Finite Group Databasebeta

← #11069613

Subgroups — #11069613

Order 768.

2206 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic, complemented
2C2×#5698C211yesmaximal, characteristic, complemented
3#83189C211yesmaximal, characteristic, complemented
4#83184C211yesmaximal, characteristic, complemented
5Dic3×#125C211yesmaximal, characteristic, complemented
6#82876C214yesmaximal, complemented
7#82861C214yesmaximal, complemented
8#82696C214yesmaximal, complemented
9#82668C212yesmaximal, complemented
10#82661C214yesmaximal, complemented
11#82646C212yesmaximal, complemented
12#82408C212yesmaximal, complemented
13#82344C212yesmaximal, complemented
14#82262C211yesmaximal, characteristic, complemented
15#82225C211yesmaximal, characteristic, complemented
16#39382—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×#936C2212yes—
27C2×#936C2212yes—
28C2×C4×#67C2211yescharacteristic
29C2×#924C2212yes—
30#5698—28no—
31C2×#898C2214yes—
32C2×#897C2212yes—
33C2×#890C2214yes—
34C2×#887C2212yes—
35C2×#872C2211yescharacteristic
36C2×#869C2211yescharacteristic
37#5223C2214yes—
38#5223C2214yes—
39#5222C2214yes—
40#5222C2214yes—
41#5221C2214yes—
42#5221C2214yes—
43#5221C2212yes—
44#5221C2214yes—
45#5221C2214yes—
46#5213C2214yes—
47#5212C2214yes—
48#5210C2214yes—
49C4×#217C2211yescharacteristic
50C4×#217C2214yes—
51#4990C2212yes—
52#4984C2212yes—
53#4981C2214yes—
54#4980C2214yes—
55#4976C2212yes—
56Dic3×C4⋊C4C2212yes—
57Dic3×C4⋊C4C2214yes—
58#4961C2214yes—
59#4957C2212yes—
60#4957C2214yes—
61#4953C2214yes—
62#4953C2214yes—
63#4949C2214yes—
64#4948C2214yes—
65#4945C2212yes—
66#4945C2214yes—
67#4944C2214yes—
68#4944C2214yes—
69C4×(C2×D3)⋊C4C2211yescharacteristic
70#4936C2211yescharacteristic
71C4×Dic3⋊C4C2211yescharacteristic
72C42×Dic3C2211yescharacteristic
73#4929C2212yes—
74C2×#541—31no—
75C2×#540D311yescharacteristic, complemented
76#2858—31no—
77#2714—34no—
78#2694—34no—
79#2621—31no—
80#2619—31no—
81#2554—34no—
82#2540—32no—
83#2490—34no—
84#2427—32no—
85#2426—32no—
86#2367—32no—
87C4×#125—31no—
88#2293—31no—
89C22×C3×D4C2312yes—
90C22×C3×D4C2314yes—
91C22×C3×D4C2314yes—
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2314yes—
94C22×#67C2311yescharacteristic
95C22×#67C2314yes—
96C22×C4×D3C2311yescharacteristic
97C22×Dic6C2312yes—
98C3×#127—24no—
99C3×#125D418yescomplemented
100C3×#123—216no—
101C3×#121—216no—
102C3×#121—28no—
103C3×C4×D4—28no—
104C3×#117—24no—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×#217C2314yes—
114C2×#217C2311yescharacteristic
115C2×#217C2314yes—
116C2×#217C2314yes—
117C2×#217C2314yes—
118C2×#217C2314yes—
119C2×#217C2312yes—
120C2×#217C2314yes—
121C2×#217C2314yes—
122C2×#217C2314yes—
123#938—44no—
124#936—28no—
125#936—44no—
126C4×#67—24no—
127C2×(C2×D3)⋊C4C2311yescharacteristic
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2314yes—
135#924—44no—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2311yescharacteristic
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2311yescharacteristic
148#898—216no—
149#897—28no—
150#890—216no—
151#887—28no—
152#872—24no—
153#869—24no—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#541—68no—
163#540—28no—
164C2×C4⋊Q8—31no—
165C2×#127C2×D311yescharacteristic
166C2×#126—34no—
167C2×#125—34no—
168C2×#125C2×D311yescharacteristic
169C2×#124—32no—
170C2×#124—32no—
171C2×#123C2×D314yes—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—32no—
174C2×#121—32no—
175C2×#121C2×D312yes—
176C2×#121C2×D314yes—
177C2×C4×Q8—32no—
178C2×C4×D4—31no—
179C2×C4×D4C2×D312yes—
180C2×#117—31no—
181C2×#117C2×D311yescharacteristic
182#399—34no—
183#398—34no—
184#397—34no—
185#397—34no—
186#396—34no—
187#394—34no—
188#394—34no—
189#394—34no—
190#392—34no—
191#392—34no—
192#391—32no—
193#390—34no—
194#390—34no—
195#390—32no—
196#390—34no—
197#390—34no—
198#390—32no—
199#390—34no—
200#389—32no—
201#389—32no—
202#388—32no—
203#388—34no—
204#388—34no—
205#388—34no—
206#388—32no—
207#387—34no—
208#386—34no—
209#385—34no—
210#382—31no—
211#380—34no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—34no—
214C4×C4⋊C4—31no—
215C4×C22⋊C4—34no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218C43—31no—
219C24×C3C2414yes—
220C24×C3C2411yescharacteristic
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×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2412yes—
240C22×C3×C4C2414yes—
241C22×C3×C4C2411yescharacteristic
242C2×#67—22no—
243C2×#67—42no—
244C2×#67—44no—
245C2×#67—44no—
246C2×#67—44no—
247C22×Dic3C2414yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2414yes—
251C22×Dic3C2411yescharacteristic
252C22×Dic3C2411yescharacteristic
253C22×Dic3C2411yescharacteristic
254C2×C4×D3—22no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4C2×D418yes—
262C3×C4⋊C4C2×D4116yes—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—28no—
265C3×C22⋊C4—28no—
266C3×C42—22no—
267C3×C42C2×D414yes—
268#217—22no—
269#217—28no—
270#217—28no—
271#217—44no—
272#217—44no—
273#217—44no—
274#217—44no—
275#217—42no—
276#217—44no—
277#217—44no—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—28no—
280(C2×D3)⋊C4—22no—
281(C2×D3)⋊C4—42no—
282C4⋊Dic3—28no—
283C4⋊Dic3—28no—
284C4⋊Dic3—24no—
285C4⋊Dic3—44no—
286Dic3⋊C4—24no—
287Dic3⋊C4—28no—
288Dic3⋊C4—22no—
289Dic3⋊C4—42no—
290Dic3⋊C4—44no—
291C4×Dic3—22no—
292C4×Dic3—28no—
293C4×Dic3—22no—
294C4×Dic3—42no—
295C4×Dic3—44no—
296C4×Dic3—42no—
297C4×Dic3#4114yes—
298C22×Q8—32no—
299C22×D4—31no—
300C22×D4—34no—
301C22×D4C22×D312yes—
302C22×D4C22×D314yes—
303C22×D4C22×D314yes—
304C23×C4—31no—
305C23×C4C22×D311yescharacteristic
306C23×C4C22×D314yes—
307C4⋊Q8—64no—
308#127—24no—
309#126—616no—
310#125—616no—
311#125#6718yescomplemented
312#124—68no—
313#124—124no—
314#123—216no—
315C22⋊Q8—68no—
316C22⋊Q8—124no—
317#121—216no—
318#121—28no—
319#121—124no—
320C4×Q8—68no—
321C4×D4—28no—
322C4×D4—64no—
323#117—24no—
324#117—64no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D314yes—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—31no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4C22×D312yes—
352C2×C22⋊C4C22×D314yes—
353C2×C22⋊C4C22×D314yes—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357C2×C42—31no—
358C2×C42—32no—
359C2×C42—32no—
360C2×C42—34no—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—34no—
366#95—32no—
367#95—34no—
368#95—34no—
369#95—34no—
370#95—34no—
371C23×C3—21no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—24no—
377C23×C3—24no—
378C23×C3—24no—
379C23×C3—22no—
380C23×C3—24no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3C2511yescharacteristic
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—22no—
388C22×D3—22no—
389C3×D4—48no—
390C3×D4—88no—
391C3×D4—88no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—44no—
403C2×C3×C4—88no—
404C2×C3×C4C22×D418yes—
405C2×C3×C4C22×D414yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2×#4112yes—
408C2×C3×C4H(5,𝔽2)12yes—
409#67—44no—
410#67—88no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—24no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—24no—
420C2×Dic3—24no—
421C2×Dic3—21no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3—22no—
428C2×Dic3C2×#4112yes—
429C2×Dic3C2×#4112yes—
430C4×D3—44no—
431Dic6—48no—
432C24—31no—
433C24C23×D314yes—
434C24C23×D311yescharacteristic
435C2×Q8—64no—
436C2×Q8—64no—
437C2×Q8—64no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—62no—
448C2×D4—122no—
449C2×D4—124no—
450C2×D4—124no—
451C2×D4—124no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—28no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—34no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—34no—
462C22×C4—31no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4C23×D312yes—
469C22×C4C23×D311yescharacteristic
470C22×C4C23×D312yes—
471C22×C4C23×D314yes—
472C22×C4C23×D311yescharacteristic
473C4⋊C4—24no—
474C4⋊C4—62no—
475C4⋊C4—64no—
476C4⋊C4—64no—
477C4⋊C4—68no—
478C4⋊C4—68no—
479C4⋊C4—68no—
480C4⋊C4—124no—
481C4⋊C4—122no—
482C4⋊C4—124no—
483C4⋊C4C2×#6718yes—
484C4⋊C4C2×#67116yes—
485C22⋊C4—24no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—62no—
489C22⋊C4—62no—
490C22⋊C4—68no—
491C22⋊C4—68no—
492C22⋊C4—64no—
493C22⋊C4—68no—
494C22⋊C4—122no—
495C22⋊C4—124no—
496C22⋊C4—124no—
497C22⋊C4—124no—
498C22⋊C4—124no—
499C22⋊C4—122no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C42—22no—
503C42—34no—
504C42—62no—
505C42—62no—
506C42—68no—
507C42—124no—
508C42—122no—
509C42—122no—
510C42C2×#6714yes—
511C22×C3—41no—
512C22×C3—44no—
513C22×C3—41no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—42no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—41no—
520C22×C3—44no—
521C22×C3—42no—
522C22×C3—44no—
523C22×C3C23×D411yescharacteristic
524C22×C3C22×#4111yescharacteristic
525C22×C3C2×H(5,𝔽2)11yescharacteristic
526C22×C3#58512yes—
527C22×C3#58512yes—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—42no—
531C2×D3—41no—
532C2×D3—42no—
533C3×C4—28no—
534C3×C4—24no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4#55014yes—
538Dic3—22no—
539Dic3—22no—
540Dic3—44no—
541Dic3—41no—
542Dic3—42no—
543Dic3—42no—
544Dic3—44no—
545C23—24no—
546C23—21no—
547C23—21no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—21no—
556C23—24no—
557C23—61no—
558C23—61no—
559C23—62no—
560C23—62no—
561C23—61no—
562C23C24×D311yescharacteristic
563Q8—128no—
564D4—48no—
565D4—88no—
566D4—88no—
567D4—124no—
568D4—248no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—44no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—61no—
586C2×C4—62no—
587C2×C4—61no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—61no—
599C2×C4—88no—
600C2×C4—124no—
601C2×C4D3×#4112yes—
602C2×C4#100912yes—
603C2×C4C22×#6718yes—
604C2×C4C22×#6714yes—
605C2×C4C22×#6712yes—
606C2×C3—81no—
607C2×C3—84no—
608C2×C3#345511yescharacteristic
609C2×C3C2×#55011yescharacteristic
610C2×C3#350012yes—
611C2×C3#351211yescharacteristic
612C2×C3#356212yes—
613D3—81no—
614C22—41no—
615C22—44no—
616C22—44no—
617C22—41no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—41no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—121no—
627C22—121no—
628C22—122no—
629C22—121no—
630C22—122no—
631C22C2×D3×#4111yescharacteristic
632C22C2×#100911yescharacteristic
633C22#596612yes—
634C22#596612yes—
635C22C23×#6711yescharacteristic
636C4—28no—
637C4—24no—
638C4—41no—
639C4—42no—
640C4—62no—
641C4—62no—
642C4—124no—
643C4—122no—
644C4—121no—
645C4—124no—
646C4—122no—
647C4#583414yes—
648C3#3938211yescharacteristic, complemented, Sylow
649C2—81no—
650C2—84no—
651C2—241no—
652C2#8646711yescharacteristic
653C2#8646911yescharacteristic
654C2#8648012yes—
655C2C2×#583411yescharacteristic
656C2#8728312yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#540C2
hypercenterC2×#540D3
hypocenterC3#39382
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×#5698C2semidirect
3. #83189C2semidirect
4. #83184C2semidirect
5. Dic3×#125C2semidirect
6. #82876C2semidirect
7. #82861C2semidirect
8. #82696C2semidirect
9. #82668C2semidirect
10. #82661C2semidirect
11. #82646C2semidirect
12. #82408C2semidirect
13. #82344C2semidirect
14. #82262C2semidirect
15. #82225C2semidirect
75. C2×#540D3semidirect
99. C3×#125D4semidirect
311. #125#67semidirect
648. C3#39382semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×#56982
#831892
#831842
Dic3×#1252
#828762
#828612
#826962
#826682
#826612
#826462
#824082
#823442
#822622
#822252
#393823