Finite Group Databasebeta

← #6449306

Subgroups — #6449306

Order 512.

1994 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#26033C214yesmaximal, complemented
4#26032C214yesmaximal, complemented
5#26025C212yesmaximal, complemented
6#26024C212yesmaximal, complemented
7#26021C212yesmaximal, complemented
8#26020C212yesmaximal, complemented
9#23618C212yesmaximal
10#23454C212yesmaximal
11#21312C212yesmaximal, complemented
12#21171C211yesmaximal, characteristic, complemented
13#21146C211yesmaximal, characteristic, complemented
14#21033C212yesmaximal, complemented
15#20764C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117—21no—
21C2×#428C2212yes—
22C2×#423C2211yescharacteristic
23C2×#423C2212yes—
24C2×#418C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2552C2211yescharacteristic, complemented
29#2552C2211yescharacteristic, complemented
30#2551C2211yescharacteristic, complemented
31#2551C2211yescharacteristic, complemented
32#2550C2212yescomplemented
33#2550C2212yescomplemented
34#2549—22no—
35#2440—21no—
36#2440—21no—
37#2399—22no—
38#2354—21no—
39#2354—21no—
40#2354—22no—
41#2354C2212yes—
42#2354C2212yes—
43#2354C2214yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8—21no—
46#2296C2212yes—
47C2×#390C2212yes—
48C2×#388—22no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C4×#117C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2067C2214yes—
56#2067C2214yes—
57#2065C2214yes—
58#2065C2214yes—
59#2025C2214yes—
60#2025C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#1989C2214yescomplemented
64#1989C2214yescomplemented
65#1988C2214yescomplemented
66#1987C2214yescomplemented
67#1948C2212yescomplemented
68#1948C2212yescomplemented
69#1947C2212yescomplemented
70#1947C2212yescomplemented
71#1940C2214yescomplemented
72#1939C2212yescomplemented
73#1939C2212yescomplemented
74#1921C2214yescomplemented
75#1920C2214yescomplemented
76#1919C2214yescomplemented
77#1918C2214yescomplemented
78#1836C2214yescomplemented
79#1834C2212yescomplemented
80#1834C2212yescomplemented
81#1804C2214yes—
82#1802C2212yes—
83#1802C2212yes—
84#1762C2212yes—
85#1762C2212yes—
86C4×#102C2212yes—
87C4×#102C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2211yescharacteristic
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1736C2212yes—
94#1728C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×M16C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2312yes—
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4Q818yescomplemented
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C2×#117—24no—
120C2×#117—22no—
121C2×#117—48no—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42—22no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#428—28no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140#423—24no—
141#423—44no—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146#418—24no—
147#418—24no—
148#418—44no—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C4×M16—24no—
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#395—22no—
163#395—22no—
164#395—22no—
165#395—22no—
166#391—22no—
167#391—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—24no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222#376—21no—
223C2×#95—22no—
224C2×#95D412yes—
225C43—21no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#336C2314yes—
239#336C2314yes—
240C25—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4C2×Q814yes—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—22no—
271C2×M16—22no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#125—24no—
282#125—24no—
283#125—42no—
284#125—42no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C4×Q8—48no—
288C4×Q8—48no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4C2×Q818yes—
294C4×D4C2×Q818yes—
295#117—22no—
296#117—22no—
297#117—24no—
298#117—24no—
299#117—48no—
300#117—416no—
301#117—48no—
302#117#4114yes—
303#117#4114yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
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—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×Q814yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4—84no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—48no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×Q814yes—
375C2×C42#4114yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#106—28no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384C4⋊C8—44no—
385C4⋊C8—44no—
386#102—24no—
387#102—24no—
388#102—24no—
389#102—24no—
390#102—44no—
391#102—44no—
392#97—24no—
393C4×C8—24no—
394#95—24no—
395#95—24no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410C24—24no—
411C24—22no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24C22×D411yescharacteristic
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4C22×Q814yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—432no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4C22⋊Q814yes—
470C22×C4C22⋊Q814yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×Q812yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487M16—44no—
488M16—48no—
489M16—44no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—81no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4AGL(1,ℤ/8)14yes—
536C4⋊C4C22×Q814yes—
537C4⋊C4C2×#4112yes—
538C4⋊C4C2×#4112yes—
539C4⋊C4C2×#4112yes—
540C4⋊C4C2×#4112yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—28no—
544C22⋊C4—28no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C42—44no—
560C42—48no—
561C42—42no—
562C42—42no—
563C42—81no—
564C42—81no—
565C42C22⋊Q814yes—
566C42C22⋊Q814yes—
567C42C22×Q812yes—
568C42C22×Q812yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—24no—
576C23—48no—
577C23—48no—
578C23—42no—
579C23—42no—
580C23—48no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23C2×C22⋊Q812yes—
590C23#54612yes—
591C23#54812yes—
592C23C23×D411yescharacteristic
593Q8—88no—
594Q8—88no—
595D4—24no—
596D4—24no—
597D4—48no—
598D4—42no—
599D4—42no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—416no—
626C2×C4—48no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—416no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4C2×C22⋊Q812yes—
650C2×C4C2×C22⋊Q812yes—
651C2×C4C2×C22⋊Q812yes—
652C2×C4C2×C22⋊Q812yes—
653C2×C4C2×C22⋊Q812yes—
654C2×C4#53314yes—
655C2×C4#54414yes—
656C2×C4#54612yes—
657C2×C4#54812yes—
658C2×C4#55514yes—
659C2×C4#55614yes—
660C2×C4C2×AGL(1,ℤ/8)14yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4#57512yes—
663C2×C4C23×Q811yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C8—42no—
670C8—44no—
671C8—42no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—816no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22#243914yes—
687C22C22×C22⋊Q811yescharacteristic
688C22C2×#54611yescharacteristic
689C22C2×#54811yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C2×#57511yescharacteristic
692C22#358511yescharacteristic
693C22#358611yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—24no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—84no—
707C4—82no—
708C4—82no—
709C4#243914yes—
710C4#307714yes—
711C4C2×#53311yescharacteristic
712C4C2×#54411yescharacteristic
713C4C2×#55511yescharacteristic
714C4C2×#55611yescharacteristic
715C2—24no—
716C2—84no—
717C2C2×#243911yescharacteristic
718C2C2×#307711yescharacteristic
719C2#6049211yescharacteristic
720C2#6116111yescharacteristic
721C2#6116611yescharacteristic
722C2#6132111yescharacteristic
723C2#6132311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #34938C2semidirect
3. #26033C2semidirect
4. #26032C2semidirect
5. #26025C2semidirect
6. #26024C2semidirect
7. #26021C2semidirect
8. #26020C2semidirect
11. #21312C2semidirect
12. #21171C2semidirect
13. #21146C2semidirect
14. #21033C2semidirect
15. #20764C2semidirect
28. #2552C22semidirect
29. #2552C22semidirect
30. #2551C22semidirect
31. #2551C22semidirect
32. #2550C22semidirect
33. #2550C22semidirect
63. #1989C22semidirect
64. #1989C22semidirect
65. #1988C22semidirect
66. #1987C22semidirect
67. #1948C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1947C22semidirect
71. #1940C22semidirect
72. #1939C22semidirect
73. #1939C22semidirect
74. #1921C22semidirect
75. #1920C22semidirect
76. #1919C22semidirect
77. #1918C22semidirect
78. #1836C22semidirect
79. #1834C22semidirect
80. #1834C22semidirect
115. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
C2×#23542
#260332
#260322
#260252
#260242
#260212
#260202
#236182
#234542
#213122
#211712
#211462
#210332
#207642
#206462
C4×#4182