Finite Group Databasebeta

← #6508491

Subgroups — #6508491

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2454C211yesmaximal, characteristic, complemented
3#29109C211yesmaximal, characteristic, complemented
4#29088C214yesmaximal, complemented
5#28966C214yesmaximal, complemented
6#28310C211yesmaximal, characteristic
7#26827C214yesmaximal, complemented
8#26488C214yesmaximal, complemented
9#25913C211yesmaximal, characteristic, complemented
10#25819C211yesmaximal, characteristic, complemented
11#24607C212yesmaximal, complemented
12#24506C212yesmaximal, complemented
13#22076C212yesmaximal, complemented
14C4×#447C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#501C2212yescomplemented
20C2×#490C2212yescomplemented
21C2×#481C2212yes—
22C2×#447C2212yes—
23C2×C4×D8C2214yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yescomplemented
33#2506C2212yescomplemented
34#2454—22no—
35#2454—22no—
36#2454C2214yes—
37#2454C2214yes—
38#2452C2212yescomplemented
39#2445C2212yescomplemented
40C2×#399—21no—
41C2×#396—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#385—21no—
49C2×#380—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2094C2212yes—
52#2088C2214yescomplemented
53#2071C2212yes—
54#2067C2214yescomplemented
55#2065C2214yes—
56#2059C2214yescomplemented
57#2054C2214yes—
58#2025C2214yescomplemented
59#2023C2212yes—
60#2011C2212yes—
61#1992C2212yescomplemented
62#1992C2212yescomplemented
63#1972C2214yes—
64#1972C2214yes—
65#1943C2214yescomplemented
66#1941C2214yescomplemented
67#1936C2214yes—
68#1936C2214yes—
69#1936C2214yescomplemented
70#1934C2214yescomplemented
71#1920C2214yescomplemented
72#1879C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1864C2214yescomplemented
76#1862C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1786C2212yes—
81#1771C2214yescomplemented
82#1771C2214yescomplemented
83C4×#102C2214yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×D8C2314yes—
90#539—48no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#125—22no—
95C2×#125—21no—
96C2×#125—21no—
97C2×#125—44no—
98C2×#125C2312yes—
99C2×#125C2312yes—
100C2×#123—22no—
101C2×#123—22no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—44no—
110C2×C4×D4C2314yes—
111C2×C4×D4C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#501—44no—
130#490—44no—
131#481—28no—
132#447—28no—
133C4×D8—48no—
134#432—28no—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×#97C2314yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#399—24no—
153#399—24no—
154#398—22no—
155#397—22no—
156#396—22no—
157#396—22no—
158#396—24no—
159#396—24no—
160#394—22no—
161#394—22no—
162#393—22no—
163#391—21no—
164#390—21no—
165#390—21no—
166#390—22no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2314yes—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—24no—
194#385—24no—
195#385—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#380—21no—
199#380—24no—
200#380—24no—
201#380C2312yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C22×Q8—21no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—24no—
229C22×D4—42no—
230C22×D4C2412yes—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×D8—28no—
248C2×D8—28no—
249C2×D8—44no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—42no—
253#127—42no—
254#126—48no—
255#125—24no—
256#125—24no—
257#125—42no—
258#125—42no—
259#125—44no—
260#125—44no—
261#125—48no—
262#123—48no—
263#123—48no—
264C22⋊Q8—48no—
265C22⋊Q8—44no—
266#121—24no—
267#121—24no—
268#121—44no—
269#121—44no—
270C4×D4—28no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—48no—
274C4×D4—48no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
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—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347#107—28no—
348#106—28no—
349#106—44no—
350C4⋊C8—28no—
351#102—28no—
352#102—28no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8#4118yes—
359#97—28no—
360#97—42no—
361#97—42no—
362C4×C8—44no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—42no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—416no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D42−1+414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—48no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—84no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—816no—
450C22×C4AGL(1,ℤ/8)14yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C2×#4114yes—
461C22×C4C2511yescharacteristic
462D8—416no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8C2×#4118yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—81no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—81no—
499C4⋊C4H(5,𝔽2)14yes—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—84no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—42no—
525C42—41no—
526C42—41no—
527C42—44no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—24no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23#53212yes—
554C23C2×AGL(1,ℤ/8)12yes—
555C23#57512yes—
556C23C23×D411yescharacteristic
557Q8—162no—
558D4—28no—
559D4—48no—
560D4—44no—
561D4—44no—
562D4—44no—
563D4—88no—
564D4—162no—
565D4—162no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—164no—
621C2×C4#53212yes—
622C2×C4C2×AGL(1,ℤ/8)14yes—
623C2×C4C22×#4112yes—
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C8—28no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—816no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22#317114yes—
650C22C2×#53211yescharacteristic
651C22#348411yescharacteristic
652C22#348611yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C2×#57511yescharacteristic
655C22#358511yescharacteristic
656C22#358511yescharacteristic
657C4—22no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—44no—
664C4—41no—
665C4—42no—
666C4—41no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#352014yes—
673C2—24no—
674C2—84no—
675C2—162no—
676C2#3539711yescharacteristic
677C2C2×#317111yescharacteristic
678C2#6070411yescharacteristic
679C2#6156911yescharacteristic
680C2#6157711yescharacteristic
681C2#6182511yescharacteristic
682C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28310C2
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
1. #35006C2semidirect
2. C2×#2454C2semidirect
3. #29109C2semidirect
4. #29088C2semidirect
5. #28966C2semidirect
7. #26827C2semidirect
8. #26488C2semidirect
9. #25913C2semidirect
10. #25819C2semidirect
11. #24607C2semidirect
12. #24506C2semidirect
13. #22076C2semidirect
14. C4×#447C2semidirect
15. #20534C2semidirect
19. C2×#501C22semidirect
20. C2×#490C22semidirect
32. #2512C22semidirect
33. #2506C22semidirect
38. #2452C22semidirect
39. #2445C22semidirect
52. #2088C22semidirect
54. #2067C22semidirect
56. #2059C22semidirect
58. #2025C22semidirect
61. #1992C22semidirect
62. #1992C22semidirect
65. #1943C22semidirect
66. #1941C22semidirect
69. #1936C22semidirect
70. #1934C22semidirect
71. #1920C22semidirect
75. #1864C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24542
#291092
#290882
#289662
#283102
#268272
#264882
#259132
#258192
#246072
#245062
#220762
C4×#4472
#205342