Finite Group Databasebeta

← #7562738

Subgroups — #7562738

Order 512.

2037 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3367C212yesmaximal, complemented
2#59821C211yesmaximal, characteristic, complemented
3D4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#30832C214yesmaximal, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28508C211yesmaximal, characteristic, complemented
7#27778C212yesmaximal, complemented
8#26188C214yesmaximal, complemented
9#26046C212yesmaximal, complemented
10#25650C212yesmaximal, complemented
11#25478C212yesmaximal, complemented
12#24731C212yesmaximal
13#24326C212yesmaximal, complemented
14#23153C211yesmaximal, characteristic, complemented
15#22671C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×D4×Q8C2212yes—
20C22×C4⋊Q8—21no—
21#3367—216no—
22C2×#489C2214yes—
23C2×#487C2212yes—
24C2×#477C2214yes—
25C2×#474C2212yes—
26C2×#474C2212yes—
27C2×#463C2214yes—
28C2×#461C2211yescharacteristic
29C2×#459C2211yescharacteristic
30C2×#446C2212yes—
31C2×#440C2214yes—
32C2×C4×SD16C2212yes—
33C2×C4×D8C2212yes—
34#2868C2211yescharacteristic
35#2857—21no—
36#2727—22no—
37#2551C2212yes—
38#2551C2212yes—
39#2548C2212yes—
40#2546C2212yes—
41#2542—21no—
42#2538C2212yes—
43#2512—22no—
44#2448C2214yes—
45#2440—22no—
46#2440—22no—
47#2434—22no—
48#2415—22no—
49#2354—21no—
50Q8×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52D4×C4⋊C4C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C42×D4C2211yescharacteristic
57#2093C2214yes—
58#2087C2214yes—
59#2072C2214yes—
60#2065C2214yes—
61#2057C2214yes—
62#2042C2214yes—
63#2042C2214yes—
64#2023C2214yes—
65#1990C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1981C2211yescharacteristic
69#1957C2212yes—
70#1948C2211yescharacteristic
71#1947C2211yescharacteristic
72#1944C2214yes—
73#1939C2212yes—
74#1936C2214yes—
75#1927C2212yes—
76#1925C2212yes—
77#1917C2214yes—
78#1879C2214yes—
79#1879C2212yes—
80#1877C2212yes—
81#1863C2214yes—
82#1863C2214yes—
83#1862C2214yes—
84#1845C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C23×Q8—22no—
95C22×SD16C2312yes—
96C22×D8C2312yes—
97#556—28no—
98#556—44no—
99D4×Q8—28no—
100D4×Q8—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8D411yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#127C2311yescharacteristic
112C2×#126—24no—
113C2×#124C2314yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2314yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C42—21no—
132#489—216no—
133#487—28no—
134#477—216no—
135#474—28no—
136#474—44no—
137#463—216no—
138#461D418yescomplemented
139#459—24no—
140#446—28no—
141#440—216no—
142C4×SD16—28no—
143C4×D8—28no—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2311yescharacteristic
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#97C2314yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168#399—24no—
169#398—24no—
170#397—24no—
171#396—24no—
172#395—24no—
173#395—24no—
174#394—24no—
175#391—22no—
176#391D412yes—
177#391C2312yes—
178#390—22no—
179#390—22no—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#390C2314yes—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—24no—
189#389D412yes—
190#389D412yes—
191#389C2314yes—
192#389C2314yes—
193#389C2314yes—
194#388—24no—
195#388—24no—
196#385—24no—
197#385—24no—
198#384—22no—
199#384—21no—
200#384—21no—
201#384D411yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#380—22no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—22no—
232C22×Q8—24no—
233C22×Q8—48no—
234C22×Q8—48no—
235C22×Q8—84no—
236C22×Q8C2×D412yes—
237C22×Q8C2412yes—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—24no—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16C2×D418yes—
250C2×D8—24no—
251C2×D8—24no—
252C2×D8—24no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—48no—
264C4⋊Q8—48no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267C4⋊Q8—82no—
268C4⋊Q8C2×D414yes—
269#127—22no—
270#127—22no—
271#126—416no—
272#124—28no—
273#124—44no—
274C22⋊Q8—416no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—84no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—28no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—82no—
294C4×D4C2×D414yes—
295C4×D4#4114yes—
296#117—48no—
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—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2414yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—24no—
355#107—44no—
356#106—24no—
357C4⋊C8—28no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8C2×D414yes—
362#103—28no—
363#103—28no—
364#103—24no—
365#103—24no—
366#103C2×D418yes—
367#102—24no—
368#102—24no—
369#102—28no—
370#102—24no—
371#102—24no—
372#102—24no—
373#102—44no—
374#97—28no—
375C4×C8—24no—
376C4×C8—24no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24—21no—
388C24—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—84no—
403C2×Q8—832no—
404C2×Q8C22×D414yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4C22×D412yes—
421C2×D4C2×#4112yes—
422C2×D42−1+412yes—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—48no—
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—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454SD16—216no—
455D8—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8C22×D414yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4C22×D412yes—
501C4⋊C4C2×#4112yes—
502C4⋊C42−1+412yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—84no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—41no—
521C42—48no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42AGL(1,ℤ/8)14yes—
527C42C22×D412yes—
528C42C2×#4112yes—
529C42H(5,𝔽2)12yes—
530C422−1+412yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—42no—
549Q8—48no—
550Q8—42no—
551Q8—816no—
552Q8—88no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—84no—
561D4—84no—
562D4#54714yes—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4—88no—
604C2×C4D4212yes—
605C2×C4D4212yes—
606C2×C4#54812yes—
607C2×C4#54812yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4#57612yes—
612C2×C4#57812yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C8—24no—
620C8—42no—
621C8—42no—
622C8—42no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22C2×D4211yescharacteristic
638C22C2×#54811yescharacteristic
639C22#348611yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22C2×#57611yescharacteristic
642C22C2×#57811yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—41no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4#327414yes—
658C4#336714yescomplemented
659C4C2×#54711yescharacteristic
660C4#349911yescharacteristic
661C4#351111yescharacteristic
662C4#355211yescharacteristic
663C2—44no—
664C2—162no—
665C2#5912511yescharacteristic
666C2C2×#327411yescharacteristic
667C2#6109711yescharacteristic
668C2C2×#336711yescharacteristic
669C2#6134511yescharacteristic
670C2#6182711yescharacteristic
671C2#6187011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3367C2semidirect
2. #59821C2semidirect
3. D4×C4⋊Q8C2semidirect
4. #30832C2semidirect
5. #28717C2semidirect
6. #28508C2semidirect
7. #27778C2semidirect
8. #26188C2semidirect
9. #26046C2semidirect
10. #25650C2semidirect
11. #25478C2semidirect
13. #24326C2semidirect
14. #23153C2semidirect
15. #22671C2semidirect
16. C4×#461C2semidirect
138. #461D4semidirect
658. C4#3367semidirect

Maximal subgroups

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

subgroupindex
C2×#33672
#598212
D4×C4⋊Q82
#308322
#287172
#285082
#277782
#261882
#260462
#256502
#254782
#247312
#243262
#231532
#226712
C4×#4612
C4×#4592