Finite Group Databasebeta

← #10827372

Subgroups — #10827372

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×#5618C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic, complemented
3#84053C211yesmaximal, characteristic, complemented
4C22×#832C211yesmaximal, characteristic, complemented
5#72043C212yesmaximal, complemented
6#72037C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#70201C212yesmaximal, complemented
9#70164C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic, complemented
11#69925C211yesmaximal, characteristic, complemented
12#23963—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×D8C2212yes—
15C2×C3×#121C2211yescharacteristic
16C2×#953—21no—
17C2×#939—21no—
18C2×#938C2212yes—
19C2×#938C2211yescharacteristic, complemented
20C2×D4×Dic3—21no—
21C2×D4×Dic3C2211yescharacteristic, complemented
22C2×C4×#67C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5618—24no—
26#5618—24no—
27C2×#892C2211yescharacteristic, complemented
28C2×#890C2211yescharacteristic, complemented
29C2×#886—21no—
30C2×#879—21no—
31C3×#447—24no—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34#5247—21no—
35#5223—21no—
36#5222—21no—
37#5221C2212yes—
38C2×#832—24no—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#832C2212yescomplemented
42#5212C2211yescharacteristic
43C2×(C2×D3)⋊C8C2212yescomplemented
44#4551C2212yescomplemented
45#4551C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C2×#553—31no—
49C2×#447D311yescharacteristic, complemented
50C22×#102—31no—
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—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—42no—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—22no—
71C22×#67D411yescharacteristic
72C23×Dic3—21no—
73C23×Dic3—21no—
74C22×C4×D3C2311yescharacteristic
75C2×C3×D8—24no—
76C2×C3×D8—24no—
77C2×C3×D8—42no—
78C22×C3×C8C2312yes—
79C3×#121—22no—
80C3×#121D414yescomplemented
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83#953—44no—
84C2×#217—21no—
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217D411yescharacteristic
91#939—28no—
92#938—22no—
93#938—24no—
94#938—22no—
95#938—42no—
96D4×Dic3—28no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99C4×#67—22no—
100C4×#67—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D414yescomplemented
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8—22no—
115C22×C3⋊C8—42no—
116C22×C3⋊C8C2312yes—
117#892—22no—
118#892—22no—
119#890—22no—
120#890—22no—
121#886—28no—
122#879—28no—
123C3×#102—24no—
124C3×C22⋊C8—22no—
125#832—24no—
126#832—24no—
127#832—42no—
128C3⋊#95—21no—
129(C2×D3)⋊C8—42no—
130C23×D4C2×D311yescharacteristic
131C22×D8C2×D312yes—
132C23×C8—31no—
133#553—64no—
134#553—64no—
135C2×#126—61no—
136C2×#125—31no—
137C2×#123—61no—
138C2×#121—31no—
139C2×#121—32no—
140C2×#121—61no—
141C2×#121C2×D311yescharacteristic
142C2×#120—61no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—61no—
146C2×#117—31no—
147C22×C4⋊C4—31no—
148#447—24no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102—64no—
153C2×#102C2×D312yes—
154C2×C22⋊C8—32no—
155C2×C22⋊C8C2×D311yescharacteristic
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—21no—
167C24×C3—44no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—42no—
171C24×C3—82no—
172C24×C3C2×D411yescharacteristic
173C23×D3C2×D411yescharacteristic
174C2×C3×D4—28no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—48no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D412yes—
191C2×C3×D4#4114yes—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—22no—
200C2×#67—22no—
201C2×#67—22no—
202C2×#67—24no—
203C2×#67—44no—
204C2×#67—44no—
205C2×#67—82no—
206C2×#67—82no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3—82no—
213C22×Dic3—82no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C3×D8—48no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4C2×D412yes—
227C3×C22⋊C4—24no—
228#217—24no—
229#217—24no—
230#217—22no—
231#217—44no—
232#217—44no—
233#217—42no—
234#217—82no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—22no—
237C4⋊Dic3—24no—
238C4⋊Dic3—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—22no—
242C4⋊Dic3—22no—
243C4⋊Dic3C2×D412yes—
244C4⋊Dic3C2×D412yes—
245Dic3⋊C4—24no—
246Dic3⋊C4—22no—
247C4×Dic3—24no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—44no—
254C25—21no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—31no—
261C22×D4—42no—
262C22×D4—61no—
263C22×D4—62no—
264C22×D4C22×D312yes—
265C22×D4C22×D311yescharacteristic
266C23×C4—31no—
267C23×C4—61no—
268C23×C4—61no—
269C23×C4C22×D311yescharacteristic
270C2×D8—24no—
271C2×D8—24no—
272C2×D8—42no—
273C22×C8—32no—
274C22×C8—62no—
275C22×C8—122no—
276C22×C8C22×D312yes—
277#126—68no—
278#125—62no—
279#125—62no—
280#123—68no—
281#121—22no—
282#121—62no—
283#121—68no—
284#121—62no—
285#121—64no—
286#121—122no—
287#121#6714yescomplemented
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—34no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
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—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—22no—
323C22⋊C8—122no—
324#95—61no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—24no—
329C23×C3—48no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—41no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—41no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—88no—
341C23×C3—88no—
342C23×C3—84no—
343C23×C3—84no—
344C23×C3#12012yes—
345C23×C3#13512yes—
346C23×C3AGL(1,ℤ/8)12yes—
347C23×C3C22×D411yescharacteristic
348C22×D3—21no—
349C22×D3—22no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—22no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—24no—
356C3×D4—48no—
357C3×D4—48no—
358C3×D4—41no—
359C3×D4—41no—
360C3×D4—42no—
361C3×D4—42no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4#12012yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4112yes—
373C2×C3×C4H(5,𝔽2)12yes—
374#67—44no—
375#67—84no—
376#67—88no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—84no—
392C2×Dic3—84no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—42no—
396C3×C8—42no—
397C3⋊C8—42no—
398C24—21no—
399C24—22no—
400C24—31no—
401C24—44no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—82no—
406C24D3×D411yescharacteristic
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—64no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—242no—
430C2×D4—242no—
431C2×D4#23714yes—
432C2×D4C2×#6712yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—242no—
449C22×C4—242no—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D412yes—
452C22×C4C2×#6712yes—
453C22×C4C23×D311yescharacteristic
454D8—48no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—124no—
462C4⋊C4—21no—
463C4⋊C4—32no—
464C4⋊C4—32no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4C2×#6712yes—
474C22⋊C4—24no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C22⋊C4—64no—
478C22⋊C4—62no—
479C22⋊C4—62no—
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—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—44no—
496C22×C3—44no—
497C22×C3—816no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3—82no—
503C22×C3—82no—
504C22×C3#44914yes—
505C22×C3C2×#12011yescharacteristic
506C22×C3#54612yes—
507C22×C3C2×#13511yescharacteristic
508C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
509C22×C3#57612yes—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—42no—
514C2×D3—42no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4#52511yescharacteristic
518C3×C4#53511yescharacteristic
519C3×C4#54612yes—
520Dic3—41no—
521Dic3—41no—
522Dic3—41no—
523Dic3—42no—
524Dic3—42no—
525Dic3—82no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—48no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—61no—
542C23—61no—
543C23—62no—
544C23—61no—
545C23—62no—
546C23—88no—
547C23—88no—
548C23—84no—
549C23—84no—
550C23#91112yes—
551C23#91212yes—
552C23#93712yes—
553C23C2×D3×D411yescharacteristic
554D4—24no—
555D4—24no—
556D4—24no—
557D4—48no—
558D4—48no—
559D4—42no—
560D4—41no—
561D4—42no—
562D4—41no—
563D4—124no—
564D4—244no—
565D4—248no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—244no—
591C2×C4—244no—
592C2×C4#93712yes—
593C2×C4C2×D3×D411yescharacteristic
594C2×C4C2×#23712yes—
595C2×C4#100412yes—
596C2×C4C22×#6711yescharacteristic
597C8—42no—
598C8—122no—
599C2×C3—24no—
600C2×C3—82no—
601C2×C3—84no—
602C2×C3#240411yescharacteristic
603C2×C3C2×#44911yescharacteristic
604C2×C3#300811yescharacteristic
605C2×C3#329712yes—
606C2×C3#331812yes—
607D3—81no—
608C22—24no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—816no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—121no—
626C22—122no—
627C22—122no—
628C22—121no—
629C22—121no—
630C22#515814yes—
631C22#559112yes—
632C22C2×#91111yescharacteristic
633C22C2×#91211yescharacteristic
634C22#577412yes—
635C22C2×#93711yescharacteristic
636C4—22no—
637C4—42no—
638C4—121no—
639C4—121no—
640C4—121no—
641C4—122no—
642C4—122no—
643C4—242no—
644C4#558611yescharacteristic
645C4#559211yescharacteristic
646C4#580612yes—
647C3#2396311yescharacteristic, complemented, Sylow
648C2—24no—
649C2—84no—
650C2—82no—
651C2—241no—
652C2#8001512yes—
653C2#8001712yes—
654C2#8258111yescharacteristic
655C2C2×#515811yescharacteristic
656C2#8394711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#447C2
socleC23×C3C22×D4
hypercenterC2×#447D3
hypocenterC3#23963
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#5618C2semidirect
2. C2×C3×#447C2semidirect
3. #84053C2semidirect
4. C22×#832C2semidirect
5. #72043C2semidirect
6. #72037C2semidirect
7. #71681C2semidirect
8. #70201C2semidirect
9. #70164C2semidirect
10. #69952C2semidirect
11. #69925C2semidirect
19. C2×#938C22semidirect
21. C2×D4×Dic3C22semidirect
27. C2×#892C22semidirect
28. C2×#890C22semidirect
41. C2×#832C22semidirect
43. C2×(C2×D3)⋊C8C22semidirect
44. #4551C22semidirect
45. #4551C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
49. C2×#447D3semidirect
80. C3×#121D4semidirect
106. C2×C4⋊Dic3D4semidirect
287. #121#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×#56182
C2×C3×#4472
#840532
C22×#8322
#720432
#720372
#716812
#702012
#701642
#699522
#699252
#239633