Finite Group Databasebeta

← #6803351

Subgroups — #6803351

Order 512.

1511 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34954C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic, complemented
4#31911C214yesmaximal, complemented
5#31211C214yesmaximal, complemented
6#29829C212yesmaximal, complemented
7#28738C212yesmaximal
8#28438C211yesmaximal, characteristic
9#28372C212yesmaximal, complemented
10#28279C211yesmaximal, characteristic
11#28254C212yesmaximal, complemented
12#26531C212yesmaximal, complemented
13#25369C212yesmaximal
14#25294C212yesmaximal, complemented
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22146C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#492C2212yes—
19C2×#475C2211yescharacteristic
20C2×#474C2212yes—
21C2×#464C2214yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#461C2212yes—
26C2×#461C2212yes—
27C2×C8×Q8C2212yes—
28#2868C2211yescharacteristic
29#2862C2212yes—
30#2593—22no—
31#2582—21no—
32#2582C2211yescharacteristic
33#2571—21no—
34#2571C2211yescharacteristic
35#2570—22no—
36#2568C2212yes—
37#2548C2211yescharacteristic
38#2548C2211yescharacteristic
39#2513—21no—
40#2513C2212yes—
41#2511—22no—
42#2448—21no—
43#2448C2212yes—
44#2446C2214yes—
45#2395—22no—
46#2365C2212yes—
47#2355—21no—
48#2355—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#127C2211yescharacteristic
52C4×#124—21no—
53C4×#124C2212yes—
54#2293—21no—
55#2293C2212yes—
56C42×Q8C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58#2097C2214yes—
59#2068C2214yes—
60#2068C2214yes—
61#2067C2214yes—
62#2056C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2055C2214yes—
66#2015C2214yes—
67#2013C2214yes—
68#1988C2211yescharacteristic
69#1982C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1969C2214yes—
74#1968C2214yes—
75#1968C2214yes—
76#1956C2212yes—
77#1948C2211yescharacteristic
78#1947C2212yes—
79#1939C2212yes—
80#1939C2214yes—
81#1933C2214yes—
82#1919C2212yes—
83#1880C2214yes—
84#1865C2212yes—
85#1865C2214yes—
86#1842C2212yes—
87#1842C2211yescharacteristic
88#1805C2214yes—
89#1803C2211yescharacteristic
90#1802C2212yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#124C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117#492—44no—
118#475—24no—
119#474—44no—
120#464—48no—
121#462—44no—
122#462—44no—
123#462—44no—
124#461—28no—
125#461—44no—
126C8×Q8—44no—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—24no—
149#400—24no—
150#400—22no—
151#399—22no—
152#398—24no—
153#398—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#394—22no—
158#394—22no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391C2314yes—
165#391C2314yes—
166#391C2314yes—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—24no—
195#387—24no—
196#387—24no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—22no—
204#382—21no—
205#382—22no—
206#382—22no—
207#382C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C43—21no—
229C43C2311yescharacteristic
230C43C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2414yes—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—22no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266#127—22no—
267#127—22no—
268#124—24no—
269#124—42no—
270#124—44no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—41no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4—82no—
288#117—84no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—44no—
352#106—24no—
353#106—42no—
354C4⋊C8—24no—
355C4⋊C8—22no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358C4⋊C8—42no—
359C4⋊C8—44no—
360C4⋊C8#4114yes—
361#103—24no—
362#103—28no—
363#103—24no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—44no—
369#102—44no—
370C4×C8—24no—
371C4×C8—44no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—24no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—21no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8#13714yes—
410C2×Q8H(5,𝔽2)12yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—81no—
422C2×D4—81no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445Q16—88no—
446SD16—88no—
447SD16—88no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8C2×#4114yes—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—81no—
524C42—84no—
525C42#13514yes—
526C42C2×#4112yes—
527C42H(5,𝔽2)12yes—
528C42H(5,𝔽2)12yes—
529C422−1+412yes—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—44no—
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8—84no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—84no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4#53712yes—
600C2×C4#53712yes—
601C2×C4#55012yes—
602C2×C4#55012yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13714yes—
607C2×C4C2×#13712yes—
608C2×C4#57712yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4#58511yescharacteristic
615C8—24no—
616C8—42no—
617C8—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C2×#53711yescharacteristic
626C22C2×#55011yescharacteristic
627C22#348811yescharacteristic
628C22C22×#13511yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358611yescharacteristic
632C4—22no—
633C4—22no—
634C4—24no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—84no—
647C4—81no—
648C4#324814yes—
649C4#339414yes—
650C4#351411yescharacteristic
651C4#352211yescharacteristic
652C4#354311yescharacteristic
653C4#355411yescharacteristic
654C2—162no—
655C2#3913711yescharacteristic
656C2C2×#324811yescharacteristic
657C2#6092311yescharacteristic
658C2C2×#339411yescharacteristic
659C2#6140411yescharacteristic
660C2#6195411yescharacteristic
661C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33163C2semidirect
4. #31911C2semidirect
5. #31211C2semidirect
6. #29829C2semidirect
9. #28372C2semidirect
11. #28254C2semidirect
12. #26531C2semidirect
14. #25294C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22146C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349542
#331632
#319112
#312112
#298292
#287382
#284382
#283722
#282792
#282542
#265312
#253692
#252942
Q8×C4⋊C82
#221462
C4×#4752