Finite Group Databasebeta

← #6624344

Subgroups — #6624344

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33150C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#29112C212yesmaximal, complemented
6#28115C212yesmaximal
7#27961C211yesmaximal, characteristic, complemented
8#27645C211yesmaximal, characteristic
9#27442C214yesmaximal, complemented
10#27380C214yesmaximal, complemented
11#26027C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#23069C212yesmaximal, complemented
14#22762C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21580C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#481C2212yes—
20C2×#477C2212yes—
21C2×#448C2211yescharacteristic
22C2×C4×SD16C2212yes—
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2589—22no—
26#2563C2211yescharacteristic
27#2552C2211yescharacteristic
28#2509C2212yes—
29#2497—21no—
30#2487—21no—
31#2454C2212yes—
32#2450—22no—
33#2369C2211yescharacteristic
34#2355—22no—
35#2355—21no—
36#2355—21no—
37#2355C2212yes—
38#2355C2214yes—
39#2355C2212yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2211yescharacteristic
42#2345—21no—
43C4×#121C2212yes—
44C4×#121C2211yescharacteristic
45C2×#391C2212yes—
46C2×#387—22no—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2084C2212yes—
54#2074C2214yes—
55#2071C2212yes—
56#2066C2214yes—
57#2055C2214yes—
58#2026C2214yes—
59#1990C2212yes—
60#1989C2214yes—
61#1949C2214yes—
62#1948C2212yes—
63#1939C2212yes—
64#1935C2214yes—
65#1935C2214yes—
66#1927C2214yes—
67#1925C2214yes—
68#1920C2214yes—
69#1920C2214yes—
70#1918C2214yes—
71#1917C2214yes—
72#1880C2212yes—
73#1877C2214yes—
74#1874C2211yescharacteristic
75#1865C2212yes—
76#1863C2214yes—
77#1853C2211yescharacteristic
78#1852C2212yes—
79#1834C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1803C2212yes—
84#1802C2212yes—
85#1797C2212yes—
86#1797C2211yescharacteristic
87#1787C2211yescharacteristic
88#1787C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×#121C2311yescharacteristic
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4—22no—
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—22no—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#481—28no—
123#477—28no—
124#448—24no—
125C4×SD16—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2311yescharacteristic
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#400—22no—
144#400—22no—
145#399—22no—
146#398—22no—
147#396—22no—
148#393—22no—
149#391—21no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2314yes—
154#391C2314yes—
155#390C2312yes—
156#390C2311yescharacteristic
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2311yescharacteristic
165#389C2312yes—
166#388—21no—
167#388—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—24no—
172#387—24no—
173#387—24no—
174#387—24no—
175#386—21no—
176#386—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2312yes—
187#382—22no—
188#382—22no—
189#382—24no—
190#382C2312yes—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—22no—
196#380C2311yescharacteristic
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—24no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—42no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C4⋊Q8—24no—
259#125—44no—
260C22⋊Q8—82no—
261#121—22no—
262#121#4114yes—
263C4×Q8—28no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8#4118yes—
269C4×D4—24no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—84no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—28no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42—84no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340#107—28no—
341#106—24no—
342#106—28no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345#103—24no—
346#103—28no—
347#103—24no—
348#103—44no—
349#103—42no—
350#102—28no—
351#102—24no—
352C22⋊C8—22no—
353C22⋊C8—24no—
354C22⋊C8—22no—
355#97—44no—
356C4×C8—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8C2×#4114yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4C2×#4112yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4#13714yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C42−1+412yes—
449C22×C4C2511yescharacteristic
450SD16—48no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4C2×#4114yes—
493C4⋊C4C2×#4112yes—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—82no—
500C22⋊C4—81no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—28no—
504C42—28no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—84no—
514C42—84no—
515C42—82no—
516C42C2×#4114yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23#54812yes—
529C23C2×#13712yes—
530C23#57712yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—24no—
534Q8—48no—
535Q8—42no—
536Q8—42no—
537Q8—162no—
538D4—24no—
539D4—48no—
540D4—42no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4—84no—
596C2×C4—164no—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4#54812yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4#57512yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4112yes—
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58512yes—
607C8—44no—
608C8—42no—
609C8—42no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#337214yes—
624C22C2×#52511yescharacteristic
625C22C2×#54811yescharacteristic
626C22#348511yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C22C2×#57711yescharacteristic
630C22#358511yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—41no—
641C4—44no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—84no—
646C4—82no—
647C4#311714yes—
648C4#347512yes—
649C4#349512yes—
650C2—24no—
651C2—82no—
652C2#3651611yescharacteristic
653C2C2×#311711yescharacteristic
654C2#6058811yescharacteristic
655C2C2×#337211yescharacteristic
656C2#6132411yescharacteristic
657C2#6171911yescharacteristic
658C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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×#2355C2semidirect
3. #33150C2semidirect
4. #32957C2semidirect
5. #29112C2semidirect
7. #27961C2semidirect
9. #27442C2semidirect
10. #27380C2semidirect
11. #26027C2semidirect
12. #24679C2semidirect
13. #23069C2semidirect
14. #22762C2semidirect
15. #22629C2semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#23552
#331502
#329572
#291122
#281152
#279612
#276452
#274422
#273802
#260272
#246792
#230692
#227622
#226292
#215802
C4×#4482