Finite Group Databasebeta

← #7562698

Subgroups — #7562698

Order 512.

1913 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3361C212yesmaximal
2#59827C211yesmaximal, characteristic, complemented
3#36180C211yesmaximal, characteristic, complemented
4#30811C214yesmaximal, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28511C211yesmaximal, characteristic, complemented
7#27768C212yesmaximal, complemented
8#26164C214yesmaximal, complemented
9#26041C212yesmaximal, complemented
10#25648C212yesmaximal, complemented
11#25471C212yesmaximal, complemented
12#24727C212yesmaximal, complemented
13#24331C212yesmaximal, complemented
14#23149C211yesmaximal, characteristic, complemented
15C4⋊C4×D8C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#559C2212yes—
19C2×#554C2212yes—
20#3361—216no—
21C2×#488C2214yes—
22C2×#486C2212yes—
23C2×#477C2212yes—
24C2×#477C2212yes—
25C2×#478C2214yes—
26C2×#463C2214yes—
27C2×#461C2211yescharacteristic
28C2×#459C2211yescharacteristic
29C2×C8×Q8C2212yes—
30C2×C4×SD16C2212yes—
31C2×C4×D8C2214yes—
32C2×C4×D8C2212yes—
33#2868C2211yescharacteristic
34#2857—21no—
35#2724—22no—
36#2582C2212yes—
37#2567C2212yes—
38#2552C2212yes—
39#2552C2212yes—
40#2547—21no—
41#2547C2212yes—
42#2502—22no—
43#2486—22no—
44#2447—22no—
45#2447C2214yes—
46#2440—22no—
47#2440—22no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4C2212yes—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C2×#384—21no—
56C42×D4C2211yescharacteristic
57#2095C2214yes—
58#2087C2214yes—
59#2071C2214yes—
60#2068C2214yes—
61#2059C2214yes—
62#2042C2214yes—
63#2042C2214yes—
64#2026C2214yes—
65#1990C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1981C2211yescharacteristic
69#1956C2212yes—
70#1948C2211yescharacteristic
71#1947C2211yescharacteristic
72#1942C2214yes—
73#1939C2212yes—
74#1933C2214yes—
75#1927C2212yes—
76#1925C2212yes—
77#1916C2214yes—
78#1879C2212yes—
79#1877C2214yes—
80#1877C2212yes—
81#1862C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1845C2212yes—
85#1802C2211yescharacteristic
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C22×SD16C2312yes—
95C22×D8C2312yes—
96#559—28no—
97#559—44no—
98#554—28no—
99#554—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8D411yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125C2314yes—
107C2×#124—22no—
108C2×#124C2314yes—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2314yes—
120C2×C4×D4C2314yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C42—21no—
128#488—216no—
129#486—28no—
130#478—216no—
131#477—28no—
132#477—44no—
133#463—216no—
134#461—24no—
135#459D418yes—
136C8×Q8—28no—
137C4×SD16—28no—
138C4×D8—216no—
139C4×D8—28no—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2314yes—
164#398—24no—
165#398—24no—
166#398—24no—
167#397—24no—
168#397—24no—
169#397—24no—
170#395—24no—
171#391—22no—
172#391—22no—
173#391D412yes—
174#391C2312yes—
175#391C2314yes—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389D412yes—
185#389D412yes—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#385—24no—
190#385—24no—
191#385—24no—
192#385—24no—
193#384—22no—
194#384—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384D411yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#381—24no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230C22×Q8—22no—
231C22×Q8C2×D412yes—
232C22×Q8C2412yes—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—24no—
242C23×C4—24no—
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—24no—
246C2×D8—24no—
247C2×D8—24no—
248C2×D8C2×D418yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—82no—
259C4⋊Q8—82no—
260#127—22no—
261#127C2×D414yes—
262#125—28no—
263#125—44no—
264#124—28no—
265#124—44no—
266#124—48no—
267#123—416no—
268C22⋊Q8—416no—
269C22⋊Q8—48no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—44no—
286C4×D4—82no—
287C4×D4C2×D414yes—
288C4×D4#4114yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2414yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#107—24no—
347#107—44no—
348#106—24no—
349C4⋊C8—28no—
350C4⋊C8—22no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8C2×D414yes—
354#103—28no—
355#103—24no—
356#103—24no—
357#103—24no—
358#103—44no—
359#102—24no—
360#102—28no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—24no—
365#102C2×D418yes—
366C4×C8—28no—
367C4×C8—24no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—21no—
382C24—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4C22×D414yes—
410C2×D4C22×D412yes—
411C2×D4C2×#4112yes—
412C2×D42−1+412yes—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4C22×D414yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D414yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447SD16—48no—
448D8—216no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8C22×D414yes—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4C22×D412yes—
494C4⋊C4C2×#4112yes—
495C4⋊C42−1+412yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—44no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42#13514yes—
518C42C22×D412yes—
519C42C2×#4112yes—
520C42H(5,𝔽2)12yes—
521C422−1+412yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C23×D411yescharacteristic
537Q8—48no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—164no—
543D4—24no—
544D4—22no—
545D4—24no—
546D4—22no—
547D4—22no—
548D4—42no—
549D4—48no—
550D4—42no—
551D4—84no—
552D4#54714yes—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—88no—
593C2×C4—168no—
594C2×C4—168no—
595C2×C4D4212yes—
596C2×C4D4212yes—
597C2×C4#54812yes—
598C2×C4#54812yes—
599C2×C4C2×#13512yes—
600C2×C4C2×#13512yes—
601C2×C4C2×#13512yes—
602C2×C4#57612yes—
603C2×C4#57812yes—
604C2×C4C23×D411yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C8—24no—
611C8—42no—
612C8—42no—
613C8—42no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×D4211yescharacteristic
629C22C2×#54811yescharacteristic
630C22#348611yescharacteristic
631C22C22×#13511yescharacteristic
632C22C2×#57611yescharacteristic
633C22C2×#57811yescharacteristic
634C22#358511yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—41no—
644C4—82no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4#328314yes—
649C4#336114yescomplemented
650C4C2×#54711yescharacteristic
651C4#349911yescharacteristic
652C4#351111yescharacteristic
653C4#355211yescharacteristic
654C2—44no—
655C2—84no—
656C2#5912511yescharacteristic
657C2C2×#328311yescharacteristic
658C2#6109311yescharacteristic
659C2C2×#336111yescharacteristic
660C2#6134811yescharacteristic
661C2#6181911yescharacteristic
662C2#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
2. #59827C2semidirect
3. #36180C2semidirect
4. #30811C2semidirect
5. #28716C2semidirect
6. #28511C2semidirect
7. #27768C2semidirect
8. #26164C2semidirect
9. #26041C2semidirect
10. #25648C2semidirect
11. #25471C2semidirect
12. #24727C2semidirect
13. #24331C2semidirect
14. #23149C2semidirect
15. C4⋊C4×D8C2semidirect
16. C4×#461C2semidirect
649. C4#3361semidirect

Maximal subgroups

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

subgroupindex
C2×#33612
#598272
#361802
#308112
#287162
#285112
#277682
#261642
#260412
#256482
#254712
#247272
#243312
#231492
C4⋊C4×D82
C4×#4612
C4×#4592