Finite Group Databasebeta

← #11234165

Subgroups — #11234165

Order 768.

2118 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
1C3×#2408C211yesmaximal, characteristic
2D4×#217C212yesmaximal, complemented
3#82643C212yesmaximal, complemented
4#82636C212yesmaximal, complemented
5#82585C214yesmaximal, complemented
6#82373C211yesmaximal, characteristic, complemented
7#82348C211yesmaximal, characteristic, complemented
8#77053C214yesmaximal, complemented
9#76737C212yesmaximal, complemented
10#76693C212yesmaximal, complemented
11#76364C212yesmaximal, complemented
12#76295C212yesmaximal, complemented
13#73919C212yesmaximal, complemented
14#73843C212yesmaximal, complemented
15#73833C212yesmaximal, complemented
16#45380—31nomaximal, Sylow
17C23×C3×D4C2211yescharacteristic
18C2×C3×#121C2212yes—
19C2×C3×#120C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C22×#217C2214yes—
22C2×#935C2212yes—
23C2×D4×Dic3C2212yes—
24C2×D4×Dic3C2212yes—
25C2×D4×Dic3C2214yescomplemented
26C22×C4⋊Dic3C2211yescharacteristic, complemented
27C2×C4×Dic6C2211yescharacteristic, complemented
28C3×#394C2212yes—
29C3×#392C2212yes—
30C3×#390C2212yes—
31C3×#380C2211yescharacteristic
32C3×#379C2212yes—
33#5247C2214yes—
34#5222C2212yes—
35#5222C2214yescomplemented
36#5221C2212yes—
37#5221C2212yes—
38#5220C2214yes—
39#5220C2214yes—
40#5220C2214yescomplemented
41#5220C2214yescomplemented
42#5212C2212yes—
43#5212C2211yescharacteristic, complemented
44C4×#217C2212yes—
45C4×#217C2212yescomplemented
46#4985C2212yescomplemented
47#4984C2212yescomplemented
48#4981C2212yescomplemented
49#4977C2212yescomplemented
50Dic3×C4⋊C4C2212yescomplemented
51#4972C2212yescomplemented
52#4953C2211yescharacteristic
53#4953C2212yes—
54#4953C2214yes—
55#4953C2211yescharacteristic, complemented
56#4952C2214yescomplemented
57#4951C2214yescomplemented
58#4950C2214yescomplemented
59#4950C2214yescomplemented
60#4949C2214yescomplemented
61#4947C2211yescharacteristic
62#4947C2214yescomplemented
63#4947C2211yescharacteristic, complemented
64#4946C2214yescomplemented
65#4945C2214yescomplemented
66#4944C2214yes—
67#4944C2214yescomplemented
68#4943C2214yescomplemented
69#4943C2214yescomplemented
70#4943C2212yescomplemented
71#4942C2214yescomplemented
72#4932C2211yescharacteristic, complemented
73#4928C2212yescomplemented
74C3⋊#401C2214yescomplemented
75#4655C2214yescomplemented
76#2760—32no—
77#2746—34no—
78#2736—32no—
79#2600—32no—
80#2580—32no—
81#2494—32no—
82#2486—31no—
83#2483—32no—
84#2420—32no—
85#2418—34no—
86#2408D311yescharacteristic
87#2398—31no—
88#2361—32no—
89D4×C22⋊C4—32no—
90#2319—32no—
91C25×C3C2312yes—
92C22×C3×D4—28no—
93C22×C3×D4—42no—
94C22×C3×D4C2312yes—
95C22×C3×D4C2314yes—
96C22×C3×D4C2311yescharacteristic
97C22×C3×D4C2312yes—
98C23×C3×C4C2311yescharacteristic
99C23×C3×C4C2311yescharacteristic
100C23×Dic3C2314yes—
101C23×Dic3C2312yes—
102C23×Dic3C2314yes—
103C22×Dic6C2311yescharacteristic
104C3×#121—28no—
105C3×#120—44no—
106C3×C4×D4—42no—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C22⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×#217—48no—
117C2×#217C2314yes—
118C2×#217C2314yes—
119C2×#217C2312yes—
120C2×#217C2312yes—
121C2×#217C2314yes—
122C2×#217C2314yes—
123C2×#217C2314yes—
124C2×#217C2312yes—
125C2×#217C2312yes—
126C2×#217C2314yes—
127C2×#217C2314yes—
128C2×#217C2312yes—
129#935—44no—
130D4×Dic3—28no—
131D4×Dic3—44no—
132D4×Dic3—48no—
133C2×C4⋊Dic3—42no—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2314yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2314yes—
145C4×Dic6—42no—
146C3×#95C2312yes—
147C3⋊#95C2312yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2311yescharacteristic
158C3⋊#95C2311yescharacteristic
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C23×D4C2×D311yescharacteristic
163C2×#123—32no—
164C2×#121C2×D312yes—
165C2×#120C2×D312yes—
166C2×C4×Q8—31no—
167C2×C4×D4—32no—
168C2×C4×D4—32no—
169C2×C4×D4—34no—
170C2×C4×D4C2×D311yescharacteristic
171C22×C4⋊C4—31no—
172C22×C22⋊C4—34no—
173#401—34no—
174#400—32no—
175#399—34no—
176#398—32no—
177#397—31no—
178#397—34no—
179#397—31no—
180#397—34no—
181#397—32no—
182#394—34no—
183#394—32no—
184#394C2×D312yes—
185#393—34no—
186#393—31no—
187#393—31no—
188#392C2×D312yes—
189#390—32no—
190#390—32no—
191#390C2×D312yes—
192#389—32no—
193#389—32no—
194#389—32no—
195#388—34no—
196#388—34no—
197#388—34no—
198#388—34no—
199#387—34no—
200#386—34no—
201#386—34no—
202#386—34no—
203#386—34no—
204#385—34no—
205#385—34no—
206#385—34no—
207#382—32no—
208#381—34no—
209#381—32no—
210#381—34no—
211#380—32no—
212#380—31no—
213#380—34no—
214#380C2×D311yescharacteristic
215#379—34no—
216#379C2×D312yes—
217C4×C4⋊C4—32no—
218C4×C22⋊C4—32no—
219C4×C22⋊C4—32no—
220#376—31no—
221C24×C3—28no—
222C24×C3—24no—
223C24×C3—44no—
224C24×C3—42no—
225C24×C3—42no—
226C24×C3C2414yes—
227C24×C3C2411yescharacteristic
228C24×C3C2411yescharacteristic
229C2×C3×D4—28no—
230C2×C3×D4—28no—
231C2×C3×D4—28no—
232C2×C3×D4—22no—
233C2×C3×D4—22no—
234C2×C3×D4—24no—
235C2×C3×D4—24no—
236C2×C3×D4—41no—
237C2×C3×D4—42no—
238C2×C3×D4—44no—
239C2×C3×D4—416no—
240C2×C3×D4#4118yes—
241C22×C3×C4—24no—
242C22×C3×C4—24no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2414yes—
249C22×C3×C4C2411yescharacteristic
250C22×C3×C4C2411yescharacteristic
251C22×Dic3—28no—
252C22×Dic3—24no—
253C22×Dic3—42no—
254C22×Dic3—44no—
255C22×Dic3—44no—
256C22×Dic3—42no—
257C22×Dic3—44no—
258C22×Dic3—44no—
259C22×Dic3—44no—
260C22×Dic3C2412yes—
261C22×Dic3C2412yes—
262C22×Dic3C2414yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2414yes—
266C2×Dic6—41no—
267C2×Dic6—42no—
268C3×C4⋊C4—24no—
269C3×C4⋊C4—41no—
270C3×C22⋊C4—28no—
271C3×C22⋊C4—44no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—41no—
274C3×C22⋊C4—42no—
275C3×C22⋊C4—44no—
276C3×C42—22no—
277#217—28no—
278#217—24no—
279#217—28no—
280#217—24no—
281#217—44no—
282#217—44no—
283#217—42no—
284#217—44no—
285#217—44no—
286#217—42no—
287#217—44no—
288#217—42no—
289C4⋊Dic3—24no—
290C4⋊Dic3—24no—
291C4⋊Dic3—41no—
292C4⋊Dic3—42no—
293C4⋊Dic3—44no—
294Dic3⋊C4—42no—
295Dic3⋊C4—44no—
296Dic3⋊C4—44no—
297C4×Dic3—24no—
298C4×Dic3—24no—
299C4×Dic3—44no—
300C25C22×D312yes—
301C22×Q8—31no—
302C22×D4—28no—
303C22×D4—42no—
304C22×D4C22×D314yes—
305C22×D4C22×D311yescharacteristic
306C22×D4C22×D312yes—
307C22×D4C22×D312yes—
308C23×C4—34no—
309C23×C4—32no—
310C23×C4—34no—
311C23×C4C22×D311yescharacteristic
312C23×C4C22×D311yescharacteristic
313#123—124no—
314#121—28no—
315#120—44no—
316C4×Q8—122no—
317C4×D4—42no—
318C4×D4—68no—
319C4×D4—124no—
320C4×D4—128no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—122no—
330C2×C4⋊C4C22×D312yes—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—32no—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—34no—
336C2×C22⋊C4—34no—
337C2×C22⋊C4—34no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—128no—
345C2×C22⋊C4C22×D314yes—
346C2×C22⋊C4C22×D311yescharacteristic
347C2×C22⋊C4C22×D311yescharacteristic
348C2×C22⋊C4C22×D312yes—
349C2×C22⋊C4C22×D314yes—
350C2×C22⋊C4C22×D314yes—
351C2×C42—32no—
352C2×C42—32no—
353C2×C42—34no—
354C2×C42C22×D311yescharacteristic
355#95—32no—
356#95—31no—
357#95—31no—
358#95—32no—
359#95—32no—
360#95—32no—
361#95—32no—
362#95—34no—
363#95—32no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95—34no—
368#95—34no—
369#95—34no—
370#95C22×D312yes—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—24no—
374C23×C3—21no—
375C23×C3—24no—
376C23×C3—24no—
377C23×C3—24no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3—22no—
384C23×C3—24no—
385C23×C3—48no—
386C23×C3—416no—
387C23×C3—88no—
388C23×C3—84no—
389C23×C3—84no—
390C23×C3C2×#4118yes—
391C23×C3H(5,𝔽2)12yes—
392C23×C3H(5,𝔽2)12yes—
393C23×C3C2511yescharacteristic
394C3×D4—216no—
395C3×D4—416no—
396C3×D4—44no—
397C3×D4—48no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—24no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—44no—
411C2×C3×C4—44no—
412C2×C3×C4C2×#4114yes—
413C2×C3×C4H(5,𝔽2)12yes—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—24no—
421C2×Dic3—24no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—24no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—88no—
433C2×Dic3—84no—
434C2×Dic3—88no—
435Dic6—44no—
436C24—28no—
437C24—24no—
438C24—42no—
439C24—44no—
440C24—42no—
441C24C23×D314yes—
442C24C23×D311yescharacteristic
443C24C23×D311yescharacteristic
444C2×Q8—121no—
445C2×Q8—122no—
446C2×D4—28no—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—41no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—416no—
457C2×D4#23718yes—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—32no—
463C22×C4—32no—
464C22×C4—34no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—34no—
468C22×C4—68no—
469C22×C4—64no—
470C22×C4—124no—
471C22×C4—124no—
472C22×C4—122no—
473C22×C4—124no—
474C22×C4—124no—
475C22×C4—122no—
476C22×C4—124no—
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D311yescharacteristic
479C22×C4C23×D311yescharacteristic
480C22×C4C23×D314yes—
481C22×C4C23×D311yescharacteristic
482C22×C4C23×D311yescharacteristic
483C4⋊C4—24no—
484C4⋊C4—41no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—122no—
488C4⋊C4—124no—
489C4⋊C4—122no—
490C4⋊C4—121no—
491C4⋊C4—124no—
492C4⋊C4—124no—
493C22⋊C4—28no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—68no—
500C22⋊C4—64no—
501C22⋊C4—64no—
502C22⋊C4—68no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—124no—
506C22⋊C4—124no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C22⋊C4—124no—
510C22⋊C4—122no—
511C42—22no—
512C42—64no—
513C42—64no—
514C42—124no—
515C22×C3—28no—
516C22×C3—28no—
517C22×C3—24no—
518C22×C3—28no—
519C22×C3—24no—
520C22×C3—22no—
521C22×C3—22no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—42no—
525C22×C3—44no—
526C22×C3—44no—
527C22×C3—42no—
528C22×C3—44no—
529C22×C3—41no—
530C22×C3—44no—
531C22×C3—41no—
532C22×C3—816no—
533C22×C3C22×#4112yes—
534C22×C3C2×H(5,𝔽2)11yescharacteristic
535C22×C3C2×H(5,𝔽2)11yescharacteristic
536C22×C3C2×H(5,𝔽2)11yescharacteristic
537C22×C3#58512yes—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—44no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4#55114yes—
544Dic3—42no—
545Dic3—44no—
546Dic3—42no—
547Dic3—42no—
548Dic3—44no—
549Dic3—42no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—21no—
554C23—24no—
555C23—21no—
556C23—21no—
557C23—24no—
558C23—21no—
559C23—24no—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—48no—
565C23—416no—
566C23—88no—
567C23—84no—
568C23—84no—
569C23C2×#23718yes—
570C23#100412yes—
571C23#100912yes—
572C23C24×D311yescharacteristic
573Q8—124no—
574D4—216no—
575D4—48no—
576D4—416no—
577D4—44no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—248no—
611C2×C4—244no—
612C2×C4—248no—
613C2×C4C2×#23714yes—
614C2×C4#100412yes—
615C2×C3—42no—
616C2×C3—48no—
617C2×C3—42no—
618C2×C3C2×#55111yescharacteristic
619C2×C3#351312yes—
620C2×C3#351312yes—
621C2×C3#353711yescharacteristic
622C2×C3#354011yescharacteristic
623C22—24no—
624C22—28no—
625C22—28no—
626C22—24no—
627C22—28no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—44no—
632C22—41no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—44no—
640C22—816no—
641C22C22×#23712yes—
642C22C2×#100411yescharacteristic
643C22C2×#100411yescharacteristic
644C22C2×#100911yescharacteristic
645C22#596612yes—
646C4—41no—
647C4—44no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—122no—
652C4—122no—
653C4—122no—
654C4—124no—
655C4—122no—
656C4—124no—
657C4#558714yes—
658C3#4538011yescharacteristic, complemented, Sylow
659C2—42no—
660C2—48no—
661C2—42no—
662C2C2×#558711yescharacteristic
663C2#8593811yescharacteristic
664C2#8632211yescharacteristic
665C2#8660912yes—
666C2#8663412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2408C2
hypercenter#2408D3
hypocenterC3#45380
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. D4×#217C2semidirect
3. #82643C2semidirect
4. #82636C2semidirect
5. #82585C2semidirect
6. #82373C2semidirect
7. #82348C2semidirect
8. #77053C2semidirect
9. #76737C2semidirect
10. #76693C2semidirect
11. #76364C2semidirect
12. #76295C2semidirect
13. #73919C2semidirect
14. #73843C2semidirect
15. #73833C2semidirect
25. C2×D4×Dic3C22semidirect
26. C22×C4⋊Dic3C22semidirect
27. C2×C4×Dic6C22semidirect
35. #5222C22semidirect
40. #5220C22semidirect
41. #5220C22semidirect
43. #5212C22semidirect
45. C4×#217C22semidirect
46. #4985C22semidirect
47. #4984C22semidirect
48. #4981C22semidirect
49. #4977C22semidirect
50. Dic3×C4⋊C4C22semidirect
51. #4972C22semidirect
55. #4953C22semidirect
56. #4952C22semidirect
57. #4951C22semidirect
58. #4950C22semidirect
59. #4950C22semidirect
60. #4949C22semidirect
62. #4947C22semidirect
63. #4947C22semidirect
64. #4946C22semidirect
65. #4945C22semidirect
67. #4944C22semidirect
68. #4943C22semidirect
69. #4943C22semidirect
70. #4943C22semidirect
71. #4942C22semidirect
72. #4932C22semidirect
73. #4928C22semidirect
74. C3⋊#401C22semidirect
75. #4655C22semidirect
658. C3#45380semidirect

Maximal subgroups

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

subgroupindex
C3×#24082
D4×#2172
#826432
#826362
#825852
#823732
#823482
#770532
#767372
#766932
#763642
#762952
#739192
#738432
#738332
#453803