Finite Group Databasebeta

← #6478193

Subgroups — #6478193

Order 512.

1776 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic, complemented
5#32979C212yesmaximal, complemented
6#29969C211yesmaximal, characteristic, complemented
7#29890C212yesmaximal, complemented
8#29755C211yesmaximal, characteristic, complemented
9#27471C214yesmaximal, complemented
10#27407C214yesmaximal, complemented
11#27343C214yesmaximal, complemented
12#27175C214yesmaximal, complemented
13#22082C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic, complemented
15#21580C212yesmaximal, complemented
16C2×#538—21no—
17C23×C4⋊C4C2211yescharacteristic
18C2×#481C2212yes—
19C2×#481C2212yes—
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2847C2212yes—
23#2639—21no—
24#2617—21no—
25#2559—22no—
26#2557—22no—
27#2496C2212yes—
28#2491C2212yes—
29#2428C2212yes—
30#2422C2212yes—
31#2345—22no—
32#2345—22no—
33#2345C2214yes—
34#2345C2214yes—
35#2325—21no—
36C4×#121C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#387—21no—
40C2×#387—21no—
41C2×#380—21no—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4—21no—
46C2×C4×C22⋊C4C2211yescharacteristic
47#2100C2212yescomplemented
48#2087C2214yescomplemented
49#2084C2212yescomplemented
50#2075C2214yes—
51#2075C2214yes—
52#2065C2214yescomplemented
53#2057C2212yescomplemented
54#2041C2212yescomplemented
55#2033C2214yes—
56#2033C2214yes—
57#1987C2212yescomplemented
58#1982C2212yescomplemented
59#1948C2212yescomplemented
60#1947C2212yescomplemented
61#1944C2214yescomplemented
62#1936C2214yescomplemented
63#1926C2214yescomplemented
64#1925C2214yescomplemented
65#1924C2214yes—
66#1924C2214yes—
67#1921C2214yescomplemented
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1874C2214yes—
71#1874C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1862C2214yescomplemented
75#1853C2214yes—
76#1853C2214yes—
77#1852C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1779C2212yes—
84#1776C2214yescomplemented
85#1762C2214yescomplemented
86#1749C2212yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89#538—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#124—22no—
94C2×#123—22no—
95C2×C22⋊Q8—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121—22no—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#121C2312yes—
102C2×#120—22no—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#481—28no—
124#481—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140#400—22no—
141#397—22no—
142#397—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#393—22no—
147#392—22no—
148#391—21no—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2314yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#387—22no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#385—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#380—21no—
178#380—22no—
179#380—24no—
180#380—24no—
181#380C2312yes—
182#380C2312yes—
183#380C2314yes—
184#380C2312yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—24no—
193C4×C22⋊C4—24no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4D412yes—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4C2312yes—
198C4×C22⋊C4C2312yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C22×Q8—21no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—28no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—82no—
237C23×C4—82no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8C2414yes—
247C22×C8C2414yes—
248C4⋊Q8—42no—
249#127—42no—
250#126—48no—
251#124—48no—
252#123—48no—
253C22⋊Q8—44no—
254#121—24no—
255#121—24no—
256#121—48no—
257#121—44no—
258#121—44no—
259#121—44no—
260#121—44no—
261#120—48no—
262C4×D4—44no—
263C4×D4—48no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—82no—
324C2×C42—82no—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—44no—
332#107—44no—
333#106—28no—
334#106—28no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337#102—28no—
338#102—28no—
339#102—44no—
340#102—44no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—44no—
344C22⋊C8—44no—
345#97—44no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24AGL(1,ℤ/8)14yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—81no—
388C2×D4—81no—
389C2×D4—81no—
390C2×D4—81no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—82no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—816no—
446C22×C4#13714yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2511yescharacteristic
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C4⋊C4—216no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4C2×#4114yes—
489C4⋊C4C2×#4114yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—42no—
512C42—44no—
513C42—82no—
514C42—82no—
515C42—84no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—28no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23#52512yes—
541C23C2×AGL(1,ℤ/8)14yes—
542C23C2×AGL(1,ℤ/8)12yes—
543C23C2×#13712yes—
544C23C23×D411yescharacteristic
545Q8—162no—
546D4—44no—
547D4—44no—
548D4—88no—
549D4—162no—
550D4—162no—
551D4—164no—
552D4—164no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
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—82no—
606C2×C4—82no—
607C2×C4—816no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—164no—
611C2×C4#52512yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4#58512yes—
617C2×C4#58512yes—
618C8—44no—
619C8—44no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—416no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22#311214yes—
642C22C2×#52511yescharacteristic
643C22#348511yescharacteristic
644C22#348511yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C22×#13711yescharacteristic
647C22#358511yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#345512yes—
665C4#349312yes—
666C2—24no—
667C2—44no—
668C2—162no—
669C2#3509311yescharacteristic
670C2C2×#311211yescharacteristic
671C2#6056411yescharacteristic
672C2#6170711yescharacteristic
673C2#6170711yescharacteristic
674C2#6170911yescharacteristic
675C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
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×#2345C2semidirect
3. #33149C2semidirect
4. #33107C2semidirect
5. #32979C2semidirect
6. #29969C2semidirect
7. #29890C2semidirect
8. #29755C2semidirect
9. #27471C2semidirect
10. #27407C2semidirect
11. #27343C2semidirect
12. #27175C2semidirect
13. #22082C2semidirect
14. #21906C2semidirect
15. #21580C2semidirect
47. #2100C22semidirect
48. #2087C22semidirect
49. #2084C22semidirect
52. #2065C22semidirect
53. #2057C22semidirect
54. #2041C22semidirect
57. #1987C22semidirect
58. #1982C22semidirect
59. #1948C22semidirect
60. #1947C22semidirect
61. #1944C22semidirect
62. #1936C22semidirect
63. #1926C22semidirect
64. #1925C22semidirect
67. #1921C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
74. #1862C22semidirect
84. #1776C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23452
#331492
#331072
#329792
#299692
#298902
#297552
#274712
#274072
#273432
#271752
#220822
#219062
#215802