Finite Group Databasebeta

← #6809658

Subgroups — #6809658

Order 512.

1997 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C212yesmaximal, complemented
2#36117C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic
5#30962C214yesmaximal, complemented
6#30824C214yesmaximal, complemented
7#29639C212yesmaximal, complemented
8#28817C212yesmaximal, complemented
9#27906C211yesmaximal, characteristic
10#27778C212yesmaximal, complemented
11#27648C211yesmaximal, characteristic, complemented
12#26460C212yesmaximal, complemented
13#26042C212yesmaximal, complemented
14#24607C212yesmaximal, complemented
15#24067C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#484C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529C2212yes—
21#3239—216no—
22C2×#501C2214yes—
23C2×#489C2214yes—
24C2×#484C2212yes—
25C2×#484C2211yescharacteristic
26C2×#481C2212yes—
27C2×#481C2212yes—
28C2×#481C2212yes—
29C2×#475C2214yes—
30C2×#474C2214yes—
31C2×#432C2212yes—
32#2856C2211yescharacteristic
33#2854C2212yes—
34#2552C2211yescharacteristic
35#2551—21no—
36#2551—22no—
37#2551—21no—
38#2551C2212yes—
39#2551C2214yes—
40#2551C2212yes—
41#2548C2211yescharacteristic
42#2512—21no—
43#2501—21no—
44#2479—22no—
45#2454C2212yes—
46#2448C2212yes—
47#2425—22no—
48C2×#395—22no—
49#2366C2211yescharacteristic
50#2361—21no—
51#2354—21no—
52Q8×C4⋊C4C2211yescharacteristic
53C4×C22⋊Q8C2211yescharacteristic
54C2×#389—21no—
55C2×#389C2212yes—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2094C2214yes—
60#2088C2214yes—
61#2083C2212yes—
62#2072C2212yes—
63#2067C2214yes—
64#2059C2214yes—
65#2058C2214yes—
66#2058C2214yes—
67#2042C2214yes—
68#2023C2214yes—
69#2023C2214yes—
70#1982C2212yes—
71#1957C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1943C2214yes—
75#1939C2212yes—
76#1936C2214yes—
77#1928C2212yes—
78#1926C2214yes—
79#1921C2212yes—
80#1917C2214yes—
81#1874C2212yes—
82#1872C2211yescharacteristic
83#1865C2214yes—
84#1862C2214yes—
85#1852C2211yescharacteristic
86#1844C2212yes—
87#1824C2212yes—
88#1824C2211yescharacteristic
89#1817C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#107C2212yes—
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2314yes—
107C2×#127C2311yescharacteristic
108#529—28no—
109#529—44no—
110C2×#126—22no—
111C2×#125C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×#123—22no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#501—216no—
138#489—216no—
139#484—28no—
140#484D418yescomplemented
141#481—28no—
142#481—28no—
143#481—44no—
144#475—216no—
145#474—216no—
146#432—28no—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2311yescharacteristic
164C2×C22⋊C8C2311yescharacteristic
165C2×#97C2314yes—
166C2×C4×C8C2312yes—
167#399—22no—
168#399—22no—
169#398—22no—
170#397—22no—
171#396—22no—
172#395—24no—
173#395—24no—
174#395—24no—
175#395—24no—
176#394—22no—
177#391C2311yescharacteristic
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—24no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2314yes—
196#389C2314yes—
197#389C2311yescharacteristic
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#387—21no—
203#386—22no—
204#386—22no—
205#385—21no—
206#385—22no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384C2312yes—
212#384C2312yes—
213#384C2311yescharacteristic
214#380—22no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—21no—
230C2×#95—21no—
231C2×#95—22no—
232C2×#95D412yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2312yes—
241#336C2314yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—48no—
247C22×Q8—48no—
248C22×Q8—84no—
249C22×Q8C2411yescharacteristic
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—416no—
273C4⋊Q8—48no—
274C4⋊Q8—48no—
275#127—24no—
276#126—48no—
277#125—24no—
278#125—42no—
279#124—24no—
280#124—42no—
281#123—48no—
282#123—84no—
283C22⋊Q8—22no—
284C22⋊Q8C2×D414yes—
285#121—24no—
286#121—24no—
287#121—44no—
288#121—82no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×D4—24no—
293C4×D4—42no—
294C4×D4—48no—
295C4×D4—44no—
296#117—48no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—24no—
373#107C2×D418yes—
374#106—28no—
375#106—28no—
376#106—44no—
377C4⋊C8—28no—
378C4⋊C8—24no—
379#103—28no—
380#103—24no—
381#103C2×D418yes—
382#102—28no—
383#102—28no—
384#102—28no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—22no—
388C22⋊C8C2×D414yes—
389#97—28no—
390C4×C8—24no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24C22×D411yescharacteristic
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—84no—
416C2×Q8—84no—
417C2×Q8—832no—
418C2×Q8C22×D412yes—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—81no—
431C2×D4—81no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4#13714yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4H(5,𝔽2)12yes—
477C22×C4C2511yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8C22×D414yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4AGL(1,ℤ/8)14yes—
523C4⋊C4C22×D414yes—
524C4⋊C4C22×D412yes—
525C4⋊C42−1+414yes—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—44no—
551C42—42no—
552C422−1+414yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23D4212yes—
565C23C2×#13712yes—
566C23#57712yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—42no—
570Q8—816no—
571Q8—816no—
572D4—44no—
573D4—44no—
574D4—88no—
575D4—84no—
576D4—162no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4#53612yes—
632C2×C4#53612yes—
633C2×C4D4212yes—
634C2×C4C2×AGL(1,ℤ/8)14yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×#13712yes—
637C2×C4C2×#13712yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+412yes—
641C2×C4C2×2−1+412yes—
642C8—24no—
643C8—44no—
644C8—42no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22#327814yes—
659C22C2×#53611yescharacteristic
660C22C2×D4211yescharacteristic
661C22#348611yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57711yescharacteristic
665C22#358511yescharacteristic
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—41no—
678C4—82no—
679C4—84no—
680C4—81no—
681C4—81no—
682C4#323914yescomplemented
683C4#349912yes—
684C4#354412yes—
685C2—24no—
686C2—162no—
687C2#3933711yescharacteristic
688C2C2×#323911yescharacteristic
689C2#6087311yescharacteristic
690C2C2×#327811yescharacteristic
691C2D4×AGL(1,ℤ/8)11yescharacteristic
692C2#6183011yescharacteristic
693C2#6184311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3239C2semidirect
3. C2×#2551C2semidirect
5. #30962C2semidirect
6. #30824C2semidirect
7. #29639C2semidirect
8. #28817C2semidirect
10. #27778C2semidirect
11. #27648C2semidirect
12. #26460C2semidirect
13. #26042C2semidirect
14. #24607C2semidirect
15. #24067C2semidirect
16. #22603C2semidirect
17. C4×#484C2semidirect
140. #484D4semidirect
682. C4#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
#361172
C2×#25512
#331072
#309622
#308242
#296392
#288172
#279062
#277782
#276482
#264602
#260422
#246072
#240672
#226032
C4×#4842