Finite Group Databasebeta

← #6852331

Subgroups — #6852331

Order 512.

1577 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32360C214yesmaximal, complemented
5#31536C214yesmaximal, complemented
6#29957C212yesmaximal
7#28542C212yesmaximal, complemented
8#27953C212yesmaximal
9#27824C212yesmaximal, complemented
10#27680C212yesmaximal
11#25811C211yesmaximal, characteristic
12#25453C212yesmaximal, complemented
13#24817C212yesmaximal, complemented
14#24302C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#451C2211yescharacteristic
24C2×C4×Q16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2590—22no—
28#2568C2211yescharacteristic
29#2567—22no—
30#2567—21no—
31#2567—21no—
32#2567C2214yes—
33#2567C2212yes—
34#2567C2212yes—
35#2552C2211yescharacteristic
36#2511—21no—
37#2508C2212yes—
38#2505—22no—
39#2502—21no—
40#2453C2212yes—
41C2×#395—22no—
42#2365C2211yescharacteristic
43#2360—21no—
44#2355—21no—
45Q8×C4⋊C4C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51C2×#384C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2214yes—
54#2097C2214yes—
55#2096C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2058C2214yes—
61#2057C2212yes—
62#2026C2214yes—
63#2024C2212yes—
64#1992C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1971C2214yes—
68#1948C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1927C2214yes—
72#1917C2214yes—
73#1878C2214yes—
74#1878C2212yes—
75#1872C2211yescharacteristic
76#1872C2212yes—
77#1865C2214yes—
78#1863C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1847C2214yes—
85#1803C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2211yescharacteristic
90#1787C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#498—44no—
127#477—44no—
128#475—44no—
129#451Q818yescomplemented
130C4×Q16—44no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#400—22no—
153#398—22no—
154#398—22no—
155#398—22no—
156#395—24no—
157#395—24no—
158#395—24no—
159#395—24no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391C2314yes—
167#391C2314yes—
168#391C2312yes—
169#391C2311yescharacteristic
170#389—22no—
171#389—22no—
172#389—24no—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—21no—
196#384—21no—
197#384—21no—
198#384—22no—
199#384—22no—
200#384C2314yes—
201#384C2314yes—
202#384C2311yescharacteristic
203#382—22no—
204#382C2312yes—
205#380—21no—
206#380C2311yescharacteristic
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2×Q814yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×Q16—42no—
257C2×Q16—42no—
258C2×Q16C2×Q818yescomplemented
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268#125—44no—
269C22⋊Q8—22no—
270C22⋊Q8—44no—
271C22⋊Q8C2×Q814yes—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#107—42no—
346#107—44no—
347#106—44no—
348#106—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—44no—
353#103—42no—
354#103—42no—
355#103—42no—
356#103—44no—
357#103—44no—
358#103C2×Q818yescomplemented
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8C2×Q814yes—
362#97—44no—
363C4×C8—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×Q8C22×Q814yes—
412C2×Q8C22×Q812yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4#13714yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×Q812yes—
455C22×C42−1+412yes—
456C22×C4C2511yescharacteristic
457Q16—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8C22×Q814yes—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4#13714yes—
505C4⋊C4C22×Q812yes—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—82no—
522C42H(5,𝔽2)14yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23D4×Q812yes—
528C23C2×#13712yes—
529C23#57712yes—
530C23C23×D411yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—24no—
534Q8—48no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—88no—
539Q8—88no—
540Q8—84no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—84no—
596C2×C4—88no—
597C2×C4#53512yes—
598C2×C4#53512yes—
599C2×C4D4×Q812yes—
600C2×C4C2×#13714yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C23×Q811yescharacteristic
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58512yes—
608C8—24no—
609C8—42no—
610C8—44no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22#338714yes—
618C22C2×#53511yescharacteristic
619C22C2×D4×Q811yescharacteristic
620C22#348511yescharacteristic
621C22C22×#13711yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358511yescharacteristic
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—84no—
642C4#321814yes—
643C4#350112yes—
644C4#354112yes—
645C2—24no—
646C2#3959411yescharacteristic
647C2C2×#321811yescharacteristic
648C2#6081211yescharacteristic
649C2C2×#338711yescharacteristic
650C2Q8×#13711yescharacteristic
651C2#6171911yescharacteristic
652C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #32360C2semidirect
5. #31536C2semidirect
7. #28542C2semidirect
9. #27824C2semidirect
12. #25453C2semidirect
13. #24817C2semidirect
129. #451Q8semidirect
258. C2×Q16C2×Q8semidirect
358. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#25672
#329542
#323602
#315362
#299572
#285422
#279532
#278242
#276802
#258112
#254532
#248172
#243022
#226292
#220502
C4×#4512