Finite Group Databasebeta

← #6433962

Subgroups — #6433962

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#30372C214yesmaximal, complemented
4#30368C214yesmaximal, complemented
5#30362C214yesmaximal, complemented
6#30360C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23640C211yesmaximal, characteristic, complemented
10#23639C211yesmaximal, characteristic, complemented
11#23551C211yesmaximal, characteristic, complemented
12#23486C211yesmaximal, characteristic
13#23486C211yesmaximal, characteristic
14#23454C211yesmaximal, characteristic, complemented
15#21158C211yesmaximal, characteristic, complemented
16#21157C211yesmaximal, characteristic, complemented
17#21089C211yesmaximal, characteristic, complemented
18#21089C211yesmaximal, characteristic, complemented
19#20378C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#408C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2854C2214yes—
36#2851C2214yescomplemented
37#2628C2214yescomplemented
38#2624C2214yes—
39#2581—22no—
40#2580—22no—
41#2564—24no—
42#2448—22no—
43#2445—22no—
44#2418—24no—
45#2316C2211yescharacteristic
46#2316C2211yescharacteristic
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2101C2214yescomplemented
56#2101C2214yescomplemented
57#2092C2212yes—
58#2092C2212yes—
59#2092C2212yes—
60#2092C2212yes—
61#2077C2214yescomplemented
62#2077C2214yescomplemented
63#2076C2214yescomplemented
64#2075C2214yescomplemented
65#2059C2212yescomplemented
66#2059C2212yescomplemented
67#2058C2212yescomplemented
68#2057C2212yescomplemented
69#2034C2214yes—
70#2034C2214yes—
71#2033C2214yes—
72#2033C2214yes—
73#2025C2214yes—
74#2023C2214yes—
75#1944C2214yescomplemented
76#1943C2214yescomplemented
77#1942C2214yescomplemented
78#1941C2214yescomplemented
79#1936C2214yes—
80#1936C2214yes—
81#1934C2214yes—
82#1934C2214yes—
83#1750C2212yescomplemented
84#1750C2212yescomplemented
85#1749C2212yescomplemented
86C4×C22⋊C8C2212yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—42no—
104#535—42no—
105#533—42no—
106#533—42no—
107C2×C4⋊Q8C2312yes—
108C2×#127—21no—
109C2×#125C2312yes—
110C2×#124—21no—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×#121C2314yes—
114C2×#120—22no—
115C4×#41—42no—
116C4×#41—42no—
117C2×C4×Q8—22no—
118C2×C4×Q8D412yes—
119C2×C4×D4D414yes—
120C2×C4×D4D412yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#427—42no—
134#427—42no—
135C2×#107—24no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#106—24no—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145#418—42no—
146#418—42no—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#408—24no—
156#408—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#405—42no—
162#405—42no—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167#400—24no—
168#399—24no—
169#399—24no—
170#397—24no—
171#395—24no—
172#394—24no—
173#394—24no—
174#393—24no—
175#391—21no—
176#391—21no—
177#391—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—22no—
182#390—22no—
183#390C2314yes—
184#390C2312yes—
185#390C2312yes—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2314yes—
192#389C2314yes—
193#389C2314yes—
194#388—24no—
195#388—24no—
196#387—24no—
197#387—24no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—24no—
202#385—24no—
203#385—24no—
204#385—22no—
205#382—22no—
206#382C2312yes—
207#382C2312yes—
208#380—22no—
209#380—22no—
210#380—22no—
211#380C2314yes—
212#380C2314yes—
213#380C2314yes—
214#379—22no—
215C4×C4⋊C4—22no—
216C2×#95—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
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×Q8—22no—
241C22×Q8C2×D411yescharacteristic
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D414yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8#4114yes—
271C22×C8#4114yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276C22×C8C2411yescharacteristic
277C22×C8C2411yescharacteristic
278C4⋊Q8—42no—
279C4⋊Q8—42no—
280#127—44no—
281#125—42no—
282#125—42no—
283#124—44no—
284#123—88no—
285C22⋊Q8—48no—
286#121—44no—
287#121—44no—
288#120—48no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×D4—44no—
292C4×D4—48no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—42no—
304#117—42no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D414yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42C2×D412yes—
355C2×C42C2×D414yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359#107—24no—
360#107—24no—
361#107—42no—
362#107—42no—
363#106—24no—
364#106—24no—
365#106—42no—
366#106—42no—
367#102—24no—
368#102—24no—
369#102—24no—
370#102—24no—
371#102—44no—
372#102—44no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#97—42no—
378#97—42no—
379C4×C8—42no—
380C4×C8—42no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24—82no—
397C24C22×D412yes—
398C24C22×D411yescharacteristic
399#41—44no—
400#41—44no—
401#41—44no—
402#41—44no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—84no—
430C2×D4—84no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468M16—44no—
469M16—44no—
470C2×C8—28no—
471C2×C8—28no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8C2×#4112yes—
487C2×C8C2×#4112yes—
488C2×C8C2×#4112yes—
489C2×C8C2×#4112yes—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—84no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—44no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—42no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—88no—
556C23—84no—
557C23—162no—
558C23—162no—
559C23—164no—
560C23#53212yes—
561C23#53412yes—
562C23#53612yes—
563C23C23×D411yescharacteristic
564Q8—44no—
565Q8—88no—
566D4—48no—
567D4—44no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—84no—
573D4—168no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
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—84no—
609C2×C4—88no—
610C2×C4—162no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4#53212yes—
614C2×C4#53412yes—
615C2×C4#53612yes—
616C2×C4#53914yes—
617C2×C4#53914yes—
618C2×C4#54214yes—
619C2×C4#54214yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+411yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—22no—
627C8—22no—
628C8—22no—
629C8—22no—
630C8—42no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22—168no—
647C22#266314yes—
648C22C2×#53211yescharacteristic
649C22C2×#53411yescharacteristic
650C22C2×#53611yescharacteristic
651C22#358412yes—
652C22#358512yes—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#266314yes—
665C4C2×#53911yescharacteristic
666C4C2×#53911yescharacteristic
667C4C2×#54211yescharacteristic
668C4C2×#54211yescharacteristic
669C2—24no—
670C2—82no—
671C2—162no—
672C2C2×#266311yescharacteristic
673C2#6071311yescharacteristic
674C2#6071911yescharacteristic
675C2#6076712yes—
676C2#6089012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35334C2semidirect
3. #30372C2semidirect
4. #30368C2semidirect
5. #30362C2semidirect
6. #30360C2semidirect
7. #23746C2semidirect
8. C2×#2092C2semidirect
9. #23640C2semidirect
10. #23639C2semidirect
11. #23551C2semidirect
14. #23454C2semidirect
15. #21158C2semidirect
16. #21157C2semidirect
17. #21089C2semidirect
18. #21089C2semidirect
19. #20378C2semidirect
36. #2851C22semidirect
37. #2628C22semidirect
55. #2101C22semidirect
56. #2101C22semidirect
61. #2077C22semidirect
62. #2077C22semidirect
63. #2076C22semidirect
64. #2075C22semidirect
65. #2059C22semidirect
66. #2059C22semidirect
67. #2058C22semidirect
68. #2057C22semidirect
75. #1944C22semidirect
76. #1943C22semidirect
77. #1942C22semidirect
78. #1941C22semidirect
83. #1750C22semidirect
84. #1750C22semidirect
85. #1749C22semidirect
86. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353342
#352542
#303722
#303682
#303622
#303602
#237462
C2×#20922
#236402
#236392
#235512
#234862
#234542
#211582
#211572
#210892
#203782