Finite Group Databasebeta

← #11025125

Subgroups — #11025125

Order 768.

2356 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic, complemented
2C2×D3×#125C211yesmaximal, characteristic, complemented
3#83192C211yesmaximal, characteristic, complemented
4#83180C211yesmaximal, characteristic, complemented
5#83023C211yesmaximal, characteristic, complemented
6#82897C214yesmaximal, complemented
7#82883C214yesmaximal, complemented
8#82695C214yesmaximal, complemented
9#82661C212yesmaximal, complemented
10#82641C214yesmaximal, complemented
11#82641C212yesmaximal, complemented
12#82399C212yesmaximal, complemented
13#82363C212yesmaximal, complemented
14#82268C211yesmaximal, characteristic, complemented
15#82265C211yesmaximal, characteristic, complemented
16#37706—31nomaximal, Sylow
17C3×#540—28no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C2×C3×#123C2214yes—
21C2×C3×#121C2214yes—
22C2×C3×#121C2212yes—
23C2×C3×C4×D4C2212yes—
24C2×C3×#117C2211yescharacteristic
25C2×#938C2212yes—
26C2×#938C2214yes—
27C2×#938C2214yes—
28C2×C4×#67C2211yescharacteristic
29C2×C4×#67C2214yes—
30D3×#125—28no—
31C2×D3×C4⋊C4C2214yes—
32C2×D3×C4⋊C4C2212yes—
33C2×#890C2212yes—
34C2×#889C2214yes—
35C2×C42×D3C2211yescharacteristic
36C2×#870C2211yescharacteristic
37#5222C2214yes—
38#5222C2214yes—
39#5222C2214yes—
40#5222C2214yes—
41#5221C2214yes—
42#5221C2214yes—
43#5221C2212yes—
44#5221C2212yes—
45#5221C2212yes—
46#5212C2212yes—
47#5212C2214yes—
48#5210C2214yes—
49#5210C2214yes—
50C4×#217C2211yescharacteristic
51#4989C2212yes—
52#4984C2212yes—
53#4984C2214yes—
54#4982C2214yes—
55#4980C2214yes—
56#4978C2212yes—
57Dic3×C4⋊C4C2212yes—
58#4956C2212yes—
59#4956C2214yes—
60#4956C2214yes—
61#4953C2214yes—
62#4949C2214yes—
63#4948C2214yes—
64#4948C2214yes—
65#4948C2214yes—
66#4945C2214yes—
67#4945C2214yes—
68#4944C2212yes—
69C4×(C2×D3)⋊C4C2211yescharacteristic
70#4935C2211yescharacteristic
71#4932C2211yescharacteristic
72C4×Dic3⋊C4C2211yescharacteristic
73#4928C2212yes—
74C2×#540D311yescharacteristic, complemented
75C22×#125—31no—
76#2856—31no—
77#2711—34no—
78#2707—34no—
79#2641—31no—
80#2629—31no—
81#2490—32no—
82#2454—32no—
83#2443—34no—
84#2416—32no—
85#2416—34no—
86#2361—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×D4C2312yes—
90C22×C3×D4C2314yes—
91C22×C3×D4C2314yes—
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2314yes—
94C22×#67C2311yescharacteristic
95C22×#67C2314yes—
96C22×C4×D3C2314yes—
97C22×C4×D3C2312yes—
98C22×C4×D3C2311yescharacteristic
99C3×#127—24no—
100C3×#125D418yescomplemented
101C3×#123—216no—
102C3×#121—216no—
103C3×#121—28no—
104C3×C4×D4—28no—
105C3×#117—24no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×#217C2311yescharacteristic
115C2×#217C2314yes—
116C2×#217C2312yes—
117C2×#217C2314yes—
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×#217C2314yes—
121C2×#217C2314yes—
122C2×#217C2314yes—
123C2×#217C2314yes—
124#938—28no—
125#938—48no—
126#938—48no—
127C4×#67—24no—
128C4×#67—48no—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2311yescharacteristic
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2311yescharacteristic
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147D3×C4⋊C4—216no—
148D3×C4⋊C4—28no—
149#890—28no—
150#889—216no—
151C42×D3—24no—
152#870—24no—
153C3⋊#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#540—28no—
163C2×#127C2×D311yescharacteristic
164C2×#125—34no—
165C2×#125—32no—
166C2×#125—31no—
167C2×#125—68no—
168C2×#125C2×D311yescharacteristic
169C2×#123C2×D314yes—
170C2×#121—32no—
171C2×#121—34no—
172C2×#121—34no—
173C2×#121C2×D314yes—
174C2×#121C2×D312yes—
175C2×C4×D4—31no—
176C2×C4×D4—34no—
177C2×C4×D4C2×D312yes—
178C2×#117C2×D311yescharacteristic
179C22×C4⋊C4—32no—
180C22×C4⋊C4—34no—
181C22×C42—31no—
182#399—34no—
183#398—34no—
184#397—34no—
185#396—34no—
186#396—34no—
187#396—34no—
188#394—34no—
189#394—34no—
190#394—34no—
191#394—34no—
192#390—34no—
193#390—32no—
194#390—32no—
195#390—32no—
196#390—34no—
197#389—34no—
198#389—32no—
199#389—32no—
200#388—34no—
201#388—32no—
202#388—34no—
203#387—34no—
204#387—32no—
205#386—34no—
206#386—32no—
207#386—34no—
208#385—34no—
209#385—34no—
210#385—32no—
211#384—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4—31no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218#376—31no—
219C24×C3C2411yescharacteristic
220C24×C3C2414yes—
221C23×D3C2411yescharacteristic
222C2×C3×D4—24no—
223C2×C3×D4—28no—
224C2×C3×D4—28no—
225C2×C3×D4—24no—
226C2×C3×D4—24no—
227C2×C3×D4—44no—
228C2×C3×D4—44no—
229C2×C3×D4—44no—
230C2×C3×D4—44no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—28no—
235C22×C3×C4—44no—
236C22×C3×C4—44no—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2412yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2414yes—
241C22×C3×C4C2411yescharacteristic
242C2×#67—22no—
243C2×#67—28no—
244C2×#67—28no—
245C2×#67—24no—
246C2×#67—28no—
247C22×Dic3C2412yes—
248C22×Dic3C2411yescharacteristic
249C22×Dic3C2411yescharacteristic
250C22×Dic3C2414yes—
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253C22×Dic3C2411yescharacteristic
254C2×C4×D3—22no—
255C2×C4×D3—22no—
256C2×C4×D3—28no—
257C2×C4×D3—24no—
258C2×C4×D3—24no—
259C2×C4×D3—28no—
260C2×C4×D3—42no—
261C2×C4×D3—44no—
262C2×C4×D3#4114yes—
263C3×C4⋊C4—24no—
264C3×C4⋊C4C2×D4116yes—
265C3×C4⋊C4C2×D418yes—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—28no—
268C3×C22⋊C4—28no—
269C3×C42—22no—
270C3×C42C2×D414yes—
271#217—22no—
272#217—28no—
273#217—44no—
274#217—44no—
275#217—44no—
276#217—44no—
277#217—44no—
278#217—44no—
279#217—42no—
280#217—44no—
281(C2×D3)⋊C4—22no—
282(C2×D3)⋊C4—24no—
283C4⋊Dic3—24no—
284C4⋊Dic3—28no—
285C4⋊Dic3—24no—
286C4⋊Dic3—44no—
287C4⋊Dic3—44no—
288Dic3⋊C4—28no—
289Dic3⋊C4—28no—
290Dic3⋊C4—28no—
291Dic3⋊C4—22no—
292Dic3⋊C4—42no—
293Dic3⋊C4—44no—
294C4×Dic3—24no—
295C4×Dic3—22no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C4×Dic3—44no—
299C22×D4—31no—
300C22×D4—34no—
301C22×D4C22×D312yes—
302C22×D4C22×D314yes—
303C22×D4C22×D314yes—
304C23×C4—31no—
305C23×C4—34no—
306C23×C4—32no—
307C23×C4C22×D311yescharacteristic
308C23×C4C22×D314yes—
309#127—24no—
310#125—616no—
311#125—68no—
312#125—64no—
313#125#6718yescomplemented
314#123—216no—
315#121—216no—
316#121—28no—
317#121—68no—
318#121—128no—
319#121—128no—
320C4×D4—28no—
321C4×D4—64no—
322C4×D4—128no—
323#117—24no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—68no—
336C2×C4⋊C4—616no—
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C4⋊C4C22×D314yes—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4C22×D312yes—
353C2×C22⋊C4C22×D314yes—
354C2×C22⋊C4C22×D314yes—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—32no—
358C2×C42—31no—
359C2×C42—34no—
360C2×C42—64no—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—32no—
366#95—32no—
367#95—32no—
368#95—34no—
369#95—34no—
370#95—34no—
371#95—34no—
372C23×C3—21no—
373C23×C3—24no—
374C23×C3—21no—
375C23×C3—24no—
376C23×C3—24no—
377C23×C3—22no—
378C23×C3—24no—
379C23×C3—21no—
380C23×C3—24no—
381C23×C3—24no—
382C23×C3—22no—
383C23×C3—24no—
384C23×C3C2511yescharacteristic
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—22no—
388C22×D3C2×#4112yes—
389C22×D3H(5,𝔽2)14yes—
390C3×D4—48no—
391C3×D4—88no—
392C3×D4—88no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—44no—
404C2×C3×C4—88no—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D418yes—
407C2×C3×C4C22×D412yes—
408C2×C3×C4C2×#4112yes—
409C2×C3×C42−1+412yes—
410#67—44no—
411#67—416no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—24no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—24no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—24no—
430C2×Dic3C2×#4112yes—
431C4×D3—28no—
432C4×D3—416no—
433C4×D3—48no—
434C24—31no—
435C24C23×D311yescharacteristic
436C24C23×D314yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—62no—
447C2×D4—68no—
448C2×D4—68no—
449C2×D4—64no—
450C2×D4—68no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—31no—
456C22×C4—34no—
457C22×C4—32no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—34no—
461C22×C4—32no—
462C22×C4—34no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—68no—
466C22×C4—64no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—64no—
470C22×C4—68no—
471C22×C4—124no—
472C22×C4—122no—
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D312yes—
475C22×C4C23×D312yes—
476C22×C4C23×D314yes—
477C22×C4C23×D311yescharacteristic
478C4⋊C4—24no—
479C4⋊C4—62no—
480C4⋊C4—64no—
481C4⋊C4—68no—
482C4⋊C4—68no—
483C4⋊C4—68no—
484C4⋊C4—68no—
485C4⋊C4—64no—
486C4⋊C4—124no—
487C4⋊C4—124no—
488C4⋊C4—124no—
489C4⋊C4—122no—
490C4⋊C4C2×#6718yes—
491C4⋊C4C2×#67116yes—
492C22⋊C4—24no—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—62no—
496C22⋊C4—64no—
497C22⋊C4—62no—
498C22⋊C4—68no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—122no—
506C22⋊C4—124no—
507C42—22no—
508C42—62no—
509C42—62no—
510C42—64no—
511C42—62no—
512C42—124no—
513C42C2×#6714yes—
514C22×C3—41no—
515C22×C3—44no—
516C22×C3—41no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—42no—
520C22×C3—44no—
521C22×C3—41no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3—42no—
525C22×C3—44no—
526C22×C3C23×D411yescharacteristic
527C22×C3C22×#4111yescharacteristic
528C22×C3C2×H(5,𝔽2)12yes—
529C22×C3C2×2−1+411yescharacteristic
530C22×C3#58512yes—
531C2×D3—22no—
532C2×D3—24no—
533C2×D3—22no—
534C2×D3—24no—
535C2×D3#54014yescomplemented
536C3×C4—28no—
537C3×C4—24no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4#54814yes—
541Dic3—22no—
542Dic3—41no—
543Dic3—44no—
544Dic3—42no—
545Dic3—44no—
546Dic3—41no—
547Dic3—42no—
548C23—21no—
549C23—24no—
550C23—21no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—24no—
558C23—21no—
559C23—24no—
560C23—32no—
561C23—34no—
562C23—61no—
563C23—62no—
564C23—61no—
565C23C24×D311yescharacteristic
566D4—48no—
567D4—88no—
568D4—88no—
569D4—124no—
570D4—1216no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—32no—
582C2×C4—44no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—68no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—64no—
602C2×C4—88no—
603C2×C4—1216no—
604C2×C4—128no—
605C2×C4D3×#4112yes—
606C2×C4#101012yes—
607C2×C4C22×#6714yes—
608C2×C4C22×#6718yes—
609C2×C4C22×#6712yes—
610C2×C3—81no—
611C2×C3—84no—
612C2×C3C2×#54011yescharacteristic
613C2×C3C2×#54811yescharacteristic
614C2×C3#348812yes—
615C2×C3#351211yescharacteristic
616C2×C3#355912yes—
617D3—24no—
618C22—34no—
619C22—41no—
620C22—44no—
621C22—41no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—41no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—62no—
632C22—64no—
633C22—62no—
634C22—64no—
635C22C2×D3×#4111yescharacteristic
636C22C2×#101011yescharacteristic
637C22D3×H(5,𝔽2)12yes—
638C22#596612yes—
639C22C23×#6711yescharacteristic
640C4—28no—
641C4—24no—
642C4—41no—
643C4—42no—
644C4—62no—
645C4—121no—
646C4—121no—
647C4—124no—
648C4—122no—
649C4—122no—
650C4—124no—
651C4#583114yes—
652C3#3770611yescharacteristic, complemented, Sylow
653C2—64no—
654C2—81no—
655C2—84no—
656C2#8646911yescharacteristic
657C2D3×#54011yescharacteristic
658C2#8649412yes—
659C2C2×#583111yescharacteristic
660C2#8728512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#540C2
hypercenterC2×#540D3
hypocenterC3#37706
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×C3×#540C2semidirect
2. C2×D3×#125C2semidirect
3. #83192C2semidirect
4. #83180C2semidirect
5. #83023C2semidirect
6. #82897C2semidirect
7. #82883C2semidirect
8. #82695C2semidirect
9. #82661C2semidirect
10. #82641C2semidirect
11. #82641C2semidirect
12. #82399C2semidirect
13. #82363C2semidirect
14. #82268C2semidirect
15. #82265C2semidirect
74. C2×#540D3semidirect
100. C3×#125D4semidirect
313. #125#67semidirect
535. C2×D3#540semidirect
652. C3#37706semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×D3×#1252
#831922
#831802
#830232
#828972
#828832
#826952
#826612
#826412
#823992
#823632
#822682
#822652
#377063