Finite Group Databasebeta

← #6628104

Subgroups — #6628104

Order 512.

1743 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33110C212yesmaximal, complemented
4#33107C211yesmaximal, characteristic
5#29330C212yesmaximal, complemented
6#29237C212yesmaximal, complemented
7#27129C214yesmaximal, complemented
8#26689C214yesmaximal, complemented
9#26513C212yesmaximal, complemented
10#25903C211yesmaximal, characteristic
11#25279C212yesmaximal, complemented
12#24301C211yesmaximal, characteristic, complemented
13#23216C212yesmaximal, complemented
14#23062C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#484C211yesmaximal, characteristic, complemented
17C4×#481C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#486C2212yes—
21C2×#484C2211yescharacteristic
22C2×#481C2212yes—
23C2×#475C2212yes—
24#2856C2211yescharacteristic
25#2850C2212yes—
26#2597—22no—
27#2571C2211yescharacteristic
28#2548C2211yescharacteristic
29#2513—21no—
30#2513C2212yes—
31#2504—21no—
32#2447C2212yes—
33#2444—22no—
34#2366C2211yescharacteristic
35#2365—22no—
36#2365—21no—
37#2365—21no—
38#2365C2212yes—
39#2365C2214yes—
40#2365C2212yes—
41#2361—21no—
42#2357—21no—
43#2355C2211yescharacteristic
44C4×C22⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46C2×#391—21no—
47C2×#389C2212yes—
48C2×#387—22no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2087C2214yes—
54#2065C2214yes—
55#2058C2212yes—
56#2042C2214yes—
57#2024C2212yes—
58#2023C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1983C2214yes—
62#1982C2212yes—
63#1949C2214yes—
64#1948C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1928C2212yes—
69#1927C2214yes—
70#1926C2214yes—
71#1926C2214yes—
72#1925C2214yes—
73#1921C2212yes—
74#1919C2214yes—
75#1919C2214yes—
76#1916C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1862C2214yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1824C2211yescharacteristic
86#1797C2212yes—
87#1797C2211yescharacteristic
88C4×#107C2212yes—
89C4×#106C2214yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2212yes—
92C4×#102C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#124C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—24no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—48no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—22no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#486—28no—
127#484—24no—
128#481—28no—
129#475—28no—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2311yescharacteristic
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#397—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#394—22no—
154#391—22no—
155#391—22no—
156#391—24no—
157#391C2312yes—
158#391C2311yescharacteristic
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389—22no—
169#389—22no—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—21no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—21no—
190#385—22no—
191#384C2311yescharacteristic
192#382—21no—
193#382—21no—
194#382C2312yes—
195#381—22no—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8#4118yes—
261#127—24no—
262#124—44no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265#121—24no—
266#121—24no—
267#121—44no—
268#121—82no—
269C4×Q8—24no—
270C4×Q8—48no—
271C4×Q8—416no—
272C4×Q8—48no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—84no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4#4114yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
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—22no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42—48no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—42no—
343#106—28no—
344#106—28no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—28no—
348#103—24no—
349#103—42no—
350#102—28no—
351#102—28no—
352#102—28no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C4×C8—24no—
358C4×C8—44no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24C22×D411yescharacteristic
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—432no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4#13514yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C42−1+412yes—
451C22×C4C2511yescharacteristic
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—48no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4AGL(1,ℤ/8)14yes—
490C4⋊C4C2×#4114yes—
491C4⋊C4C2×#4118yes—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—48no—
513C42—42no—
514C42—48no—
515C42—44no—
516C42—42no—
517C42—82no—
518C42C2×#4114yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23#54812yes—
531C23C2×#13512yes—
532C23#57712yes—
533C23C23×D411yescharacteristic
534Q8—416no—
535Q8—416no—
536Q8—42no—
537Q8—42no—
538D4—44no—
539D4—44no—
540D4—88no—
541D4—84no—
542D4—162no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—164no—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4#54812yes—
600C2×C4C2×#13512yes—
601C2×C4C2×#13512yes—
602C2×C4C2×AGL(1,ℤ/8)14yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4112yes—
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58512yes—
608C8—42no—
609C8—44no—
610C8—42no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22#336514yes—
625C22C2×#52511yescharacteristic
626C22C2×#54811yescharacteristic
627C22#348511yescharacteristic
628C22C22×#13511yescharacteristic
629C22C22×AGL(1,ℤ/8)11yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358611yescharacteristic
632C4—24no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—48no—
640C4—44no—
641C4—41no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4#310514yes—
649C4#348012yes—
650C4#351612yes—
651C2—24no—
652C2—162no—
653C2#3658211yescharacteristic
654C2C2×#310511yescharacteristic
655C2#6058211yescharacteristic
656C2C2×#336511yescharacteristic
657C2#6131711yescharacteristic
658C2#6170111yescharacteristic
659C2#6172811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#2365C2semidirect
3. #33110C2semidirect
5. #29330C2semidirect
6. #29237C2semidirect
7. #27129C2semidirect
8. #26689C2semidirect
9. #26513C2semidirect
11. #25279C2semidirect
12. #24301C2semidirect
13. #23216C2semidirect
14. #23062C2semidirect
15. #22625C2semidirect
16. C4×#484C2semidirect
17. C4×#481C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#23652
#331102
#331072
#293302
#292372
#271292
#266892
#265132
#259032
#252792
#243012
#232162
#230622
#226252
C4×#4842
C4×#4812