Finite Group Databasebeta

← #11807599

Subgroups — #11807599

Order 1152.

1505 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#937C211yesmaximal, characteristic
2#10592920C211yesmaximal, characteristic, complemented
3C3×#5220C211yesmaximal, characteristic, complemented
4#10591710C212yesmaximal, complemented
5#10591707C212yesmaximal, complemented
6#10591706C212yesmaximal, complemented
7#10591702C212yesmaximal, complemented
8#10591316C212yesmaximal, complemented
9#10591286C212yesmaximal, complemented
10#82580—31nomaximal
11#74882—31nomaximal
12C24×C3×D3C2211yescharacteristic
13C22×C32×D4C2211yescharacteristic
14C22×C3×#67C2212yes—
15C23×C3×Dic3C2211yescharacteristic, complemented
16C2×#3757C2212yes—
17C2×#3757C2212yescomplemented
18C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C2×C4×C3⋊Dic3C2211yescharacteristic, complemented
20C2×C3×#217C2212yes—
21C2×C3×#217C2211yescharacteristic
22C2×C3×#217C2211yescharacteristic, complemented
23C3×#937C418yescomplemented
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×Dic3⋊Dic3C2212yescomplemented
26C2×(C2×D3)⋊Dic3C2212yes—
27C2×(C2×D3)⋊Dic3C2212yes—
28C2×(C2×D3)⋊Dic3C2212yes—
29C2×(C2×D3)⋊Dic3C2212yes—
30C2×Dic32C2212yescomplemented
31C3×C3⋊#95C2212yescomplemented
32#63579C2212yescomplemented
33#63579C2212yescomplemented
34C2×C3×#120—31no—
35C2×#937D311yescharacteristic
36C3×#386—31no—
37#5247—32no—
38#5221—31no—
39#5221—31no—
40#5221—32no—
41#5221—61no—
42#5220—32no—
43#5220D311yescharacteristic, complemented
44#5213—32no—
45#4957—32no—
46#4955—32no—
47#4951—32no—
48#4951—32no—
49#4951—32no—
50#4945—32no—
51#4670—32no—
52#4669—32no—
53C24×C32C2311yescharacteristic
54C24×C32C2311yescharacteristic
55C23×C3×D3—41no—
56C23×C3×D3—41no—
57C23×C3×D3—42no—
58C23×C3×D3—42no—
59C23×C3×D3C2×C414yescomplemented
60C23×C3×D3C2312yes—
61C2×C32×D4—41no—
62C2×C32×D4—41no—
63C2×C32×D4C2×C414yes—
64C22×C32×C4C2311yescharacteristic
65C22×C3⋊Dic3C2312yes—
66C22×C3⋊Dic3C2312yes—
67C2×C3×#67—42no—
68C2×C3×#67—42no—
69C2×C3×#67C2×C418yes—
70C22×C3×Dic3—41no—
71C22×C3×Dic3—42no—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2312yes—
75#3757—42no—
76#3757—42no—
77C4⋊(C3⋊Dic3)—41no—
78C4×C3⋊Dic3—41no—
79C3×#217—42no—
80C3×#217—41no—
81C3×#217C2×C414yescomplemented
82C3×(C2×D3)⋊C4C2×C418yescomplemented
83Dic3⋊Dic3—42no—
84(C2×D3)⋊Dic3—42no—
85(C2×D3)⋊Dic3—42no—
86(C2×D3)⋊Dic3—42no—
87(C2×D3)⋊Dic3—42no—
88Dic32—42no—
89#2375—91noSylow
90C25×C3—31no—
91C24×D3C2×D311yescharacteristic
92C22×C3×D4—21no—
93C22×C3×D4—32no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C2×D311yescharacteristic
96C23×C3×C4—31no—
97C22×#67C2×D312yes—
98C23×Dic3C2×D311yescharacteristic, complemented
99C3×#120—38no—
100C2×C3×C22⋊C4—31no—
101C2×C3×C22⋊C4—32no—
102C2×C3×C22⋊C4—32no—
103C2×C3×C22⋊C4—31no—
104C2×#217—32no—
105C2×#217—32no—
106C2×#217—32no—
107C2×#217—32no—
108C2×#217—32no—
109C2×#217—32no—
110C2×#217—32no—
111C2×#217—32no—
112C2×#217—62no—
113C2×#217—62no—
114C2×#217C2×D312yes—
115C2×#217C2×D311yescharacteristic
116C2×#217C2×D311yescharacteristic, complemented
117#937Dic318yescomplemented
118C2×(C2×D3)⋊C4—32no—
119C2×(C2×D3)⋊C4—32no—
120C2×(C2×D3)⋊C4—32no—
121C2×(C2×D3)⋊C4—32no—
122C2×(C2×D3)⋊C4C2×D312yes—
123C2×C4⋊Dic3—31no—
124C2×C4⋊Dic3—31no—
125C2×C4⋊Dic3—32no—
126C2×C4⋊Dic3—61no—
127C2×Dic3⋊C4—32no—
128C2×C4×Dic3—31no—
129C2×C4×Dic3—31no—
130C2×C4×Dic3—32no—
131C2×C4×Dic3—32no—
132C2×C4×Dic3—61no—
133C3×#95—32no—
134C3⋊#95—32no—
135C3⋊#95—32no—
136C3⋊#95—32no—
137C3⋊#95—32no—
138C3⋊#95C2×D312yescomplemented
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—21no—
142C23×C32—21no—
143C23×C32—21no—
144C23×C32—22no—
145C23×C32—22no—
146C23×C32—21no—
147C23×C32C22×C412yes—
148C23×C32C22×C412yes—
149C23×C32C2411yescharacteristic
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—22no—
156C22×C3×D3—48no—
157C22×C3×D3—44no—
158C22×C3×D3—44no—
159C22×C3×D3—48no—
160C22×C3×D3—44no—
161C22×C3×D3C22×C414yes—
162C32×D4—44no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4C22×C412yes—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—22no—
172C3×#67—48no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—44no—
182C2×C3×Dic3C22×C414yes—
183C2×#120—31no—
184#390—92no—
185#390—91no—
186#388—92no—
187#388—92no—
188#388—92no—
189#386—31no—
190#386—92no—
191#379—92no—
192C24×C3—21no—
193C24×C3—21no—
194C24×C3—32no—
195C24×C3—34no—
196C24×C3—122no—
197C24×C3—121no—
198C24×C3—121no—
199C24×C3—122no—
200C24×C3C22×D311yescharacteristic
201C24×C3C22×D311yescharacteristic
202C24×C3C22×D311yescharacteristic
203C24×C3C22×D311yescharacteristic
204C23×D3—41no—
205C23×D3—41no—
206C23×D3—42no—
207C23×D3—42no—
208C23×D3C2×Dic314yescomplemented
209C23×D3C22×D312yes—
210C2×C3×D4—24no—
211C2×C3×D4—38no—
212C2×C3×D4—41no—
213C2×C3×D4—41no—
214C2×C3×D4—41no—
215C2×C3×D4—41no—
216C2×C3×D4—42no—
217C2×C3×D4—42no—
218C2×C3×D4—122no—
219C2×C3×D4—122no—
220C2×C3×D4C4×D314yes—
221C2×C3×D4C2×Dic314yes—
222C22×C3×C4—21no—
223C22×C3×C4—32no—
224C22×C3×C4—32no—
225C22×C3×C4—32no—
226C22×C3×C4—121no—
227C22×C3×C4—122no—
228C22×C3×C4C22×D311yescharacteristic
229C22×C3×C4C22×D311yescharacteristic
230C2×#67—42no—
231C2×#67—42no—
232C2×#67C2×Dic318yes—
233C22×Dic3—32no—
234C22×Dic3—32no—
235C22×Dic3—32no—
236C22×Dic3—32no—
237C22×Dic3—42no—
238C22×Dic3—41no—
239C22×Dic3—62no—
240C22×Dic3—62no—
241C22×Dic3C22×D312yes—
242C22×Dic3C22×D312yes—
243C22×Dic3C22×D312yes—
244C3×C22⋊C4—34no—
245C3×C22⋊C4—38no—
246C3×C22⋊C4—122no—
247C3×C22⋊C4—121no—
248#217—42no—
249#217—41no—
250#217—122no—
251#217—122no—
252#217—122no—
253#217—122no—
254#217—122no—
255#217—122no—
256#217—122no—
257#217—122no—
258#217—124no—
259#217—124no—
260#217C2×Dic314yescomplemented
261(C2×D3)⋊C4—122no—
262(C2×D3)⋊C4—122no—
263(C2×D3)⋊C4—122no—
264(C2×D3)⋊C4—122no—
265(C2×D3)⋊C4C2×Dic318yescomplemented
266C4⋊Dic3—122no—
267C4⋊Dic3—121no—
268C4⋊Dic3—121no—
269C4⋊Dic3—122no—
270Dic3⋊C4—122no—
271C4×Dic3—122no—
272C4×Dic3—122no—
273C4×Dic3—121no—
274C4×Dic3—121no—
275C4×Dic3—122no—
276C22×C32—24no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—24no—
280C22×C32—41no—
281C22×C32—42no—
282C22×C32—41no—
283C22×C32—42no—
284C22×C32—41no—
285C22×C32—41no—
286C22×C32C23×C411yescharacteristic
287C22×C32H(5,𝔽2)11yescharacteristic
288C22×C32H(5,𝔽2)12yes—
289C22×C32H(5,𝔽2)11yescharacteristic
290C22×C32H(5,𝔽2)12yes—
291C2×C3×D3—24no—
292C2×C3×D3—24no—
293C2×C3×D3—24no—
294C2×C3×D3—416no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C32×C4—22no—
300C3⋊Dic3—42no—
301C3⋊Dic3—42no—
302C3×Dic3—24no—
303C3×Dic3—42no—
304C3×Dic3—42no—
305C25—31no—
306C22×D4—32no—
307C22×D4D3211yescharacteristic
308C23×C4—31no—
309#120—38no—
310C2×C4⋊C4—92no—
311C2×C4⋊C4—91no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—32no—
314C2×C22⋊C4—32no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—92no—
317C2×C22⋊C4—92no—
318C2×C22⋊C4—92no—
319C2×C22⋊C4—92no—
320C2×C22⋊C4—92no—
321C2×C22⋊C4—92no—
322C2×C42—92no—
323C2×C42—91no—
324#95—32no—
325#95—92no—
326#95—92no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—21no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—34no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3—62no—
356C23×C3—62no—
357C23×C3—62no—
358C23×C3—62no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—124no—
362C23×C3—128no—
363C23×C3—124no—
364C23×C3—124no—
365C23×C3—128no—
366C23×C3C2×C4×D312yes—
367C23×C3C2×C4×D312yes—
368C23×C3C22×Dic312yes—
369C23×C3C22×Dic312yes—
370C23×C3C23×D311yescharacteristic
371C23×C3C23×D311yescharacteristic
372C22×D3—22no—
373C22×D3—22no—
374C22×D3—22no—
375C22×D3—22no—
376C22×D3—22no—
377C22×D3—22no—
378C22×D3—48no—
379C22×D3—44no—
380C22×D3—44no—
381C22×D3—48no—
382C22×D3—44no—
383C22×D3C22×Dic314yes—
384C3×D4—44no—
385C3×D4—44no—
386C3×D4—48no—
387C3×D4—128no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—34no—
396C2×C3×C4—62no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—62no—
402C2×C3×C4—62no—
403C2×C3×C4—62no—
404C2×C3×C4—124no—
405C2×C3×C4C2×C4×D312yes—
406C2×C3×C4C22×Dic312yes—
407#67—48no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—44no—
417C2×Dic3—64no—
418C2×Dic3—62no—
419C2×Dic3—62no—
420C2×Dic3—62no—
421C2×Dic3—62no—
422C2×Dic3—62no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425C2×Dic3—62no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—62no—
430C2×Dic3—122no—
431C2×Dic3—122no—
432C2×Dic3—122no—
433C2×Dic3—122no—
434C2×Dic3—122no—
435C2×Dic3C22×Dic314yes—
436C2×C32—42no—
437C2×C32—42no—
438C2×C32#51811yescharacteristic
439C2×C32#51812yes—
440C2×C32#56011yescharacteristic
441C2×C32#56012yes—
442C2×C32H(3,𝔽4)11yescharacteristic
443C3×D3—44no—
444C24—32no—
445C24—34no—
446C24—122no—
447C24—122no—
448C24—121no—
449C24—121no—
450C24C2×D3211yescharacteristic
451C24C2×D3211yescharacteristic
452C2×D4—38no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4D3×Dic314yes—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—92no—
462C22×C4—92no—
463C22×C4—122no—
464C22×C4—121no—
465C22×C4C2×D3211yescharacteristic
466C4⋊C4—361no—
467C4⋊C4—362no—
468C22⋊C4—34no—
469C22⋊C4—38no—
470C22⋊C4—122no—
471C22⋊C4—121no—
472C22⋊C4—362no—
473C22⋊C4—362no—
474C22⋊C4—362no—
475C22⋊C4—362no—
476C22⋊C4—362no—
477C22⋊C4—362no—
478C42—361no—
479C42—362no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—24no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—24no—
486C22×C3—24no—
487C22×C3—24no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—44no—
496C22×C3—44no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—42no—
504C22×C3—41no—
505C22×C3—42no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—64no—
510C22×C3—64no—
511C22×C3—64no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—1216no—
519C22×C3—122no—
520C22×C3—122no—
521C22×C3—122no—
522C22×C3—122no—
523C22×C3C22×C4×D311yescharacteristic
524C22×C3#100412yes—
525C22×C3#100412yes—
526C22×C3#100412yes—
527C22×C3#100411yescharacteristic
528C22×C3#100411yescharacteristic
529C22×C3#100411yescharacteristic
530C22×C3#100411yescharacteristic
531C22×C3#100912yes—
532C22×C3C23×Dic311yescharacteristic
533C2×D3—24no—
534C2×D3—24no—
535C2×D3—24no—
536C2×D3—416no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C3×C4—24no—
542C3×C4—22no—
543C3×C4—22no—
544C3×C4—64no—
545C3×C4—122no—
546C3×C4—122no—
547Dic3—24no—
548Dic3—42no—
549Dic3—42no—
550Dic3—124no—
551Dic3—122no—
552Dic3—122no—
553Dic3—122no—
554Dic3—122no—
555Dic3—242no—
556C32#237511yescharacteristic, complemented, Sylow
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—34no—
566C23—62no—
567C23—62no—
568C23—62no—
569C23—62no—
570C23—62no—
571C23—62no—
572C23—128no—
573C23—124no—
574C23—124no—
575C23—128no—
576C23—124no—
577C23C2×D3×Dic312yes—
578C23C2×D3×Dic312yes—
579C23C22×D3211yescharacteristic
580D4—44no—
581D4—128no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—34no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—124no—
594C2×C4—182no—
595C2×C4—182no—
596C2×C4—182no—
597C2×C4—182no—
598C2×C4—182no—
599C2×C4—182no—
600C2×C4C2×D3×Dic312yes—
601C2×C3—22no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—22no—
605C2×C3—21no—
606C2×C3—42no—
607C2×C3—42no—
608C2×C3—42no—
609C2×C3—42no—
610C2×C3—82no—
611C2×C3—82no—
612C2×C3—124no—
613C2×C3#548611yescharacteristic
614C2×C3#554512yes—
615C2×C3#559411yescharacteristic
616C2×C3#559411yescharacteristic
617C2×C3#559512yes—
618C2×C3#559511yescharacteristic
619C2×C3#562012yes—
620C2×C3#566611yescharacteristic
621C2×C3#579812yes—
622C2×C3#579811yescharacteristic
623D3—44no—
624C22—24no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—64no—
635C22—64no—
636C22—64no—
637C22—1216no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22#6431312yes—
643C22#6431412yes—
644C22C22×D3×Dic311yescharacteristic
645C22#6433011yescharacteristic
646C22#6433011yescharacteristic
647C4—22no—
648C4—64no—
649C4—122no—
650C4—122no—
651C4—362no—
652C4—362no—
653C3—21no—
654C3#7488211yescharacteristic, complemented
655C3#8258011yescharacteristic, complemented
656C2—42no—
657C2—42no—
658C2—124no—
659C2#1059521512yes—
660C2#1059525611yescharacteristic
661C2#1059525812yes—
662C2#1059526111yescharacteristic
663C2#1059550611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC22×C32×D4C22
socle, cosocleC23×C32C24
prime radicalC2×C3×#937C2
hypercenterC22×D4D32
hypocenterC32#2375
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #10592920C2semidirect
3. C3×#5220C2semidirect
4. #10591710C2semidirect
5. #10591707C2semidirect
6. #10591706C2semidirect
7. #10591702C2semidirect
8. #10591316C2semidirect
9. #10591286C2semidirect
15. C23×C3×Dic3C22semidirect
17. C2×#3757C22semidirect
18. C2×C4⋊(C3⋊Dic3)C22semidirect
19. C2×C4×C3⋊Dic3C22semidirect
22. C2×C3×#217C22semidirect
23. C3×#937C4semidirect
25. C2×Dic3⋊Dic3C22semidirect
30. C2×Dic32C22semidirect
31. C3×C3⋊#95C22semidirect
32. #63579C22semidirect
33. #63579C22semidirect
43. #5220D3semidirect
59. C23×C3×D3C2×C4semidirect
81. C3×#217C2×C4semidirect
82. C3×(C2×D3)⋊C4C2×C4semidirect
98. C23×Dic3C2×D3semidirect
116. C2×#217C2×D3semidirect
117. #937Dic3semidirect
138. C3⋊#95C2×D3semidirect
208. C23×D3C2×Dic3semidirect
260. #217C2×Dic3semidirect
265. (C2×D3)⋊C4C2×Dic3semidirect
556. C32#2375semidirect
654. C3#74882semidirect
655. C3#82580semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9372
#105929202
C3×#52202
#105917102
#105917072
#105917062
#105917022
#105913162
#105912862
#825803
#748823