Finite Group Databasebeta

← #7563371

Subgroups — #7563371

Order 512.

1905 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3380C212yesmaximal, complemented
2#59823C211yesmaximal, characteristic, complemented
3#36180C211yesmaximal, characteristic, complemented
4#30951C214yesmaximal, complemented
5#28811C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28509C211yesmaximal, characteristic, complemented
8#26463C212yesmaximal, complemented
9#26382C214yesmaximal, complemented
10#26045C212yesmaximal, complemented
11#25839C212yesmaximal, complemented
12#25470C212yesmaximal, complemented
13#24327C212yesmaximal, complemented
14#23149C211yesmaximal, characteristic, complemented
15#22995C212yesmaximal
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#554C2212yes—
20#3380—216no—
21C2×#500C2212yes—
22C2×#499C2214yes—
23C2×#478C2214yes—
24C2×#478C2214yes—
25C2×#476C2212yes—
26C2×#476C2212yes—
27C2×#476C2211yescharacteristic
28C2×#459C2211yescharacteristic
29C2×C8×Q8C2212yes—
30C2×C4×D8C2212yes—
31C2×C4×D8C2214yes—
32#2868C2211yescharacteristic
33#2857—21no—
34#2732—22no—
35#2567C2212yes—
36#2552C2212yes—
37#2552C2212yes—
38#2552C2212yes—
39#2542—21no—
40#2538—22no—
41#2538C2212yes—
42#2502—22no—
43#2454C2214yes—
44#2440—22no—
45#2440—22no—
46#2417—22no—
47#2366C2212yes—
48D4×C4⋊C4—21no—
49D4×C4⋊C4C2212yes—
50D4×C4⋊C4C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127C2211yescharacteristic
54C2×#384—21no—
55C42×D4C2211yescharacteristic
56#2094C2214yes—
57#2088C2214yes—
58#2083C2214yes—
59#2065C2214yes—
60#2059C2214yes—
61#2059C2214yes—
62#2059C2214yes—
63#2025C2214yes—
64#1990C2212yes—
65#1987C2211yescharacteristic
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1957C2212yes—
69#1948C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1947C2211yescharacteristic
73#1941C2214yes—
74#1939C2212yes—
75#1934C2214yes—
76#1925C2212yes—
77#1920C2212yes—
78#1916C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1864C2214yes—
82#1864C2212yes—
83#1862C2214yes—
84#1862C2214yes—
85#1844C2212yes—
86#1802C2211yescharacteristic
87#1802C2211yescharacteristic
88C4×#106C2212yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94C22×D8C2312yes—
95#554—28no—
96#554—28no—
97#554—44no—
98#554—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8D411yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#125C2314yes—
106C2×#125C2314yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C4×Q8—21no—
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#500—28no—
129#499—216no—
130#478—216no—
131#478—216no—
132#476—28no—
133#476—44no—
134#476D418yescomplemented
135#459Q818yescomplemented
136C8×Q8—28no—
137C4×D8—28no—
138C4×D8—216no—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×C4×C8C2312yes—
160C2×C4×C8C2314yes—
161C2×C4×C8C2312yes—
162#399—24no—
163#398—24no—
164#398—24no—
165#398—24no—
166#396—24no—
167#396—24no—
168#395—24no—
169#391—22no—
170#391—21no—
171#391D412yes—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2314yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—21no—
181#389D412yes—
182#389D412yes—
183#389C2312yes—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#386—24no—
188#385—24no—
189#385—24no—
190#385—24no—
191#384—22no—
192#384—22no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384D411yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—24no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8C2×D412yes—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—24no—
241C23×C4—24no—
242C2×D8—24no—
243C2×D8—24no—
244C2×D8C2×Q818yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—82no—
254C4⋊Q8—82no—
255C4⋊Q8C2×D414yes—
256#127—22no—
257#127C2×Q814yes—
258#125—28no—
259#125—28no—
260#125—44no—
261#125—44no—
262C22⋊Q8—416no—
263C22⋊Q8—48no—
264C22⋊Q8—416no—
265C22⋊Q8—48no—
266C22⋊Q8—84no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—82no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—82no—
282C4×D4—84no—
283C4×D4C2×D414yes—
284C4×D4C2×Q814yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—48no—
305C2×C4⋊C4—84no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2414yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—24no—
344#107—28no—
345#107—28no—
346#106—28no—
347#106—24no—
348#106—44no—
349#106C2×D418yes—
350C4⋊C8—28no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8C2×D414yes—
354C4⋊C8C2×Q814yes—
355#102—24no—
356#102—24no—
357#102—28no—
358#102—24no—
359#102—24no—
360#102—44no—
361#102C2×D418yes—
362#102C2×Q818yes—
363C4×C8—28no—
364C4×C8—24no—
365C4×C8—24no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95C2×D414yes—
375#95C2×D414yes—
376C24—21no—
377C24—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—44no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4C22×D412yes—
402C2×D4C22×Q814yes—
403C2×D4C22×Q812yes—
404C2×D42−1+412yes—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4C22×D414yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D414yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439D8—216no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8C22×D414yes—
451C2×C8C22×Q814yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4C22×D412yes—
487C4⋊C4C22×D414yes—
488C4⋊C4C22×Q812yes—
489C4⋊C42−1+412yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—84no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—24no—
504C42—21no—
505C42—21no—
506C42—41no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—44no—
511C42#13514yes—
512C42C22×D412yes—
513C42C22×Q812yes—
514C42H(5,𝔽2)12yes—
515C422−1+412yes—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×D411yescharacteristic
531Q8—48no—
532Q8—84no—
533Q8—164no—
534D4—24no—
535D4—22no—
536D4—22no—
537D4—24no—
538D4—22no—
539D4—48no—
540D4—42no—
541D4—42no—
542D4—84no—
543D4D4×Q814yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—88no—
584C2×C4—168no—
585C2×C4—168no—
586C2×C4D4212yes—
587C2×C4D4212yes—
588C2×C4D4×Q812yes—
589C2×C4D4×Q812yes—
590C2×C4C2×#13512yes—
591C2×C4C2×#13512yes—
592C2×C4C2×#13512yes—
593C2×C4#57612yes—
594C2×C4#57712yes—
595C2×C4C23×D411yescharacteristic
596C2×C4C23×Q811yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C8—24no—
602C8—24no—
603C8—42no—
604C8—42no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×D4211yescharacteristic
620C22C2×D4×Q811yescharacteristic
621C22#348611yescharacteristic
622C22C22×#13511yescharacteristic
623C22C2×#57611yescharacteristic
624C22C2×#57711yescharacteristic
625C22#358511yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—42no—
635C4—41no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4#328414yes—
640C4#338014yescomplemented
641C4C2×D4×Q811yescharacteristic
642C4#349911yescharacteristic
643C4#350311yescharacteristic
644C4#355211yescharacteristic
645C2—44no—
646C2—84no—
647C2#5912711yescharacteristic
648C2C2×#328411yescharacteristic
649C2#6109311yescharacteristic
650C2C2×#338011yescharacteristic
651C2#6138211yescharacteristic
652C2#6181911yescharacteristic
653C2#6186811yescharacteristic
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×#3380C2semidirect
2. #59823C2semidirect
3. #36180C2semidirect
4. #30951C2semidirect
5. #28811C2semidirect
6. #28689C2semidirect
7. #28509C2semidirect
8. #26463C2semidirect
9. #26382C2semidirect
10. #26045C2semidirect
11. #25839C2semidirect
12. #25470C2semidirect
13. #24327C2semidirect
14. #23149C2semidirect
16. C4×#476C2semidirect
134. #476D4semidirect
135. #459Q8semidirect
640. C4#3380semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
#598232
#361802
#309512
#288112
#286892
#285092
#264632
#263822
#260452
#258392
#254702
#243272
#231492
#229952
C4×#4762
C4×#4592