Finite Group Databasebeta

← #6433322

Subgroups — #6433322

Order 512.

1768 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#35279C211yesmaximal, characteristic
2#34250C211yesmaximal, characteristic
3#29977C214yesmaximal, complemented
4#29975C214yesmaximal, complemented
5#29971C212yesmaximal, complemented
6#29970C212yesmaximal, complemented
7#29968C212yesmaximal, complemented
8#29967C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23744C211yesmaximal, characteristic, complemented
11#23671C212yesmaximal, complemented
12#23668C212yesmaximal, complemented
13#23552C211yesmaximal, characteristic, complemented
14C2×#2058C211yesmaximal, characteristic, complemented
15#20848C212yesmaximal
16#20703C212yesmaximal
17#20630C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C22×#106C2212yes—
24C2×#424C2212yes—
25C2×#408C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2628C2214yes—
29#2568—21no—
30#2567—21no—
31#2563—21no—
32#2542—21no—
33#2487—22no—
34#2409—22no—
35#2351—21no—
36Q8×C4⋊C4—21no—
37#2349—22no—
38#2349—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41#2317C2212yes—
42#2317C2211yescharacteristic
43#2317C2212yes—
44#2316C2212yes—
45#2316C2211yescharacteristic
46#2311C2212yes—
47#2296C2211yescharacteristic
48#2296C2212yes—
49C2×#389C2212yes—
50C2×#389C2211yescharacteristic
51#2285C2211yescharacteristic
52#2101C2212yes—
53#2101C2214yes—
54#2101C2212yes—
55#2100C2211yescharacteristic
56#2100C2212yes—
57#2100C2211yescharacteristic
58#2099C2211yescharacteristic
59#2099C2212yes—
60#2099C2211yescharacteristic
61#2088C2214yes—
62#2088C2214yes—
63#2088C2214yes—
64#2088C2214yes—
65#2088C2214yes—
66#2088C2214yes—
67#2087C2214yes—
68#2086C2214yes—
69#2058C2212yes—
70#2058C2212yes—
71#2058C2212yes—
72#2058C2212yes—
73#2058C2214yes—
74#2057C2212yes—
75#2057C2212yes—
76#1854C2214yes—
77#1854C2214yes—
78#1853C2214yes—
79#1853C2214yes—
80#1825C2214yes—
81#1825C2214yes—
82#1824C2214yes—
83#1823C2214yes—
84#1798C2214yes—
85#1797C2212yes—
86#1797C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106#520—24no—
107#520—44no—
108#520Q818yescomplemented
109C4×#41—44no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4D412yes—
121C2×C4×D4D411yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#428—44no—
140C2×#107C2314yes—
141C2×#106—44no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145#424—44no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#408—28no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151#400—24no—
152#399—24no—
153#395—24no—
154#393—24no—
155#391—22no—
156#391—22no—
157#391—22no—
158#390—22no—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#389C2314yes—
181#386—24no—
182#385—21no—
183#385—22no—
184#385—21no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#384—22no—
191#384—21no—
192#384—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—22no—
196#382—21no—
197#382—22no—
198#382C2312yes—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380C2314yes—
207#380C2314yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C2×#95—21no—
215C2×#95—22no—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41C2×Q814yes—
240C2×#41#4114yes—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—82no—
245C22×Q8—82no—
246C22×Q8C2×D411yescharacteristic
247C22×D4C2×D412yes—
248C22×D4C2×D411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C22⋊Q8—84no—
274C22⋊Q8—84no—
275C4×Q8—28no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8—44no—
279C4×D4—28no—
280C4×D4—216no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287#117—22no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—28no—
292#117—44no—
293#117—44no—
294#117C2×Q814yes—
295#117C2×Q818yescomplemented
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×Q814yes—
333C2×C4⋊C4C2×Q818yescomplemented
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367#107—44no—
368#106—44no—
369#106—44no—
370#106—44no—
371C4⋊C8—44no—
372C4⋊C8—44no—
373C22⋊C8—28no—
374C22⋊C8—28no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—216no—
394#41—216no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—816no—
404C2×Q8C22⋊Q814yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C22⋊Q818yes—
410C2×D4C22⋊Q814yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—81no—
446C22×C4—82no—
447C22×C4—81no—
448C22×C4—82no—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C42−1+414yes—
463C22×C4C2511yescharacteristic
464M16—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—48no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4C22×Q814yes—
513C4⋊C4C22×Q812yes—
514C4⋊C4C22×Q812yes—
515C22⋊C4—24no—
516C22⋊C4—28no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—24no—
526C42—28no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—42no—
536C42—44no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23C2×C22⋊Q814yes—
548C23C2×C22⋊Q812yes—
549C23#53412yes—
550C23#53612yes—
551C23C23×D411yescharacteristic
552Q8—28no—
553Q8—88no—
554Q8—88no—
555D4—216no—
556D4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—88no—
610C2×C4—88no—
611C2×C4—162no—
612C2×C4—164no—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C22⋊Q812yes—
615C2×C4#53412yes—
616C2×C4#53612yes—
617C2×C4#53918yes—
618C2×C4#54418yes—
619C2×C4#57712yes—
620C2×C4#57712yes—
621C2×C4C23×Q811yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×2−1+412yes—
625C8—42no—
626C8—42no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22#262714yes—
642C22C22×C22⋊Q811yescharacteristic
643C22C2×#53411yescharacteristic
644C22C2×#53611yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57711yescharacteristic
647C22#358511yescharacteristic
648C22#358611yescharacteristic
649C4—22no—
650C4—22no—
651C4—24no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—81no—
660C4—84no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4#262714yes—
666C4C2×#53912yes—
667C4C2×#54412yes—
668C2—24no—
669C2—44no—
670C2C2×#262711yescharacteristic
671C2#6049511yescharacteristic
672C2#6049711yescharacteristic
673C2#6075711yescharacteristic
674C2#6075811yescharacteristic
675C2#6088111yescharacteristic
676C2#6088611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #29977C2semidirect
4. #29975C2semidirect
5. #29971C2semidirect
6. #29970C2semidirect
7. #29968C2semidirect
8. #29967C2semidirect
9. #23747C2semidirect
10. #23744C2semidirect
11. #23671C2semidirect
12. #23668C2semidirect
13. #23552C2semidirect
14. C2×#2058C2semidirect
108. #520Q8semidirect
295. #117C2×Q8semidirect
333. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#342502
#299772
#299752
#299712
#299702
#299682
#299672
#237472
#237442
#236712
#236682
#235522
C2×#20582
#208482
#207032
#206302