Finite Group Databasebeta

← #227691

Subgroups — #227691

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23569C211yesmaximal, characteristic
2#22985C211yesmaximal, characteristic, complemented
3#22531C211yesmaximal, characteristic, complemented
4C2×C8×SD16C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic, complemented
6#9665C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8830C211yesmaximal, characteristic, complemented
14#8213C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C8×D4—21no—
22C2×C8×D4C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2059C411yescharacteristic
27#2042C411yescharacteristic
28#2036C411yescharacteristic
29#2034C411yescharacteristic
30#2025C411yescharacteristic
31#2012C411yescharacteristic
32#2000C411yescharacteristic
33#1999C411yescharacteristic
34#1955—21no—
35#1954—21no—
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1954C2211yescharacteristic
39#1939C2211yescharacteristic
40#1923—21no—
41#1869—21no—
42#1850—21no—
43#1798—21no—
44#1797C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C8×SD16—24no—
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1483—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74#529—22no—
75#529—22no—
76#529—41no—
77#529—41no—
78C2×#126—21no—
79C2×#125C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×#123—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×#117—21no—
91C22×C42C2311yescharacteristic
92#446—22no—
93#446—41no—
94C8×Q8—22no—
95C8×Q8—41no—
96C4×SD16—22no—
97C4×SD16—22no—
98C4×D8—41no—
99C4×D8—41no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—44no—
103C2×#107C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×C4⋊C8—22no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—41no—
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C4×M16—22no—
136C4×M16—41no—
137C2×#97—21no—
138C2×#97—21no—
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#390C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#388—21no—
150#386—21no—
151#380C2×C411yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C22⋊C4—21no—
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#336—21no—
159#336—21no—
160#336—21no—
161#336C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336D411yescharacteristic
164#336D411yescharacteristic
165#334—22no—
166#334—41no—
167Q8⋊C8—22no—
168Q8⋊C8D414yescomplemented
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173#322—41no—
174C82—22no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C2×SD16—22no—
186C2×SD16—22no—
187C2×SD16—22no—
188C2×D8—22no—
189C2×D8—41no—
190C2×D8—41no—
191C2×M16—24no—
192C2×M16—42no—
193C2×M16—42no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—81no—
205C22×C8—81no—
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214#126—42no—
215#126—42no—
216#125—22no—
217#125—41no—
218#124—22no—
219#124—41no—
220#123—42no—
221#123—42no—
222C22⋊Q8—22no—
223C22⋊Q8—41no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—41no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C2×D412yes—
231C4×Q8#4112yes—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244#117—42no—
245#117—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
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×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276#107—41no—
277#106—22no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288C4⋊C8—42no—
289C4⋊C8C2×D412yes—
290#103—22no—
291#103—22no—
292#103—41no—
293#102—22no—
294#102—22no—
295#102—22no—
296#102—41no—
297#102—41no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310#97—42no—
311#97—42no—
312C4×C8—21no—
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—42no—
324C4×C8—42no—
325C4×C8C2×D412yes—
326#95—21no—
327#95—21no—
328#95—21no—
329C24—21no—
330C24—21no—
331C24C4×D411yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C4×D412yes—
338C2×D4—24no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—81no—
355C2×D4#13112yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
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—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396SD16—28no—
397D8—44no—
398M16—48no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—21no—
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—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—84no—
440C2×C8—84no—
441C2×C8#13112yes—
442C2×C8AGL(1,ℤ/8)12yes—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4C4×D412yes—
458C4⋊C4#13112yes—
459C22⋊C4—24no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—81no—
471C42—21no—
472C42—21no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—41no—
480C42—41no—
481C42C22×D411yescharacteristic
482C42C2×#4111yescharacteristic
483C42H(5,𝔽2)11yescharacteristic
484C42H(5,𝔽2)11yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23#44312yes—
506C23C2×C4×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—22no—
512D4—22no—
513D4—48no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—84no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4C2×C4×D411yescharacteristic
569C2×C4#51811yescharacteristic
570C2×C4D4211yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4#54611yescharacteristic
573C2×C4#55111yescharacteristic
574C2×C4C2×#13111yescharacteristic
575C2×C4C2×#13111yescharacteristic
576C2×C4C2×#13111yescharacteristic
577C2×C4#56811yescharacteristic
578C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
579C2×C4#57511yescharacteristic
580C2×C4#57711yescharacteristic
581C2×C4#57711yescharacteristic
582C8—24no—
583C8—22no—
584C8—22no—
585C8—42no—
586C8—41no—
587C8—41no—
588C8—41no—
589C8—41no—
590C8—42no—
591C8—41no—
592C8—41no—
593C8—81no—
594C8—81no—
595C8—81no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#235811yescharacteristic
618C22#292911yescharacteristic
619C22#293511yescharacteristic
620C22C4×AGL(1,ℤ/8)11yescharacteristic
621C22#295211yescharacteristic
622C22C2×#44311yescharacteristic
623C22#295611yescharacteristic
624C4—21no—
625C4—21no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4#292911yescharacteristic
636C4#293511yescharacteristic
637C4#297611yescharacteristic
638C4#297911yescharacteristic
639C4#326511yescharacteristic
640C4#327111yescharacteristic
641C4#332611yescharacteristic
642C4#333211yescharacteristic
643C2—42no—
644C2—82no—
645C2—161no—
646C2#1791011yescharacteristic
647C2#1792711yescharacteristic
648C2#1807811yescharacteristic
649C2#1808811yescharacteristic
650C2#2284311yescharacteristic
651C2#2288311yescharacteristic
652C2#2288811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
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. #22985C2semidirect
3. #22531C2semidirect
4. C2×C8×SD16C2semidirect
5. C2×#1483C2semidirect
6. #9665C2semidirect
7. #9554C2semidirect
8. #9542C2semidirect
9. #9078C2semidirect
10. #8952C2semidirect
11. #8947C2semidirect
12. #8846C2semidirect
13. #8830C2semidirect
14. #8213C2semidirect
15. #8146C2semidirect
168. Q8⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235692
#229852
#225312
C2×C8×SD162
C2×#14832
#96652
#95542
#95422
#90782
#89522
#89472
#88462
#88302
#82132
#81462