Finite Group Databasebeta

← #6433324

Subgroups — #6433324

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34250C211yesmaximal, characteristic
3#29979C214yesmaximal, complemented
4#29978C214yesmaximal, complemented
5#29971C212yesmaximal, complemented
6#29971C212yesmaximal, complemented
7#29967C212yesmaximal, complemented
8#29966C212yesmaximal, complemented
9#23745C211yesmaximal, characteristic, complemented
10C2×#2099C211yesmaximal, characteristic, complemented
11#23671C212yesmaximal, complemented
12#23669C212yesmaximal, complemented
13#23554C211yesmaximal, characteristic, complemented
14#23551C211yesmaximal, characteristic, complemented
15#20849C212yesmaximal
16#20703C212yesmaximal
17#20630C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#424C2212yes—
25C2×#408C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2629C2214yes—
29#2582—21no—
30#2582—21no—
31#2538—21no—
32#2538—21no—
33#2417—22no—
34#2417—22no—
35#2351—21no—
36Q8×C4⋊C4—21no—
37#2349—22no—
38#2349—21no—
39D4×C4⋊C4—22no—
40D4×C4⋊C4—21no—
41#2317C2212yes—
42#2317C2211yescharacteristic
43#2316C2212yes—
44#2316C2211yescharacteristic
45#2316C2212yes—
46#2311C2212yes—
47#2296C2212yes—
48#2296C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2212yes—
51C2×#384C2211yescharacteristic
52#2100C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2099C2212yes—
57#2099C2212yes—
58#2099C2214yes—
59#2088C2214yes—
60#2088C2214yes—
61#2088C2214yes—
62#2088C2214yes—
63#2087C2214yes—
64#2087C2214yes—
65#2086C2214yes—
66#2086C2214yes—
67#2059C2214yes—
68#2059C2212yes—
69#2059C2212yes—
70#2058C2212yes—
71#2058C2211yescharacteristic
72#2058C2211yescharacteristic
73#2057C2212yes—
74#2057C2211yescharacteristic
75#2057C2211yescharacteristic
76#1854C2214yes—
77#1854C2214yes—
78#1853C2214yes—
79#1852C2214yes—
80#1825C2214yes—
81#1825C2214yes—
82#1824C2214yes—
83#1823C2214yes—
84#1798C2214yes—
85#1797C2212yes—
86#1797C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#533—44no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
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×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4D412yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4D411yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#428—44no—
143#427—44no—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150#424—44no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153#408—28no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#398—24no—
157#398—24no—
158#396—24no—
159#396—24no—
160#391—22no—
161#391—22no—
162#391—22no—
163#390—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2311yescharacteristic
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2314yes—
186#389C2314yes—
187#389C2312yes—
188#386—24no—
189#385—21no—
190#385—22no—
191#385—21no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#384—21no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384C2312yes—
201#384C2312yes—
202#382—21no—
203#382—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—24no—
210#380C2314yes—
211#380C2314yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C2×#95—21no—
217C2×#95—22no—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41C2×Q814yes—
240C2×#41#4114yes—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8C2×D411yescharacteristic
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—24no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268#125—42no—
269#125—42no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—84no—
273C22⋊Q8—84no—
274C4×Q8—28no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—216no—
281C4×D4—28no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—44no—
288#117—24no—
289#117—22no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—44no—
295#117—44no—
296#117C2×Q818yes—
297#117C2×Q814yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4—84no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×Q818yes—
337C2×C4⋊C4C2×Q814yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371#107—42no—
372#107—42no—
373#107—44no—
374#106—42no—
375#106—42no—
376#106—44no—
377C4⋊C8—44no—
378C4⋊C8—44no—
379C22⋊C8—28no—
380C22⋊C8—28no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397#41—216no—
398#41—216no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8C22⋊Q814yes—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4C22⋊Q818yes—
413C2×D4C22⋊Q814yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—81no—
443C22×C4—82no—
444C22×C4—81no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—81no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—81no—
451C22×C4—82no—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×Q814yes—
461C22×C4C22×Q812yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C22×C4C2511yescharacteristic
467M16—48no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—48no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—28no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4C22×Q812yes—
519C4⋊C4C22×Q814yes—
520C4⋊C4C22×Q812yes—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—28no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—42no—
542C42—44no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23C2×C22⋊Q814yes—
554C23C2×C22⋊Q812yes—
555C23#53412yes—
556C23#53612yes—
557C23C23×D411yescharacteristic
558Q8—28no—
559Q8—84no—
560Q8—84no—
561D4—216no—
562D4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4—162no—
617C2×C4—164no—
618C2×C4—164no—
619C2×C4C2×C22⋊Q814yes—
620C2×C4C2×C22⋊Q812yes—
621C2×C4#53412yes—
622C2×C4#53612yes—
623C2×C4#53918yes—
624C2×C4#54418yes—
625C2×C4#57612yes—
626C2×C4#57712yes—
627C2×C4C23×Q811yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4C2×2−1+412yes—
631C8—42no—
632C8—42no—
633C8—44no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22#262714yes—
648C22C22×C22⋊Q811yescharacteristic
649C22C2×#53411yescharacteristic
650C22C2×#53611yescharacteristic
651C22C2×#57611yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358611yescharacteristic
654C22#358611yescharacteristic
655C4—22no—
656C4—22no—
657C4—24no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—42no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—84no—
671C4#262714yes—
672C4C2×#53912yes—
673C4C2×#54412yes—
674C2—24no—
675C2—44no—
676C2C2×#262711yescharacteristic
677C2#6049411yescharacteristic
678C2#6049711yescharacteristic
679C2#6076211yescharacteristic
680C2#6076311yescharacteristic
681C2#6087711yescharacteristic
682C2#6088411yescharacteristic
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. #29979C2semidirect
4. #29978C2semidirect
5. #29971C2semidirect
6. #29971C2semidirect
7. #29967C2semidirect
8. #29966C2semidirect
9. #23745C2semidirect
10. C2×#2099C2semidirect
11. #23671C2semidirect
12. #23669C2semidirect
13. #23554C2semidirect
14. #23551C2semidirect
108. #520Q8semidirect

Maximal subgroups

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

subgroupindex
#352802
#342502
#299792
#299782
#299712
#299672
#299662
#237452
C2×#20992
#236712
#236692
#235542
#235512
#208492
#207032
#206302