Finite Group Databasebeta

← #6468614

Subgroups — #6468614

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#29352C214yesmaximal, complemented
4#29336C214yesmaximal, complemented
5#29220C212yesmaximal, complemented
6#28750C212yesmaximal, complemented
7#27973C211yesmaximal, characteristic
8#27972C211yesmaximal, characteristic, complemented
9#25903C211yesmaximal, characteristic
10#25902C211yesmaximal, characteristic, complemented
11#23821C214yesmaximal, complemented
12#21647C212yesmaximal, complemented
13#21646C212yesmaximal, complemented
14#21391C214yesmaximal, complemented
15C4×#432C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529C2212yes—
18C22×C4×Q8—21no—
19C2×#487C2212yes—
20C2×#486C2212yes—
21C2×#432C2212yes—
22#2861C2212yes—
23#2860—21no—
24#2860—21no—
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2854—21no—
30#2854—21no—
31#2850C2212yes—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×#127C2211yescharacteristic
42C4×#124C2212yes—
43C4×#121—21no—
44C4×#121—21no—
45C2×#391—21no—
46#2293—21no—
47#2293—21no—
48C2×#389—21no—
49C2×#389C2212yes—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2212yes—
53C2×C43C2211yescharacteristic
54#2101C2218yes—
55#2072C2214yes—
56#2071C2214yes—
57#2042C2218yes—
58#2024C2214yes—
59#2023C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1989C2218yes—
63#1983C2218yes—
64#1957C2212yes—
65#1956C2212yes—
66#1936C2218yes—
67#1935C2218yes—
68#1874C2214yes—
69#1874C2214yes—
70#1872C2214yes—
71#1872C2214yes—
72#1834C2212yes—
73#1834C2212yes—
74#1767C2212yes—
75C4×C4⋊C8C2212yes—
76#1763C2218yes—
77#1762C2218yes—
78#1749C2212yes—
79C4×C22⋊C8C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81#1745C2212yes—
82C4×#97C2212yes—
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91#529—216no—
92C2×#126—22no—
93C2×#125C2312yes—
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×#123—22no—
97C2×C22⋊Q8C2312yes—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—48no—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C42—22no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#487—28no—
127#486—28no—
128#432—28no—
129C2×C4⋊C8C2314yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×#102C2318yes—
133C2×#102C2318yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2314yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2312yes—
145#391C2312yes—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389C2314yes—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#386—22no—
181#386—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#382C2312yes—
189#380—21no—
190#380—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2314yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C2×#95—22no—
204C2×#95—22no—
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2318yes—
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—28no—
243C4⋊Q8—28no—
244C4⋊Q8—28no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8#4118yes—
248#127—24no—
249#126—48no—
250#125—28no—
251#124—28no—
252#124—44no—
253#123—48no—
254C22⋊Q8—28no—
255#121—28no—
256#121—44no—
257#121—44no—
258C4×Q8—28no—
259C4×Q8—48no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×D4—28no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—48no—
267#117—48no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—48no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4118yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C4⋊C8—28no—
349#103—216no—
350#103—48no—
351#102—216no—
352#102—48no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355#97—28no—
356#97—44no—
357C4×C8—24no—
358C4×C8—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369C24—22no—
370C24C22×D411yescharacteristic
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—44no—
377C2×Q8—432no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4#12414yes—
424C22×C4#12414yes—
425C22×C4AGL(1,ℤ/8)14yes—
426C22×C4#13714yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4114yes—
441C22×C4C2511yescharacteristic
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4C2×#4118yes—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—48no—
508C42—48no—
509C42—44no—
510C42#12414yes—
511C42#12414yes—
512C42C2×#4114yes—
513C42C2×#4114yes—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C2×#12412yes—
527C23C2×AGL(1,ℤ/8)12yes—
528C23C2×#13712yes—
529C23C23×D411yescharacteristic
530Q8—416no—
531Q8—416no—
532Q8—48no—
533D4—48no—
534D4—88no—
535D4—84no—
536D4—84no—
537C2×C4—28no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4#52512yes—
589C2×C4#52512yes—
590C2×C4C2×#12412yes—
591C2×C4C2×#12412yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#12412yes—
594C2×C4C2×#12412yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57512yes—
600C2×C4#57512yes—
601C2×C4C22×#4112yes—
602C2×C4C22×#4112yes—
603C2×C4#58512yes—
604C8—44no—
605C8—44no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22#314114yes—
620C22C2×#52511yescharacteristic
621C22C22×#12411yescharacteristic
622C22#344911yescharacteristic
623C22C22×AGL(1,ℤ/8)11yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57511yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—48no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#311414yes—
644C4#311714yes—
645C4#348014yes—
646C2—24no—
647C2—162no—
648C2#3495511yescharacteristic
649C2C2×#311411yescharacteristic
650C2C2×#311711yescharacteristic
651C2C2×#314111yescharacteristic
652C2#6061811yescharacteristic
653C2#6096511yescharacteristic
654C2#6096611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #29352C2semidirect
4. #29336C2semidirect
5. #29220C2semidirect
6. #28750C2semidirect
8. #27972C2semidirect
10. #25902C2semidirect
11. #23821C2semidirect
12. #21647C2semidirect
13. #21646C2semidirect
14. #21391C2semidirect
15. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#293522
#293362
#292202
#287502
#279732
#279722
#259032
#259022
#238212
#216472
#216462
#213912
C4×#4322