Finite Group Databasebeta

← #6463621

Subgroups — #6463621

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#29267C214yesmaximal, complemented
4#29264C214yesmaximal, complemented
5#29262C212yesmaximal, complemented
6#29260C212yesmaximal, complemented
7#29259C212yesmaximal, complemented
8#29255C212yesmaximal, complemented
9#23732C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic
11#23452C211yesmaximal, characteristic
12C2×#2024C211yesmaximal, characteristic, complemented
13#21262C212yesmaximal
14#21175C212yesmaximal, complemented
15#20880C212yesmaximal, complemented
16#20878C212yesmaximal
17#20754C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#424C2212yes—
25C2×#417C2212yes—
26C2×#419C2212yes—
27C2×#416C2212yes—
28C22×#103C2212yes—
29#2369—21no—
30#2369—21no—
31#2365—21no—
32#2365—21no—
33#2362—21no—
34#2362—21no—
35#2360—21no—
36#2360—21no—
37#2359—22no—
38#2359—22no—
39#2345—22no—
40Q8×C22⋊C4—22no—
41#2331C2214yes—
42#2327C2212yes—
43#2326C2212yes—
44#2317C2211yescharacteristic
45#2316C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×#124C2212yes—
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52#2285C2211yescharacteristic
53#2274C2212yes—
54C4×#117C2211yescharacteristic
55#2097C2212yes—
56#2097C2214yes—
57#2096C2211yescharacteristic
58#2096C2212yes—
59#2095C2211yescharacteristic
60#2095C2212yes—
61#2094C2211yescharacteristic
62#2093C2211yescharacteristic
63#2092C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2024C2214yes—
69#2023C2212yes—
70#2023C2212yes—
71#1983C2214yes—
72#1983C2214yes—
73#1981C2214yes—
74#1981C2214yes—
75#1949C2214yes—
76#1949C2214yes—
77#1948C2214yes—
78#1947C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1865C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1863C2214yes—
85#1863C2214yes—
86#1862C2214yes—
87#1832C2214yes—
88#1831C2212yes—
89#1831C2212yes—
90#1736C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#536—24no—
101#536Q818yescomplemented
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#124C2312yes—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112#520—44no—
113#519—44no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—22no—
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134#428—44no—
135#427—44no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#419—44no—
146#417—28no—
147#416—44no—
148C2×#103—28no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159#391—22no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391—24no—
164#391—24no—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#390C2311yescharacteristic
170#390C2311yescharacteristic
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2314yes—
180#389C2314yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—24no—
190#387—24no—
191#386—21no—
192#386—21no—
193#385—22no—
194#385—22no—
195#385—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#382C2312yes—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—24no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376C2312yes—
226C2×#95—21no—
227C2×#95—21no—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239#336C2312yes—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41C2×Q814yes—
244C22×Q8—28no—
245C22×Q8—21no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—82no—
252C22×Q8—82no—
253C22×Q8C2×Q814yes—
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—24no—
280C4⋊Q8C2×Q818yescomplemented
281#124—24no—
282#124C2×Q818yescomplemented
283#123—216no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—84no—
292C4×Q8—84no—
293C4×D4—44no—
294C4×D4—48no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—48no—
300#117—44no—
301#117C2×Q814yes—
302#117#4114yes—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4#4118yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364#107—42no—
365#107—42no—
366#107—44no—
367#106—42no—
368#106—42no—
369#106—44no—
370C4⋊C8—28no—
371C4⋊C8—28no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—42no—
376#103—42no—
377#103—44no—
378#103—44no—
379#102—28no—
380#102—42no—
381#102—42no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—21no—
403C24C22×D411yescharacteristic
404#41—216no—
405#41—48no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—48no—
420C2×Q8—44no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8—816no—
424C2×Q8C22×Q814yes—
425C2×Q8C22×Q812yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C22×Q812yes—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q812yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4C2511yescharacteristic
476M16—48no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—48no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22⋊Q814yes—
515C4⋊C4C22⋊Q814yes—
516C4⋊C4C2×#4118yes—
517C22⋊C4—24no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4C22⋊Q818yes—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42—42no—
542C42—84no—
543C42—84no—
544C42C22×Q814yes—
545C42C2×#4114yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×C22⋊Q812yes—
558C23#52512yes—
559C23#53512yes—
560C23C23×D411yescharacteristic
561Q8—28no—
562Q8—24no—
563Q8—416no—
564Q8—48no—
565Q8—44no—
566Q8—42no—
567Q8—88no—
568D4—24no—
569D4—48no—
570D4—42no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—88no—
620C2×C4—164no—
621C2×C4—162no—
622C2×C4C2×C22⋊Q814yes—
623C2×C4C2×C22⋊Q814yes—
624C2×C4C2×C22⋊Q812yes—
625C2×C4#52512yes—
626C2×C4#53512yes—
627C2×C4#55418yes—
628C2×C4#55918yes—
629C2×C4#57712yes—
630C2×C4#57812yes—
631C2×C4C23×Q811yescharacteristic
632C2×C4C22×#4112yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)12yes—
635C8—42no—
636C8—42no—
637C8—44no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22#258014yes—
652C22C22×C22⋊Q811yescharacteristic
653C22C2×#52511yescharacteristic
654C22C2×#53511yescharacteristic
655C22C2×#57711yescharacteristic
656C22C2×#57811yescharacteristic
657C22#358511yescharacteristic
658C22#358611yescharacteristic
659C4—24no—
660C4—24no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—48no—
669C4—41no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4#258014yes—
676C4C2×#55412yes—
677C4C2×#55912yes—
678C2—24no—
679C2—82no—
680C2C2×#258011yescharacteristic
681C2#6049711yescharacteristic
682C2#6049811yescharacteristic
683C2#6057911yescharacteristic
684C2#6058311yescharacteristic
685C2#6082411yescharacteristic
686C2#6082911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #29267C2semidirect
4. #29264C2semidirect
5. #29262C2semidirect
6. #29260C2semidirect
7. #29259C2semidirect
8. #29255C2semidirect
9. #23732C2semidirect
12. C2×#2024C2semidirect
14. #21175C2semidirect
15. #20880C2semidirect
17. #20754C2semidirect
101. #536Q8semidirect
280. C4⋊Q8C2×Q8semidirect
282. #124C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#342762
#342162
#292672
#292642
#292622
#292602
#292592
#292552
#237322
#237302
#234522
C2×#20242
#212622
#211752
#208802
#208782
#207542