Finite Group Databasebeta

← #6448210

Subgroups — #6448210

Order 512.

1765 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35556C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#26332C214yesmaximal, complemented
4#26330C214yesmaximal, complemented
5#26326C214yesmaximal, complemented
6#26316C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8C2×#2093C211yesmaximal, characteristic, complemented
9#23632C211yesmaximal, characteristic, complemented
10#23630C211yesmaximal, characteristic, complemented
11#21304C211yesmaximal, characteristic, complemented
12#21263C211yesmaximal, characteristic
13#21206C211yesmaximal, characteristic
14#21173C211yesmaximal, characteristic, complemented
15#21067C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic, complemented
17#21060C211yesmaximal, characteristic, complemented
18#21058C211yesmaximal, characteristic, complemented
19#20455C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#405C2211yescharacteristic
36#2592C2214yes—
37#2548C2214yes—
38#2491—22no—
39#2490—22no—
40#2472—24no—
41#2360—24no—
42#2355C2214yescomplemented
43Q8×C4⋊C4C2214yescomplemented
44#2330—22no—
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2311C2211yescharacteristic
48#2309C2211yescharacteristic
49C4×C4⋊Q8—22no—
50#2297C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2095C2212yescomplemented
59#2093C2214yes—
60#2093C2214yes—
61#2093C2212yescomplemented
62#2092C2212yescomplemented
63#2073C2214yes—
64#2072C2214yescomplemented
65#2071C2214yes—
66#2071C2214yescomplemented
67#1985C2214yescomplemented
68#1984C2214yescomplemented
69#1983C2214yes—
70#1982C2214yes—
71#1961C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1957C2212yes—
75#1949C2214yescomplemented
76#1947C2214yescomplemented
77#1928C2212yescomplemented
78#1928C2214yescomplemented
79#1927C2212yescomplemented
80#1926C2214yes—
81#1926C2214yes—
82#1926C2212yescomplemented
83#1925C2214yes—
84#1925C2214yes—
85#1925C2212yescomplemented
86#1925C2214yescomplemented
87#1776C2214yescomplemented
88C4×#107C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—24no—
104#535—24no—
105#533—24no—
106#533—24no—
107C2×C4⋊Q8D412yes—
108C2×#127C2312yes—
109C2×#125D412yes—
110C2×#124—24no—
111C2×#124C2312yes—
112C2×C22⋊Q8D414yes—
113C2×#121—22no—
114C2×#120—22no—
115C2×C4×Q8—24no—
116C2×C4×Q8C2314yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—22no—
127C22×C42C2311yescharacteristic
128#428—42no—
129#427—24no—
130C2×#107—24no—
131C2×#107—42no—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139#424—24no—
140C2×C4⋊C8C2314yes—
141#418—24no—
142#418—42no—
143#417—42no—
144#417—42no—
145#416—24no—
146#416—24no—
147C2×#103C2311yescharacteristic
148C2×#103C2311yescharacteristic
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#102—24no—
153C2×#102—42no—
154C2×#102C2311yescharacteristic
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2311yescharacteristic
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165#405—24no—
166C2×#97C2312yes—
167C2×C4×C8C2312yes—
168#397—24no—
169#394—24no—
170#394—24no—
171#393—24no—
172#391—24no—
173#391C2312yes—
174#391C2312yes—
175#390—22no—
176#390—24no—
177#390—21no—
178#390—21no—
179#390C2314yes—
180#390C2312yes—
181#390C2312yes—
182#390C2314yes—
183#389—22no—
184#389—22no—
185#389—24no—
186#389—24no—
187#389—24no—
188#388—24no—
189#388—24no—
190#387—24no—
191#387—24no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2312yes—
200#384C2314yes—
201#382C2312yes—
202#382C2312yes—
203#380—22no—
204#379—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220#376—21no—
221C43—21no—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C22×Q8C2×D414yes—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—24no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4C2412yes—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276C4⋊Q8—28no—
277#127—24no—
278#127—24no—
279#125—28no—
280#124—24no—
281#124—24no—
282#124—88no—
283C22⋊Q8—216no—
284#121—48no—
285#120—48no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—88no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117#4114yes—
297#117#4114yes—
298#117#4114yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—84no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—24no—
355#107—24no—
356#107—24no—
357#107—44no—
358#107—42no—
359#106—24no—
360#106—42no—
361C4⋊C8—28no—
362#103—22no—
363#103—22no—
364#103—24no—
365#103—42no—
366#103—42no—
367#102—24no—
368#102—24no—
369#102—22no—
370#102—22no—
371#102—24no—
372#102—28no—
373#102—44no—
374#102—42no—
375#102—42no—
376#102—42no—
377#102—42no—
378#97—24no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390C24—22no—
391C24—21no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24—82no—
397C24C22×D411yescharacteristic
398#41—44no—
399#41—44no—
400#41—44no—
401#41—44no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—44no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—48no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—84no—
435C2×D4—84no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468M16—44no—
469M16—44no—
470C2×C8—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—44no—
485C2×C8C2×#4112yes—
486C2×C8C2×#4112yes—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4#12318yes—
512C4⋊C4C2×#4114yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4#12318yes—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—84no—
538C42—84no—
539C42—84no—
540C42C2×#4114yes—
541C42C2×#4114yes—
542C23—22no—
543C23—22no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—88no—
555C23—84no—
556C23—164no—
557C23—162no—
558C23—162no—
559C23C2×#12312yes—
560C23#54812yes—
561C23#55012yes—
562C23C23×D411yescharacteristic
563Q8—416no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—41no—
572D4—42no—
573D4—41no—
574D4—42no—
575D4—41no—
576D4—84no—
577D4—84no—
578D4—168no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—162no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4C2×#12318yes—
614C2×C4C2×#12312yes—
615C2×C4#52914yes—
616C2×C4#54214yes—
617C2×C4#54812yes—
618C2×C4#55012yes—
619C2×C4#55614yes—
620C2×C4#55914yes—
621C2×C4#57614yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C8—22no—
629C8—22no—
630C8—42no—
631C8—42no—
632C8—42no—
633C22—24no—
634C22—24no—
635C22—22no—
636C22—22no—
637C22—82no—
638C22—84no—
639C22—81no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—81no—
644C22—82no—
645C22—81no—
646C22—82no—
647C22—168no—
648C22#244814yes—
649C22C22×#12311yescharacteristic
650C22C2×#54811yescharacteristic
651C22C2×#55011yescharacteristic
652C22C2×#57612yes—
653C22#358612yes—
654C4—24no—
655C4—28no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4#244814yes—
667C4C2×#52911yescharacteristic
668C4C2×#54211yescharacteristic
669C4C2×#55611yescharacteristic
670C4C2×#55911yescharacteristic
671C2—24no—
672C2—161no—
673C2—162no—
674C2C2×#244811yescharacteristic
675C2#6052112yes—
676C2#6134512yes—
677C2#6140911yescharacteristic
678C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #26332C2semidirect
4. #26330C2semidirect
5. #26326C2semidirect
6. #26316C2semidirect
7. #23737C2semidirect
8. C2×#2093C2semidirect
9. #23632C2semidirect
10. #23630C2semidirect
11. #21304C2semidirect
14. #21173C2semidirect
15. #21067C2semidirect
16. #21065C2semidirect
17. #21060C2semidirect
18. #21058C2semidirect
19. #20455C2semidirect
42. #2355C22semidirect
43. Q8×C4⋊C4C22semidirect
57. #2097C22semidirect
58. #2095C22semidirect
61. #2093C22semidirect
62. #2092C22semidirect
64. #2072C22semidirect
66. #2071C22semidirect
67. #1985C22semidirect
68. #1984C22semidirect
75. #1949C22semidirect
76. #1947C22semidirect
77. #1928C22semidirect
78. #1928C22semidirect
79. #1927C22semidirect
82. #1926C22semidirect
85. #1925C22semidirect
86. #1925C22semidirect
87. #1776C22semidirect
88. C4×#107C22semidirect

Maximal subgroups

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

subgroupindex
#355562
#342142
#263322
#263302
#263262
#263162
#237372
C2×#20932
#236322
#236302
#213042
#212632
#212062
#211732
#210672
#210652
#210602
#210582
#204552