Finite Group Databasebeta

← #6651604

Subgroups — #6651604

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#32690C214yesmaximal, complemented
5#32640C214yesmaximal, complemented
6#32585C214yesmaximal, complemented
7#32581C214yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#27611C211yesmaximal, characteristic, complemented
10#27610C211yesmaximal, characteristic, complemented
11#24608C212yesmaximal, complemented
12#24607C212yesmaximal, complemented
13#24061C211yesmaximal, characteristic
14#24060C211yesmaximal, characteristic
15#21822C211yesmaximal, characteristic, complemented
16C2×#536—21no—
17C2×#532C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#481C2212yes—
21C2×#480C2212yes—
22C2×#432C2212yes—
23#2854C2212yes—
24#2854C2212yes—
25#2618—21no—
26#2618—21no—
27#2584—22no—
28#2584—22no—
29#2584C2214yes—
30#2584C2214yes—
31#2548C2212yes—
32#2546C2212yes—
33#2545—22no—
34#2538C2212yes—
35#2538C2212yes—
36#2536—22no—
37C2×#398—21no—
38C2×#395—21no—
39#2311—21no—
40C2×#389—21no—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2099C2212yes—
49#2099C2212yes—
50#2088C2214yes—
51#2087C2214yes—
52#2084C2212yes—
53#2083C2212yes—
54#2067C2214yes—
55#2065C2214yes—
56#2059C2214yes—
57#2059C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2041C2212yes—
61#2041C2212yes—
62#2025C2214yes—
63#2025C2214yes—
64#1985C2214yes—
65#1985C2214yes—
66#1944C2214yes—
67#1943C2214yes—
68#1936C2214yes—
69#1936C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1920C2214yes—
75#1918C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1864C2214yes—
79#1862C2214yes—
80#1853C2212yes—
81#1852C2212yes—
82#1824C2212yes—
83#1823C2212yes—
84#1817C2212yes—
85#1817C2211yescharacteristic
86#1817C2211yescharacteristic
87C22×#41—21no—
88C23×Q8D411yescharacteristic
89#536—48no—
90#532—44no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#125—22no—
95C2×#125—44no—
96C2×#125C2314yes—
97C2×#124—22no—
98C2×#124—21no—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—44no—
106C2×C4×Q8D412yes—
107C2×C4×Q8D412yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#481—44no—
131#480—44no—
132#432—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145#398—24no—
146#398—24no—
147#398—22no—
148#398—22no—
149#396—22no—
150#396—22no—
151#396—22no—
152#396—22no—
153#395—24no—
154#395—24no—
155#394—22no—
156#394—22no—
157#391—22no—
158#390—22no—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#389—24no—
164#389—24no—
165#389—24no—
166#389—24no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#386—22no—
188#386—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C2×#41—44no—
220C2×#41—44no—
221C2×#41—44no—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×D412yes—
227C22×Q8C2×D412yes—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—42no—
247C4⋊Q8—48no—
248#127—42no—
249#125—44no—
250#125—44no—
251#125—48no—
252#124—44no—
253#124—48no—
254#123—48no—
255#123—48no—
256C22⋊Q8—48no—
257C22⋊Q8—48no—
258#121—24no—
259#121—24no—
260#121—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—42no—
264C4×Q8—42no—
265C4×Q8—42no—
266C4×Q8—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—44no—
270#117—44no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—82no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343#107—28no—
344#107—44no—
345#106—28no—
346#106—44no—
347C4⋊C8—44no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102—44no—
352C22⋊C8—24no—
353C22⋊C8—24no—
354#97—44no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368C24—22no—
369C24C22×D411yescharacteristic
370#41—816no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—416no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4AGL(1,ℤ/8)14yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4H(5,𝔽2)14yes—
446C22×C4C2511yescharacteristic
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C42−1+414yes—
489C4⋊C42−1+414yes—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—44no—
512C42—84no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#53412yes—
525C23C2×AGL(1,ℤ/8)12yes—
526C23#57512yes—
527C23C23×D411yescharacteristic
528Q8—48no—
529Q8—48no—
530Q8—84no—
531D4—44no—
532D4—44no—
533D4—84no—
534D4—84no—
535D4—88no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#53412yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×2−1+412yes—
598C2×C4C2×2−1+412yes—
599C8—44no—
600C8—44no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22#319414yes—
615C22C2×#53411yescharacteristic
616C22#348611yescharacteristic
617C22#348611yescharacteristic
618C22C22×AGL(1,ℤ/8)11yescharacteristic
619C22C2×#57511yescharacteristic
620C22#358511yescharacteristic
621C22#358611yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—44no—
632C4—44no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#354414yes—
639C2—24no—
640C2—162no—
641C2#3725811yescharacteristic
642C2C2×#319411yescharacteristic
643C2#6075511yescharacteristic
644C2#6182511yescharacteristic
645C2#6182811yescharacteristic
646C2#6183111yescharacteristic
647C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2584C2semidirect
3. #33096C2semidirect
4. #32690C2semidirect
5. #32640C2semidirect
6. #32585C2semidirect
7. #32581C2semidirect
8. #29640C2semidirect
9. #27611C2semidirect
10. #27610C2semidirect
11. #24608C2semidirect
12. #24607C2semidirect
15. #21822C2semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#25842
#330962
#326902
#326402
#325852
#325812
#296402
#276112
#276102
#246082
#246072
#240612
#240602
#218222