Finite Group Databasebeta

← #6852762

Subgroups — #6852762

Order 512.

1583 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32134C214yesmaximal, complemented
5#30954C214yesmaximal
6#29981C212yesmaximal
7#29287C211yesmaximal, characteristic, complemented
8#29273C212yesmaximal, complemented
9#29206C211yesmaximal, characteristic, complemented
10#28809C212yesmaximal, complemented
11#28687C212yesmaximal, complemented
12#27813C212yesmaximal
13#26278C212yesmaximal, complemented
14#25058C212yesmaximal
15#22821C211yesmaximal, characteristic, complemented
16#22111C212yesmaximal
17C4×#476C211yesmaximal, characteristic
18C2×Q82—21no—
19C2×#533—21no—
20C2×#498C2214yes—
21C2×#476C2212yes—
22C2×#476C2211yescharacteristic
23#2868C2212yes—
24#2868C2211yescharacteristic
25#2567—21no—
26#2567—22no—
27#2556—22no—
28#2551—22no—
29#2551—21no—
30#2551C2211yescharacteristic
31#2551C2214yes—
32#2551C2212yes—
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2538C2211yescharacteristic
36#2506C2212yes—
37#2492—21no—
38#2452C2212yes—
39#2429—21no—
40#2396—22no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8—21no—
48#2293C2212yes—
49C42×Q8C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2094C2214yes—
54#2088C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2058C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2023C2214yes—
61#1990C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2211yescharacteristic
65#1987C2212yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1941C2214yes—
76#1934C2214yes—
77#1926C2214yes—
78#1920C2212yes—
79#1919C2214yes—
80#1919C2214yes—
81#1919C2214yes—
82#1916C2214yes—
83#1864C2212yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95Q82—48no—
96#533—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2314yes—
110C2×#125—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C22×C42—21no—
128#498—48no—
129#476—44no—
130#476Q818yescomplemented
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#399—22no—
149#398—22no—
150#398—22no—
151#396—22no—
152#396—22no—
153#395—24no—
154#395—24no—
155#395—24no—
156#395—22no—
157#393—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—21no—
162#391—22no—
163#390C2312yes—
164#390C2312yes—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#385—22no—
185#384—21no—
186#384—22no—
187#384—22no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2314yes—
195#380—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C43C2312yes—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—24no—
238C22×Q8C2×D411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256C4⋊Q8—44no—
257C4⋊Q8—416no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—82no—
262C4⋊Q8—84no—
263C4⋊Q8C2×Q814yes—
264#125—48no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—44no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—48no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×Q8—44no—
277C4×Q8SD1618yes—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—41no—
283C4×D4C2×Q814yes—
284#117—48no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
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⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#106—28no—
351#106—24no—
352#106—44no—
353#106—44no—
354#106—44no—
355#106C2×Q818yes—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—24no—
359C4⋊C8—42no—
360C4⋊C8C2×Q814yes—
361#102—42no—
362#102—44no—
363#102C2×Q818yes—
364C4×C8—24no—
365C4×C8—44no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—21no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—48no—
395C2×Q8—82no—
396C2×Q8—84no—
397C2×Q8—82no—
398C2×Q8—84no—
399C2×Q8—82no—
400C2×Q8C2×SD1614yes—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22×Q812yes—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—81no—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2511yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8C22×Q814yes—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4C2×SD1618yes—
485C4⋊C4C2×SD1614yes—
486C4⋊C4AGL(1,ℤ/8)14yes—
487C4⋊C4C22×Q814yes—
488C4⋊C4C22×Q812yes—
489C4⋊C4H(5,𝔽2)12yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—21no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—22no—
507C42—41no—
508C42—42no—
509C42—48no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42C2×SD1614yes—
516C42C2×SD1618yes—
517C42C22×Q812yes—
518C422−1+412yes—
519C422−1+412yes—
520C422−1+412yes—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—48no—
531Q8—88no—
532Q8—84no—
533Q8—168no—
534Q8—164no—
535D4—22no—
536D4—22no—
537D4—41no—
538D4—41no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4#53612yes—
585C2×C4#53612yes—
586C2×C4D4×Q812yes—
587C2×C4D4×Q812yes—
588C2×C4C22×SD1612yes—
589C2×C4C22×SD1614yes—
590C2×C4C22×SD1614yes—
591C2×C4C22×SD1612yes—
592C2×C4C22×SD1612yes—
593C2×C4C2×AGL(1,ℤ/8)14yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4#57712yes—
596C2×C4C23×Q811yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C2×C4#58511yescharacteristic
602C8—24no—
603C8—44no—
604C8—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#53611yescharacteristic
613C22C2×D4×Q811yescharacteristic
614C22#348811yescharacteristic
615C22C23×SD1611yescharacteristic
616C22C22×AGL(1,ℤ/8)11yescharacteristic
617C22C2×#57711yescharacteristic
618C22#358611yescharacteristic
619C4—24no—
620C4—24no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—24no—
625C4—41no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—84no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4#322714yes—
636C4Q8×SD1614yes—
637C4#350211yescharacteristic
638C4#350311yescharacteristic
639C4#355011yescharacteristic
640C4#355511yescharacteristic
641C2—82no—
642C2#3959711yescharacteristic
643C2C2×#322711yescharacteristic
644C2#6088911yescharacteristic
645C2C2×Q8×SD1611yescharacteristic
646C2Q8×AGL(1,ℤ/8)11yescharacteristic
647C2#6194811yescharacteristic
648C2#6198311yescharacteristic
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. #32134C2semidirect
7. #29287C2semidirect
8. #29273C2semidirect
9. #29206C2semidirect
10. #28809C2semidirect
11. #28687C2semidirect
13. #26278C2semidirect
15. #22821C2semidirect
130. #476Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349292
#331972
#321342
#309542
#299812
#292872
#292732
#292062
#288092
#286872
#278132
#262782
#250582
#228212
#221112
C4×#4762