Finite Group Databasebeta

← #192474

Subgroups — #192474

Order 512.

1041 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2976C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21933C211yesmaximal, characteristic, complemented
4#21773C211yesmaximal, characteristic
5#17572C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8051C211yesmaximal, characteristic, complemented
9C2×#1282C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11#2976—22no—
12#2976—22no—
13#2976—22no—
14#2976C2214yes—
15C2×#436—21no—
16C2×#435—21no—
17C2×#435—21no—
18C2×C8×D4—21no—
19C2×#433—21no—
20C2×#408—21no—
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2280C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1896—22no—
28#1871C2212yes—
29#1871C2212yes—
30#1855—21no—
31#1855—21no—
32#1855C2212yescomplemented
33#1855C2212yescomplemented
34#1851C2212yescomplemented
35#1851C2212yescomplemented
36#1850—21no—
37#1850—21no—
38#1826—21no—
39#1810—22no—
40#1798—22no—
41#1789—21no—
42#1520C411yescharacteristic, complemented
43#1520C411yescharacteristic, complemented
44#1519C411yescharacteristic, complemented
45#1519C411yescharacteristic, complemented
46#1516C414yescomplemented
47#1501C2214yes—
48#1297C2214yescomplemented
49#1282C2214yescomplemented
50C22×#41D411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C22×M16D411yescharacteristic
54C22×M16D411yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C23×C8—21no—
58C23×C8—21no—
59C23×C8D411yescharacteristic
60C23×C8D411yescharacteristic
61#525—21no—
62#525—21no—
63#525—21no—
64#525—21no—
65#525C2×C412yes—
66#525D412yes—
67#525Q812yes—
68#525C2312yes—
69C2×#123—21no—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8Q811yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121Q811yescharacteristic
74C2×#120—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C42—21no—
81#436—24no—
82#436—24no—
83#435—24no—
84#435—24no—
85#435—24no—
86#435—24no—
87C8×D4—24no—
88C8×D4—24no—
89#433—24no—
90#433—24no—
91C2×#108—22no—
92C2×#108—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×#104—22no—
98C2×#104—22no—
99#408—24no—
100#408—24no—
101#407—21no—
102#407—21no—
103#407—21no—
104#407—21no—
105#407—21no—
106#407—21no—
107#407D412yes—
108#407C2312yes—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8C2×C412yes—
123C2×C22⋊C8C2×C411yescharacteristic
124C2×C22⋊C8C2×C411yescharacteristic
125C2×C22⋊C8D412yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×#97—21no—
129C2×C4×C8—21no—
130#381C2×C412yes—
131#380—21no—
132#380C2×C411yescharacteristic
133#380C2×C411yescharacteristic
134#379—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4C2312yes—
140C4×C22⋊C4C2312yes—
141C2×#95—21no—
142#341—22no—
143#341—22no—
144#339—22no—
145#339—22no—
146#336—22no—
147#336—22no—
148#328C2×C414yescomplemented
149#328C2×C414yescomplemented
150#327C2×C414yescomplemented
151#327C2×C414yescomplemented
152C25#4111yescharacteristic
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—81no—
156C2×#41—81no—
157C2×#41C4⋊C412yes—
158C2×#41C2×D412yes—
159C22×Q8C4⋊C411yescharacteristic
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C4⋊C411yescharacteristic
164C23×C4—21no—
165C23×C4—41no—
166C23×C4—81no—
167C23×C4—81no—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q811yescharacteristic
177C23×C4C2×Q811yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4C2411yescharacteristic
182C2×M16—24no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—81no—
190C2×M16—81no—
191C2×M16—81no—
192C2×M16—81no—
193C2×M16—82no—
194C2×M16C2×D412yes—
195C2×M16C2×D412yes—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—24no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—81no—
210C22×C8—81no—
211C22×C8—81no—
212C22×C8—81no—
213C22×C8—82no—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216#123—22no—
217#123—22no—
218#123—22no—
219#123—22no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8C4⋊C412yes—
225C22⋊Q8C4⋊C412yes—
226C22⋊Q8C22×C412yes—
227C22⋊Q8C2×Q812yes—
228#121—21no—
229#121—21no—
230#121—21no—
231#121—21no—
232#121C4⋊C412yes—
233#121C4⋊C412yes—
234#121C22×C412yes—
235#121C2×Q812yes—
236#120—22no—
237#120—22no—
238#120—22no—
239#120—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248#117—21no—
249#117—21no—
250#117—81no—
251#117—81no—
252#117C4⋊C412yes—
253#117C2×D412yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C4⋊C411yescharacteristic
259C2×C4⋊C4C4⋊C411yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—41no—
277C2×C42—81no—
278C2×C42—81no—
279#108—82no—
280#108—82no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283C4⋊C8—82no—
284C4⋊C8—82no—
285#104—82no—
286#104—82no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C22⋊C8C22×C412yes—
306C22⋊C8C22×C412yes—
307#97—24no—
308C4×C8—24no—
309#95—22no—
310#95—22no—
311#95—41no—
312#95C22×C412yes—
313#95C22×C412yes—
314C24—22no—
315C24—22no—
316C24—21no—
317C24—81no—
318C24—81no—
319C24C4×D411yescharacteristic
320C24C4×D411yescharacteristic
321C24C22⋊Q811yescharacteristic
322C24C22⋊Q811yescharacteristic
323C24#12311yescharacteristic
324C24#12311yescharacteristic
325C24C2×#4111yescharacteristic
326#41—42no—
327#41—42no—
328#41—81no—
329#41—81no—
330#41—82no—
331#41—82no—
332C2×Q8—21no—
333C2×Q8—81no—
334C2×Q8C2×C4⋊C412yes—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—81no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4C2×C4⋊C412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C2×C4⋊C411yescharacteristic
383C22×C4C2×C4⋊C411yescharacteristic
384C22×C4C2×C4⋊C411yescharacteristic
385C22×C4C2×C4⋊C411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4#12011yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4C22⋊Q811yescharacteristic
397C22×C4C22⋊Q811yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4C22⋊Q811yescharacteristic
400C22×C4C22⋊Q811yescharacteristic
401C22×C4C22⋊Q811yescharacteristic
402C22×C4#12311yescharacteristic
403C22×C4#12311yescharacteristic
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4111yescharacteristic
410M16—42no—
411M16—42no—
412M16—42no—
413M16—42no—
414M16—42no—
415M16—42no—
416M16—81no—
417M16—81no—
418M16—84no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—81no—
438C2×C8—81no—
439C2×C8—82no—
440C2×C8—81no—
441C2×C8—81no—
442C2×C8—84no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4C2×C4⋊C412yes—
452C4⋊C4C2×C4⋊C412yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—22no—
471C42—22no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—82no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—84no—
504C23—84no—
505C23#38511yescharacteristic
506C23#38511yescharacteristic
507C23#38511yescharacteristic
508C23#38511yescharacteristic
509C23#44312yes—
510C23#44412yes—
511C23C2×C4×D411yescharacteristic
512C23C2×C22⋊Q811yescharacteristic
513C23C2×#12311yescharacteristic
514Q8—42no—
515Q8—81no—
516D4—42no—
517D4—44no—
518D4—42no—
519D4—42no—
520D4—81no—
521D4—82no—
522D4—82no—
523D4—82no—
524D4—82no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
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—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—88no—
579C2×C4#38511yescharacteristic
580C2×C4#38511yescharacteristic
581C2×C4#38511yescharacteristic
582C2×C4#38511yescharacteristic
583C2×C4C22×C4⋊C411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×#12011yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—42no—
593C8—81no—
594C8—81no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22#186612yes—
628C22#186612yes—
629C22C2×#38511yescharacteristic
630C22C2×#44311yescharacteristic
631C22C2×#44411yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—82no—
643C4—82no—
644C4—84no—
645C4#302611yescharacteristic
646C4#302611yescharacteristic
647C2—21no—
648C2—21no—
649C2—22no—
650C2—22no—
651C2—41no—
652C2—41no—
653C2—41no—
654C2—41no—
655C2—82no—
656C2#2086711yescharacteristic
657C2C2×#186611yescharacteristic
658C2#2088911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21933C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8051C2semidirect
9. C2×#1282C2semidirect
32. #1855C22semidirect
33. #1855C22semidirect
34. #1851C22semidirect
35. #1851C22semidirect
42. #1520C4semidirect
43. #1520C4semidirect
44. #1519C4semidirect
45. #1519C4semidirect
46. #1516C4semidirect
48. #1297C22semidirect
49. #1282C22semidirect
148. #328C2×C4semidirect
149. #328C2×C4semidirect
150. #327C2×C4semidirect
151. #327C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29762
#219482
#219332
#217732
#175722
#88752
#88742
#80512
C2×#12822