Finite Group Databasebeta

← #198029

Subgroups — #198029

Order 512.

1491 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#34141C211yesmaximal, characteristic, complemented
3#21209C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5#17485C211yesmaximal, characteristic, complemented
6#9237C214yesmaximal, complemented
7#9233C214yesmaximal, complemented
8C2×#1303C211yesmaximal, characteristic, complemented
9#7866C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×C4×#41—21no—
13#3245C414yescomplemented
14#3226—21no—
15#3226C412yescomplemented
16#3205—21no—
17#3205C412yescomplemented
18#3186—22no—
19#3169—22no—
20#3100—22no—
21#3100C2214yes—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421—21no—
26C2×#421—21no—
27#2367—22no—
28#2366—21no—
29#2365—21no—
30#2364—22no—
31#2361—22no—
32#2297—21no—
33#2293C2212yescomplemented
34#2290—22no—
35C42×D4C2212yescomplemented
36C4×#117C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1963—22no—
39#1962C2214yes—
40#1502—22no—
41#1502C2212yes—
42#1502C2212yes—
43C2×#327—22no—
44#1492—22no—
45#1486C2214yescomplemented
46#1480—22no—
47#1303C2212yescomplemented
48#1303C2212yescomplemented
49#1276C2214yescomplemented
50C22×#41—21no—
51C22×#41—21no—
52C22×M16—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532C2×C412yes—
64#532C2×C412yes—
65#532C2×C412yescomplemented
66#532D412yes—
67#532D412yes—
68#532C2312yes—
69C2×C4⋊Q8C2×C411yescharacteristic, complemented
70C2×C4⋊Q8D411yescharacteristic
71C2×#127C2×C411yescharacteristic, complemented
72C2×#127D411yescharacteristic
73C2×#124—21no—
74C2×#124—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×#121D412yes—
79C4×#41—44no—
80C4×#41—44no—
81C2×C4×Q8—22no—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2312yes—
89C2×#117—22no—
90C2×#117D411yescharacteristic
91C2×#117D411yescharacteristic
92C22×C4⋊C4—21no—
93C22×C42—21no—
94C22×C42—22no—
95C22×C42C2311yescharacteristic
96#500—22no—
97#493—22no—
98#486—22no—
99#486—22no—
100#480—24no—
101#480—24no—
102#474—24no—
103#474C2×C418yescomplemented
104#466—28no—
105#459—24no—
106#459C2×C418yescomplemented
107#447—24no—
108#447—24no—
109C4×D8—28no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2×C412yescomplemented
116#431C2312yes—
117#431C2312yes—
118#423—21no—
119#423—21no—
120#423—21no—
121#423C2312yes—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8C2×C412yescomplemented
126#421—44no—
127#421—44no—
128C2×#104—24no—
129C2×#104—24no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134#391—22no—
135#390—22no—
136#390—22no—
137#389—22no—
138#388—24no—
139#388—24no—
140#387—24no—
141#386—24no—
142#386—24no—
143#385—24no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4D412yes—
150C4×C4⋊C4C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—24no—
157#376D412yes—
158C2×#95D412yes—
159C43C2312yes—
160#327—24no—
161#327—24no—
162D4⋊C8—28no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—41no—
170C2×#41—41no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C2×#41—42no—
175C2×#41—42no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—41no—
188C22×D4C2×D412yes—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—41no—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×D8—28no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C2×M16—42no—
201C22×C8—22no—
202C22×C8—41no—
203C22×C8—41no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C22⋊C412yes—
210C4⋊Q8C22⋊C412yes—
211C4⋊Q8C22×C412yes—
212#127—21no—
213#127—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127C22⋊C412yes—
218#127C22⋊C412yes—
219#127C22×C412yes—
220#124—22no—
221#124—22no—
222#124—22no—
223#124—22no—
224#124—22no—
225#124—22no—
226#124—41no—
227#124—41no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—41no—
239#121—41no—
240#121—41no—
241#121—41no—
242#121C22⋊C414yes—
243#121C22⋊C414yes—
244C4×Q8—48no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—416no—
256C4×D4—48no—
257C4×D4—48no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4C22×C414yes—
263C4×D4C2×D414yes—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—44no—
268#117—44no—
269#117D814yescomplemented
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D412yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—44no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42#4111yescharacteristic
312C2×C42#4111yescharacteristic
313C2×C42C2411yescharacteristic
314#107—28no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—41no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8C22×C414yes—
323#104—48no—
324#104—48no—
325#102—28no—
326#102—28no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—41no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—22no—
334C4×C8—41no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95C2×D412yes—
341#95C2×D412yes—
342C24—22no—
343C24—41no—
344C24#12011yescharacteristic
345#41—44no—
346#41—44no—
347#41—44no—
348#41—44no—
349#41—44no—
350#41—44no—
351#41—44no—
352#41—44no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—44no—
358#41—44no—
359#41—82no—
360#41—82no—
361#41—82no—
362#41—82no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4C2×C22⋊C414yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—84no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4#12011yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4#12011yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4#12111yescharacteristic
436C22×C4#12311yescharacteristic
437C22×C4#12311yescharacteristic
438C22×C4C2×D812yes—
439C22×C4#13512yes—
440C22×C4AGL(1,ℤ/8)12yes—
441C22×C4AGL(1,ℤ/8)12yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445D8—88no—
446M16—42no—
447M16—42no—
448M16—82no—
449C2×C8—24no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—82no—
453C2×C8—82no—
454C2×C8—82no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4C2×C22⋊C414yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—84no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—48no—
486C42—44no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—44no—
492C42—82no—
493C42C2×C22⋊C411yescharacteristic
494C42C2×C22⋊C411yescharacteristic
495C42C4×D411yescharacteristic
496C42C4×D411yescharacteristic
497C42C4×D411yescharacteristic
498C42C4×D411yescharacteristic
499C42#12111yescharacteristic
500C42#12111yescharacteristic
501C42#12111yescharacteristic
502C42#12111yescharacteristic
503C42#12311yescharacteristic
504C42#12311yescharacteristic
505C42C2×D814yes—
506C42C23×C411yescharacteristic
507C42C22×D411yescharacteristic
508C42C2×#4111yescharacteristic
509C42C2×#4111yescharacteristic
510C23—24no—
511C23—41no—
512C23—41no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23#38611yescharacteristic
522C23#38611yescharacteristic
523C23#44712yes—
524C23#44912yes—
525C23C2×#12011yescharacteristic
526Q8—44no—
527Q8—44no—
528Q8—42no—
529Q8—42no—
530Q8—82no—
531D4—28no—
532D4—44no—
533D4—48no—
534D4—48no—
535D4—44no—
536D4—44no—
537D4—44no—
538D4—42no—
539D4—42no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545D4—84no—
546D4—82no—
547D4—82no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—816no—
582C2×C4—88no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#38611yescharacteristic
588C2×C4#38611yescharacteristic
589C2×C4#38611yescharacteristic
590C2×C4#38611yescharacteristic
591C2×C4#38611yescharacteristic
592C2×C4#38611yescharacteristic
593C2×C4C4×D812yes—
594C2×C4C4×D812yes—
595C2×C4#44212yes—
596C2×C4#44212yes—
597C2×C4#44712yes—
598C2×C4#44912yes—
599C2×C4#45912yes—
600C2×C4#45912yes—
601C2×C4#46312yes—
602C2×C4#46312yes—
603C2×C4C22×C22⋊C411yescharacteristic
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12311yescharacteristic
609C2×C4C22×D811yescharacteristic
610C2×C4C2×#13511yescharacteristic
611C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
612C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
613C8—44no—
614C8—82no—
615C8—82no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—84no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—164no—
633C22#187912yes—
634C22#187912yes—
635C22C2×#38611yescharacteristic
636C22C2×#44711yescharacteristic
637C22C2×#44911yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#187912yes—
652C4#187912yes—
653C4#290112yes—
654C4C2×C4×D811yescharacteristic
655C4C2×#44211yescharacteristic
656C4C2×#45911yescharacteristic
657C4C2×#46311yescharacteristic
658C2—21no—
659C2—21no—
660C2—22no—
661C2—22no—
662C2—84no—
663C2—162no—
664C2C2×#187911yescharacteristic
665C2#2098011yescharacteristic
666C2#2100011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#60940C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34141C2semidirect
3. #21209C2semidirect
4. C2×#1502C2semidirect
5. #17485C2semidirect
6. #9237C2semidirect
7. #9233C2semidirect
8. C2×#1303C2semidirect
9. #7866C2semidirect
13. #3245C4semidirect
15. #3226C4semidirect
17. #3205C4semidirect
33. #2293C22semidirect
35. C42×D4C22semidirect
45. #1486C22semidirect
47. #1303C22semidirect
48. #1303C22semidirect
49. #1276C22semidirect
65. #532C2×C4semidirect
69. C2×C4⋊Q8C2×C4semidirect
71. C2×#127C2×C4semidirect
103. #474C2×C4semidirect
106. #459C2×C4semidirect
115. #431C2×C4semidirect
125. C2×C4⋊C8C2×C4semidirect
269. #117D8semidirect

Maximal subgroups

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

subgroupindex
#609402
#341412
#212092
C2×#15022
#174852
#92372
#92332
C2×#13032
#78662