Finite Group Databasebeta

← #227759

Subgroups — #227759

Order 512.

890 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22985C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3C4×#448C211yesmaximal, characteristic
4C2×#1585C211yesmaximal, characteristic, complemented
5C22×Q8⋊C8C211yesmaximal, characteristic, complemented
6#9666C211yesmaximal, characteristic, complemented
7#9559C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9072C211yesmaximal, characteristic
10#8952C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8848C211yesmaximal, characteristic, complemented
13#8829C211yesmaximal, characteristic, complemented
14#8213C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#448C411yescharacteristic
18C2×#446C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C8×D4—21no—
23C2×C8×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C4×#121C2211yescharacteristic
27#1985—21no—
28#1985—21no—
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1954C2211yescharacteristic
32#1939C2211yescharacteristic
33#1923—21no—
34#1880C411yescharacteristic
35#1880C411yescharacteristic
36#1874C411yescharacteristic
37#1869—21no—
38#1865C411yescharacteristic
39#1865C411yescharacteristic
40#1853C411yescharacteristic
41#1850—21no—
42#1797—21no—
43#1797C2211yescharacteristic
44#1787C411yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#103C2211yescharacteristic
48#1749—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1585—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×Q8⋊C8—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C23×Q8C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—21no—
75C2×#121C2×C411yescharacteristic
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—41no—
81C2×C4×Q8—41no—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#448—24no—
93#446—22no—
94#446—22no—
95#446—41no—
96#446—41no—
97C4×SD16—22no—
98C4×SD16—22no—
99C4×SD16—41no—
100C4×SD16—41no—
101C8×D4—22no—
102C8×D4—41no—
103C8×D4—44no—
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—41no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#103C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8C2×C411yescharacteristic
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8D411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×#97—21no—
133C2×#97—21no—
134C2×C4×C8—22no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—41no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#390C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#388—21no—
147#386—21no—
148#380C2×C411yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C22⋊C4—21no—
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—22no—
167#335—41no—
168Q8⋊C8—22no—
169Q8⋊C8—22no—
170Q8⋊C8—22no—
171Q8⋊C8D414yescomplemented
172D4⋊C8—22no—
173D4⋊C8—41no—
174#322—22no—
175#322—41no—
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8C22×C411yescharacteristic
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193C2×SD16—22no—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—41no—
198C2×SD16—41no—
199C22×C8—21no—
200C22×C8—24no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—81no—
213C22×C8—81no—
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222#121—22no—
223#121M1614yes—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—42no—
233C4×Q8—42no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×Q8C2×D412yes—
239C4×Q8#4112yes—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—42no—
245C4×D4—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#106—22no—
282#106—41no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8C2×D412yes—
295#103—22no—
296#103—22no—
297#103—22no—
298#103—22no—
299#103—41no—
300#103—41no—
301#102—22no—
302#102—41no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—81no—
311#97—42no—
312#97—42no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—42no—
325C4×C8—42no—
326C4×C8C2×D412yes—
327#95—21no—
328#95—21no—
329#95—21no—
330C24—21no—
331C24C4×D411yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—28no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8C4×D412yes—
350C2×Q8#13112yes—
351C2×D4—24no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C2×M1612yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—81no—
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C2×M1612yes—
395C22×C4#13112yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4111yescharacteristic
401SD16—44no—
402SD16—44no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—48no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—84no—
447C2×C8—84no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4C4×D412yes—
465C4⋊C4C2×M1612yes—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—42no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42C22×D411yescharacteristic
487C42C2×#4111yescharacteristic
488C42H(5,𝔽2)11yescharacteristic
489C42H(5,𝔽2)11yescharacteristic
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23#43512yes—
502C23#44012yes—
503C23#44312yes—
504C23C2×C4×D411yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—48no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514D4—22no—
515D4—22no—
516D4—48no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4#43512yes—
566C2×C4#44012yes—
567C2×C4C2×C4×D411yescharacteristic
568C2×C4#51811yescharacteristic
569C2×C4D4211yescharacteristic
570C2×C4#54611yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4#55111yescharacteristic
573C2×C4C22×M1611yescharacteristic
574C2×C4C2×#13111yescharacteristic
575C2×C4C2×#13111yescharacteristic
576C2×C4#56811yescharacteristic
577C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
578C2×C4C2×#13711yescharacteristic
579C2×C4#57711yescharacteristic
580C2×C4#57811yescharacteristic
581C8—24no—
582C8—42no—
583C8—41no—
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8—42no—
589C8—41no—
590C8—41no—
591C8—41no—
592C8—81no—
593C8—81no—
594C8—81no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22#158514yescomplemented
609C22#235811yescharacteristic
610C22C2×#43511yescharacteristic
611C22#293311yescharacteristic
612C22C2×#44011yescharacteristic
613C22#294711yescharacteristic
614C22C2×#44311yescharacteristic
615C22#295611yescharacteristic
616C4—22no—
617C4—22no—
618C4—21no—
619C4—21no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4#292911yescharacteristic
629C4D4×M1611yescharacteristic
630C4#297411yescharacteristic
631C4#297611yescharacteristic
632C4#326511yescharacteristic
633C4#327011yescharacteristic
634C4#330811yescharacteristic
635C4#332011yescharacteristic
636C2—24no—
637C2—82no—
638C2C2×#158511yescharacteristic
639C2#1787611yescharacteristic
640C2#1800411yescharacteristic
641C2#1801711yescharacteristic
642C2#2284611yescharacteristic
643C2#2285511yescharacteristic
644C2#2288811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
4. C2×#1585C2semidirect
5. C22×Q8⋊C8C2semidirect
6. #9666C2semidirect
7. #9559C2semidirect
8. #9544C2semidirect
10. #8952C2semidirect
11. #8945C2semidirect
12. #8848C2semidirect
13. #8829C2semidirect
14. #8213C2semidirect
15. #8144C2semidirect
171. Q8⋊C8D4semidirect
608. C22#1585semidirect

Maximal subgroups

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

subgroupindex
#229852
#225102
C4×#4482
C2×#15852
C22×Q8⋊C82
#96662
#95592
#95442
#90722
#89522
#89452
#88482
#88292
#82132
#81442