Finite Group Databasebeta

← #6434098

Subgroups — #6434098

Order 512.

2178 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#29914C214yesmaximal, complemented
4#29912C214yesmaximal, complemented
5#29910C214yesmaximal, complemented
6#29907C212yesmaximal, complemented
7#29906C212yesmaximal, complemented
8#23747C212yesmaximal, complemented
9#23731C212yesmaximal, complemented
10#23657C212yesmaximal, complemented
11#23628C212yesmaximal, complemented
12#20912C212yesmaximal
13#20846C212yesmaximal, complemented
14#20682C211yesmaximal, characteristic, complemented
15C22×#123—21no—
16C22×#121C2211yescharacteristic
17C22×#117C2212yes—
18C23×C42C2211yescharacteristic
19C2×#428C2214yes—
20C2×#418C2214yes—
21C2×#407C2212yes—
22#2854—21no—
23#2854—21no—
24#2854—21no—
25#2854—21no—
26#2854C2212yes—
27#2854C2212yes—
28#2854C2212yes—
29#2854C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#388—21no—
35C2×#388—21no—
36C2×#386—21no—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285—21no—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2212yescomplemented
44#2285C2212yescomplemented
45C2×#384C2211yescharacteristic, complemented
46C2×#384C2211yescharacteristic, complemented
47C2×#382C2212yescomplemented
48C2×#380—21no—
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4—21no—
53#2101C2218yescomplemented
54#2100C2214yescomplemented
55#2099C2214yescomplemented
56#2094C2214yescomplemented
57#2093C2214yescomplemented
58#2092C2218yescomplemented
59#2085C2218yescomplemented
60#2084C2214yescomplemented
61#2083C2214yescomplemented
62#2073C2218yescomplemented
63#2071C2214yescomplemented
64#2071C2214yescomplemented
65#1875C2218yes—
66#1874C2214yes—
67#1874C2214yes—
68#1854C2218yescomplemented
69#1853C2214yescomplemented
70#1852C2214yescomplemented
71#1817C2212yescomplemented
72#1817C2212yescomplemented
73#1817C2212yescomplemented
74#1817C2212yescomplemented
75#1736C2214yes—
76#1736C2214yes—
77#1728C2212yes—
78#1728C2212yes—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16C2314yes—
85C2×#123—22no—
86C2×#123—22no—
87C2×#123—22no—
88C2×#123—22no—
89C2×#123—48no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—28no—
93C2×#121C2312yes—
94C2×#121C2312yes—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—48no—
98C2×#117C2314yes—
99C2×#117C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—28no—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#428—48no—
133C2×#107C2318yes—
134C2×#106C2318yes—
135#418—48no—
136C2×#102C2318yes—
137C2×#102C2318yes—
138#407—28no—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141#390—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#384—21no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384C2312yes—
174#384C2312yes—
175#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382—22no—
179#382C2314yes—
180#382C2314yes—
181#380—21no—
182#380—21no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C2×#95—22no—
202C2×#95—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25—21no—
215C25C2×D411yescharacteristic
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4C2412yes—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4#4118yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×M16—28no—
259C2×M16—28no—
260C2×M16—28no—
261C22×C8C2418yes—
262#123—48no—
263#123—48no—
264#123—48no—
265#123—48no—
266#121—28no—
267#121—28no—
268#121—44no—
269#121—44no—
270#117—28no—
271#117—28no—
272#117—44no—
273#117—44no—
274#117—48no—
275#117—48no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—48no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—48no—
362#106—48no—
363#102—48no—
364#102—48no—
365C22⋊C8—28no—
366C22⋊C8—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24#12314yes—
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—416no—
408C2×D4—816no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—232no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4#12314yes—
478C22×C4#12314yes—
479C22×C4#12314yes—
480C22×C4AGL(1,ℤ/8)14yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4114yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C42−1+414yes—
497C22×C4C2511yescharacteristic
498M16—416no—
499C2×C8—28no—
500C2×C8—28no—
501C2×C8—28no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—48no—
557C42—48no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—28no—
567C23—28no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23C2×#12312yes—
594C23C2×#12312yes—
595C23C2×#12312yes—
596C23#53512yes—
597C23#53612yes—
598C23C2×AGL(1,ℤ/8)12yes—
599C23#57512yes—
600C23C23×D411yescharacteristic
601D4—48no—
602D4—88no—
603D4—88no—
604D4—84no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—88no—
662C2×C4—84no—
663C2×C4—88no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4C2×#12312yes—
667C2×C4C2×#12312yes—
668C2×C4C2×#12312yes—
669C2×C4#53512yes—
670C2×C4#53612yes—
671C2×C4#539116yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4#57512yes—
675C2×C4#57512yes—
676C2×C4C22×#4112yes—
677C2×C4C2×H(5,𝔽2)12yes—
678C2×C4C2×2−1+412yes—
679C8—48no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—416no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—168no—
708C22#262314yes—
709C22#309414yes—
710C22C22×#12311yescharacteristic
711C22C2×#53511yescharacteristic
712C22C2×#53611yescharacteristic
713C22C22×AGL(1,ℤ/8)11yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57511yescharacteristic
716C22C2×#57511yescharacteristic
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—84no—
729C4—82no—
730C4—84no—
731C4—82no—
732C4#262314yes—
733C4C2×#53914yes—
734C2—24no—
735C2—24no—
736C2—42no—
737C2—162no—
738C2C2×#262311yescharacteristic
739C2C2×#309411yescharacteristic
740C2#6051111yescharacteristic
741C2#6080311yescharacteristic
742C2#6080511yescharacteristic
743C2#6086411yescharacteristic
744C2#6086611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #29914C2semidirect
4. #29912C2semidirect
5. #29910C2semidirect
6. #29907C2semidirect
7. #29906C2semidirect
8. #23747C2semidirect
9. #23731C2semidirect
10. #23657C2semidirect
11. #23628C2semidirect
13. #20846C2semidirect
14. #20682C2semidirect
43. #2285C22semidirect
44. #2285C22semidirect
45. C2×#384C22semidirect
46. C2×#384C22semidirect
47. C2×#382C22semidirect
53. #2101C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2094C22semidirect
57. #2093C22semidirect
58. #2092C22semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2083C22semidirect
62. #2073C22semidirect
63. #2071C22semidirect
64. #2071C22semidirect
68. #1854C22semidirect
69. #1853C22semidirect
70. #1852C22semidirect
71. #1817C22semidirect
72. #1817C22semidirect
73. #1817C22semidirect
74. #1817C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#22852
#299142
#299122
#299102
#299072
#299062
#237472
#237312
#236572
#236282
#209122
#208462
#206822