Finite Group Databasebeta

← #6461285

Subgroups — #6461285

Order 512.

1962 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34268C211yesmaximal, characteristic
3#29171C214yesmaximal, complemented
4#29170C214yesmaximal, complemented
5#29166C212yesmaximal, complemented
6#29165C212yesmaximal, complemented
7#29160C212yesmaximal, complemented
8#29159C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23732C211yesmaximal, characteristic, complemented
11#23552C211yesmaximal, characteristic, complemented
12C2×#2024C211yesmaximal, characteristic, complemented
13#21035C212yesmaximal, complemented
14#20974C212yesmaximal
15#20877C212yesmaximal, complemented
16#20664C212yesmaximal
17#20336C211yesmaximal, characteristic
18C22×#131C2212yes—
19C2×#536C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#428C2212yes—
24C2×#428C2212yes—
25C2×#419C2212yes—
26C22×#103C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2628C2214yes—
30#2568—21no—
31#2567—21no—
32#2563—21no—
33#2542—21no—
34#2487—22no—
35#2409—22no—
36#2367—21no—
37#2367—21no—
38#2365—21no—
39#2365—21no—
40#2359—22no—
41#2359—22no—
42#2323C2212yes—
43#2323C2212yes—
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yes—
49#2296C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285C2211yescharacteristic
52#2274C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2100C2211yescharacteristic
56#2100C2211yescharacteristic
57#2099C2211yescharacteristic
58#2099C2211yescharacteristic
59#2097C2214yes—
60#2096C2212yes—
61#2095C2212yes—
62#2058C2212yes—
63#2058C2212yes—
64#2057C2212yes—
65#2057C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2024C2214yes—
69#1921C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1918C2214yes—
73#1894C2214yes—
74#1894C2214yes—
75#1892C2214yes—
76#1892C2214yes—
77#1865C2214yes—
78#1865C2214yes—
79#1863C2214yes—
80#1863C2214yes—
81#1809C2214yes—
82#1809C2214yes—
83#1808C2214yes—
84#1807C2214yes—
85#1736C2212yes—
86#1735C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C2×#131—28no—
99C2×#131—28no—
100C2×#131—44no—
101C22×M16C2314yes—
102C22×M16C2312yes—
103C23×C8C2312yes—
104C23×C8C2314yes—
105#536—24no—
106#536Q818yescomplemented
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—44no—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×#124—21no—
114C2×#123—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119#519—44no—
120C4×#41—44no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D412yes—
129C2×C4×D4D412yes—
130C2×C4×D4D411yescharacteristic
131C2×#117—22no—
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#428—28no—
143#428—44no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148#419—28no—
149C2×#103—44no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154#400—24no—
155#399—24no—
156#395—24no—
157#393—24no—
158#391—22no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2312yes—
166#390—22no—
167#390—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#389C2314yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—24no—
188#385—21no—
189#385—21no—
190#385—22no—
191#385—22no—
192#385—24no—
193#384—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382—22no—
197#382C2312yes—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2314yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C2×#95—21no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41C2×Q814yes—
243C2×#41#4114yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—44no—
251C22×Q8—82no—
252C22×Q8—82no—
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2×Q814yes—
255C22×Q8C2412yes—
256C22×D4C2×D412yes—
257C22×D4C2×D411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267#131—432no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—28no—
271C2×M16—28no—
272C2×M16—44no—
273C2×M16#4118yes—
274C22×C8—28no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8—28no—
279C22×C8—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283C4⋊Q8—28no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288#124—28no—
289#123—216no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—84no—
293C22⋊Q8—84no—
294C22⋊Q8C2×Q818yescomplemented
295C22⋊Q8C2×Q818yescomplemented
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4—44no—
305C4×D4—44no—
306#117—22no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117—24no—
311#117—44no—
312#117—48no—
313#117C2×Q814yes—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—21no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—28no—
377#107—44no—
378#106—28no—
379#106—44no—
380#103—28no—
381#103—28no—
382#103—44no—
383#103—44no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404#41—216no—
405#41—216no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—216no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—816no—
423C2×Q8C22⋊Q814yes—
424C2×Q8C22×Q814yes—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4C22⋊Q818yes—
430C2×D4C22⋊Q814yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q814yes—
472C22×C4C22×Q812yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4C2511yescharacteristic
478M16—216no—
479M16—416no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—416no—
489C2×C8—48no—
490C2×C8C2×#4118yes—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4C22×Q812yes—
525C4⋊C4C22×Q812yes—
526C22⋊C4—24no—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—24no—
544C42—42no—
545C42—42no—
546C42—44no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—82no—
551C42—82no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23C2×C22⋊Q814yes—
563C23C2×C22⋊Q812yes—
564C23#53212yes—
565C23#53412yes—
566C23C23×D411yescharacteristic
567Q8—28no—
568Q8—28no—
569Q8—48no—
570Q8—44no—
571Q8—88no—
572Q8—88no—
573D4—216no—
574D4—28no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4—162no—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×C22⋊Q812yes—
631C2×C4#53212yes—
632C2×C4#53412yes—
633C2×C4#55118yes—
634C2×C4#55418yes—
635C2×C4#57712yes—
636C2×C4#57812yes—
637C2×C4C23×Q811yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4112yes—
640C2×C4C2×H(5,𝔽2)12yes—
641C8—28no—
642C8—42no—
643C8—42no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22#256414yes—
658C22C22×C22⋊Q811yescharacteristic
659C22C2×#53211yescharacteristic
660C22C2×#53411yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57811yescharacteristic
663C22#358511yescharacteristic
664C22#358611yescharacteristic
665C4—24no—
666C4—22no—
667C4—22no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—81no—
679C4—84no—
680C4—81no—
681C4#256414yes—
682C4C2×#55112yes—
683C4C2×#55412yes—
684C2—24no—
685C2—44no—
686C2C2×#256411yescharacteristic
687C2#6049711yescharacteristic
688C2#6049811yescharacteristic
689C2#6072311yescharacteristic
690C2#6072711yescharacteristic
691C2#6075711yescharacteristic
692C2#6075911yescharacteristic
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. #29171C2semidirect
4. #29170C2semidirect
5. #29166C2semidirect
6. #29165C2semidirect
7. #29160C2semidirect
8. #29159C2semidirect
9. #23744C2semidirect
10. #23732C2semidirect
11. #23552C2semidirect
12. C2×#2024C2semidirect
13. #21035C2semidirect
15. #20877C2semidirect
106. #536Q8semidirect
294. C22⋊Q8C2×Q8semidirect
295. C22⋊Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#342682
#291712
#291702
#291662
#291652
#291602
#291592
#237442
#237322
#235522
C2×#20242
#210352
#209742
#208772
#206642
#203362