Finite Group Databasebeta

← #6464153

Subgroups — #6464153

Order 512.

1803 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35264C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29363C214yesmaximal, complemented
4#29360C214yesmaximal, complemented
5#29354C212yesmaximal, complemented
6#29352C212yesmaximal, complemented
7#29350C212yesmaximal, complemented
8#29346C212yesmaximal, complemented
9#23630C211yesmaximal, characteristic, complemented
10C2×#2071C211yesmaximal, characteristic, complemented
11#23454C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic, complemented
13#21315C212yesmaximal, complemented
14#21263C212yesmaximal
15#21091C212yesmaximal, complemented
16#21087C212yesmaximal, complemented
17#20430C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#535C2211yescharacteristic
20C2×#533C2211yescharacteristic
21C2×#424C2212yes—
22C2×#417C2212yes—
23C2×#418C2212yes—
24C2×#416C2212yes—
25C22×#102C2212yes—
26C2×#405C2212yes—
27#2861C2212yes—
28#2858C2212yes—
29#2641C2212yes—
30#2640C2212yes—
31#2491—21no—
32#2491—21no—
33#2490—21no—
34#2490—21no—
35#2472—22no—
36#2472—22no—
37#2360—22no—
38#2360—22no—
39#2355—21no—
40#2355—21no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4—21no—
43#2330C2214yescomplemented
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2309C2211yescharacteristic
47C4×C4⋊Q8C2212yescomplemented
48C4×C4⋊Q8C2212yescomplemented
49#2297C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53C4×#117C2211yescharacteristic
54#2072C2212yescomplemented
55#2072C2212yescomplemented
56#2071C2212yes—
57#2071C2212yes—
58#2071C2214yes—
59#2071C2212yescomplemented
60#2071C2212yescomplemented
61#2026C2212yescomplemented
62#2025C2214yes—
63#2025C2212yescomplemented
64#2024C2211yescharacteristic, complemented
65#2024C2211yescharacteristic, complemented
66#2023C2212yes—
67#2023C2212yes—
68#2023C2211yescharacteristic, complemented
69#2023C2211yescharacteristic, complemented
70#1989C2214yescomplemented
71#1989C2214yescomplemented
72#1988C2214yescomplemented
73#1987C2214yescomplemented
74#1983C2214yes—
75#1983C2214yes—
76#1982C2214yes—
77#1982C2214yes—
78#1936C2214yes—
79#1936C2214yes—
80#1936C2214yescomplemented
81#1935C2214yescomplemented
82#1934C2214yes—
83#1934C2214yes—
84#1934C2214yescomplemented
85#1933C2214yescomplemented
86#1768C2214yescomplemented
87#1767C2212yescomplemented
88C4×C4⋊C8C2212yescomplemented
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#535—28no—
101#533—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2312yes—
107C2×#125—21no—
108C2×#125C2312yes—
109C2×#125C2312yes—
110C2×#124—22no—
111C2×#124—22no—
112C2×#124C2312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#120—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42C2311yescharacteristic
134#424—28no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137#418—28no—
138#417—44no—
139#416—28no—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#102—44no—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#405—44no—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#397—22no—
155#397—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#393—22no—
161#393—22no—
162#391—22no—
163#391—22no—
164#391C2312yes—
165#391C2311yescharacteristic
166#391C2311yescharacteristic
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#389—22no—
181#389—22no—
182#389—24no—
183#389—24no—
184#389C2314yes—
185#389C2314yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—24no—
191#387—24no—
192#386—21no—
193#386—21no—
194#386—22no—
195#385—24no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#380—21no—
203#380—21no—
204#379—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—24no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220#376C2312yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—22no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—24no—
251C22×D4—24no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4C2412yes—
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—28no—
274C4⋊Q8—44no—
275#127—28no—
276#125—24no—
277#125—24no—
278#125—28no—
279#125—44no—
280#124—28no—
281#124—84no—
282#124—84no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—48no—
286#121—44no—
287#121—44no—
288#120—48no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—84no—
292C4×Q8—84no—
293#117—24no—
294#117—22no—
295#117—22no—
296#117—24no—
297#117#4118yes—
298#117#4118yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
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—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—82no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C4⋊C8—28no—
367C4⋊C8—28no—
368#103—24no—
369#103—24no—
370#103—44no—
371#102—24no—
372#102—24no—
373#102—28no—
374#102—28no—
375#102—44no—
376#102—44no—
377#102—44no—
378#97—44no—
379C4×C8—44no—
380#95—24no—
381#95—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24—21no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24—81no—
403C24C22×D411yescharacteristic
404#41—48no—
405#41—48no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477M16—48no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—48no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4#12414yes—
513C4⋊C4#12414yes—
514C4⋊C4C2×#4114yes—
515C4⋊C4C2×#4114yes—
516C22⋊C4—28no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—82no—
526C22⋊C4—81no—
527C22⋊C4—82no—
528C22⋊C4—81no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4#12418yes—
536C42—22no—
537C42—22no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—84no—
549C42—84no—
550C42C2×#4118yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—162no—
573C23—162no—
574C23—162no—
575C23—162no—
576C23#52512yes—
577C23C2×#12412yes—
578C23#53712yes—
579C23C23×D411yescharacteristic
580Q8—48no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—164no—
585D4—44no—
586D4—42no—
587D4—44no—
588D4—42no—
589D4—84no—
590D4—84no—
591D4—164no—
592D4—164no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—84no—
637C2×C4—162no—
638C2×C4—164no—
639C2×C4#52512yes—
640C2×C4C2×#12414yes—
641C2×C4C2×#12414yes—
642C2×C4C2×#12412yes—
643C2×C4#53712yes—
644C2×C4#55618yes—
645C2×C4#55918yes—
646C2×C4#57612yes—
647C2×C4#57712yes—
648C2×C4C22×#4112yes—
649C2×C4C22×#4112yes—
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C8—42no—
653C8—42no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—81no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—81no—
668C22—82no—
669C22—81no—
670C22—82no—
671C22—81no—
672C22—82no—
673C22—81no—
674C22—82no—
675C22—168no—
676C22#259214yes—
677C22C2×#52511yescharacteristic
678C22C22×#12411yescharacteristic
679C22C2×#53711yescharacteristic
680C22C2×#57611yescharacteristic
681C22C2×#57711yescharacteristic
682C22#358511yescharacteristic
683C22#358611yescharacteristic
684C4—22no—
685C4—22no—
686C4—24no—
687C4—24no—
688C4—41no—
689C4—44no—
690C4—41no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—81no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—84no—
699C4#259214yes—
700C4C2×#55612yes—
701C4C2×#55912yes—
702C2—24no—
703C2—161no—
704C2—162no—
705C2C2×#259211yescharacteristic
706C2#6058411yescharacteristic
707C2#6058811yescharacteristic
708C2#6062211yescharacteristic
709C2#6062511yescharacteristic
710C2#6092211yescharacteristic
711C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #34214C2semidirect
3. #29363C2semidirect
4. #29360C2semidirect
5. #29354C2semidirect
6. #29352C2semidirect
7. #29350C2semidirect
8. #29346C2semidirect
9. #23630C2semidirect
10. C2×#2071C2semidirect
11. #23454C2semidirect
12. #23451C2semidirect
13. #21315C2semidirect
15. #21091C2semidirect
16. #21087C2semidirect
17. #20430C2semidirect
43. #2330C22semidirect
47. C4×C4⋊Q8C22semidirect
48. C4×C4⋊Q8C22semidirect
54. #2072C22semidirect
55. #2072C22semidirect
59. #2071C22semidirect
60. #2071C22semidirect
61. #2026C22semidirect
63. #2025C22semidirect
64. #2024C22semidirect
65. #2024C22semidirect
68. #2023C22semidirect
69. #2023C22semidirect
70. #1989C22semidirect
71. #1989C22semidirect
72. #1988C22semidirect
73. #1987C22semidirect
80. #1936C22semidirect
81. #1935C22semidirect
84. #1934C22semidirect
85. #1933C22semidirect
86. #1768C22semidirect
87. #1767C22semidirect
88. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#342142
#293632
#293602
#293542
#293522
#293502
#293462
#236302
C2×#20712
#234542
#234512
#213152
#212632
#210912
#210872
#204302