Finite Group Databasebeta

← #6971551

Subgroups — #6971551

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36463C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32739C212yesmaximal, complemented
4#31756C214yesmaximal, complemented
5#31096C214yesmaximal, complemented
6#30203C211yesmaximal, characteristic, complemented
7#30172C212yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#29354C211yesmaximal, characteristic, complemented
10#29213C211yesmaximal, characteristic, complemented
11#26460C212yesmaximal, complemented
12#26442C212yesmaximal, complemented
13#26302C212yesmaximal, complemented
14#26232C212yesmaximal
15#24298C212yesmaximal, complemented
16#23838C211yesmaximal, characteristic, complemented
17#22012C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#481C2214yes—
21C2×#474C2214yes—
22#2868C2211yescharacteristic
23#2858C2211yescharacteristic
24#2857C2212yes—
25#2854C2212yes—
26#2563C2211yescharacteristic
27#2557—22no—
28#2551—21no—
29#2551C2212yes—
30#2546—21no—
31#2541C2211yescharacteristic
32#2513—21no—
33#2490C2212yes—
34#2459—22no—
35#2445—21no—
36#2440—22no—
37#2440C2214yes—
38#2434—22no—
39#2420C2212yes—
40#2369C2211yescharacteristic
41#2366—21no—
42#2366C2211yescharacteristic
43#2357—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C4×#121C2212yes—
49C42×D4C2211yescharacteristic
50#2088C2214yes—
51#2085C2212yes—
52#2084C2211yescharacteristic
53#2083C2211yescharacteristic
54#2076C2214yes—
55#2076C2214yes—
56#2071C2214yes—
57#2071C2212yes—
58#2067C2214yes—
59#2059C2212yes—
60#2058C2212yes—
61#2058C2211yescharacteristic
62#2058C2214yes—
63#2057C2212yes—
64#2057C2211yescharacteristic
65#2034C2214yes—
66#2034C2214yes—
67#2023C2214yes—
68#2023C2212yes—
69#2023C2212yes—
70#2023C2212yes—
71#1992C2212yes—
72#1990C2212yes—
73#1988C2214yes—
74#1987C2212yes—
75#1982C2212yes—
76#1957C2212yes—
77#1954C2212yes—
78#1948C2212yes—
79#1947C2212yes—
80#1939C2212yes—
81#1936C2214yes—
82#1924C2214yes—
83#1919C2214yes—
84#1874C2214yes—
85#1862C2214yes—
86#1852C2214yes—
87#1847C2212yes—
88#1824C2214yes—
89#1817C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C4×#106C2212yes—
93#1767C2211yescharacteristic
94#1744C2211yescharacteristic
95C23×Q8—22no—
96D4×Q8—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2312yes—
107C2×#127—21no—
108C2×#127C2311yescharacteristic
109C2×#125C2311yescharacteristic
110C2×#124—21no—
111C2×#124—21no—
112C2×#123—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×#121C2312yes—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2314yes—
127C22×C42C2312yes—
128#481—48no—
129#474—48no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146#398—24no—
147#397—22no—
148#397—22no—
149#395—24no—
150#395—24no—
151#394—22no—
152#394—22no—
153#393—22no—
154#392—22no—
155#391—21no—
156#391—22no—
157#391—22no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—21no—
182#387—21no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—24no—
188#385—24no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#382C2311yescharacteristic
196#381—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—84no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—24no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—28no—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—48no—
261C4⋊Q8—48no—
262C4⋊Q8—82no—
263#127—24no—
264#127—44no—
265#125—22no—
266#125—41no—
267#124—82no—
268#124—82no—
269#123—84no—
270C22⋊Q8—416no—
271C22⋊Q8—48no—
272#121—24no—
273#121—42no—
274#121—44no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—84no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—24no—
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×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—42no—
347#107—44no—
348#106—24no—
349#106—24no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#102—28no—
354#102—44no—
355#102—44no—
356#102—44no—
357C22⋊C8—44no—
358C22⋊C8—44no—
359#97—42no—
360C4×C8—24no—
361#95—21no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—24no—
373#95—22no—
374#95—24no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24C22×D412yes—
380C2×Q8—24no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—84no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—832no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4#13714yes—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—84no—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4AGL(1,ℤ/8)14yes—
487C4⋊C4H(5,𝔽2)14yes—
488C4⋊C42−1+412yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—21no—
511C42—21no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42—42no—
517C42—82no—
518C422−1+414yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C2×#13714yes—
533C23C23×D411yescharacteristic
534Q8—48no—
535Q8—816no—
536Q8—88no—
537Q8—162no—
538D4—28no—
539D4—44no—
540D4—44no—
541D4—88no—
542D4—84no—
543D4—162no—
544D4—164no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—816no—
594C2×C4—164no—
595C2×C4#53612yes—
596C2×C4#53612yes—
597C2×C4C2×AGL(1,ℤ/8)14yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×#13712yes—
600C2×C4#57612yes—
601C2×C4#57712yes—
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4C2×2−1+411yescharacteristic
604C2×C4C2×2−1+412yes—
605C2×C4#58511yescharacteristic
606C8—42no—
607C8—44no—
608C8—42no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#53611yescharacteristic
624C22#348411yescharacteristic
625C22#348911yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57611yescharacteristic
629C22C2×#57711yescharacteristic
630C4—22no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—44no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—84no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4#323914yes—
647C4#355012yes—
648C4#355412yes—
649C2—44no—
650C2—162no—
651C2#4248111yescharacteristic
652C2C2×#323911yescharacteristic
653C2#6087711yescharacteristic
654C2#6159111yescharacteristic
655C2#6159311yescharacteristic
656C2#6212711yescharacteristic
657C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Q8C2semidirect
3. #32739C2semidirect
4. #31756C2semidirect
5. #31096C2semidirect
6. #30203C2semidirect
7. #30172C2semidirect
8. #29639C2semidirect
9. #29354C2semidirect
10. #29213C2semidirect
11. #26460C2semidirect
12. #26442C2semidirect
13. #26302C2semidirect
15. #24298C2semidirect
16. #23838C2semidirect
17. #22012C2semidirect

Maximal subgroups

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

subgroupindex
#364632
D4×C4⋊Q82
#327392
#317562
#310962
#302032
#301722
#296392
#293542
#292132
#264602
#264422
#263022
#262322
#242982
#238382
#220122