Finite Group Databasebeta

← #6561633

Subgroups — #6561633

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30364C212yesmaximal, complemented
4#30277C211yesmaximal, characteristic, complemented
5#29346C211yesmaximal, characteristic, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29162C212yesmaximal, complemented
8#28865C214yesmaximal, complemented
9#28713C212yesmaximal, complemented
10#28605C212yesmaximal
11#28091C214yesmaximal, complemented
12#27863C212yesmaximal, complemented
13#23835C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23223C212yesmaximal, complemented
16#23163C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#498C2212yes—
20C2×#486C2212yes—
21C2×#436C2212yes—
22#2868C2211yescharacteristic
23#2858C2211yescharacteristic
24#2857C2212yes—
25#2854C2212yes—
26#2602—22no—
27#2592—21no—
28#2571—21no—
29#2563C2212yes—
30#2557—22no—
31#2547C2211yescharacteristic
32#2544—21no—
33#2541C2211yescharacteristic
34#2538—21no—
35#2497C2214yes—
36#2492—22no—
37#2491C2212yes—
38#2439—22no—
39#2422C2212yes—
40#2369C2212yes—
41#2367—21no—
42#2366C2211yescharacteristic
43#2365C2211yescharacteristic
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2094C2212yes—
54#2094C2214yes—
55#2094C2212yes—
56#2083C2212yes—
57#2083C2211yescharacteristic
58#2083C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2071C2212yes—
62#2057C2211yescharacteristic
63#2057C2212yes—
64#2057C2211yescharacteristic
65#2033C2214yes—
66#2033C2214yes—
67#2023C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1987C2212yes—
72#1982C2212yes—
73#1944C2214yes—
74#1936C2214yes—
75#1934C2214yes—
76#1925C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1895C2214yes—
80#1895C2214yes—
81#1862C2214yes—
82#1834C2212yes—
83#1809C2214yes—
84#1809C2214yes—
85#1776C2214yes—
86C4×#107C2212yes—
87C4×#107C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×M16C2314yes—
96#559—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127—21no—
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×#124—22no—
107C2×#124—21no—
108C2×#124—21no—
109C2×#123—22no—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C22×C42C2312yes—
125#498—28no—
126#486—28no—
127#436—28no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#397—22no—
145#397—22no—
146#396—24no—
147#394—22no—
148#394—22no—
149#393—24no—
150#393—22no—
151#392—22no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—24no—
158#390—22no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#388—24no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—24no—
182#386—22no—
183#386—22no—
184#386—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#380—22no—
195#380—22no—
196#380C2314yes—
197#380C2314yes—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×M16—28no—
247C2×M16—44no—
248C2×M16—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—41no—
257C4⋊Q8—42no—
258C4⋊Q8—82no—
259C4⋊Q8—82no—
260#127—24no—
261#127—44no—
262#124—48no—
263#124—44no—
264#124—44no—
265#124—44no—
266#123—48no—
267#123—48no—
268C22⋊Q8—84no—
269#121—24no—
270#121—42no—
271#121—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281#117—84no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—24no—
343#107—42no—
344#107—42no—
345#107—42no—
346#106—28no—
347#106—28no—
348C4⋊C8—24no—
349#102—28no—
350#102—28no—
351#102—44no—
352#102—44no—
353C22⋊C8—28no—
354#97—44no—
355C4×C8—42no—
356C4×C8#4118yes—
357#95—21no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24C22×D412yes—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—81no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438M16—88no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8C2×#4114yes—
447C2×C8C2×#4114yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—81no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C42−1+412yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—21no—
498C42—21no—
499C42—24no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—41no—
505C42—44no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—84no—
510C42AGL(1,ℤ/8)14yes—
511C42#13714yes—
512C42C2×#4114yes—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23C23×D411yescharacteristic
528Q8—84no—
529Q8—84no—
530Q8—84no—
531Q8—164no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—88no—
536D4—84no—
537D4—162no—
538D4—164no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—84no—
586C2×C4—164no—
587C2×C4#53212yes—
588C2×C4#53212yes—
589C2×C4C2×AGL(1,ℤ/8)12yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×#13712yes—
593C2×C4C2×#13712yes—
594C2×C4C2×#13712yes—
595C2×C4C22×#4112yes—
596C2×C4C2×2−1+411yescharacteristic
597C2×C4#58512yes—
598C2×C4#58511yescharacteristic
599C8—24no—
600C8—24no—
601C8—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×#53211yescharacteristic
617C22#348511yescharacteristic
618C22#348911yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22C22×#13711yescharacteristic
621C22#358511yescharacteristic
622C22#358511yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—82no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—81no—
639C4#318214yes—
640C4#350712yes—
641C4#351012yes—
642C2—82no—
643C2—162no—
644C2#3595911yescharacteristic
645C2C2×#318211yescharacteristic
646C2#6071911yescharacteristic
647C2#6171911yescharacteristic
648C2#6172011yescharacteristic
649C2#6211011yescharacteristic
650C2#6211211yescharacteristic
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. #36194C2semidirect
3. #30364C2semidirect
4. #30277C2semidirect
5. #29346C2semidirect
6. #29218C2semidirect
7. #29162C2semidirect
8. #28865C2semidirect
9. #28713C2semidirect
11. #28091C2semidirect
12. #27863C2semidirect
13. #23835C2semidirect
14. #23760C2semidirect
15. #23223C2semidirect
16. #23163C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#361942
#303642
#302772
#293462
#292182
#291622
#288652
#287132
#286052
#280912
#278632
#238352
#237602
#232232
#231632
#205682