Finite Group Databasebeta

← #6561643

Subgroups — #6561643

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36463C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic, complemented
3#30364C212yesmaximal, complemented
4#30277C211yesmaximal, characteristic, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29225C211yesmaximal, characteristic, complemented
7#29156C212yesmaximal, complemented
8#28850C214yesmaximal, complemented
9#28706C212yesmaximal, complemented
10#28597C212yesmaximal
11#28082C214yesmaximal, complemented
12#27857C212yesmaximal, complemented
13#23838C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23220C212yesmaximal, complemented
16#23163C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#498C2212yes—
20C2×#486C2212yes—
21C2×C8×D4C2212yes—
22#2868C2211yescharacteristic
23#2858C2211yescharacteristic
24#2857C2212yes—
25#2854C2212yes—
26#2568—21no—
27#2563C2211yescharacteristic
28#2557—22no—
29#2547—21no—
30#2547C2212yes—
31#2541C2211yescharacteristic
32#2513—21no—
33#2499—22no—
34#2496—22no—
35#2496C2214yes—
36#2490C2212yes—
37#2459—22no—
38#2445—21no—
39#2420C2212yes—
40#2369C2211yescharacteristic
41#2366—21no—
42#2366C2211yescharacteristic
43#2365—21no—
44#2365C2212yes—
45#2357—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2094C2212yes—
54#2094C2214yes—
55#2094C2212yes—
56#2085C2212yes—
57#2084C2211yescharacteristic
58#2083C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2071C2212yes—
62#2059C2212yes—
63#2058C2211yescharacteristic
64#2057C2211yescharacteristic
65#2033C2214yes—
66#2033C2214yes—
67#2023C2212yes—
68#1992C2212yes—
69#1988C2212yes—
70#1988C2212yes—
71#1987C2212yes—
72#1982C2212yes—
73#1944C2214yes—
74#1936C2214yes—
75#1936C2214yes—
76#1926C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1893C2214yes—
80#1893C2214yes—
81#1864C2214yes—
82#1836C2212yes—
83#1807C2214yes—
84#1807C2214yes—
85#1776C2214yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89#1767C2211yescharacteristic
90#1762C2214yes—
91#1752C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127—21no—
103C2×#127C2311yescharacteristic
104C2×#126—24no—
105C2×#125C2311yescharacteristic
106C2×#124—22no—
107C2×#124—21no—
108C2×#124—21no—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C22×C42C2312yes—
123#498—28no—
124#486—28no—
125C8×D4—28no—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#400—24no—
142#397—24no—
143#397—24no—
144#397—22no—
145#397—22no—
146#394—22no—
147#394—22no—
148#393—22no—
149#392—22no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—22no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—21no—
177#387—21no—
178#387—24no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—22no—
191#382C2311yescharacteristic
192#381—22no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#380C2314yes—
198#380C2314yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
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—22no—
234C22×Q8—21no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C22×C8—28no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257#127—24no—
258#127—44no—
259#126—416no—
260#125—22no—
261#125—41no—
262#124—48no—
263#124—82no—
264#124—82no—
265#123—84no—
266C22⋊Q8—48no—
267#121—24no—
268#121—42no—
269#121—44no—
270C4×Q8—44no—
271C4×Q8—48no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—42no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—84no—
278#117—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
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—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
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—22no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—44no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—24no—
339#107—42no—
340#106—28no—
341#106—28no—
342#106—42no—
343#106—42no—
344C4⋊C8—24no—
345#102—28no—
346#102—28no—
347#102—44no—
348#102—44no—
349C22⋊C8—28no—
350#97—42no—
351#97—44no—
352C4×C8#4118yes—
353#95—21no—
354#95—22no—
355#95—24no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—22no—
373C24C22×D412yes—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—81no—
382C2×Q8—81no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4AGL(1,ℤ/8)14yes—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—88no—
441C2×C8C2×#4114yes—
442C2×C8C2×#4114yes—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—82no—
474C4⋊C42−1+412yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—24no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—48no—
502C42—44no—
503C42—42no—
504C42—42no—
505C42—82no—
506C42C2×#4114yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C2×AGL(1,ℤ/8)14yes—
521C23C23×D411yescharacteristic
522Q8—84no—
523Q8—88no—
524Q8—162no—
525D4—28no—
526D4—44no—
527D4—44no—
528D4—88no—
529D4—84no—
530D4—162no—
531D4—164no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—164no—
582C2×C4#53212yes—
583C2×C4#53212yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4#57512yes—
586C2×C4#57712yes—
587C2×C4#57712yes—
588C2×C4C22×#4112yes—
589C2×C4C2×2−1+411yescharacteristic
590C2×C4#58512yes—
591C2×C4#58511yescharacteristic
592C8—24no—
593C8—24no—
594C8—44no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C2×#53211yescharacteristic
610C22#348511yescharacteristic
611C22#348911yescharacteristic
612C22C22×AGL(1,ℤ/8)11yescharacteristic
613C22C2×#57511yescharacteristic
614C22C2×#57711yescharacteristic
615C22C2×#57711yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—84no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—81no—
632C4#318014yes—
633C4#350712yes—
634C4#351012yes—
635C2—44no—
636C2—162no—
637C2#3595911yescharacteristic
638C2C2×#318011yescharacteristic
639C2#6071211yescharacteristic
640C2#6172811yescharacteristic
641C2#6173911yescharacteristic
642C2#6212811yescharacteristic
643C2#6213511yescharacteristic
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. #36188C2semidirect
3. #30364C2semidirect
4. #30277C2semidirect
5. #29354C2semidirect
6. #29225C2semidirect
7. #29156C2semidirect
8. #28850C2semidirect
9. #28706C2semidirect
11. #28082C2semidirect
12. #27857C2semidirect
13. #23838C2semidirect
14. #23760C2semidirect
15. #23220C2semidirect
16. #23163C2semidirect
17. #20558C2semidirect

Maximal subgroups

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

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