Finite Group Databasebeta

← #6800898

Subgroups — #6800898

Order 512.

1499 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34954C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32242C214yesmaximal, complemented
5#30824C214yesmaximal, complemented
6#29981C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic
8#28799C212yesmaximal, complemented
9#28726C212yesmaximal, complemented
10#28279C211yesmaximal, characteristic
11#28203C212yesmaximal, complemented
12#27778C212yesmaximal
13#26318C212yesmaximal
14#24996C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22116C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×Q82—21no—
19C2×#489C2214yes—
20C2×#475C2211yescharacteristic
21C2×#475C2212yes—
22C2×#474C2212yes—
23#2868C2212yes—
24#2868C2211yescharacteristic
25#2593—22no—
26#2582C2211yescharacteristic
27#2571C2211yescharacteristic
28#2567—22no—
29#2567—21no—
30#2551—22no—
31#2551—21no—
32#2551C2214yes—
33#2551C2212yes—
34#2548C2211yescharacteristic
35#2548C2211yescharacteristic
36#2513—21no—
37#2513C2212yes—
38#2448—21no—
39#2448C2212yes—
40#2395—22no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#124—21no—
47C4×#124C2212yes—
48#2293—21no—
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2100C2214yes—
53#2094C2214yes—
54#2085C2214yes—
55#2072C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2058C2214yes—
59#2042C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#1993C2214yes—
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1984C2212yes—
66#1982C2212yes—
67#1957C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1943C2214yes—
73#1936C2214yes—
74#1928C2214yes—
75#1926C2214yes—
76#1926C2214yes—
77#1921C2214yes—
78#1919C2212yes—
79#1917C2214yes—
80#1865C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2211yescharacteristic
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95Q82—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2314yes—
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×#124C2312yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×#117—22no—
121#489—48no—
122#475—24no—
123#475—44no—
124#474—28no—
125C2×#107C2314yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#400—22no—
144#399—22no—
145#398—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#395—24no—
150#395—24no—
151#395—24no—
152#394—22no—
153#394—22no—
154#391—21no—
155#391—22no—
156#391—21no—
157#391C2312yes—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—24no—
169#389—22no—
170#389—24no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384—22no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2314yes—
196#382—21no—
197#382C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C43—21no—
218C43C2311yescharacteristic
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—22no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—22no—
246C4⋊Q8—42no—
247C4⋊Q8—44no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—48no—
251C4⋊Q8—416no—
252C4⋊Q8—48no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C4⋊Q8—84no—
256C4⋊Q8#4114yes—
257#127—22no—
258#127—22no—
259#124—24no—
260#124—42no—
261#124—44no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—41no—
272C4×Q8—48no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—82no—
279#117—84no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
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×C42—21no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#106—24no—
347#106—42no—
348#106—44no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—42no—
354#103—28no—
355#103—24no—
356#103—24no—
357#102—28no—
358#102—44no—
359#102—44no—
360#102—44no—
361#97—44no—
362C4×C8—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8—84no—
394C2×Q8—82no—
395C2×Q8—84no—
396C2×Q8#13714yes—
397C2×Q8H(5,𝔽2)12yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—81no—
409C2×D4—81no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2511yescharacteristic
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—81no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4C2×#4114yes—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—22no—
503C42—41no—
504C42—42no—
505C42—42no—
506C42—48no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—42no—
512C42—81no—
513C42AGL(1,ℤ/8)14yes—
514C42C2×#4112yes—
515C422−1+412yes—
516C422−1+412yes—
517C422−1+412yes—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—22no—
528Q8—22no—
529Q8—48no—
530Q8—41no—
531Q8—41no—
532Q8—164no—
533Q8—168no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—84no—
539D4—84no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—88no—
585C2×C4#53612yes—
586C2×C4#53612yes—
587C2×C4#54812yes—
588C2×C4#54812yes—
589C2×C4C2×AGL(1,ℤ/8)12yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×#13714yes—
593C2×C4C2×#13712yes—
594C2×C4#57712yes—
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4#58511yescharacteristic
601C8—42no—
602C8—44no—
603C8—42no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22C2×#53611yescharacteristic
612C22C2×#54811yescharacteristic
613C22#348811yescharacteristic
614C22C22×AGL(1,ℤ/8)11yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57711yescharacteristic
617C22#358511yescharacteristic
618C4—22no—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—84no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4#323914yes—
635C4#337314yes—
636C4#351811yescharacteristic
637C4#352211yescharacteristic
638C4#354411yescharacteristic
639C4#355511yescharacteristic
640C2—162no—
641C2#3912511yescharacteristic
642C2C2×#323911yescharacteristic
643C2#6089211yescharacteristic
644C2C2×#337311yescharacteristic
645C2#6131911yescharacteristic
646C2#6196311yescharacteristic
647C2#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. Q8×C4⋊Q8C2semidirect
3. #33196C2semidirect
4. #32242C2semidirect
5. #30824C2semidirect
6. #29981C2semidirect
8. #28799C2semidirect
9. #28726C2semidirect
11. #28203C2semidirect
14. #24996C2semidirect
15. #22825C2semidirect
16. #22116C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349542
#331962
#322422
#308242
#299812
#291392
#287992
#287262
#282792
#282032
#277782
#263182
#249962
#228252
#221162
C4×#4752