Finite Group Databasebeta

← #6578572

Subgroups — #6578572

Order 512.

2333 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C212yesmaximal, complemented
2#36126C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic, complemented
5#33005C212yesmaximal, complemented
6#26902C214yesmaximal, complemented
7#26588C214yesmaximal, complemented
8#26462C212yesmaximal, complemented
9#25718C211yesmaximal, characteristic, complemented
10#25366C212yesmaximal, complemented
11#24503C212yesmaximal, complemented
12#24209C211yesmaximal, characteristic, complemented
13#23931C212yesmaximal, complemented
14#23182C212yesmaximal, complemented
15#22996C212yesmaximal, complemented
16#22599C211yesmaximal, characteristic, complemented
17C4×#481C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C2×#492C2212yes—
23#3103—216no—
24C2×#481C2212yes—
25C2×#481C2211yescharacteristic
26C2×#475C2214yes—
27C2×#460C2214yes—
28C2×#461C2212yes—
29C2×#448C2212yes—
30C2×#448C2212yes—
31C2×#448C2212yes—
32C2×C4×SD16C2214yes—
33C2×C4×SD16C2214yes—
34C2×#431C2212yes—
35#2856C2211yescharacteristic
36#2854C2212yes—
37#2551C2211yescharacteristic
38#2541C2211yescharacteristic
39#2491—21no—
40#2461—22no—
41#2452C2212yes—
42#2445C2212yes—
43#2430—22no—
44#2416—21no—
45#2366C2211yescharacteristic
46#2365—21no—
47#2365—22no—
48#2365—21no—
49#2365C2212yes—
50#2365C2214yes—
51#2365C2212yes—
52#2361—21no—
53D4×C4⋊C4C2211yescharacteristic
54C4×#121C2211yescharacteristic
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#387—22no—
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61#2066C2214yes—
62#2055C2214yes—
63#2043C2212yes—
64#2024C2214yes—
65#2014C2214yes—
66#2013C2214yes—
67#2013C2212yes—
68#1982C2212yes—
69#1981C2212yes—
70#1968C2214yes—
71#1956C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1933C2214yes—
75#1926C2214yes—
76#1926C2212yes—
77#1919C2212yes—
78#1916C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1877C2214yes—
83#1874C2211yescharacteristic
84#1874C2212yes—
85#1865C2214yes—
86#1865C2214yes—
87#1853C2211yescharacteristic
88#1824C2211yescharacteristic
89#1817C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92C4×#106C2212yes—
93C8×C4⋊C4C2212yes—
94C4×#102C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C22×#41—22no—
97C23×Q8C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×SD16C2314yes—
101#547—48no—
102#532—28no—
103#532—44no—
104C2×C4⋊Q8—28no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#127C2311yescharacteristic
112C2×#127C2312yes—
113C2×#124—22no—
114C2×#123—22no—
115C2×#123—22no—
116C2×C22⋊Q8—22no—
117C2×#121—21no—
118C2×#121—22no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×#121C2311yescharacteristic
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×C4×Q8—28no—
127C2×C4×Q8—44no—
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2314yes—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—22no—
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4D411yescharacteristic
141C22×C4⋊C4D411yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C42—22no—
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148#492—28no—
149#481—28no—
150#481D418yescomplemented
151#475—216no—
152#461—28no—
153#460—216no—
154#448—28no—
155#448—28no—
156#448—44no—
157C4×SD16—216no—
158C4×SD16—216no—
159#431—28no—
160C2×#106C2314yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2311yescharacteristic
176C2×C22⋊C8C2311yescharacteristic
177C2×C4×C8C2312yes—
178C2×C4×C8C2314yes—
179#396—22no—
180#395—22no—
181#394—22no—
182#394—22no—
183#393—22no—
184#392—22no—
185#391—22no—
186#391—22no—
187#391C2314yes—
188#391C2314yes—
189#390—21no—
190#390—21no—
191#390—21no—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#390C2311yescharacteristic
196#389—21no—
197#389—22no—
198#389—22no—
199#389—24no—
200#389C2312yes—
201#389C2312yes—
202#389C2311yescharacteristic
203#388—22no—
204#388—22no—
205#388—21no—
206#387—24no—
207#387—24no—
208#387—24no—
209#387—24no—
210#386—22no—
211#386—21no—
212#386—22no—
213#386—22no—
214#385—22no—
215#385—22no—
216#384C2311yescharacteristic
217#380—21no—
218#380C2311yescharacteristic
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—24no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D412yes—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2311yescharacteristic
233C4×C22⋊C4—21no—
234C2×#95—22no—
235C2×#95—21no—
236C2×#95—22no—
237C2×#95D412yes—
238#340C2312yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#340C2312yes—
243#336C2312yes—
244#336C2314yes—
245#336C2314yes—
246#336C2312yes—
247C2×#41—48no—
248C2×#41—48no—
249C2×#41—84no—
250C22×Q8—28no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—44no—
256C22×Q8C2414yes—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4C2412yes—
264C22×D4C2411yescharacteristic
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D412yes—
274C23×C4C2×D412yes—
275C23×C4C2411yescharacteristic
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16—28no—
280C2×SD16—28no—
281C2×SD16—44no—
282C2×SD16—44no—
283C2×SD16—44no—
284C2×SD16—44no—
285C22×C8C2412yes—
286C22×C8C2414yes—
287C22×C8C2412yes—
288C4⋊Q8—28no—
289C4⋊Q8—28no—
290C4⋊Q8—28no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—42no—
294C4⋊Q8—44no—
295C4⋊Q8#4118yescomplemented
296#127—24no—
297#127—24no—
298#127—42no—
299#124—48no—
300#123—48no—
301#123—48no—
302C22⋊Q8—48no—
303#121—22no—
304#121—24no—
305#121—24no—
306#121—44no—
307#121—48no—
308#121—44no—
309#121—44no—
310#121—82no—
311#121C2×D414yes—
312C4×Q8—28no—
313C4×Q8—28no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×Q8—48no—
317C4×Q8—48no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—48no—
322C4×D4—42no—
323C4×D4—42no—
324C4×D4—44no—
325C4×D4—44no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—48no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4SD1618yescomplemented
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—21no—
379C2×C42—24no—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—48no—
384C2×C42—44no—
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#106—28no—
394#106—42no—
395#106C2×D418yes—
396C4⋊C8—28no—
397C4⋊C8—24no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—44no—
402#103—44no—
403#102—28no—
404#102—44no—
405#102—42no—
406#102C2×D418yes—
407C22⋊C8—24no—
408C22⋊C8—22no—
409C22⋊C8C2×D414yes—
410C4×C8—28no—
411C4×C8—24no—
412#95—24no—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—22no—
422#95—22no—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427C24—21no—
428C24—22no—
429C24C22×D411yescharacteristic
430#41—832no—
431C2×Q8—28no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—44no—
438C2×Q8—416no—
439C2×Q8—416no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—82no—
444C2×Q8C2×#4118yes—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—82no—
467C2×D4—82no—
468C2×D4—81no—
469C2×D4—81no—
470C2×D4C22×D412yes—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—42no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4C2×SD1614yes—
509C22×C4C2×SD1618yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4H(5,𝔽2)12yes—
521C22×C4C2511yescharacteristic
522SD16—416no—
523SD16—416no—
524SD16—88no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—24no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—22no—
531C2×C8—24no—
532C2×C8—22no—
533C2×C8C22×D414yes—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—42no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—81no—
559C4⋊C4C2×SD1614yes—
560C4⋊C4C2×SD1614yes—
561C4⋊C4C2×SD1618yes—
562C4⋊C4C22×D414yes—
563C4⋊C4C22×D412yes—
564C22⋊C4—24no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—44no—
571C22⋊C4—42no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—81no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—22no—
579C42—22no—
580C42—24no—
581C42—24no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—48no—
588C42—48no—
589C42—44no—
590C42—42no—
591C42—44no—
592C42C2×#4114yes—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—22no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23D4212yes—
612C23C22×SD1612yes—
613C23#57712yes—
614C23C23×D411yescharacteristic
615Q8—28no—
616Q8—28no—
617Q8—416no—
618Q8—44no—
619Q8—44no—
620Q8—88no—
621D4—24no—
622D4—48no—
623D4—44no—
624D4—42no—
625D4—44no—
626D4—88no—
627D4—88no—
628D4—88no—
629D4—84no—
630D4—162no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—216no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—28no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—44no—
658C2×C4—48no—
659C2×C4—416no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4—88no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4#52512yes—
684C2×C4#52512yes—
685C2×C4D4212yes—
686C2×C4C22×SD1612yes—
687C2×C4C22×SD1614yes—
688C2×C4C22×SD1614yes—
689C2×C4C22×SD1612yes—
690C2×C4#57512yes—
691C2×C4C23×D411yescharacteristic
692C2×C4C22×#4112yes—
693C2×C4C2×H(5,𝔽2)11yescharacteristic
694C2×C4#58512yes—
695C8—24no—
696C8—44no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22D4×SD1614yes—
719C22C2×#52511yescharacteristic
720C22C2×D4211yescharacteristic
721C22#348711yescharacteristic
722C22C23×SD1611yescharacteristic
723C22C2×#57511yescharacteristic
724C22C2×#57711yescharacteristic
725C22#358611yescharacteristic
726C4—22no—
727C4—24no—
728C4—24no—
729C4—22no—
730C4—22no—
731C4—48no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—44no—
736C4—41no—
737C4—82no—
738C4—81no—
739C4—81no—
740C4—82no—
741C4#310314yescomplemented
742C4D4×#4112yes—
743C4#351312yes—
744C2—24no—
745C2—82no—
746C2—162no—
747C2#3610211yescharacteristic
748C2C2×#310311yescharacteristic
749C2#6058211yescharacteristic
750C2C2×D4×SD1611yescharacteristic
751C2#6106711yescharacteristic
752C2#6188311yescharacteristic
753C2#6189911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36126C2
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×#3103C2semidirect
3. C2×#2365C2semidirect
4. #33107C2semidirect
5. #33005C2semidirect
6. #26902C2semidirect
7. #26588C2semidirect
8. #26462C2semidirect
9. #25718C2semidirect
10. #25366C2semidirect
11. #24503C2semidirect
12. #24209C2semidirect
13. #23931C2semidirect
14. #23182C2semidirect
15. #22996C2semidirect
16. #22599C2semidirect
17. C4×#481C2semidirect
150. #481D4semidirect
295. C4⋊Q8#41semidirect
347. C2×C4⋊C4SD16semidirect
741. C4#3103semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
#361262
C2×#23652
#331072
#330052
#269022
#265882
#264622
#257182
#253662
#245032
#242092
#239312
#231822
#229962
#225992
C4×#4812