Finite Group Databasebeta

← #6800731

Subgroups — #6800731

Order 512.

1541 conjugacy classes of subgroups.

The subgroup lattice has 667 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#33198C211yesmaximal, characteristic, complemented
4#32442C214yesmaximal, complemented
5#31652C214yesmaximal, complemented
6#29980C212yesmaximal, complemented
7#29148C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic
9#28418C212yesmaximal
10#28252C211yesmaximal, characteristic
11#28159C212yesmaximal, complemented
12#27738C212yesmaximal, complemented
13#25932C212yesmaximal
14#24333C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×Q82C2212yes—
19C2×#486C2212yes—
20C2×#477C2212yes—
21C2×#462C2211yescharacteristic
22C2×#460C2212yes—
23C2×#461C2212yes—
24C2×C4×Q16C2212yes—
25#2868C2212yes—
26#2868C2211yescharacteristic
27#2593—22no—
28#2582C2211yescharacteristic
29#2571C2211yescharacteristic
30#2567C2212yes—
31#2567C2214yes—
32#2551—22no—
33#2551—22no—
34#2551—21no—
35#2551—21no—
36#2548C2211yescharacteristic
37#2548C2211yescharacteristic
38#2513—21no—
39#2513C2212yes—
40#2448—21no—
41#2448C2212yes—
42#2395—22no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#124—21no—
49C4×#124C2212yes—
50#2293—21no—
51#2293C2212yes—
52C42×Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2099C2214yes—
55#2093C2214yes—
56#2074C2214yes—
57#2071C2214yes—
58#2057C2214yes—
59#2055C2214yes—
60#2024C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#2015C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1973C2214yes—
69#1959C2214yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1935C2214yes—
74#1928C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1878C2214yes—
79#1878C2212yes—
80#1863C2214yes—
81#1842C2211yescharacteristic
82#1836C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97Q82—44no—
98Q82—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#127C2311yescharacteristic
109C2×#124—21no—
110C2×#124C2312yes—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124#486—44no—
125#477—44no—
126#462—24no—
127#461—28no—
128#460—44no—
129C4×Q16—44no—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#395—24no—
155#395—24no—
156#395—24no—
157#394—22no—
158#394—22no—
159#391—21no—
160#391C2312yes—
161#391C2314yes—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—24no—
174#389—22no—
175#389—24no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#384—22no—
193#384—21no—
194#384—22no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2314yes—
201#382—21no—
202#382C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C43—21no—
223C43C2311yescharacteristic
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—22no—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—416no—
260C4⋊Q8—48no—
261C4⋊Q8—84no—
262C4⋊Q8—82no—
263C4⋊Q8—82no—
264C4⋊Q8—84no—
265C4⋊Q8#4114yes—
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—24no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—41no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—48no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—82no—
290#117—84no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
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×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—42no—
357#107—44no—
358C4⋊C8—24no—
359C4⋊C8—22no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—42no—
363#103—28no—
364#103—24no—
365#103—42no—
366#103—42no—
367#103—42no—
368#103—44no—
369#102—44no—
370#102—44no—
371#102—44no—
372#97—44no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—22no—
385#95—24no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—84no—
408C2×Q8—84no—
409C2×Q8#13714yes—
410C2×Q8C2×#4114yes—
411C2×Q82−1+412yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446Q16—48no—
447SD16—88no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C42—24no—
508C42—22no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—41no—
519C42—42no—
520C42—42no—
521C42—48no—
522C42—42no—
523C42—44no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—81no—
528C42#13714yes—
529C42C2×#4112yes—
530C42H(5,𝔽2)12yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C23×D411yescharacteristic
541Q8—28no—
542Q8—24no—
543Q8—22no—
544Q8—24no—
545Q8—22no—
546Q8—41no—
547Q8—42no—
548Q8—41no—
549Q8—42no—
550Q8—164no—
551Q8—168no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—84no—
557D4—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—88no—
603C2×C4#53512yes—
604C2×C4#53512yes—
605C2×C4#54812yes—
606C2×C4#54812yes—
607C2×C4C2×#13714yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4#57612yes—
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4#58511yescharacteristic
619C8—44no—
620C8—42no—
621C8—42no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22C2×#53511yescharacteristic
630C22C2×#54811yescharacteristic
631C22#348911yescharacteristic
632C22C22×#13711yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57611yescharacteristic
635C22#358511yescharacteristic
636C4—24no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—44no—
645C4—42no—
646C4—82no—
647C4—81no—
648C4—81no—
649C4—84no—
650C4—82no—
651C4—81no—
652C4#321814yes—
653C4#337514yes—
654C4#351411yescharacteristic
655C4#352011yescharacteristic
656C4#354711yescharacteristic
657C4#355211yescharacteristic
658C2—162no—
659C2#3912411yescharacteristic
660C2C2×#321811yescharacteristic
661C2#6083211yescharacteristic
662C2C2×#337511yescharacteristic
663C2#6131911yescharacteristic
664C2#6211211yescharacteristic
665C2#6214911yescharacteristic
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. #33198C2semidirect
4. #32442C2semidirect
5. #31652C2semidirect
6. #29980C2semidirect
7. #29148C2semidirect
11. #28159C2semidirect
12. #27738C2semidirect
14. #24333C2semidirect
15. #22825C2semidirect
16. #22152C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349542
#331982
#324422
#316522
#299802
#291482
#287162
#284182
#282522
#281592
#277382
#259322
#243332
#228252
#221522
C4×#4622