Finite Group Databasebeta

← #6635201

Subgroups — #6635201

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2451C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32903C211yesmaximal, characteristic
5#31868C214yesmaximal, complemented
6#31804C214yesmaximal, complemented
7#31240C214yesmaximal, complemented
8#31156C214yesmaximal, complemented
9#29967C211yesmaximal, characteristic, complemented
10#29755C211yesmaximal, characteristic
11#28115C212yesmaximal, complemented
12#24603C212yesmaximal, complemented
13#24503C212yesmaximal, complemented
14#24299C212yesmaximal, complemented
15#21951C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#481C2212yes—
20C2×#475C2212yes—
21C2×#476C2214yes—
22C2×#460C2212yes—
23C2×#448C2212yes—
24C2×#431C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yes—
33#2506C2212yes—
34#2452C2212yes—
35#2451—22no—
36#2451—22no—
37#2451C2214yes—
38#2451C2214yes—
39#2445C2212yes—
40C2×#398—21no—
41C2×#395—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#391—21no—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2099C2212yes—
52#2088C2214yes—
53#2088C2214yes—
54#2084C2212yes—
55#2066C2214yes—
56#2066C2214yes—
57#2065C2214yes—
58#2065C2214yes—
59#2058C2212yes—
60#2058C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2041C2212yes—
64#2023C2214yes—
65#1987C2214yes—
66#1968C2214yes—
67#1941C2214yes—
68#1939C2214yes—
69#1939C2214yes—
70#1934C2214yes—
71#1933C2214yes—
72#1927C2214yes—
73#1880C2214yes—
74#1877C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1865C2214yes—
78#1853C2212yes—
79#1853C2212yes—
80#1824C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1787C2212yes—
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×SD16C2314yes—
90#539—48no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#125—22no—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—44no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#481—44no—
126#476—48no—
127#475—44no—
128#460—44no—
129#448—44no—
130#431—44no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#398—22no—
149#398—24no—
150#398—24no—
151#397—22no—
152#396—22no—
153#396—22no—
154#395—24no—
155#395—24no—
156#394—22no—
157#394—22no—
158#393—22no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—24no—
163#391—24no—
164#391C2314yes—
165#391C2314yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#387—24no—
189#387—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#380—21no—
197#380C2312yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2314yes—
220#336C2312yes—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—42no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8—82no—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—28no—
254C4⋊Q8—44no—
255C4⋊Q8—42no—
256C4⋊Q8—48no—
257#127—42no—
258#126—48no—
259#125—48no—
260#123—48no—
261#123—48no—
262C22⋊Q8—44no—
263C22⋊Q8—48no—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—44no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—42no—
271C4×Q8—42no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×D4—44no—
275C4×D4—48no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4SD1618yescomplemented
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
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×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341#106—28no—
342#106—44no—
343#106—44no—
344C4⋊C8—44no—
345C4⋊C8—42no—
346C4⋊C8—42no—
347C4⋊C8—44no—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—28no—
352#102—44no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C4×C8—44no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—42no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—84no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—816no—
393C2×Q82−1+414yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—81no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4C2×SD1614yes—
447C22×C4C2×SD1614yes—
448C22×C4C2×SD1614yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461SD16—88no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C4C2×SD1618yes—
497C4⋊C4C2×SD1618yes—
498C4⋊C4H(5,𝔽2)14yes—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—24no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—44no—
520C42—44no—
521C42—44no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#53712yes—
534C23C22×SD1612yes—
535C23#57512yes—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—24no—
539Q8—48no—
540Q8—42no—
541Q8—42no—
542Q8—816no—
543Q8—162no—
544D4—44no—
545D4—44no—
546D4—88no—
547D4—162no—
548D4—162no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—164no—
605C2×C4#53712yes—
606C2×C4C22×SD1614yes—
607C2×C4C22×SD1614yes—
608C2×C4C22×SD1612yes—
609C2×C4C22×SD1612yes—
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+412yes—
613C2×C4C2×2−1+411yescharacteristic
614C8—44no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#324614yes—
630C22C2×#53711yescharacteristic
631C22#348411yescharacteristic
632C22#348611yescharacteristic
633C22C23×SD1611yescharacteristic
634C22C2×#57511yescharacteristic
635C22#358511yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—41no—
645C4—41no—
646C4—44no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4#353612yes—
654C4#353812yes—
655C2—24no—
656C2—162no—
657C2#3671911yescharacteristic
658C2C2×#324611yescharacteristic
659C2#6091911yescharacteristic
660C2#6156211yescharacteristic
661C2#6158111yescharacteristic
662C2#6182011yescharacteristic
663C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2451C2semidirect
3. #32954C2semidirect
5. #31868C2semidirect
6. #31804C2semidirect
7. #31240C2semidirect
8. #31156C2semidirect
9. #29967C2semidirect
11. #28115C2semidirect
12. #24603C2semidirect
13. #24503C2semidirect
14. #24299C2semidirect
15. #21951C2semidirect
303. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24512
#329542
#329032
#318682
#318042
#312402
#311562
#299672
#297552
#281152
#246032
#245032
#242992
#219512