Finite Group Databasebeta

← #6446235

Subgroups — #6446235

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#26294C214yesmaximal, complemented
4#26292C214yesmaximal, complemented
5#26283C212yesmaximal, complemented
6#26281C212yesmaximal, complemented
7#26278C212yesmaximal, complemented
8#26276C212yesmaximal, complemented
9#23747C212yesmaximal
10#23657C212yesmaximal
11#21324C211yesmaximal, characteristic
12#21313C211yesmaximal, characteristic
13#21303C211yesmaximal, characteristic
14#21170C211yesmaximal, characteristic
15#21034C212yesmaximal
16#21034C212yesmaximal
17C4×#428C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2546—21no—
27#2546—22no—
28#2546—21no—
29#2546C2212yes—
30#2546C2212yes—
31#2546C2214yes—
32C2×#398—22no—
33#2355C2211yescharacteristic
34#2355C2211yescharacteristic
35#2355C2212yes—
36#2355C2212yes—
37Q8×C4⋊C4C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2345—22no—
40#2345—21no—
41Q8×C22⋊C4—21no—
42#2295—22no—
43#2295—21no—
44C2×#391—21no—
45C2×#389C2212yes—
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C42×Q8—21no—
50C4×#117C2212yes—
51C4×#117C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2085C2214yes—
58#2085C2214yes—
59#2084C2214yes—
60#2083C2214yes—
61#1993C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1985C2214yes—
69#1984C2212yes—
70#1984C2212yes—
71#1949C2214yes—
72#1948C2212yes—
73#1947C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1776C2212yes—
83#1776C2212yes—
84C4×#107C2212yes—
85C4×#106C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97C22×M16C2311yescharacteristic
98C22×M16C2312yes—
99C22×M16C2311yescharacteristic
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D412yes—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#428—24no—
135#428—28no—
136#428Q818yescomplemented
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143#423—24no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C4×M16—24no—
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149#398—24no—
150#398—24no—
151#398—24no—
152#398—24no—
153#391—21no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—24no—
159#391—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#384C2312yes—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#382C2312yes—
183#382C2312yes—
184#382C2312yes—
185#380—22no—
186#380—21no—
187#380—22no—
188#380—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213#376—22no—
214#376—21no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95—22no—
218C43—22no—
219C43—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D412yes—
248C22×Q8C2×D411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—24no—
265C2×M16C2×Q814yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—416no—
274C4⋊Q8—48no—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4×Q8—24no—
279C4×Q8—28no—
280C4×Q8—24no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—44no—
289#117—22no—
290#117—24no—
291#117—22no—
292#117—24no—
293#117—22no—
294#117C2×Q814yes—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×Q814yes—
329C2×C4⋊C4#4114yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42#4114yes—
363C2×C42#4114yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—24no—
369#107—28no—
370#107C2×Q818yescomplemented
371#106—24no—
372#106—28no—
373#106C2×Q818yescomplemented
374C4⋊C8—24no—
375C4⋊C8—24no—
376#97—24no—
377C4×C8—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398C24C22×D411yescharacteristic
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—416no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—816no—
415C2×Q8AGL(1,ℤ/8)14yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4#12314yes—
445C22×C4#12314yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q812yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C42−1+412yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462M16—28no—
463M16—44no—
464M16—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8C22×Q814yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C22×Q812yes—
513C4⋊C4C22×Q814yes—
514C4⋊C4C22×Q812yes—
515C4⋊C4C2×#4114yes—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C42—24no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—44no—
527C42—44no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—42no—
534C42—81no—
535C42—81no—
536C42—82no—
537C42—82no—
538C42#12314yes—
539C42#12314yes—
540C42C2×#4112yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23C2×#12312yes—
549C23#54812yes—
550C23D4×Q812yes—
551C23C23×D411yescharacteristic
552Q8—28no—
553Q8—48no—
554Q8—44no—
555Q8—88no—
556Q8—88no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4C2×#12312yes—
609C2×C4C2×#12312yes—
610C2×C4C2×#12312yes—
611C2×C4C2×#12312yes—
612C2×C4C2×#12312yes—
613C2×C4#53314yes—
614C2×C4#54114yes—
615C2×C4#54812yes—
616C2×C4D4×Q812yes—
617C2×C4#55614yes—
618C2×C4#55714yes—
619C2×C4C2×AGL(1,ℤ/8)14yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4#57512yes—
622C2×C4C23×Q811yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C8—24no—
629C8—42no—
630C8—44no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22#244614yes—
638C22C22×#12311yescharacteristic
639C22C2×#54811yescharacteristic
640C22C2×D4×Q811yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C2×#57511yescharacteristic
643C22#358611yescharacteristic
644C22#358611yescharacteristic
645C4—22no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—84no—
657C4—81no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4#244614yes—
662C4#309414yes—
663C4C2×#53311yescharacteristic
664C4C2×#54111yescharacteristic
665C4C2×#55611yescharacteristic
666C4C2×#55711yescharacteristic
667C2—24no—
668C2C2×#244611yescharacteristic
669C2C2×#309411yescharacteristic
670C2#6051911yescharacteristic
671C2#6131711yescharacteristic
672C2#6132011yescharacteristic
673C2Q8×AGL(1,ℤ/8)11yescharacteristic
674C2#6136911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26294C2semidirect
4. #26292C2semidirect
5. #26283C2semidirect
6. #26281C2semidirect
7. #26278C2semidirect
8. #26276C2semidirect
136. #428Q8semidirect
370. #107C2×Q8semidirect
373. #106C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
Q8×#1172
#262942
#262922
#262832
#262812
#262782
#262762
#237472
#236572
#213242
#213132
#213032
#211702
#210342
C4×#4282