Finite Group Databasebeta

← #6793521

Subgroups — #6793521

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31424C214yesmaximal, complemented
5#31352C214yesmaximal, complemented
6#29938C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#28791C212yesmaximal, complemented
10#27813C212yesmaximal, complemented
11#26298C212yesmaximal, complemented
12#24967C212yesmaximal
13#24701C212yesmaximal, complemented
14#24299C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21834C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529—21no—
21C2×#476C2211yescharacteristic
22C2×#476C2212yes—
23C2×#431C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2553—22no—
27#2552C2211yescharacteristic
28#2551—21no—
29#2551C2212yes—
30#2547C2211yescharacteristic
31#2546—21no—
32#2542C2211yescharacteristic
33#2537C2211yescharacteristic
34#2496C2212yes—
35#2489—21no—
36#2440—22no—
37#2440C2214yes—
38#2434—22no—
39#2427—21no—
40#2422C2212yes—
41#2395—22no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8—21no—
47C4×#121C2212yes—
48#2285C2212yes—
49C42×D4C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2088C2214yes—
54#2084C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2058C2214yes—
59#2034C2214yes—
60#2023C2214yes—
61#1990C2212yes—
62#1990C2214yes—
63#1990C2214yes—
64#1988C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2211yescharacteristic
68#1987C2212yes—
69#1987C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1948C2212yes—
75#1948C2212yes—
76#1941C2214yes—
77#1934C2214yes—
78#1923C2214yes—
79#1920C2212yes—
80#1874C2214yes—
81#1864C2212yes—
82#1853C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1824C2214yes—
86#1824C2214yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×Q8—22no—
96D4×Q8—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108#529—48no—
109C2×#126—22no—
110C2×#125—21no—
111C2×#124—21no—
112C2×#123—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—24no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2314yes—
132C22×C42—21no—
133C22×C42C2312yes—
134#476—24no—
135#476—44no—
136#431—44no—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#399—22no—
155#398—22no—
156#398—24no—
157#397—22no—
158#397—22no—
159#397—22no—
160#395—24no—
161#395—24no—
162#394—22no—
163#394—22no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#388—22no—
180#387—22no—
181#386—22no—
182#385—24no—
183#385—24no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#382—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380C2314yes—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—48no—
240C22×Q8—48no—
241C22×Q8—84no—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—24no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—28no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4⋊Q8—82no—
267C4⋊Q8#4114yes—
268#126—48no—
269#125—44no—
270#124—44no—
271#123—48no—
272C22⋊Q8—416no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276#121—24no—
277#121—42no—
278#121—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—41no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4SD1618yes—
293#117—48no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#106—24no—
361#106—42no—
362#106—44no—
363#106—44no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—42no—
369#102—24no—
370#102—24no—
371#102—44no—
372C22⋊C8—44no—
373C22⋊C8—44no—
374C4×C8—24no—
375C4×C8—44no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24C22×D412yes—
394C2×Q8—24no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—48no—
404C2×Q8—42no—
405C2×Q8—84no—
406C2×Q8—832no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—82no—
417C2×D4C2×SD1614yes—
418C2×D42−1+412yes—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—84no—
450C22×C4C2×SD1618yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C2×SD1614yes—
508C4⋊C4AGL(1,ℤ/8)14yes—
509C4⋊C4C2×#4114yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4C2×SD1618yes—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—41no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—44no—
537C42C2×SD1614yes—
538C42C2×#4112yes—
539C42H(5,𝔽2)12yes—
540C422−1+412yes—
541C422−1+412yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C22×SD1614yes—
556C23C23×D411yescharacteristic
557Q8—48no—
558Q8—816no—
559Q8—88no—
560Q8—84no—
561D4—28no—
562D4—22no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—88no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—216no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—816no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4#53612yes—
618C2×C4#53612yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1614yes—
623C2×C4C22×SD1612yes—
624C2×C4C22×SD1612yes—
625C2×C4C2×AGL(1,ℤ/8)14yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4#57812yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4#58511yescharacteristic
634C8—42no—
635C8—42no—
636C8—44no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22C2×#53611yescharacteristic
652C22C2×#54611yescharacteristic
653C22#348911yescharacteristic
654C22C23×SD1611yescharacteristic
655C22C22×AGL(1,ℤ/8)11yescharacteristic
656C22C2×#57811yescharacteristic
657C22#358511yescharacteristic
658C4—22no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—24no—
663C4—41no—
664C4—44no—
665C4—42no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4—84no—
673C4#322714yes—
674C4#329814yes—
675C4#351811yescharacteristic
676C4#352311yescharacteristic
677C4#355211yescharacteristic
678C4#355411yescharacteristic
679C2—44no—
680C2—82no—
681C2#3891111yescharacteristic
682C2C2×#322711yescharacteristic
683C2#6089311yescharacteristic
684C2C2×#329811yescharacteristic
685C2#6115511yescharacteristic
686C2#6209611yescharacteristic
687C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. D4×C4⋊Q8C2semidirect
3. #33197C2semidirect
4. #31424C2semidirect
5. #31352C2semidirect
6. #29938C2semidirect
7. #29287C2semidirect
8. #29218C2semidirect
9. #28791C2semidirect
10. #27813C2semidirect
11. #26298C2semidirect
13. #24701C2semidirect
14. #24299C2semidirect
15. #22795C2semidirect
16. #21834C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#349322
#331972
#314242
#313522
#299382
#292872
#292182
#287912
#278132
#262982
#249672
#247012
#242992
#227952
#218342
C4×#4762