Finite Group Databasebeta

← #6448195

Subgroups — #6448195

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#26330C214yesmaximal, complemented
4#26329C214yesmaximal, complemented
5#26322C212yesmaximal, complemented
6#26321C212yesmaximal, complemented
7#26319C212yesmaximal, complemented
8#26316C212yesmaximal, complemented
9#23737C212yesmaximal, complemented
10#23634C212yesmaximal, complemented
11#21302C211yesmaximal, characteristic, complemented
12#21257C211yesmaximal, characteristic
13#21200C211yesmaximal, characteristic
14C2×#1947C211yesmaximal, characteristic, complemented
15#21067C212yesmaximal, complemented
16#21062C212yesmaximal, complemented
17C4×#428C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#428C2211yescharacteristic
21C2×#428C2211yescharacteristic
22C22×#107C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C2×#417C2212yes—
25C2×#417C2212yes—
26C2×#416C2212yes—
27C2×#416C2212yes—
28C2×C4×M16C2211yescharacteristic
29#2850—22no—
30#2639—22no—
31#2592C2214yes—
32#2568C2212yes—
33#2563—21no—
34#2562—22no—
35#2548C2212yes—
36#2547—21no—
37#2360—22no—
38#2360—22no—
39#2355C2212yes—
40#2355C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43#2330—22no—
44#2317C2212yes—
45#2311C2212yes—
46#2311C2212yes—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8—21no—
49C2×#384C2211yescharacteristic
50#2274C2212yes—
51C4×#117C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2095C2214yes—
55#2093C2214yes—
56#2092C2214yes—
57#2074C2214yes—
58#2073C2214yes—
59#2072C2214yes—
60#2071C2214yes—
61#1985C2214yes—
62#1984C2212yes—
63#1984C2212yes—
64#1983C2212yes—
65#1983C2212yes—
66#1982C2211yescharacteristic
67#1982C2211yescharacteristic
68#1981C2211yescharacteristic
69#1981C2211yescharacteristic
70#1959C2214yes—
71#1957C2212yes—
72#1956C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2214yes—
76#1928C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1925C2214yes—
83#1925C2214yes—
84#1776C2212yes—
85#1776C2212yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93C22×#41C2312yes—
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C23×C8C2312yes—
97#533—24no—
98#533—24no—
99#532—24no—
100#532—24no—
101C2×C4⋊Q8D411yescharacteristic
102C2×C4⋊Q8D411yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#125D412yes—
106C2×#124—22no—
107C2×#124C2312yes—
108C2×C22⋊Q8D412yes—
109C2×C22⋊Q8D412yes—
110C2×#121—21no—
111C2×#121—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128#428—24no—
129#428—24no—
130C2×#107—44no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×C4⋊C8—24no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141#417—28no—
142#417—44no—
143#416—28no—
144#416—44no—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C4×M16—24no—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#400—24no—
161#397—24no—
162#394—24no—
163#393—24no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#391C2314yes—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—24no—
178#388—24no—
179#388—24no—
180#387—24no—
181#387—24no—
182#386—24no—
183#386—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384C2314yes—
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382C2314yes—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—24no—
213#376—21no—
214C2×#95—21no—
215C43—21no—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#336C2314yes—
223#336C2314yes—
224#336C2314yes—
225#336C2314yes—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—24no—
232C22×Q8C2×D412yes—
233C22×Q8C2×D412yes—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—22no—
251C2×M16#4114yes—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263#127—22no—
264#127—22no—
265#125—28no—
266#124—24no—
267#124—24no—
268#124—84no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271#121—44no—
272#121—44no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—84no—
278C4×Q8—84no—
279C4×D4—44no—
280C4×D4—44no—
281#117—24no—
282#117—24no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117#4114yes—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4C2×D414yes—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—82no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4114yes—
342C2×C42#4114yes—
343C2×C42C2414yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—24no—
347#107—24no—
348#107—44no—
349#107—44no—
350#106—24no—
351#106—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355#103—28no—
356#103—24no—
357#103—24no—
358#103—42no—
359#103—42no—
360#103—44no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—42no—
365#102—42no—
366#102—44no—
367#97—24no—
368C4×C8—24no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24—22no—
384C24C22×D411yescharacteristic
385#41—48no—
386#41—48no—
387#41—48no—
388#41—48no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4#12314yes—
441C22×C4#12314yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456M16—28no—
457M16—44no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—48no—
470C2×C8C2×#4114yes—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4C2×#4112yes—
501C4⋊C4C2×#4112yes—
502C22⋊C4—28no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—24no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—82no—
521C42—84no—
522C42—84no—
523C42#12314yes—
524C42#12314yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C2×#12312yes—
541C23#54812yes—
542C23#55012yes—
543C23C23×D411yescharacteristic
544Q8—48no—
545Q8—48no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—88no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4—164no—
599C2×C4C2×#12312yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4C2×#12312yes—
603C2×C4C2×#12312yes—
604C2×C4#52914yes—
605C2×C4#54214yes—
606C2×C4#54812yes—
607C2×C4#55012yes—
608C2×C4#55614yes—
609C2×C4#55914yes—
610C2×C4#57612yes—
611C2×C4#57812yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C8—24no—
619C8—42no—
620C8—42no—
621C8—42no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#244814yes—
636C22C22×#12311yescharacteristic
637C22C2×#54811yescharacteristic
638C22C2×#55011yescharacteristic
639C22C2×#57611yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358511yescharacteristic
642C22#358611yescharacteristic
643C4—22no—
644C4—22no—
645C4—22no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—81no—
653C4—82no—
654C4—84no—
655C4—81no—
656C4#244814yes—
657C4#309814yes—
658C4C2×#52911yescharacteristic
659C4C2×#54211yescharacteristic
660C4C2×#55611yescharacteristic
661C4C2×#55911yescharacteristic
662C2—24no—
663C2—162no—
664C2C2×#244811yescharacteristic
665C2C2×#309811yescharacteristic
666C2#6051711yescharacteristic
667C2#6134511yescharacteristic
668C2#6134811yescharacteristic
669C2#6141911yescharacteristic
670C2#6142211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #26330C2semidirect
4. #26329C2semidirect
5. #26322C2semidirect
6. #26321C2semidirect
7. #26319C2semidirect
8. #26316C2semidirect
9. #23737C2semidirect
10. #23634C2semidirect
11. #21302C2semidirect
14. C2×#1947C2semidirect
15. #21067C2semidirect
16. #21062C2semidirect
17. C4×#428C2semidirect

Maximal subgroups

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

subgroupindex
#351802
#342142
#263302
#263292
#263222
#263212
#263192
#263162
#237372
#236342
#213022
#212572
#212002
C2×#19472
#210672
#210622
C4×#4282