Finite Group Databasebeta

← #6971528

Subgroups — #6971528

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#32739C212yesmaximal, complemented
4#31739C214yesmaximal, complemented
5#31092C214yesmaximal, complemented
6#30203C211yesmaximal, characteristic, complemented
7#30173C212yesmaximal, complemented
8#29644C212yesmaximal, complemented
9#29346C211yesmaximal, characteristic, complemented
10#29206C211yesmaximal, characteristic, complemented
11#26467C212yesmaximal, complemented
12#26447C212yesmaximal, complemented
13#26303C212yesmaximal, complemented
14#26237C212yesmaximal
15#24304C212yesmaximal, complemented
16#23835C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#533C2212yes—
20C2×#482C2214yes—
21C2×#478C2214yes—
22#2868C2211yescharacteristic
23#2858C2211yescharacteristic
24#2857C2212yes—
25#2854C2212yes—
26#2602—22no—
27#2592—21no—
28#2557—22no—
29#2552C2212yes—
30#2547C2211yescharacteristic
31#2544—21no—
32#2542—21no—
33#2541C2211yescharacteristic
34#2538—21no—
35#2491C2212yes—
36#2441C2214yes—
37#2429—22no—
38#2423—22no—
39#2422C2212yes—
40#2367—21no—
41#2366C2211yescharacteristic
42#2365—21no—
43#2365C2211yescharacteristic
44#2354C2212yes—
45Q8×C4⋊C4—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C42×D4C2211yescharacteristic
50#2088C2214yes—
51#2083C2212yes—
52#2083C2211yescharacteristic
53#2083C2211yescharacteristic
54#2077C2214yes—
55#2076C2214yes—
56#2071C2214yes—
57#2071C2212yes—
58#2067C2214yes—
59#2059C2214yes—
60#2058C2212yes—
61#2058C2212yes—
62#2057C2211yescharacteristic
63#2057C2211yescharacteristic
64#2057C2212yes—
65#2034C2214yes—
66#2033C2214yes—
67#2025C2214yes—
68#2023C2212yes—
69#2023C2212yes—
70#2023C2212yes—
71#1990C2212yes—
72#1990C2212yes—
73#1989C2214yes—
74#1988C2212yes—
75#1982C2212yes—
76#1957C2212yes—
77#1954C2212yes—
78#1948C2212yes—
79#1948C2212yes—
80#1939C2212yes—
81#1934C2214yes—
82#1923C2214yes—
83#1921C2214yes—
84#1875C2214yes—
85#1864C2214yes—
86#1854C2214yes—
87#1844C2212yes—
88#1825C2214yes—
89#1817C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1744C2211yescharacteristic
95#559—48no—
96#533—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127—21no—
104C2×#127C2311yescharacteristic
105C2×#125C2314yes—
106C2×#124—21no—
107C2×#124—22no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×#121C2312yes—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—22no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117C2314yes—
129C22×C42C2312yes—
130#482—48no—
131#478—48no—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#399—24no—
149#398—24no—
150#397—22no—
151#397—22no—
152#396—24no—
153#394—22no—
154#394—22no—
155#393—22no—
156#392—22no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—22no—
162#390—22no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—24no—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#388—24no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—41no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256#127—24no—
257#127—44no—
258#125—28no—
259#125—44no—
260#124—48no—
261#124—44no—
262#124—44no—
263#124—44no—
264#123—48no—
265#123—48no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—84no—
269#121—24no—
270#121—42no—
271#121—44no—
272C4×Q8—48no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—48no—
283#117—28no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—24no—
348#107—24no—
349#107—24no—
350#107—42no—
351#106—44no—
352#106—44no—
353C4⋊C8—24no—
354C4⋊C8—44no—
355#102—28no—
356#102—44no—
357#102—44no—
358#102—44no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361C4×C8—24no—
362C4×C8—42no—
363#95—21no—
364#95—22no—
365#95—24no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24C22×D412yes—
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4H(5,𝔽2)14yes—
485C4⋊C42−1+412yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—84no—
502C42—24no—
503C42—22no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—42no—
508C42—41no—
509C42—41no—
510C42—48no—
511C42—44no—
512C42—42no—
513C42—44no—
514C42—42no—
515C42#13514yes—
516C42AGL(1,ℤ/8)14yes—
517C422−1+414yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C23×D411yescharacteristic
533Q8—48no—
534Q8—84no—
535Q8—88no—
536Q8—84no—
537D4—44no—
538D4—44no—
539D4—44no—
540D4—88no—
541D4—84no—
542D4—162no—
543D4—164no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—164no—
592C2×C4#53612yes—
593C2×C4#53612yes—
594C2×C4C2×#13512yes—
595C2×C4C2×#13512yes—
596C2×C4C2×#13512yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×H(5,𝔽2)12yes—
601C2×C4C2×2−1+411yescharacteristic
602C2×C4C2×2−1+412yes—
603C2×C4#58511yescharacteristic
604C8—42no—
605C8—44no—
606C8—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C2×#53611yescharacteristic
622C22#348411yescharacteristic
623C22#348911yescharacteristic
624C22C22×#13511yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—84no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4#322814yes—
645C4#355012yes—
646C4#355412yes—
647C2—82no—
648C2—162no—
649C2#4248111yescharacteristic
650C2C2×#322811yescharacteristic
651C2#6088511yescharacteristic
652C2#6156611yescharacteristic
653C2#6158511yescharacteristic
654C2#6210111yescharacteristic
655C2#6211011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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
2. #36134C2semidirect
3. #32739C2semidirect
4. #31739C2semidirect
5. #31092C2semidirect
6. #30203C2semidirect
7. #30173C2semidirect
8. #29644C2semidirect
9. #29346C2semidirect
10. #29206C2semidirect
11. #26467C2semidirect
12. #26447C2semidirect
13. #26303C2semidirect
15. #24304C2semidirect
16. #23835C2semidirect
17. #22008C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#361342
#327392
#317392
#310922
#302032
#301732
#296442
#293462
#292062
#264672
#264472
#263032
#262372
#243042
#238352
#220082