Finite Group Databasebeta

← #7545558

Subgroups — #7545558

Order 512.

1549 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#32554C214yesmaximal, complemented
4#29219C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#27736C212yesmaximal, complemented
7#27735C212yesmaximal, complemented
8#26509C212yesmaximal, complemented
9#26110C212yesmaximal, complemented
10#25318C212yesmaximal
11#25002C214yesmaximal, complemented
12#24677C212yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#23220C212yesmaximal, complemented
15#22758C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#486C2212yes—
20C2×#476C2211yescharacteristic
21C2×#459C2211yescharacteristic
22C2×C4×D8C2212yes—
23#2868C2211yescharacteristic
24#2857—21no—
25#2804—22no—
26#2580—22no—
27#2580C2214yes—
28#2571C2212yes—
29#2563—21no—
30#2563C2212yes—
31#2552C2212yes—
32#2542C2212yes—
33#2538C2212yes—
34#2487—22no—
35#2441—22no—
36#2429—22no—
37#2423—22no—
38#2366C2212yes—
39#2365C2212yes—
40#2354C2212yes—
41Q8×C4⋊C4C2212yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C42×D4C2211yescharacteristic
48#2101C2214yes—
49#2100C2214yes—
50#2099C2214yes—
51#2088C2214yes—
52#2073C2214yes—
53#2071C2214yes—
54#2071C2214yes—
55#2054C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1988C2212yes—
59#1988C2212yes—
60#1987C2211yescharacteristic
61#1985C2214yes—
62#1982C2211yescharacteristic
63#1949C2214yes—
64#1948C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1936C2214yes—
69#1936C2214yes—
70#1926C2214yes—
71#1925C2212yes—
72#1921C2214yes—
73#1920C2212yes—
74#1879C2212yes—
75#1879C2214yes—
76#1864C2212yes—
77#1864C2214yes—
78#1836C2214yes—
79#1834C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1802C2211yescharacteristic
83#1802C2211yescharacteristic
84#1802C2214yes—
85C4×#106C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#533—48no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#125—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2314yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C22×C4⋊C4—22no—
119C22×C42—21no—
120#486—28no—
121#476—24no—
122#459—24no—
123C4×D8—28no—
124C2×#107C2312yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2314yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#399—24no—
145#399—24no—
146#398—24no—
147#396—24no—
148#393—24no—
149#393—24no—
150#391—22no—
151#391—21no—
152#391D412yes—
153#390—22no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—21no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#388—24no—
170#387—24no—
171#386—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384—21no—
176#384—21no—
177#384D411yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2314yes—
182#384C2312yes—
183#382—22no—
184#382C2314yes—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C43C2311yescharacteristic
202#340C2314yes—
203#340C2312yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8C2×D412yes—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2412yes—
224C22×D4C2411yescharacteristic
225C23×C4—21no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C2×D8—24no—
231C2×D8—24no—
232C2×D8—42no—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—44no—
243C4⋊Q8—82no—
244#127—22no—
245#127#4114yes—
246#125—48no—
247C22⋊Q8—48no—
248C22⋊Q8—48no—
249C22⋊Q8—84no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—82no—
255C4×D4—24no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—24no—
259C4×D4—22no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—82no—
266C4×D4—84no—
267C4×D4#4114yes—
268#117—48no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—84no—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—44no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2414yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—24no—
324#106—24no—
325#106—42no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—44no—
331#102—24no—
332#102—24no—
333#102—24no—
334#102—28no—
335#102—28no—
336#102—24no—
337#102—42no—
338#102—42no—
339#97—44no—
340C4×C8—24no—
341C4×C8—24no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—24no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—21no—
355C24—22no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4C2×#4114yes—
382C2×D4C2×#4112yes—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2511yescharacteristic
414D8—48no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—84no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4C2×#4112yes—
462C22⋊C4—24no—
463C22⋊C4—48no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—82no—
472C22⋊C4—84no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—24no—
477C42—21no—
478C42—21no—
479C42—41no—
480C42—48no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42AGL(1,ℤ/8)14yes—
486C42C2×#4112yes—
487C42C2×#4112yes—
488C422−1+412yes—
489C422−1+412yes—
490C23—22no—
491C23—24no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23C23×D411yescharacteristic
505Q8—48no—
506Q8—84no—
507Q8—88no—
508D4—22no—
509D4—24no—
510D4—22no—
511D4—24no—
512D4—48no—
513D4—42no—
514D4—41no—
515D4—41no—
516D4—42no—
517D4—84no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—88no—
558C2×C4—168no—
559C2×C4#54812yes—
560C2×C4#54812yes—
561C2×C4#54812yes—
562C2×C4#54812yes—
563C2×C4C2×AGL(1,ℤ/8)12yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C2×C4C2×AGL(1,ℤ/8)12yes—
566C2×C4#57612yes—
567C2×C4#57712yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C2×2−1+411yescharacteristic
571C2×C4C2×2−1+411yescharacteristic
572C2×C4#58511yescharacteristic
573C2×C4#58511yescharacteristic
574C8—42no—
575C8—42no—
576C8—42no—
577C8—42no—
578C22—44no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—82no—
592C22C2×#54811yescharacteristic
593C22C2×#54811yescharacteristic
594C22#348511yescharacteristic
595C22C22×AGL(1,ℤ/8)11yescharacteristic
596C22C2×#57611yescharacteristic
597C22C2×#57711yescharacteristic
598C22#358611yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—44no—
603C4—41no—
604C4—44no—
605C4—42no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4#336614yes—
613C4#337314yes—
614C4#347511yescharacteristic
615C4#350711yescharacteristic
616C4#351511yescharacteristic
617C4#355511yescharacteristic
618C2—84no—
619C2—82no—
620C2#5907911yescharacteristic
621C2C2×#336611yescharacteristic
622C2C2×#337311yescharacteristic
623C2#6134011yescharacteristic
624C2#6134511yescharacteristic
625C2#6171711yescharacteristic
626C2#6178811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
2. #36134C2semidirect
3. #32554C2semidirect
4. #29219C2semidirect
5. #28716C2semidirect
6. #27736C2semidirect
7. #27735C2semidirect
8. #26509C2semidirect
9. #26110C2semidirect
11. #25002C2semidirect
12. #24677C2semidirect
13. #23250C2semidirect
14. #23220C2semidirect
15. #22758C2semidirect
16. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#361342
#325542
#292192
#287162
#277362
#277352
#265092
#261102
#253182
#250022
#246772
#232502
#232202
#227582
C4×#4762
C4×#4592