Finite Group Databasebeta

← #6479615

Subgroups — #6479615

Order 512.

1936 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#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32957C212yesmaximal, complemented
4#32903C212yesmaximal
5#29906C212yesmaximal, complemented
6#29624C212yesmaximal, complemented
7#29500C214yesmaximal, complemented
8#29369C214yesmaximal, complemented
9#29132C212yesmaximal, complemented
10#28386C212yesmaximal, complemented
11#28206C212yesmaximal, complemented
12#28160C212yesmaximal, complemented
13#22047C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—21no—
17C22×C4×Q8C2212yes—
18C2×#477C2214yes—
19C2×#460C2214yes—
20C2×#448C2214yes—
21#2861—21no—
22#2860—21no—
23#2860C2212yes—
24#2859C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854—21no—
29#2854C2212yes—
30#2850—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#127C2211yescharacteristic
41C4×#124—21no—
42C4×#121—21no—
43C4×#121C2212yes—
44C2×#391C2212yes—
45#2293—21no—
46#2293C2212yes—
47C2×#389—21no—
48C2×#389—21no—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4—21no—
52C2×C43C2211yescharacteristic
53#2099C2214yes—
54#2093C2214yes—
55#2083C2214yes—
56#2071C2214yes—
57#2043C2214yes—
58#2043C2214yes—
59#2013C2214yes—
60#2013C2214yes—
61#1973C2218yes—
62#1935C2218yes—
63#1928C2218yes—
64#1880C2218yes—
65#1877C2218yes—
66#1874C2214yes—
67#1863C2218yes—
68#1852C2214yes—
69#1839C2212yes—
70#1831C2212yes—
71#1817C2212yes—
72#1803C2214yes—
73#1803C2214yes—
74#1797C2212yes—
75#1787C2214yes—
76#1767C2212yes—
77#1749C2212yes—
78#1745C2212yes—
79C23×Q8C2312yes—
80C22×SD16C2318yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127C2311yescharacteristic
88#529—48no—
89C2×#126—22no—
90C2×#125—21no—
91C2×#124—21no—
92C2×#124—21no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2312yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—48no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C42—22no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#477—48no—
124#460—48no—
125#448—216no—
126C2×#107C2318yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#102C2318yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2314yes—
134#391—22no—
135#391—22no—
136#391—21no—
137#391—21no—
138#391C2314yes—
139#391C2314yes—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#386—22no—
175#386—22no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#384C2311yescharacteristic
182#382—21no—
183#380—21no—
184#380C2312yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C2×#95—22no—
198C2×#95—22no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C22×Q8—22no—
214C22×Q8—21no—
215C22×Q8—24no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8#4118yes—
219C22×Q8C2414yes—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2412yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×SD16—216no—
235C2×SD16—48no—
236C2×SD16—48no—
237C22×C8C2418yes—
238C4⋊Q8—28no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242C4⋊Q8—24no—
243#127—24no—
244#126—48no—
245#125—44no—
246#124—44no—
247#124—44no—
248#123—48no—
249C22⋊Q8—44no—
250#121—28no—
251#121—44no—
252#121—44no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—48no—
258C4×Q8—48no—
259C4×Q8—44no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—48no—
263C4×D4—44no—
264#117—48no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
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×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—48no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—48no—
346C4⋊C8—28no—
347#103—216no—
348#103—48no—
349#103—48no—
350#102—48no—
351C22⋊C8—28no—
352#97—44no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363C24—22no—
364C24C22×D411yescharacteristic
365C2×Q8—28no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—416no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—416no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8C2×#4118yes—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
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×C4AGL(1,ℤ/8)14yes—
424C22×C4#13714yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
439C22×C4C2511yescharacteristic
440SD16—816no—
441C2×C8—28no—
442C2×C8—28no—
443C2×C8—28no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
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—22no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—48no—
507C42—44no—
508C42—48no—
509C42#13714yes—
510C42H(5,𝔽2)14yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#52512yes—
523C23C2×AGL(1,ℤ/8)12yes—
524C23C2×#13712yes—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—28no—
528Q8—416no—
529Q8—44no—
530Q8—44no—
531Q8—84no—
532D4—48no—
533D4—84no—
534D4—88no—
535D4—84no—
536C2×C4—28no—
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—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
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—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—416no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—88no—
588C2×C4—88no—
589C2×C4#52512yes—
590C2×C4#53512yes—
591C2×C4#53512yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57512yes—
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)12yes—
602C2×C4#58512yes—
603C8—48no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#311214yes—
618C22C2×#52511yescharacteristic
619C22C2×#53511yescharacteristic
620C22#344911yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C22×#13711yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—48no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4#321814yes—
643C4#350514yes—
644C2—24no—
645C2—162no—
646C2#3510211yescharacteristic
647C2C2×#311211yescharacteristic
648C2#6056011yescharacteristic
649C2C2×#321811yescharacteristic
650C2#6080311yescharacteristic
651C2#6095211yescharacteristic
652C2#6096911yescharacteristic
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. #32957C2semidirect
5. #29906C2semidirect
6. #29624C2semidirect
7. #29500C2semidirect
8. #29369C2semidirect
9. #29132C2semidirect
10. #28386C2semidirect
11. #28206C2semidirect
12. #28160C2semidirect
13. #22047C2semidirect
14. #20778C2semidirect

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
#329572
#329032
#299062
#296242
#295002
#293692
#291322
#283862
#282062
#281602
#220472
#207782