Finite Group Databasebeta

← #6506558

Subgroups — #6506558

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic, complemented
3#29335C211yesmaximal, characteristic, complemented
4#29334C211yesmaximal, characteristic, complemented
5#29330C212yesmaximal, complemented
6#27977C212yesmaximal, complemented
7#27710C214yesmaximal, complemented
8#27708C214yesmaximal, complemented
9#27284C214yesmaximal, complemented
10#27280C214yesmaximal, complemented
11#25913C211yesmaximal, characteristic, complemented
12#25912C211yesmaximal, characteristic, complemented
13#21769C212yesmaximal, complemented
14#20909C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#525C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C2×#491C2212yescomplemented
20C2×#490C2212yescomplemented
21C2×#417C2214yes—
22C2×#407C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2850C2212yes—
26#2617—21no—
27#2617—21no—
28#2602—22no—
29#2602—22no—
30#2500C2212yes—
31#2499C2212yes—
32#2442—22no—
33#2442—22no—
34#2442C2214yes—
35#2442C2214yes—
36#2432C2212yes—
37#2431C2212yes—
38C2×#400—21no—
39C2×#398—21no—
40#2325—21no—
41#2325—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#382C2211yescharacteristic
46C2×#380—21no—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2101C2214yescomplemented
50#2100C2212yescomplemented
51#2099C2212yescomplemented
52#2074C2214yescomplemented
53#2073C2214yescomplemented
54#2043C2212yescomplemented
55#2042C2214yescomplemented
56#2041C2212yescomplemented
57#2026C2214yescomplemented
58#2025C2214yescomplemented
59#1992C2212yescomplemented
60#1992C2212yescomplemented
61#1958C2214yescomplemented
62#1936C2214yescomplemented
63#1935C2214yescomplemented
64#1934C2214yescomplemented
65#1933C2214yescomplemented
66#1895C2214yes—
67#1894C2214yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1874C2214yes—
72#1874C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2214yes—
77#1872C2212yes—
78#1835C2214yescomplemented
79#1825C2214yes—
80#1788C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2211yescharacteristic
84#1749C2211yescharacteristic
85C4×C22⋊C8C2212yes—
86#1736C2214yes—
87C22×#41C2312yes—
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×M16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#125—21no—
94C2×#125—21no—
95C2×#125—44no—
96C2×#125C2312yes—
97C2×#125C2312yes—
98C2×#124C2312yes—
99#525—28no—
100#525—28no—
101C2×#123—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#120—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×#117C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—44no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#491—44no—
130#490—44no—
131#417—48no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140#407—28no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#398—24no—
151#397—22no—
152#397—22no—
153#394—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#392—22no—
158#392—22no—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—24no—
184#385—24no—
185#385—22no—
186#385—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382C2314yes—
191#382C2312yes—
192#380—21no—
193#380—21no—
194#380—22no—
195#380—24no—
196#380—24no—
197#380C2314yes—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4C2312yes—
201C4×C22⋊C4C2312yes—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C25C2×D411yescharacteristic
221C2×#41—24no—
222C2×#41—24no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4#4118yes—
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C2×M16—28no—
253C2×M16—44no—
254C2×M16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258#127—42no—
259#125—24no—
260#125—24no—
261#125—42no—
262#125—42no—
263#125—44no—
264#125—44no—
265#124—44no—
266#123—48no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271#121—24no—
272#121—24no—
273#121—24no—
274#121—24no—
275#121—44no—
276#121—44no—
277#120—48no—
278C4×D4—48no—
279C4×D4—48no—
280#117—24no—
281#117—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
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—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#103—28no—
349#103—44no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—44no—
354#102—44no—
355#102—44no—
356C22⋊C8—28no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359#97—42no—
360#97—42no—
361C4×C8—44no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24#12414yes—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388#41—416no—
389#41—88no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4—82no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—416no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—816no—
460C22×C4#12414yes—
461C22×C4#12418yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4C2511yescharacteristic
474M16—88no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4H(5,𝔽2)14yes—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—42no—
532C42—41no—
533C42—41no—
534C42—44no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—28no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C2×#12414yes—
560C23C2×#12412yes—
561C23#57512yes—
562C23#57512yes—
563C23C23×D411yescharacteristic
564Q8—44no—
565Q8—44no—
566D4—44no—
567D4—44no—
568D4—88no—
569D4—88no—
570D4—88no—
571D4—162no—
572D4—162no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
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—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—88no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—816no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C2×#12414yes—
632C2×C4C2×#12412yes—
633C2×C4C22×#4112yes—
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4#58512yes—
636C8—44no—
637C8—44no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—416no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22#314214yes—
660C22C22×#12411yescharacteristic
661C22#348411yescharacteristic
662C22#348511yescharacteristic
663C22C2×#57511yescharacteristic
664C22C2×#57511yescharacteristic
665C22#358511yescharacteristic
666C22#358611yescharacteristic
667C4—24no—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—41no—
673C4—44no—
674C4—41no—
675C4—41no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#350514yes—
683C2—24no—
684C2—44no—
685C2—162no—
686C2#3537711yescharacteristic
687C2C2×#314211yescharacteristic
688C2#6062111yescharacteristic
689C2#6158111yescharacteristic
690C2#6158311yescharacteristic
691C2#6171111yescharacteristic
692C2#6171311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2442C2semidirect
3. #29335C2semidirect
4. #29334C2semidirect
5. #29330C2semidirect
6. #27977C2semidirect
7. #27710C2semidirect
8. #27708C2semidirect
9. #27284C2semidirect
10. #27280C2semidirect
11. #25913C2semidirect
12. #25912C2semidirect
13. #21769C2semidirect
14. #20909C2semidirect
15. #20524C2semidirect
19. C2×#491C22semidirect
20. C2×#490C22semidirect
49. #2101C22semidirect
50. #2100C22semidirect
51. #2099C22semidirect
52. #2074C22semidirect
53. #2073C22semidirect
54. #2043C22semidirect
55. #2042C22semidirect
56. #2041C22semidirect
57. #2026C22semidirect
58. #2025C22semidirect
59. #1992C22semidirect
60. #1992C22semidirect
61. #1958C22semidirect
62. #1936C22semidirect
63. #1935C22semidirect
64. #1934C22semidirect
65. #1933C22semidirect
78. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#350322
C2×#24422
#293352
#293342
#293302
#279772
#277102
#277082
#272842
#272802
#259132
#259122
#217692
#209092
#205242