C22×C4C4⋊S411yescharacteristic
463C22×C4C2×SL(2,ℤ/4)11yescharacteristic
464C22×C4C2×SL(2,ℤ/4)13yes—
465C22×C4C22×D1211yescharacteristic
466C22×C4C23×Dic311yescharacteristic
467C22×C4C22×S413yes—
468M16—68no—
469M16—68no—
470M16—68no—
471M16—68no—
472M16—1212no—
473M16—1212no—
474M16—1212no—
475M16—1212no—
476C2×C8—312no—
477C2×C8—312no—
478C2×C8—34no—
479C2×C8—38no—
480C2×C8—38no—
481C2×C8—324no—
482C2×C8—324no—
483C2×C8—38no—
484C2×C8—64no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—68no—
488C2×C8—68no—
489C2×C8—1212no—
490C2×C8—1212no—
491C2×C8—1212no—
492C2×C8—1212no—
493C2×C8C2×SL(2,ℤ/4)112yes—
494C2×C8C2×SL(2,ℤ/4)112yes—
495C2×C8C22×S414yes—
496C22×C3—44no—
497C22×C3—41no—
498C22×C3—43no—
499C22×C3—43no—
500C22×C3—86no—
501C22×C3—86no—
502A4C22×#10811yescharacteristic, complemented
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—43no—
506C3×C4—43no—
507C23—24no—
508C23—23no—
509C23—21no—
510C23—34no—
511C23—34no—
512C23—33no—
513C23—33no—
514C23—33no—
515C23—33no—
516C23—31no—
517C23—312no—
518C23—36no—
519C23—312no—
520C23—312no—
521C23—31no—
522C23—33no—
523C23—33no—
524C23—33no—
525C23—36no—
526C23—33no—
527C23—33no—
528C23—66no—
529C23—66no—
530C23—66no—
531C23—612no—
532C23—66no—
533C23—612no—
534C23—63no—
535C23—61no—
536C23—61no—
537C23—63no—
538C23—63no—
539C23—66no—
540C23—612no—
541C23—66no—
542C23—66no—
543C23—66no—
544C23—66no—
545C23—624no—
546C23—612no—
547C23—66no—
548C23—66no—
549C23—68no—
550C23—624no—
551C23—64no—
552C23C2×#81916yescomplemented
553C23C4⋊SL(2,ℤ/4)13yes—
554C23C4⋊SL(2,ℤ/4)11yescharacteristic
555C23C22×C4⋊Dic311yescharacteristic
556C23C2×A4⋊Q813yes—
557C2×C4—31no—
558C2×C4—33no—
559C2×C4—33no—
560C2×C4—33no—
561C2×C4—33no—
562C2×C4—36no—
563C2×C4—33no—
564C2×C4—33no—
565C2×C4—33no—
566C2×C4—31no—
567C2×C4—33no—
568C2×C4—31no—
569C2×C4—33no—
570C2×C4—31no—
571C2×C4—31no—
572C2×C4—33no—
573C2×C4—33no—
574C2×C4—33no—
575C2×C4—33no—
576C2×C4—33no—
577C2×C4—33no—
578C2×C4—33no—
579C2×C4—36no—
580C2×C4—36no—
581C2×C4—33no—
582C2×C4—33no—
583C2×C4—33no—
584C2×C4—36no—
585C2×C4—36no—
586C2×C4—33no—
587C2×C4—33no—
588C2×C4—31no—
589C2×C4—31no—
590C2×C4—63no—
591C2×C4—63no—
592C2×C4—61no—
593C2×C4—63no—
594C2×C4—63no—
595C2×C4—63no—
596C2×C4—63no—
597C2×C4—66no—
598C2×C4—63no—
599C2×C4C4⋊SL(2,ℤ/4)16yes—
600C2×C4C4⋊SL(2,ℤ/4)13yes—
601C2×C4C4⋊SL(2,ℤ/4)13yes—
602C2×C4C2×A4⋊Q813yes—
603C2×C4C2×C4⋊S413yes—
604C2×C4C2×C4⋊S413yes—
605C2×C4C22×SL(2,ℤ/4)13yes—
606C2×C4C22×SL(2,ℤ/4)13yes—
607C2×C4C23×S411yescharacteristic
608C8—38no—
609C8—128no—
610C8—128no—
611C8C22×SL(2,ℤ/4)18yes—
612C2×C3—46no—
613C2×C3—41no—
614C2×C3—81no—
615C2×C3—83no—
616C22—26no—
617C22—26no—
618C22—31no—
619C22—36no—
620C22—36no—
621C22—36no—
622C22—312no—
623C22—31no—
624C22—33no—
625C22—36no—
626C22—32no—
627C22—31no—
628C22—33no—
629C22—61no—
630C22—63no—
631C22—61no—
632C22—66no—
633C22—63no—
634C22—66no—
635C22—612no—
636C22—63no—
637C22—64no—
638C22—63no—
639C22—61no—
640C22—612no—
641C22—612no—
642C22—66no—
643C22—66no—
644C22#7310514yesdirect product factor
645C22C22×#81911yescharacteristic, complemented
646C22C2×C4⋊SL(2,ℤ/4)13yes—
647C22C2×C4⋊SL(2,ℤ/4)13yes—
648C22C22×A4⋊Q811yescharacteristic
649C4—31no—
650C4—31no—
651C4—33no—
652C4—33no—
653C4C2×C4⋊SL(2,ℤ/4)13yes—
654C4C2×C4⋊SL(2,ℤ/4)13yes—
655C4C22×C4⋊S411yescharacteristic
656C4C23×SL(2,ℤ/4)11yescharacteristic
657C3—41noSylow
658C2—21no—
659C2—23no—
660C2—31no—
661C2—31no—
662C2—36no—
663C2—63no—
664C2—61no—
665C2C2×#7310516yesdirect product factor
666C2C22×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C4C4⋊S4
commutatorC4×A4C23×C4
FrattiniC2×C4C23×S4
FittingC25×C8D3
socleC25C4⋊Dic3
prime radicalC24×A4Q8
cosocleC2×C4×A4C24
hypercenterC23×C8S4
hypocenterA4C22×#108
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C2×#73105C2direct (N × H)
13. #73105C22direct (N × H)
162. C22×A4#108semidirect
322. C2×A4C2×#108semidirect
363. C24#819semidirect
502. A4C22×#108semidirect
552. C23C2×#819semidirect
644. C22#73105direct (N × H)
645. C22C22×#819semidirect
665. C2C2×#73105direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#54092
C2×#731052
C23×C8×A42
C22×#18103
C22×#8194