Finite Group Databasebeta

← #6446034

Subgroups — #6446034

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#25069C214yesmaximal, complemented
4#25068C214yesmaximal, complemented
5#25059C212yesmaximal, complemented
6#25058C212yesmaximal, complemented
7#25057C212yesmaximal, complemented
8#25056C212yesmaximal, complemented
9#23735C212yesmaximal
10#23662C212yesmaximal, complemented
11#21302C211yesmaximal, characteristic, complemented
12#21169C211yesmaximal, characteristic, complemented
13#21063C212yesmaximal
14#21035C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#427C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C2×#418C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C22×#102C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2848—22no—
29#2626—22no—
30#2580C2214yes—
31#2566—22no—
32#2563—21no—
33#2542—21no—
34#2538C2212yes—
35#2538C2212yes—
36#2360—22no—
37#2360—22no—
38#2355C2212yescomplemented
39#2355C2212yescomplemented
40Q8×C4⋊C4C2212yescomplemented
41Q8×C4⋊C4C2212yescomplemented
42#2330—22no—
43#2317C2212yes—
44#2317C2212yes—
45#2309C2212yes—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8—21no—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2274C2212yes—
51C4×#117C2211yescharacteristic
52#2094C2214yes—
53#2093C2214yes—
54#2092C2214yes—
55#2092C2214yes—
56#2085C2214yescomplemented
57#2085C2214yescomplemented
58#2084C2214yescomplemented
59#2083C2214yescomplemented
60#1985C2214yescomplemented
61#1984C2212yescomplemented
62#1984C2212yescomplemented
63#1949C2212yescomplemented
64#1949C2212yescomplemented
65#1948C2211yescharacteristic, complemented
66#1948C2211yescharacteristic, complemented
67#1947C2211yescharacteristic, complemented
68#1947C2211yescharacteristic, complemented
69#1926C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yescomplemented
74#1920C2214yescomplemented
75#1919C2214yescomplemented
76#1918C2214yescomplemented
77#1847C2214yescomplemented
78#1845C2212yescomplemented
79#1844C2212yescomplemented
80#1802C2212yes—
81#1802C2212yes—
82#1802C2214yes—
83#1762C2212yes—
84#1762C2212yes—
85C4×#102C2212yes—
86C4×#102C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8D411yescharacteristic
102C2×#125D412yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8D412yes—
106C2×C22⋊Q8D412yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—21no—
126C22×C42C2311yescharacteristic
127#428—28no—
128#428—44no—
129#427—44no—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8—24no—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142#418—24no—
143#418Q818yescomplemented
144C2×#102—44no—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C4×M16—24no—
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154#399—24no—
155#396—24no—
156#396—24no—
157#393—24no—
158#391—22no—
159#391—22no—
160#390—22no—
161#390—22no—
162#390C2314yes—
163#390C2314yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—24no—
170#389C2314yes—
171#389C2314yes—
172#387—24no—
173#387—24no—
174#385—24no—
175#385—24no—
176#385—24no—
177#385—24no—
178#384—22no—
179#384—21no—
180#384—21no—
181#384C2312yes—
182#384C2312yes—
183#384C2314yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—22no—
187#382C2314yes—
188#382C2312yes—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#379—24no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—24no—
209#376—21no—
210C43—21no—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#336C2314yes—
222#336C2314yes—
223#336C2314yes—
224#336C2314yes—
225C25—21no—
226C22×Q8C2×D412yes—
227C22×Q8C2×D412yes—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4C2×Q814yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16C2×Q814yes—
251C2×M16#4114yes—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261#125—28no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—84no—
265C22⋊Q8—84no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—84no—
271C4×Q8—84no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—44no—
277C4×D4—44no—
278#117—24no—
279#117—24no—
280#117—22no—
281#117—22no—
282#117—24no—
283#117—22no—
284#117—24no—
285#117C2×Q814yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×D414yes—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4114yes—
343C2×C42#4114yes—
344C2×C42C2414yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#107—42no—
349#107—42no—
350#107—44no—
351#106—28no—
352#106—42no—
353#106—42no—
354#106—44no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—24no—
358#102—24no—
359#102—24no—
360#102—44no—
361#102—44no—
362#102C2×Q818yes—
363#102C2×Q818yes—
364#97—24no—
365C4×C8—24no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378C24—22no—
379C24—24no—
380C24—42no—
381C24—42no—
382C24—81no—
383C24—81no—
384C24—81no—
385C24—81no—
386C24C22×D411yescharacteristic
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—28no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4C22×Q814yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—816no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#12314yes—
432C22×C4#12314yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×Q812yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447M16—28no—
448M16—28no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—48no—
460C2×C8C22×Q814yes—
461C2×C8C2×#4114yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4C22×Q812yes—
497C4⋊C4C22×Q812yes—
498C22⋊C4—28no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—44no—
515C42—42no—
516C42—42no—
517C42—84no—
518C42—84no—
519C42#12314yes—
520C42#12314yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23C2×#12312yes—
544C23D4×Q812yes—
545C23#55012yes—
546C23C23×D411yescharacteristic
547Q8—48no—
548Q8—48no—
549D4—24no—
550D4—24no—
551D4—48no—
552D4—42no—
553D4—42no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4—164no—
596C2×C4C2×#12312yes—
597C2×C4C2×#12312yes—
598C2×C4C2×#12312yes—
599C2×C4C2×#12312yes—
600C2×C4C2×#12312yes—
601C2×C4#53314yes—
602C2×C4#54214yes—
603C2×C4D4×Q812yes—
604C2×C4#55012yes—
605C2×C4#55214yes—
606C2×C4#55414yes—
607C2×C4#57612yes—
608C2×C4#57712yes—
609C2×C4C23×Q811yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C8—24no—
616C8—24no—
617C8—42no—
618C8—42no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—816no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#242314yes—
634C22C22×#12311yescharacteristic
635C22C2×D4×Q811yescharacteristic
636C22C2×#55011yescharacteristic
637C22C2×#57611yescharacteristic
638C22C2×#57711yescharacteristic
639C22#358511yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4—81no—
653C4—82no—
654C4#242314yes—
655C4#309614yes—
656C4C2×#53311yescharacteristic
657C4C2×#54211yescharacteristic
658C4C2×#55211yescharacteristic
659C4C2×#55411yescharacteristic
660C2—24no—
661C2—84no—
662C2C2×#242311yescharacteristic
663C2C2×#309611yescharacteristic
664C2#6051711yescharacteristic
665C2#6138111yescharacteristic
666C2#6138211yescharacteristic
667C2#6141911yescharacteristic
668C2#6142011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #25069C2semidirect
4. #25068C2semidirect
5. #25059C2semidirect
6. #25058C2semidirect
7. #25057C2semidirect
8. #25056C2semidirect
10. #23662C2semidirect
11. #21302C2semidirect
12. #21169C2semidirect
14. #21035C2semidirect
15. #20800C2semidirect
38. #2355C22semidirect
39. #2355C22semidirect
40. Q8×C4⋊C4C22semidirect
41. Q8×C4⋊C4C22semidirect
56. #2085C22semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #1985C22semidirect
61. #1984C22semidirect
62. #1984C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1947C22semidirect
68. #1947C22semidirect
73. #1921C22semidirect
74. #1920C22semidirect
75. #1919C22semidirect
76. #1918C22semidirect
77. #1847C22semidirect
78. #1845C22semidirect
79. #1844C22semidirect
143. #418Q8semidirect

Maximal subgroups

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

subgroupindex
#351722
#342142
#250692
#250682
#250592
#250582
#250572
#250562
#237352
#236622
#213022
#211692
#210632
#210352
#208002
C2×#18022
C4×#4182