Finite Group Databasebeta

← #10827378

Subgroups — #10827378

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×#5565C211yesmaximal, characteristic, complemented
2C3×#2408C211yesmaximal, characteristic, complemented
3C2×#5217C211yesmaximal, characteristic, complemented
4C22×#832C211yesmaximal, characteristic, complemented
5#71684C212yesmaximal, complemented
6#71678C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69657C212yesmaximal, complemented
9#69639C212yesmaximal, complemented
10#69515C211yesmaximal, characteristic, complemented
11#69502C211yesmaximal, characteristic, complemented
12#23963—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×C3×#120—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C2×#939—21no—
18C22×#213C2212yes—
19C2×#930C2211yescharacteristic
20C2×#930C2211yescharacteristic, complemented
21C2×#926C2211yescharacteristic, complemented
22C22×C4⋊Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24#5565—24no—
25#5565—24no—
26C2×#898—21no—
27C2×#883—21no—
28C2×#67⋊C4—21no—
29C2×C4×D12C2211yescharacteristic, complemented
30C2×#870C2211yescharacteristic, complemented
31C3×#394—21no—
32C3×#392—21no—
33C3×#390C2212yes—
34C3×#380C2211yescharacteristic
35C3×#379—21no—
36#5217—24no—
37C2×#832—24no—
38C2×#832C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2212yescomplemented
43C2×#809C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C2×#553—31no—
49C2×#447—31no—
50C22×#102—31no—
51#2408D311yescharacteristic, complemented
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—22no—
64C22×C3×D4—21no—
65C22×C3×D4—42no—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C22×#67—21no—
72C23×Dic3—21no—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C3×#121—24no—
76C3×#121—42no—
77C3×#120—44no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90#939—28no—
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—42no—
94#930—22no—
95#930—22no—
96#930—22no—
97#930D414yescomplemented
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101#926—22no—
102#926—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3D414yescomplemented
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C22×C3⋊C8—22no—
113C22×C3⋊C8—42no—
114C22×C3⋊C8C2312yes—
115C22×C3⋊C8C2312yes—
116#898—28no—
117#883—28no—
118#67⋊C4—28no—
119C4×D12—22no—
120C4×D12—22no—
121#870—22no—
122#870—22no—
123C3×#95—21no—
124#832—24no—
125#832—24no—
126#832—24no—
127#830—22no—
128#830—42no—
129#809—42no—
130C23×D4C2×D311yescharacteristic
131C22×D8—32no—
132C23×C8—31no—
133#553—64no—
134#553—64no—
135C2×#126—61no—
136C2×#125—31no—
137C2×#123—61no—
138C2×#121—31no—
139C2×#121—31no—
140C2×#121—61no—
141C2×#121C2×D312yes—
142C2×#120—21no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4C2×D311yescharacteristic
146C2×#117—31no—
147C22×C4⋊C4—31no—
148#447—64no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—64no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#394—21no—
157#392—21no—
158#390C2×D312yes—
159#380C2×D311yescharacteristic
160#379—21no—
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
172C24×C3C2×D411yescharacteristic
173C23×D3—21no—
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—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—48no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—42no—
189C2×C3×D4—82no—
190C2×C3×D4#4114yes—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C2×#67—24no—
206C2×#67—24no—
207C2×#67—42no—
208C2×#67—42no—
209C2×#67—82no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—24no—
214C22×Dic3—82no—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—21no—
218C2×D12C2×D412yes—
219C2×C4×D3—24no—
220C2×C4×D3—82no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—44no—
227C3×C22⋊C4—44no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—82no—
230C3×C42—21no—
231C3×C42—21no—
232#217—24no—
233#213—48no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3C2×D412yes—
242C4⋊Dic3C2×D412yes—
243C4⋊Dic3C2×D412yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246C4×Dic3—24no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—44no—
254C25—21no—
255C22×D4—24no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—31no—
261C22×D4—42no—
262C22×D4—61no—
263C22×D4—61no—
264C22×D4#6711yescharacteristic
265C22×D4C22×D312yes—
266C23×C4—61no—
267C23×C4—61no—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—122no—
273C22×C8—32no—
274C22×C8—32no—
275C22×C8—62no—
276C22×C8—122no—
277#126—68no—
278#125—62no—
279#125—62no—
280#123—68no—
281#121—24no—
282#121—34no—
283#121—42no—
284#121—62no—
285#121—62no—
286#121—68no—
287#121—62no—
288#120—44no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—62no—
292C4×D4—62no—
293C4×D4—68no—
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—62no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D312yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4#6711yescharacteristic
315C2×C22⋊C4#6711yescharacteristic
316C2×C42—61no—
317C2×C42C22×D311yescharacteristic
318#102—64no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—62no—
323C22⋊C8—122no—
324#95—21no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—21no—
328C23×C3—24no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—48no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
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—22no—
349C22×D3—42no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—42no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—48no—
356C3×D4—44no—
357C3×D4—42no—
358C3×D4—42no—
359C3×D4—88no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4#12012yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4112yes—
381C2×C3×C4H(5,𝔽2)12yes—
382#67—48no—
383#67—84no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—84no—
392D12—24no—
393D12—41no—
394D12—41no—
395C4×D3—84no—
396C3⋊C8—42no—
397C3⋊C8—42no—
398C24—22no—
399C24—44no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—61no—
404C24—82no—
405C24C2×#6711yescharacteristic
406C24C2×#6711yescharacteristic
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—32no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—61no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—82no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4—242no—
432C2×D4#23714yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—64no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—64no—
445C22×C4—242no—
446C22×C4—242no—
447C22×C4D3×D412yes—
448C22×C4D3×D412yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454D8—128no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—124no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—32no—
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—64no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—82no—
484C42—21no—
485C42—21no—
486C42—64no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—41no—
492C22×C3—44no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3—44no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—816no—
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—42no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—82no—
514C2×D3—82no—
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—82no—
520C3×C4#52511yescharacteristic
521C3×C4#53511yescharacteristic
522C3×C4#54612yes—
523Dic3—42no—
524Dic3—42no—
525Dic3—42no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—48no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—62no—
542C23—88no—
543C23—88no—
544C23—84no—
545C23—84no—
546C23—121no—
547C23—122no—
548C23—121no—
549C23—122no—
550C23#93212yes—
551C23#93712yes—
552C23#95012yes—
553C23C22×#6711yescharacteristic
554D4—24no—
555D4—24no—
556D4—48no—
557D4—44no—
558D4—42no—
559D4—42no—
560D4—64no—
561D4—88no—
562D4—128no—
563D4—121no—
564D4—121no—
565D4—244no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—244no—
591C2×C4—244no—
592C2×C4#93712yes—
593C2×C4C2×D3×D411yescharacteristic
594C2×C4C2×D3×D411yescharacteristic
595C2×C4C2×#23712yes—
596C2×C4#100412yes—
597C8—122no—
598C8—122no—
599C2×C3—24no—
600C2×C3—81no—
601C2×C3—84no—
602C2×C3#240411yescharacteristic
603C2×C3C2×#44911yescharacteristic
604C2×C3#300811yescharacteristic
605C2×C3#329712yes—
606C2×C3#331812yes—
607D3—82no—
608C22—24no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—41no—
617C22—44no—
618C22—41no—
619C22—41no—
620C22—816no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22—242no—
629C22—242no—
630C22#503714yes—
631C22C2×#93211yescharacteristic
632C22C2×#93711yescharacteristic
633C22#580612yes—
634C22C2×#95011yescharacteristic
635C22#583512yes—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—82no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4#558611yescharacteristic
645C4#559112yes—
646C4#559211yescharacteristic
647C3#2396311yescharacteristic, complemented, Sylow
648C2—24no—
649C2—81no—
650C2—84no—
651C2—242no—
652C2#8258111yescharacteristic
653C2C2×#503711yescharacteristic
654C2#8338311yescharacteristic
655C2#8348512yes—
656C2#8349512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#23963
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#5565C2semidirect
2. C3×#2408C2semidirect
3. C2×#5217C2semidirect
4. C22×#832C2semidirect
5. #71684C2semidirect
6. #71678C2semidirect
7. #71610C2semidirect
8. #69657C2semidirect
9. #69639C2semidirect
10. #69515C2semidirect
11. #69502C2semidirect
20. C2×#930C22semidirect
21. C2×#926C22semidirect
29. C2×C4×D12C22semidirect
30. C2×#870C22semidirect
42. C2×#830C22semidirect
43. C2×#809C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
51. #2408D3semidirect
97. #930D4semidirect
105. C2×C4⋊Dic3D4semidirect
647. C3#23963semidirect

Maximal subgroups

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

subgroupindex
C2×#55652
C3×#24082
C2×#52172
C22×#8322
#716842
#716782
#716102
#696572
#696392
#695152
#695022
#239633