Finite Group Databasebeta

← #6848159

Subgroups — #6848159

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#32230C214yesmaximal, complemented
5#32134C214yesmaximal, complemented
6#29639C212yesmaximal
7#28791C212yesmaximal, complemented
8#27961C211yesmaximal, characteristic, complemented
9#27813C212yesmaximal, complemented
10#27645C211yesmaximal, characteristic
11#27611C212yesmaximal, complemented
12#26444C212yesmaximal, complemented
13#26024C212yesmaximal, complemented
14#24603C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21834C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C2×#552—21no—
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#481C2211yescharacteristic
22C2×#481C2212yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2854C2212yes—
26#2552C2211yescharacteristic
27#2551—21no—
28#2551—21no—
29#2551—22no—
30#2551C2212yes—
31#2551C2212yes—
32#2551C2214yes—
33#2542C2211yescharacteristic
34#2519—22no—
35#2512C2212yes—
36#2506C2212yes—
37#2498—22no—
38#2489—21no—
39#2417—21no—
40C2×#395—22no—
41#2365C2211yescharacteristic
42#2362—21no—
43#2354C2211yescharacteristic
44C4×#121C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47#2285C2212yes—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2088C2214yes—
54#2084C2214yes—
55#2084C2212yes—
56#2071C2212yes—
57#2065C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#1990C2212yes—
63#1990C2214yes—
64#1990C2214yes—
65#1988C2212yes—
66#1987C2214yes—
67#1987C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1941C2214yes—
71#1939C2212yes—
72#1934C2214yes—
73#1926C2212yes—
74#1920C2214yes—
75#1920C2214yes—
76#1919C2212yes—
77#1919C2214yes—
78#1916C2214yes—
79#1874C2211yescharacteristic
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1834C2212yes—
83#1824C2212yes—
84#1824C2212yes—
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1824C2212yes—
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96#552—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2314yes—
108C2×#126—22no—
109C2×#125—21no—
110C2×#125—22no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×#121C2311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#481—24no—
134#481—44no—
135#431—44no—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#399—22no—
152#398—22no—
153#398—22no—
154#397—22no—
155#396—22no—
156#395—24no—
157#395—24no—
158#395—24no—
159#395—24no—
160#391—21no—
161#391—21no—
162#390C2312yes—
163#390C2311yescharacteristic
164#390C2312yes—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2311yescharacteristic
176#389C2314yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—21no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#382—22no—
195#380—21no—
196#380—22no—
197#380—21no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—48no—
239C22×Q8—48no—
240C22×Q8—84no—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—24no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—48no—
262C4⋊Q8—416no—
263C4⋊Q8—48no—
264C4⋊Q8—48no—
265#126—48no—
266#125—48no—
267#125—44no—
268#123—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272#121—22no—
273#121—22no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—44no—
277#117—48no—
278#117—44no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4SD1618yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#106—24no—
354#106—28no—
355#106—24no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#102—24no—
360#102—24no—
361#102—44no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—24no—
365C4×C8—24no—
366C4×C8—44no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24C22×D411yescharacteristic
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—48no—
393C2×Q8—84no—
394C2×Q8—84no—
395C2×Q8—832no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C2×SD1618yes—
436C22×C4C2×SD1614yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4C2511yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4C2×SD1614yes—
493C4⋊C4C2×SD1614yes—
494C4⋊C4C2×SD1618yes—
495C4⋊C4AGL(1,ℤ/8)14yes—
496C4⋊C4C2×#4114yes—
497C4⋊C42−1+414yes—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—81no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42—42no—
521C42—44no—
522C422−1+414yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#54612yes—
535C23C22×SD1612yes—
536C23#57812yes—
537C23C23×D411yescharacteristic
538Q8—816no—
539Q8—816no—
540Q8—84no—
541D4—42no—
542D4—42no—
543D4—84no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—216no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—88no—
595C2×C4—84no—
596C2×C4#53612yes—
597C2×C4#53612yes—
598C2×C4#54612yes—
599C2×C4C22×SD1614yes—
600C2×C4C22×SD1612yes—
601C2×C4C22×SD1614yes—
602C2×C4C22×SD1612yes—
603C2×C4C2×AGL(1,ℤ/8)14yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+412yes—
608C2×C4C2×2−1+412yes—
609C8—42no—
610C8—42no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22#329814yes—
626C22C2×#53611yescharacteristic
627C22C2×#54611yescharacteristic
628C22#348611yescharacteristic
629C22C23×SD1611yescharacteristic
630C22C22×AGL(1,ℤ/8)11yescharacteristic
631C22C2×#57811yescharacteristic
632C22#358511yescharacteristic
633C4—22no—
634C4—22no—
635C4—24no—
636C4—24no—
637C4—22no—
638C4—24no—
639C4—44no—
640C4—41no—
641C4—42no—
642C4—44no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4—81no—
648C4—84no—
649C4#322714yes—
650C4#350912yes—
651C4#354412yes—
652C2—24no—
653C2—161no—
654C2#3946511yescharacteristic
655C2C2×#322711yescharacteristic
656C2#6087411yescharacteristic
657C2C2×#329811yescharacteristic
658C2#6115511yescharacteristic
659C2#6182011yescharacteristic
660C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2551C2semidirect
3. #33147C2semidirect
4. #32230C2semidirect
5. #32134C2semidirect
7. #28791C2semidirect
8. #27961C2semidirect
9. #27813C2semidirect
11. #27611C2semidirect
12. #26444C2semidirect
13. #26024C2semidirect
15. #22599C2semidirect
16. #21834C2semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#25512
#331472
#322302
#321342
#296392
#287912
#279612
#278132
#276452
#276112
#264442
#260242
#246032
#225992
#218342
C4×#4812