Finite Group Databasebeta

← #6435201

Subgroups — #6435201

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34058C211yesmaximal, characteristic
3#29991C214yesmaximal, complemented
4#29990C214yesmaximal, complemented
5#29986C212yesmaximal, complemented
6#29986C212yesmaximal, complemented
7#29981C212yesmaximal, complemented
8#29980C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23735C212yesmaximal
11#23557C212yesmaximal, complemented
12#23456C212yesmaximal
13#21171C211yesmaximal, characteristic, complemented
14#21169C211yesmaximal, characteristic, complemented
15#20755C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C2×#1802C211yesmaximal, characteristic
18C2×#544C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#418C2212yes—
25C22×#102C2212yes—
26#2868C2212yescomplemented
27#2859C2212yescomplemented
28#2660C2214yescomplemented
29#2552—21no—
30#2552—21no—
31#2551—21no—
32#2551—21no—
33#2440—22no—
34#2440—22no—
35#2319—21no—
36#2319—21no—
37#2319C2212yes—
38#2319C2212yes—
39#2317C2212yes—
40#2316C2211yescharacteristic
41#2316C2211yescharacteristic
42#2312—22no—
43#2309C2211yescharacteristic
44#2309C2211yescharacteristic
45#2293—21no—
46#2293—21no—
47#2293C2212yes—
48#2293C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2280—22no—
52C4×#117C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2101C2214yescomplemented
55#2101C2214yescomplemented
56#2100C2214yescomplemented
57#2099C2214yescomplemented
58#2094C2214yes—
59#2093C2214yes—
60#2092C2214yes—
61#2092C2214yes—
62#2059C2214yescomplemented
63#2059C2214yescomplemented
64#2058C2214yescomplemented
65#2057C2214yescomplemented
66#2025C2214yes—
67#2025C2214yes—
68#2023C2214yes—
69#2023C2214yes—
70#1949C2212yescomplemented
71#1949C2212yescomplemented
72#1948C2211yescharacteristic, complemented
73#1948C2212yescomplemented
74#1948C2211yescharacteristic, complemented
75#1948C2212yescomplemented
76#1947C2211yescharacteristic, complemented
77#1947C2212yescomplemented
78#1947C2211yescharacteristic, complemented
79#1947C2212yescomplemented
80#1832C2214yescomplemented
81#1831C2212yescomplemented
82#1831C2212yescomplemented
83#1804C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2214yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—44no—
100#544—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2314yes—
105C2×#125—22no—
106C2×#125C2314yes—
107C2×C22⋊Q8—24no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—24no—
111C2×C4×D4Q818yescomplemented
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C42C2311yescharacteristic
127#428—44no—
128#427—44no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135#423—24no—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#418—44no—
143C2×#102—44no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148#398—24no—
149#398—24no—
150#395—24no—
151#395—24no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#385—24no—
183#384—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2314yes—
192#382C2312yes—
193#382C2314yes—
194#381—22no—
195#381—22no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#379—22no—
203#379—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376—21no—
221#376C2312yes—
222C43—21no—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C22×Q8—24no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4C2×Q814yes—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—28no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—82no—
254C23×C4—82no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—84no—
271C4⋊Q8—84no—
272#125—44no—
273#125—44no—
274#125—48no—
275C22⋊Q8—416no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4C2×Q818yes—
281C4×D4C2×Q818yes—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—24no—
287#117—24no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—84no—
293#117—84no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×Q814yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×Q814yes—
361C2×C42#4114yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—42no—
367#107—42no—
368#107—44no—
369#106—42no—
370#106—42no—
371#106—44no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377#102—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—81no—
402C24—81no—
403C24—81no—
404C24—81no—
405C24C22×D411yescharacteristic
406C2×Q8—48no—
407C2×Q8—84no—
408C2×Q8—84no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4C22×Q814yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—816no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C22⋊Q814yes—
448C22×C4C22⋊Q814yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q812yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465M16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—48no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×Q814yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—81no—
544C42—81no—
545C42C22⋊Q814yes—
546C42C22⋊Q814yes—
547C42C22×Q812yes—
548C42C22×Q812yes—
549C42C2×#4112yes—
550C42C2×#4112yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—24no—
556C23—42no—
557C23—42no—
558C23—48no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23C2×C22⋊Q812yes—
570C23#53512yes—
571C23#53612yes—
572C23C23×D411yescharacteristic
573Q8—168no—
574D4—24no—
575D4—24no—
576D4—48no—
577D4—42no—
578D4—42no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—416no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—164no—
627C2×C4C2×C22⋊Q812yes—
628C2×C4C2×C22⋊Q812yes—
629C2×C4C2×C22⋊Q812yes—
630C2×C4C2×C22⋊Q812yes—
631C2×C4C2×C22⋊Q812yes—
632C2×C4#53512yes—
633C2×C4#53612yes—
634C2×C4#54214yes—
635C2×C4#54214yes—
636C2×C4#54414yes—
637C2×C4#54414yes—
638C2×C4#57612yes—
639C2×C4#57712yes—
640C2×C4C23×Q811yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C8—42no—
647C8—42no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—816no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22#262814yes—
664C22C22×C22⋊Q811yescharacteristic
665C22C2×#53511yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#57611yescharacteristic
668C22C2×#57711yescharacteristic
669C22#358511yescharacteristic
670C22#358611yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—81no—
680C4—84no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4#262814yes—
687C4#308114yes—
688C4C2×#54211yescharacteristic
689C4C2×#54211yescharacteristic
690C4C2×#54411yescharacteristic
691C4C2×#54411yescharacteristic
692C2—24no—
693C2—84no—
694C2C2×#262811yescharacteristic
695C2C2×#308111yescharacteristic
696C2#6049311yescharacteristic
697C2#6083211yescharacteristic
698C2#6083611yescharacteristic
699C2#6088811yescharacteristic
700C2#6089211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #35275C2semidirect
3. #29991C2semidirect
4. #29990C2semidirect
5. #29986C2semidirect
6. #29986C2semidirect
7. #29981C2semidirect
8. #29980C2semidirect
9. #23749C2semidirect
11. #23557C2semidirect
13. #21171C2semidirect
14. #21169C2semidirect
15. #20755C2semidirect
26. #2868C22semidirect
27. #2859C22semidirect
28. #2660C22semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
62. #2059C22semidirect
63. #2059C22semidirect
64. #2058C22semidirect
65. #2057C22semidirect
70. #1949C22semidirect
71. #1949C22semidirect
72. #1948C22semidirect
73. #1948C22semidirect
74. #1948C22semidirect
75. #1948C22semidirect
76. #1947C22semidirect
77. #1947C22semidirect
78. #1947C22semidirect
79. #1947C22semidirect
80. #1832C22semidirect
81. #1831C22semidirect
82. #1831C22semidirect
111. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#340582
#299912
#299902
#299862
#299812
#299802
#237492
#237352
#235572
#234562
#211712
#211692
#207552
#206462
C2×#18022