Finite Group Databasebeta

← #6852811

Subgroups — #6852811

Order 512.

1469 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic
4#32148C214yesmaximal, complemented
5#30962C214yesmaximal
6#29981C212yesmaximal
7#29287C211yesmaximal, characteristic, complemented
8#29274C212yesmaximal, complemented
9#29206C211yesmaximal, characteristic, complemented
10#28817C212yesmaximal, complemented
11#28694C212yesmaximal, complemented
12#27821C212yesmaximal
13#26282C212yesmaximal, complemented
14#25058C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22116C212yesmaximal
17C4×#474C211yesmaximal, characteristic
18C2×Q82—21no—
19C2×#501C2214yes—
20C2×#474C2212yes—
21C2×#474C2211yescharacteristic
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2574—22no—
25#2567—22no—
26#2567—21no—
27#2567C2211yescharacteristic
28#2563C2211yescharacteristic
29#2552C2211yescharacteristic
30#2551—22no—
31#2551—21no—
32#2551C2212yes—
33#2551C2214yes—
34#2538C2211yescharacteristic
35#2508C2212yes—
36#2497—21no—
37#2454C2212yes—
38#2423—21no—
39#2396—22no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#125C2212yes—
46C4×C22⋊Q8—21no—
47#2293C2212yes—
48#2285—21no—
49C42×Q8C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2214yes—
53#2094C2214yes—
54#2088C2214yes—
55#2084C2214yes—
56#2083C2214yes—
57#2059C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2023C2214yes—
61#1993C2214yes—
62#1990C2212yes—
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1988C2212yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1936C2214yes—
77#1926C2214yes—
78#1921C2214yes—
79#1921C2214yes—
80#1919C2214yes—
81#1918C2212yes—
82#1917C2214yes—
83#1862C2212yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90C4×#107C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
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⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×#125C2312yes—
107C2×C22⋊Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C22×C42—21no—
120#501—48no—
121#474—44no—
122#474Q818yescomplemented
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#399—22no—
141#398—22no—
142#398—22no—
143#396—22no—
144#396—22no—
145#395—22no—
146#395—24no—
147#395—24no—
148#395—24no—
149#393—22no—
150#391—21no—
151#391—22no—
152#391—21no—
153#391—21no—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—24no—
158#389—22no—
159#389—24no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384—22no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2314yes—
188#382—22no—
189#382C2312yes—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C43C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—42no—
250C4⋊Q8—44no—
251C4⋊Q8—48no—
252C4⋊Q8—416no—
253C4⋊Q8—48no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8—84no—
257C4⋊Q8C2×Q814yes—
258#125—24no—
259#125—42no—
260C22⋊Q8—44no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×Q8—44no—
268C4×Q8—82no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—41no—
274C4×D4C2×Q814yes—
275#117—84no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—44no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—28no—
341#107—24no—
342#107C2×Q818yes—
343#106—44no—
344#106—44no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—22no—
348C4⋊C8—24no—
349C4⋊C8—42no—
350C4⋊C8C2×Q814yes—
351#102—42no—
352#102—44no—
353#102C2×Q818yes—
354#97—44no—
355C4×C8—24no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—48no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8—82no—
386C2×Q8—84no—
387C2×Q8—82no—
388C2×Q8—84no—
389C2×Q8#13714yes—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C22×Q812yes—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—81no—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2511yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8C22×Q814yes—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—84no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4C22×Q814yes—
476C4⋊C4C22×Q812yes—
477C4⋊C4H(5,𝔽2)12yes—
478C22⋊C4—24no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—22no—
496C42—41no—
497C42—42no—
498C42—48no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—44no—
504C42AGL(1,ℤ/8)14yes—
505C42C22×Q812yes—
506C422−1+412yes—
507C422−1+412yes—
508C422−1+412yes—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C23×D411yescharacteristic
517Q8—28no—
518Q8—48no—
519Q8—84no—
520Q8—164no—
521Q8—168no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#53612yes—
574C2×C4#53612yes—
575C2×C4D4×Q812yes—
576C2×C4D4×Q812yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×#13714yes—
581C2×C4C2×#13712yes—
582C2×C4#57712yes—
583C2×C4C23×Q811yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C2×C4C2×2−1+411yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C2×C4#58511yescharacteristic
589C8—24no—
590C8—44no—
591C8—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22C2×#53611yescharacteristic
600C22C2×D4×Q811yescharacteristic
601C22#348811yescharacteristic
602C22C22×AGL(1,ℤ/8)11yescharacteristic
603C22C22×#13711yescharacteristic
604C22C2×#57711yescharacteristic
605C22#358511yescharacteristic
606C4—22no—
607C4—24no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—42no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4—81no—
621C4—82no—
622C4#323914yes—
623C4#338414yes—
624C4#350211yescharacteristic
625C4#350311yescharacteristic
626C4#355011yescharacteristic
627C4#355511yescharacteristic
628C2—82no—
629C2#3959711yescharacteristic
630C2C2×#323911yescharacteristic
631C2#6089211yescharacteristic
632C2C2×#338411yescharacteristic
633C2Q8×#13711yescharacteristic
634C2#6195911yescharacteristic
635C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
4. #32148C2semidirect
7. #29287C2semidirect
8. #29274C2semidirect
9. #29206C2semidirect
10. #28817C2semidirect
11. #28694C2semidirect
13. #26282C2semidirect
15. #22825C2semidirect
122. #474Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349412
#331952
#321482
#309622
#299812
#292872
#292742
#292062
#288172
#286942
#278212
#262822
#250582
#228252
#221162
C4×#4742