Finite Group Databasebeta

← #10827375

Subgroups — #10827375

Order 768.

1500 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×#5514C211yesmaximal, characteristic, complemented
2C22×C3×#102C211yesmaximal, characteristic, complemented
3#84053C211yesmaximal, characteristic, complemented
4C2×#5217C211yesmaximal, characteristic, complemented
5#71975C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#71554C211yesmaximal, characteristic, complemented
8#70208C212yesmaximal, complemented
9#70204C212yesmaximal, complemented
10#70201C212yesmaximal, complemented
11#70164C212yesmaximal, complemented
12#23963—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#953—21no—
17C2×#938C2212yes—
18C22×#213C2212yes—
19C2×#930—21no—
20C2×#930C2211yescharacteristic
21C2×C4×#67—21no—
22C2×C4×#67C2211yescharacteristic
23#5514—24no—
24#5514—24no—
25C2×#898—21no—
26C2×#895C2211yescharacteristic, complemented
27C2×#894—21no—
28C2×#893C2211yescharacteristic, complemented
29C2×#892C2211yescharacteristic, complemented
30C2×#890C2211yescharacteristic, complemented
31C2×C3×#102—24no—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34#5247—21no—
35#5223—21no—
36#5222—21no—
37#5221C2212yes—
38#5217—24no—
39C2×#832C2212yes—
40#5212C2211yescharacteristic
41C2×#830C2211yescharacteristic
42C2×#821C2212yescomplemented
43C2×(C2×D3)⋊C8C2212yescomplemented
44#4551C2212yescomplemented
45#4550C2212yescomplemented
46#4529C2212yescomplemented
47#4529C2212yescomplemented
48C2×#553—31no—
49C2×#447—31no—
50C22×#102D311yescharacteristic, complemented
51#2408—31no—
52#2025—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1875—32no—
57#1869—31no—
58#1824—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—42no—
65C22×C3×D4C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C22×#67—21no—
69C22×#67—22no—
70C22×#67D411yescharacteristic
71C22×D12—21no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3C2311yescharacteristic
75C22×C3×C8—22no—
76C22×C3×C8—42no—
77C22×C3×C8C2312yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D414yescomplemented
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83#953—44no—
84C2×#217—21no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217D411yescharacteristic
89#938—24no—
90#938—42no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—42no—
94#930—28no—
95#930—22no—
96#930D414yescomplemented
97C4×#67—28no—
98C4×#67—22no—
99C4×#67—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2312yes—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113#898—28no—
114#895—22no—
115#895—22no—
116#894—28no—
117#893—22no—
118#893—22no—
119#892—22no—
120#892—22no—
121#890—22no—
122#890—22no—
123C3×#102—24no—
124C3×#102—24no—
125#832—24no—
126C3⋊#95—21no—
127#830—22no—
128#821—42no—
129(C2×D3)⋊C8—42no—
130C23×D4C2×D311yescharacteristic
131C22×D8—32no—
132C23×C8C2×D311yescharacteristic
133#553—64no—
134#553—64no—
135C2×#126—61no—
136C2×#125—31no—
137C2×#123—61no—
138C2×#121—31no—
139C2×#121—32no—
140C2×#121—31no—
141C2×#121—61no—
142C2×#120—61no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×#117—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148#447—64no—
149C2×#102—24no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102C2×D312yes—
153C2×#102C2×D312yes—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#394—61no—
157#392—61no—
158#390—32no—
159#380—31no—
160#379—61no—
161#340—32no—
162#340—32no—
163#340—32no—
164#336—32no—
165C24×C3—22no—
166C24×C3—44no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—82no—
171C24×C3C2×D411yescharacteristic
172C23×D3—21no—
173C23×D3C2×D411yescharacteristic
174C2×C3×D4—28no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—48no—
181C2×C3×D4—44no—
182C2×C3×D4—44no—
183C2×C3×D4—42no—
184C2×C3×D4#4114yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—24no—
190C22×C3×C4—82no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2411yescharacteristic
195C2×#67—22no—
196C2×#67—24no—
197C2×#67—22no—
198C2×#67—22no—
199C2×#67—42no—
200C2×#67—42no—
201C2×#67—44no—
202C2×#67—44no—
203C2×#67—82no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—22no—
210C2×D12—22no—
211C2×D12—21no—
212C2×D12—24no—
213C2×D12—82no—
214C2×D12C2×D412yes—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—24no—
220C2×C4×D3—82no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—44no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D412yes—
230#217—22no—
231#217—44no—
232#217—44no—
233#217—42no—
234#217—82no—
235#213—48no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—24no—
241C4⋊Dic3—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3—22no—
244C4⋊Dic3—22no—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—22no—
248C4×Dic3—24no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C25—21no—
255C22×D4—24no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—31no—
260C22×D4—31no—
261C22×D4—42no—
262C22×D4—61no—
263C22×D4—61no—
264C22×D4—62no—
265C22×D4C22×D312yes—
266C23×C4—21no—
267C23×C4—31no—
268C23×C4—61no—
269C23×C4C22×D311yescharacteristic
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—122no—
273C22×C8—22no—
274C22×C8—32no—
275C22×C8—42no—
276C22×C8C22×D312yes—
277#126—68no—
278#125—62no—
279#125—62no—
280#123—68no—
281#121—34no—
282#121—62no—
283#121—68no—
284#121—62no—
285#121—62no—
286#121—64no—
287#121—122no—
288#120—124no—
289C4×D4—62no—
290C4×D4—68no—
291C4×D4—62no—
292C4×D4—62no—
293C4×D4—62no—
294#117—62no—
295#117—62no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6714yescomplemented
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—62no—
316C2×C42—31no—
317C2×C42—61no—
318#102—24no—
319#102—24no—
320#102—64no—
321#102—122no—
322C22⋊C8—62no—
323C22⋊C8—122no—
324#95—61no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—24no—
328C23×C3—48no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—88no—
336C23×C3—88no—
337C23×C3—84no—
338C23×C3—84no—
339C23×C3#12012yes—
340C23×C3#13512yes—
341C23×C3AGL(1,ℤ/8)12yes—
342C23×C3C22×D411yescharacteristic
343C22×D3—21no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—22no—
349C22×D3—42no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—42no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—48no—
356C3×D4—42no—
357C3×D4—42no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—84no—
367C2×C3×C4#12012yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4112yes—
371C2×C3×C4H(5,𝔽2)12yes—
372#67—44no—
373#67—48no—
374#67—88no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—22no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388D12—24no—
389D12—41no—
390D12—41no—
391D12—84no—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—42no—
395C4×D3—84no—
396C3×C8—42no—
397C3⋊C8—42no—
398C24—22no—
399C24—31no—
400C24—44no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—61no—
405C24—82no—
406C24D3×D411yescharacteristic
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—32no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—64no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—64no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—61no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4—124no—
429C2×D4—124no—
430C2×D4—242no—
431C2×D4—242no—
432C2×D4#23514yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—64no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—82no—
449C22×C4—242no—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D412yes—
452C22×C4C2×#6712yes—
453C22×C4C23×D311yescharacteristic
454D8—128no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—44no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—62no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—32no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4C2×#6712yes—
473C4⋊C4C2×#6712yes—
474C22⋊C4—64no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—64no—
480C22⋊C4—124no—
481C22⋊C4—124no—
482C22⋊C4—122no—
483C22⋊C4—242no—
484C42—64no—
485C42—61no—
486C42—61no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—816no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3#44914yes—
500C22×C3C2×#12011yescharacteristic
501C22×C3#54612yes—
502C22×C3C2×#13511yescharacteristic
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3#57612yes—
505C2×D3—41no—
506C2×D3—42no—
507C2×D3—41no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—41no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—82no—
514C2×D3—82no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4#52511yescharacteristic
519C3×C4#53511yescharacteristic
520C3×C4#54612yes—
521Dic3—41no—
522Dic3—41no—
523Dic3—42no—
524Dic3—41no—
525Dic3—82no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—62no—
537C23—62no—
538C23—61no—
539C23—61no—
540C23—62no—
541C23—61no—
542C23—88no—
543C23—88no—
544C23—84no—
545C23—84no—
546C23—122no—
547C23—122no—
548C23—121no—
549C23—121no—
550C23#91412yes—
551C23#91612yes—
552C23#93712yes—
553C23C2×D3×D411yescharacteristic
554D4—24no—
555D4—24no—
556D4—48no—
557D4—42no—
558D4—42no—
559D4—64no—
560D4—124no—
561D4—128no—
562D4—121no—
563D4—121no—
564D4—244no—
565D4—248no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—84no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—122no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—244no—
592C2×C4#93712yes—
593C2×C4C2×#23512yes—
594C2×C4C2×D3×D411yescharacteristic
595C2×C4#100412yes—
596C2×C4C22×#6711yescharacteristic
597C8—42no—
598C8—122no—
599C2×C3—24no—
600C2×C3—84no—
601C2×C3#240411yescharacteristic
602C2×C3C2×#44911yescharacteristic
603C2×C3#300811yescharacteristic
604C2×C3#329712yes—
605C2×C3#331812yes—
606D3—81no—
607D3—82no—
608C22—22no—
609C22—24no—
610C22—24no—
611C22—22no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—816no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—122no—
621C22—121no—
622C22—121no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—121no—
627C22—122no—
628C22—242no—
629C22—242no—
630C22#517214yes—
631C22#549012yes—
632C22#575712yes—
633C22C2×#91411yescharacteristic
634C22C2×#91611yescharacteristic
635C22C2×#93711yescharacteristic
636C4—22no—
637C4—42no—
638C4—42no—
639C4—121no—
640C4—121no—
641C4—122no—
642C4—121no—
643C4—242no—
644C4#549012yes—
645C4#579111yescharacteristic
646C4#580511yescharacteristic
647C3#2396311yescharacteristic, complemented, Sylow
648C2—24no—
649C2—84no—
650C2—241no—
651C2—242no—
652C2#7876012yes—
653C2#7876412yes—
654C2#8212511yescharacteristic
655C2#8383311yescharacteristic
656C2C2×#517211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#102C2
socleC23×C3C22×D4
hypercenterC22×#102D3
hypocenterC3#23963
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C2×#5514C2semidirect
2. C22×C3×#102C2semidirect
3. #84053C2semidirect
4. C2×#5217C2semidirect
5. #71975C2semidirect
6. #71973C2semidirect
7. #71554C2semidirect
8. #70208C2semidirect
9. #70204C2semidirect
10. #70201C2semidirect
11. #70164C2semidirect
26. C2×#895C22semidirect
28. C2×#893C22semidirect
29. C2×#892C22semidirect
30. C2×#890C22semidirect
42. C2×#821C22semidirect
43. C2×(C2×D3)⋊C8C22semidirect
44. #4551C22semidirect
45. #4550C22semidirect
46. #4529C22semidirect
47. #4529C22semidirect
50. C22×#102D3semidirect
80. C2×C3×C4⋊C4D4semidirect
96. #930D4semidirect
303. C2×C4⋊C4#67semidirect
647. C3#23963semidirect

Maximal subgroups

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

subgroupindex
C2×#55142
C22×C3×#1022
#840532
C2×#52172
#719752
#719732
#715542
#702082
#702042
#702012
#701642
#239633