Finite Group Databasebeta

← #6458364

Subgroups — #6458364

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#27829C214yesmaximal, complemented
4#27825C214yesmaximal, complemented
5#27819C214yesmaximal, complemented
6#27817C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23745C211yesmaximal, characteristic, complemented
9#23454C211yesmaximal, characteristic
10#23454C211yesmaximal, characteristic
11#21325C211yesmaximal, characteristic, complemented
12#21323C211yesmaximal, characteristic, complemented
13#21322C211yesmaximal, characteristic, complemented
14#21314C211yesmaximal, characteristic, complemented
15#21095C211yesmaximal, characteristic
16#21095C211yesmaximal, characteristic
17#20646C211yesmaximal, characteristic
18C4⋊C4×M16C211yesmaximal, characteristic, complemented
19C4×#418C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2566—22no—
33#2566—22no—
34#2566C2212yes—
35#2566C2212yes—
36#2566C2212yes—
37#2566C2212yes—
38#2552C2212yescomplemented
39#2551C2212yescomplemented
40#2550C2212yescomplemented
41#2550C2212yescomplemented
42#2549—22no—
43#2440—22no—
44#2399—22no—
45C2×#396—22no—
46C4×C4⋊Q8—22no—
47#2296C2211yescharacteristic
48#2296C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#385—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C4×#117C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2101C2214yescomplemented
58#2101C2214yescomplemented
59#2100C2214yescomplemented
60#2099C2214yescomplemented
61#2025C2214yes—
62#2025C2214yes—
63#2023C2214yes—
64#2023C2214yes—
65#1993C2214yescomplemented
66#1992C2214yescomplemented
67#1992C2214yescomplemented
68#1991C2214yescomplemented
69#1991C2214yescomplemented
70#1990C2214yescomplemented
71#1989C2214yescomplemented
72#1989C2214yescomplemented
73#1936C2214yes—
74#1936C2214yes—
75#1934C2214yes—
76#1934C2214yes—
77#1804C2214yes—
78#1802C2214yes—
79#1771C2212yescomplemented
80#1771C2212yescomplemented
81#1770C2212yescomplemented
82C8×C4⋊C4C2212yescomplemented
83#1762C2214yes—
84C4×#102C2214yes—
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C23×D4C2311yescharacteristic
100C24×C4—21no—
101C22×M16C2311yescharacteristic
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105#533—24no—
106#533—24no—
107#533—42no—
108#533—42no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—24no—
115C2×#125—24no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×#125C2312yes—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—24no—
127C2×C4×D4—24no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4D412yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C22⋊C4—21no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#423—24no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150#418—24no—
151#418—42no—
152#418—42no—
153#418Q818yescomplemented
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C4×M16—24no—
159C4×M16—42no—
160C4×M16—42no—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#396—24no—
172#396—24no—
173#396—24no—
174#396—24no—
175#395—22no—
176#395—22no—
177#395—22no—
178#395—22no—
179#391—22no—
180#391—22no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#385—22no—
196#385—22no—
197#385—24no—
198#385—24no—
199#385—24no—
200#385—24no—
201#384—22no—
202#384—22no—
203#384C2312yes—
204#384C2312yes—
205#384C2312yes—
206#382—22no—
207#382C2312yes—
208#382C2312yes—
209#382C2312yes—
210#380—21no—
211#380—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226#376—21no—
227C2×#95—22no—
228C2×#95D412yes—
229C43—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#336C2312yes—
243#336C2312yes—
244#336C2312yes—
245#336C2312yes—
246C25—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4C2×Q814yes—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—44no—
263C23×C4—84no—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16C2×Q814yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—42no—
288C4⋊Q8—42no—
289C4⋊Q8—48no—
290#125—24no—
291#125—24no—
292#125—24no—
293#125—24no—
294#125—24no—
295#125—24no—
296#125—42no—
297#125—42no—
298#125—44no—
299#125—44no—
300C22⋊Q8—48no—
301C22⋊Q8—48no—
302C4×Q8—48no—
303C4×Q8—48no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—44no—
309C4×D4—44no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117C2×Q814yes—
317#117#4114yes—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—84no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—48no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—84no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42#4114yes—
388C2×C42#4114yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C4⋊C8—24no—
396C4⋊C8—24no—
397#102—28no—
398#102—44no—
399#102—44no—
400#102C2×Q8116yes—
401#97—24no—
402#97—42no—
403#97—42no—
404C4×C8—24no—
405C4×C8—42no—
406C4×C8—42no—
407#95—24no—
408#95—24no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95C2×D412yes—
418#95C2×D412yes—
419#95C2×D412yes—
420#95C2×D412yes—
421C24—24no—
422C24—22no—
423C24—44no—
424C24—44no—
425C24—42no—
426C24—42no—
427C24C22×D411yescharacteristic
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—82no—
433C2×Q8—82no—
434C2×D4—28no—
435C2×D4—24no—
436C2×D4—28no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4C22×Q814yes—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—816no—
473C22×C4#12414yes—
474C22×C4#12414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×Q812yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491M16—28no—
492M16—44no—
493M16—44no—
494M16—44no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8C22×Q814yes—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4C22×Q814yes—
545C4⋊C4C2×#4114yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—48no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—24no—
558C42—24no—
559C42—24no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—82no—
572C42#12414yes—
573C42#12414yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C23—22no—
579C23—22no—
580C23—24no—
581C23—416no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—48no—
586C23—44no—
587C23—84no—
588C23—84no—
589C23—88no—
590C23C2×#12412yes—
591C23#54612yes—
592C23D4×Q812yes—
593C23C23×D411yescharacteristic
594Q8—88no—
595Q8—88no—
596D4—24no—
597D4—24no—
598D4—42no—
599D4—42no—
600D4—48no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4C2×#12412yes—
644C2×C4C2×#12412yes—
645C2×C4C2×#12412yes—
646C2×C4C2×#12412yes—
647C2×C4C2×#12412yes—
648C2×C4#53314yes—
649C2×C4#54214yes—
650C2×C4#54612yes—
651C2×C4D4×Q812yes—
652C2×C4#55414yes—
653C2×C4#55614yes—
654C2×C4#57514yes—
655C2×C4C23×Q811yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—24no—
662C8—42no—
663C8—42no—
664C8—42no—
665C22—24no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—48no—
670C22—44no—
671C22—816no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22#249214yes—
676C22C22×#12411yescharacteristic
677C22C2×#54611yescharacteristic
678C22C2×D4×Q811yescharacteristic
679C22C2×#57512yes—
680C22#358512yes—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—24no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#249214yes—
697C4C2×#53311yescharacteristic
698C4C2×#54211yescharacteristic
699C4C2×#55411yescharacteristic
700C4C2×#55611yescharacteristic
701C2—24no—
702C2—84no—
703C2C2×#249211yescharacteristic
704C2#6061811yescharacteristic
705C2#6061911yescharacteristic
706C2#6116412yes—
707C2#6136712yes—
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. #27829C2semidirect
4. #27825C2semidirect
5. #27819C2semidirect
6. #27817C2semidirect
7. #23746C2semidirect
8. #23745C2semidirect
11. #21325C2semidirect
12. #21323C2semidirect
13. #21322C2semidirect
14. #21314C2semidirect
18. C4⋊C4×M16C2semidirect
38. #2552C22semidirect
39. #2551C22semidirect
40. #2550C22semidirect
41. #2550C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
65. #1993C22semidirect
66. #1992C22semidirect
67. #1992C22semidirect
68. #1991C22semidirect
69. #1991C22semidirect
70. #1990C22semidirect
71. #1989C22semidirect
72. #1989C22semidirect
79. #1771C22semidirect
80. #1771C22semidirect
81. #1770C22semidirect
82. C8×C4⋊C4C22semidirect
153. #418Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
C2×#25662
#278292
#278252
#278192
#278172
#237462
#237452
#234542
#213252
#213232
#213222
#213142
#210952
#206462
C4⋊C4×M162
C4×#4182