Finite Group Databasebeta

← #6435208

Subgroups — #6435208

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29991C214yesmaximal, complemented
4#29990C214yesmaximal, complemented
5#29982C212yesmaximal, complemented
6#29982C212yesmaximal, complemented
7#29981C212yesmaximal, complemented
8#29980C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23735C212yesmaximal
11#23552C211yesmaximal, characteristic, complemented
12#23551C211yesmaximal, characteristic, complemented
13#23454C211yesmaximal, characteristic
14C2×#2023C211yesmaximal, characteristic
15#21175C212yesmaximal, complemented
16#20754C211yesmaximal, characteristic, complemented
17#20650C212yesmaximal
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#518C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#424C2212yes—
24C2×#418C2212yes—
25C22×#102C2212yes—
26#2868C2212yescomplemented
27#2862C2212yescomplemented
28#2660C2214yescomplemented
29#2567—21no—
30#2567—21no—
31#2546—21no—
32#2546—21no—
33#2502—22no—
34#2434—22no—
35#2357—21no—
36#2357—21no—
37#2354—21no—
38#2354—21no—
39#2342—22no—
40#2342—22no—
41#2319C2212yes—
42#2319C2212yes—
43#2317C2211yescharacteristic
44#2317C2212yes—
45#2317C2212yes—
46#2309C2211yescharacteristic
47#2296C2211yescharacteristic
48#2293C2212yes—
49#2293C2212yes—
50C2×#390C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2100C2214yescomplemented
56#2099C2214yescomplemented
57#2094C2214yes—
58#2093C2214yes—
59#2092C2214yes—
60#2092C2214yes—
61#2059C2212yescomplemented
62#2059C2212yescomplemented
63#2058C2212yescomplemented
64#2058C2212yescomplemented
65#2058C2211yescharacteristic, complemented
66#2058C2211yescharacteristic, complemented
67#2057C2212yescomplemented
68#2057C2212yescomplemented
69#2057C2211yescharacteristic, complemented
70#2057C2211yescharacteristic, complemented
71#2025C2214yes—
72#2023C2212yes—
73#2023C2212yes—
74#2023C2212yes—
75#2023C2212yes—
76#2023C2214yes—
77#1949C2214yescomplemented
78#1949C2214yescomplemented
79#1948C2214yescomplemented
80#1947C2214yescomplemented
81#1832C2214yescomplemented
82#1831C2212yescomplemented
83#1831C2212yescomplemented
84#1804C2214yes—
85#1804C2214yes—
86#1802C2214yes—
87#1802C2214yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112#518—24no—
113#518Q818yescomplemented
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×#117—22no—
121C2×#117—22no—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42C2311yescharacteristic
134#428—44no—
135#427—44no—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#418—44no—
146C2×#102—44no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#398—24no—
152#398—24no—
153#395—24no—
154#395—24no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#385—21no—
186#385—21no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—22no—
192#384—22no—
193#384C2314yes—
194#384C2314yes—
195#382—21no—
196#382C2314yes—
197#382C2314yes—
198#381—22no—
199#381—22no—
200#380—22no—
201#380—22no—
202#379—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376C2312yes—
219C43C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4C2×Q814yes—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
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—42no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—84no—
274C4⋊Q8—84no—
275#125—42no—
276#125—42no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4C2×Q818yescomplemented
286C4×D4C2×Q818yescomplemented
287#117—22no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—84no—
293#117—84no—
294#117C2×Q814yes—
295#117C2×Q814yes—
296#117#4114yes—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—216no—
335C2×C22⋊C4—22no—
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⋊C4—82no—
346C2×C22⋊C4—82no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—21no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365#107—42no—
366#107—42no—
367#107—44no—
368#106—42no—
369#106—42no—
370#106—44no—
371C4⋊C8—28no—
372C4⋊C8—28no—
373#102—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24—24no—
395C24—42no—
396C24—42no—
397C24—81no—
398C24—81no—
399C24—81no—
400C24—81no—
401C24C22×D411yescharacteristic
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—816no—
415C2×D4—24no—
416C2×D4—216no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4C22×Q814yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q812yes—
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—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—48no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4C22⋊Q814yes—
507C4⋊C4C22⋊Q814yes—
508C4⋊C4C22×Q812yes—
509C4⋊C4C22×Q812yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4C22⋊Q818yes—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—82no—
538C42—82no—
539C42C22×Q814yes—
540C42C2×#4114yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—24no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—48no—
553C23—84no—
554C23—84no—
555C23—162no—
556C23—162no—
557C23—162no—
558C23—162no—
559C23C2×C22⋊Q812yes—
560C23#53512yes—
561C23#53612yes—
562C23C23×D411yescharacteristic
563Q8—84no—
564Q8—84no—
565Q8—88no—
566Q8—88no—
567D4—28no—
568D4—48no—
569D4—44no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—88no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4—162no—
620C2×C4C2×C22⋊Q814yes—
621C2×C4C2×C22⋊Q814yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4#53512yes—
624C2×C4#53612yes—
625C2×C4#54218yes—
626C2×C4#54418yes—
627C2×C4#57612yes—
628C2×C4#57712yes—
629C2×C4C23×Q811yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4C2×2−1+412yes—
633C8—42no—
634C8—42no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—168no—
650C22#262814yes—
651C22C22×C22⋊Q811yescharacteristic
652C22C2×#53511yescharacteristic
653C22C2×#53611yescharacteristic
654C22C2×#57611yescharacteristic
655C22C2×#57711yescharacteristic
656C22#358511yescharacteristic
657C22#358611yescharacteristic
658C4—24no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—81no—
669C4—84no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4#262814yes—
675C4C2×#54212yes—
676C4C2×#54412yes—
677C2—24no—
678C2—84no—
679C2C2×#262811yescharacteristic
680C2#6049411yescharacteristic
681C2#6049511yescharacteristic
682C2#6081111yescharacteristic
683C2#6081311yescharacteristic
684C2#6087711yescharacteristic
685C2#6088511yescharacteristic
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. #35281C2semidirect
3. #29991C2semidirect
4. #29990C2semidirect
5. #29982C2semidirect
6. #29982C2semidirect
7. #29981C2semidirect
8. #29980C2semidirect
9. #23748C2semidirect
11. #23552C2semidirect
12. #23551C2semidirect
15. #21175C2semidirect
16. #20754C2semidirect
26. #2868C22semidirect
27. #2862C22semidirect
28. #2660C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2099C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2058C22semidirect
64. #2058C22semidirect
65. #2058C22semidirect
66. #2058C22semidirect
67. #2057C22semidirect
68. #2057C22semidirect
69. #2057C22semidirect
70. #2057C22semidirect
77. #1949C22semidirect
78. #1949C22semidirect
79. #1948C22semidirect
80. #1947C22semidirect
81. #1832C22semidirect
82. #1831C22semidirect
83. #1831C22semidirect
113. #518Q8semidirect
285. C4×D4C2×Q8semidirect
286. C4×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352812
#342552
#299912
#299902
#299822
#299812
#299802
#237482
#237352
#235522
#235512
#234542
C2×#20232
#211752
#207542
#206502