Finite Group Databasebeta

← #224828

Subgroups — #224828

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36576C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#13931C211yesmaximal, characteristic, complemented
4#9715C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic, complemented
6#9684C212yesmaximal, complemented
7#9680C211yesmaximal, characteristic, complemented
8#8575C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic, complemented
10#8216C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic, complemented
13C2×#548—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#446—21no—
16#2856C2211yescharacteristic
17#2856C2212yes—
18#2548C411yescharacteristic, complemented
19#2548C412yescomplemented
20#2538C411yescharacteristic, complemented
21#2538C412yescomplemented
22#2426—21no—
23#2426—22no—
24#2416—21no—
25#2416—22no—
26Q8×C4⋊C4C411yescharacteristic, complemented
27#2341—21no—
28C4×#121C2211yescharacteristic
29C4×#121C2212yes—
30C2×#391—21no—
31#1985—22no—
32#1984—21no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#327C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2212yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44#1445—22no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47#1434C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C22×#41—21no—
54C23×Q8D411yescharacteristic
55#550—44no—
56#548—44no—
57#548—44no—
58C2×#127C2×C411yescharacteristic
59C2×#127C2×C412yes—
60C2×#124—21no—
61C2×#124—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×#121—22no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—21no—
70C2×#121C2312yes—
71C2×#121C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—44no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2×C412yes—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D412yes—
80C2×C4×D4—22no—
81C2×C4×D4—21no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C412yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D412yes—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90#446—44no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—22no—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2311yescharacteristic
96C2×#97—22no—
97C2×C4×C8—21no—
98#396—22no—
99#396—22no—
100#396—22no—
101#394—22no—
102#394—22no—
103#394—22no—
104#391—44no—
105#390—22no—
106#390—21no—
107#390—21no—
108#390—22no—
109#390—21no—
110#390—22no—
111#390C2×C412yes—
112#390C2×C412yes—
113#390C2×C412yes—
114#390C2×C411yescharacteristic
115#390C2×C412yes—
116#389—22no—
117#389—21no—
118#389C2×C411yescharacteristic
119#389C2×C412yes—
120#388—21no—
121#388—22no—
122#386—21no—
123#386—22no—
124#384C2×C411yescharacteristic
125#384C2×C412yes—
126#380—22no—
127#380—21no—
128#380C2311yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146#327—22no—
147#327—24no—
148#327—22no—
149#327—24no—
150#327—42no—
151#327—42no—
152#327—42no—
153C2×#41—44no—
154C2×#41—44no—
155C2×#41—44no—
156C22×Q8—24no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—42no—
160C22×Q8C22⋊C411yescharacteristic
161C22×Q8C22⋊C412yes—
162C22×Q8C22⋊C411yescharacteristic
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C412yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184#127—44no—
185#127—42no—
186#124—48no—
187#124—44no—
188C22⋊Q8—44no—
189C22⋊Q8—48no—
190#121—22no—
191#121—21no—
192#121—22no—
193#121—21no—
194#121—22no—
195#121—21no—
196#121—22no—
197#121—21no—
198#121—48no—
199#121—44no—
200#121—44no—
201#121—44no—
202#121—42no—
203#121—42no—
204#121—81no—
205#121—82no—
206#121—81no—
207#121—82no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—44no—
211C4×Q8—42no—
212C4×Q8—42no—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×D4—44no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—42no—
220C4×D4—42no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C412yes—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—42no—
261C2×C42—44no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42C22×C412yes—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C412yes—
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C4⋊C8—42no—
274C4⋊C8—44no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—21no—
279C22⋊C8—22no—
280C22⋊C8#4112yescomplemented
281#97—44no—
282C4×C8—42no—
283#95—21no—
284#95—22no—
285#95—21no—
286#95—22no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—21no—
292#95—22no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—82no—
298#95—81no—
299#95#4112yescomplemented
300C24—22no—
301C24—21no—
302C24C22×D411yescharacteristic
303#41—816no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—42no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8#9914yes—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—81no—
344C2×D4—82no—
345C2×D4—81no—
346C2×D4—82no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4AGL(1,ℤ/8)12yes—
403C22×C4AGL(1,ℤ/8)12yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C42−1+411yescharacteristic
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—41no—
418C2×C8—42no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—82no—
438C4⋊C4—84no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—42no—
447C22⋊C4—41no—
448C22⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—81no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—22no—
458C42—24no—
459C42—24no—
460C42—22no—
461C42—41no—
462C42—42no—
463C42—41no—
464C42—42no—
465C42—44no—
466C42—42no—
467C42—84no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23C22×C22⋊C411yescharacteristic
487C23#54811yescharacteristic
488C23#55011yescharacteristic
489C23C2×AGL(1,ℤ/8)11yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23#57611yescharacteristic
492C23#57711yescharacteristic
493Q8—28no—
494Q8—48no—
495Q8—44no—
496D4—82no—
497D4—81no—
498D4—82no—
499D4—81no—
500D4—88no—
501D4—82no—
502D4—81no—
503D4—82no—
504D4—81no—
505D4—82no—
506D4—81no—
507D4—82no—
508D4—81no—
509D4—84no—
510D4—164no—
511D4—162no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—88no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—84no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4C2×#9914yes—
583C2×C4C2×#9912yes—
584C2×C4#41812yes—
585C2×C4#41812yes—
586C2×C4#42112yes—
587C2×C4C4×#4111yescharacteristic
588C2×C4#52011yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4#55011yescharacteristic
591C8—42no—
592C8—81no—
593C8—82no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—82no—
607C22—81no—
608C22—82no—
609C22—81no—
610C22—82no—
611C22—81no—
612C22—82no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22—82no—
619C22—81no—
620C22#234111yescharacteristic
621C22C22×#9911yescharacteristic
622C22#288411yescharacteristic
623C22C2×#41811yescharacteristic
624C22#289711yescharacteristic
625C22C2×#42111yescharacteristic
626C22#290111yescharacteristic
627C22#336611yescharacteristic
628C22#336811yescharacteristic
629C22#339711yescharacteristic
630C22#339911yescharacteristic
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—22no—
636C4—24no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—162no—
649C4#151014yes—
650C2—21no—
651C2—21no—
652C2—21no—
653C2—21no—
654C2—162no—
655C2—161no—
656C2C2×#151011yescharacteristic
657C2#1760211yescharacteristic
658C2#1761111yescharacteristic
659C2#1763111yescharacteristic
660C2#2253411yescharacteristic
661C2#2254511yescharacteristic
662C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36576C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. #13931C2semidirect
4. #9715C2semidirect
5. #9712C2semidirect
6. #9684C2semidirect
7. #9680C2semidirect
8. #8575C2semidirect
9. #8571C2semidirect
10. #8216C2semidirect
11. C4×#327C2semidirect
12. C2×#550C4semidirect
18. #2548C4semidirect
19. #2548C4semidirect
20. #2538C4semidirect
21. #2538C4semidirect
26. Q8×C4⋊C4C4semidirect
280. C22⋊C8#41semidirect
299. #95#41semidirect

Maximal subgroups

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

subgroupindex
#365762
#225102
#139312
#97152
#97122
#96842
#96802
#85752
#85712
#82162
C4×#3272