Finite Group Databasebeta

← #6851923

Subgroups — #6851923

Order 512.

1813 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#32134C214yesmaximal, complemented
5#30954C214yesmaximal, complemented
6#29967C212yesmaximal, complemented
7#28809C212yesmaximal, complemented
8#27948C211yesmaximal, characteristic, complemented
9#27813C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic
11#26444C212yesmaximal, complemented
12#26044C212yesmaximal, complemented
13#25811C212yesmaximal, complemented
14#24299C212yesmaximal
15#22625C211yesmaximal, characteristic, complemented
16#22076C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#498C2214yes—
22C2×#481C2211yescharacteristic
23C2×#476C2214yes—
24#2857C2212yes—
25#2857C2211yescharacteristic
26#2564—22no—
27#2556—22no—
28#2551—21no—
29#2551—21no—
30#2551—22no—
31#2551C2212yes—
32#2551C2214yes—
33#2551C2212yes—
34#2551C2211yescharacteristic
35#2542C2211yescharacteristic
36#2506C2212yes—
37#2492—21no—
38#2452C2212yes—
39#2417—21no—
40C2×#395—22no—
41#2365C2211yescharacteristic
42#2360—21no—
43D4×C4⋊C4C2211yescharacteristic
44C4×#121C2211yescharacteristic
45C4×#121C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2212yes—
52#2099C2214yes—
53#2094C2212yes—
54#2088C2214yes—
55#2088C2214yes—
56#2084C2214yes—
57#2058C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2058C2214yes—
61#2023C2214yes—
62#1990C2212yes—
63#1990C2214yes—
64#1988C2212yes—
65#1987C2214yes—
66#1948C2212yes—
67#1948C2212yes—
68#1941C2214yes—
69#1939C2212yes—
70#1934C2214yes—
71#1926C2212yes—
72#1920C2214yes—
73#1919C2212yes—
74#1919C2214yes—
75#1916C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1864C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1853C2212yes—
84#1844C2212yes—
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8—22no—
96#554—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×#121C2311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#498—48no—
135#481Q818yescomplemented
136#476—48no—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#399—22no—
155#398—22no—
156#396—22no—
157#395—22no—
158#395—24no—
159#395—24no—
160#395—24no—
161#395—24no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391—22no—
166#390C2311yescharacteristic
167#390C2312yes—
168#389—21no—
169#389—22no—
170#389—24no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#389C2314yes—
182#389C2311yescharacteristic
183#388—22no—
184#388—21no—
185#387—22no—
186#387—22no—
187#386—21no—
188#386—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2312yes—
197#380—21no—
198#380—22no—
199#380—21no—
200#380C2312yes—
201#380C2311yescharacteristic
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C2×#95—21no—
217C2×#95—22no—
218C2×#95D412yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—48no—
239C22×Q8—48no—
240C22×Q8—84no—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264C4⋊Q8—416no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268#125—48no—
269C22⋊Q8—48no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—22no—
275#121C2×Q814yes—
276C4×Q8—44no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—44no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4SD1618yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×Q814yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#106—28no—
353#106—24no—
354#106—44no—
355#106—44no—
356#106—44no—
357#106C2×Q818yescomplemented
358C4⋊C8—24no—
359C4⋊C8—44no—
360#102—24no—
361#102—44no—
362#102C2×Q818yescomplemented
363C22⋊C8—22no—
364C22⋊C8—24no—
365C22⋊C8C2×Q814yes—
366C4×C8—24no—
367C4×C8—44no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D411yescharacteristic
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—48no—
395C2×Q8—84no—
396C2×Q8—84no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—832no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4C22×Q812yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4C2×SD1614yes—
443C22×C4C2×SD1618yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×Q812yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8C22×Q814yes—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4C2×SD1614yes—
499C4⋊C4C2×SD1618yes—
500C4⋊C4C2×SD1614yes—
501C4⋊C4AGL(1,ℤ/8)14yes—
502C4⋊C4C22×Q814yes—
503C4⋊C4C22×Q812yes—
504C4⋊C4H(5,𝔽2)14yes—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42—44no—
525C422−1+414yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23D4×Q812yes—
538C23C22×SD1612yes—
539C23#57712yes—
540C23C23×D411yescharacteristic
541Q8—816no—
542Q8—816no—
543Q8—84no—
544Q8—164no—
545D4—24no—
546D4—48no—
547D4—42no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—164no—
601C2×C4#53612yes—
602C2×C4#53612yes—
603C2×C4D4×Q812yes—
604C2×C4C22×SD1614yes—
605C2×C4C22×SD1612yes—
606C2×C4C22×SD1614yes—
607C2×C4C22×SD1612yes—
608C2×C4C2×AGL(1,ℤ/8)14yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C23×Q811yescharacteristic
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+412yes—
613C2×C4C2×2−1+411yescharacteristic
614C8—24no—
615C8—44no—
616C8—42no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22Q8×SD1614yes—
631C22C2×#53611yescharacteristic
632C22C2×D4×Q811yescharacteristic
633C22#348411yescharacteristic
634C22C23×SD1611yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57711yescharacteristic
637C22#358611yescharacteristic
638C4—24no—
639C4—24no—
640C4—24no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—41no—
649C4—82no—
650C4—81no—
651C4—84no—
652C4—81no—
653C4—82no—
654C4#322714yes—
655C4#350312yes—
656C4#355212yes—
657C2—24no—
658C2—82no—
659C2#3959211yescharacteristic
660C2C2×#322711yescharacteristic
661C2#6087211yescharacteristic
662C2C2×Q8×SD1611yescharacteristic
663C2Q8×AGL(1,ℤ/8)11yescharacteristic
664C2#6156211yescharacteristic
665C2#6160011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2551C2semidirect
3. #33147C2semidirect
4. #32134C2semidirect
5. #30954C2semidirect
6. #29967C2semidirect
7. #28809C2semidirect
8. #27948C2semidirect
9. #27813C2semidirect
11. #26444C2semidirect
12. #26044C2semidirect
13. #25811C2semidirect
15. #22625C2semidirect
135. #481Q8semidirect
357. #106C2×Q8semidirect
362. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25512
#331472
#321342
#309542
#299672
#288092
#279482
#278132
#276782
#264442
#260442
#258112
#242992
#226252
#220762
C4×#4812