Finite Group Databasebeta

← #7537806

Subgroups — #7537806

Order 512.

1911 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
1C2×#3400C212yesmaximal
2#59827C211yesmaximal, characteristic
3#36560C211yesmaximal, characteristic, complemented
4#30757C214yesmaximal, complemented
5#28471C212yesmaximal, complemented
6#28440C211yesmaximal, characteristic, complemented
7#28438C211yesmaximal, characteristic
8#26604C214yesmaximal, complemented
9#25629C212yesmaximal
10#25365C212yesmaximal, complemented
11#25302C212yesmaximal, complemented
12#24730C212yesmaximal, complemented
13#24191C212yesmaximal, complemented
14#23153C211yesmaximal, characteristic, complemented
15#23002C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic
18C2×#559C2212yes—
19C2×#557C2212yes—
20#3400—216no—
21C2×#497C2214yes—
22C2×#494C2212yes—
23C2×#492C2212yes—
24C2×#464C2214yes—
25C2×#462C2212yes—
26C2×#462C2211yescharacteristic
27C2×#462C2212yes—
28C2×#460C2211yescharacteristic
29C2×#461C2212yes—
30C2×#461C2214yes—
31C2×#461C2212yes—
32C2×#446C2212yes—
33C2×#440C2214yes—
34C2×C4×Q16C2212yes—
35C2×C4×SD16C2212yes—
36#2868C2211yescharacteristic
37#2859—21no—
38#2724—22no—
39#2582C2212yes—
40#2568C2212yes—
41#2565—22no—
42#2547—21no—
43#2547C2212yes—
44#2546C2212yes—
45#2546C2212yes—
46#2506—22no—
47#2506—22no—
48#2486—22no—
49#2447—22no—
50#2447C2214yes—
51#2365C2212yes—
52#2365C2212yes—
53D4×C4⋊C4—21no—
54D4×C4⋊C4C2212yes—
55C4×C4⋊Q8—21no—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2066C2214yes—
61#2056C2214yes—
62#2043C2214yes—
63#2016C2214yes—
64#2016C2214yes—
65#2015C2214yes—
66#2015C2214yes—
67#2013C2214yes—
68#1984C2212yes—
69#1982C2211yescharacteristic
70#1981C2211yescharacteristic
71#1971C2214yes—
72#1956C2212yes—
73#1956C2212yes—
74#1948C2211yescharacteristic
75#1947C2211yescharacteristic
76#1939C2212yes—
77#1935C2214yes—
78#1928C2212yes—
79#1926C2212yes—
80#1917C2214yes—
81#1880C2212yes—
82#1878C2214yes—
83#1878C2212yes—
84#1877C2214yes—
85#1877C2214yes—
86#1863C2214yes—
87#1803C2211yescharacteristic
88#1803C2211yescharacteristic
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C22×Q16C2312yes—
95C22×Q16C2314yes—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2312yes—
100#559—28no—
101#559—44no—
102#557—28no—
103#557—44no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#125—22no—
112C2×#125—24no—
113C2×#124—22no—
114C2×#124C2314yes—
115C2×#123—24no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2314yes—
128C22×C4⋊C4—22no—
129#497—216no—
130#494—44no—
131#492—28no—
132#464—216no—
133#462—28no—
134#462—24no—
135#462—44no—
136#461—216no—
137#461—28no—
138#461—44no—
139#460D418yes—
140#446—28no—
141#440—216no—
142C4×Q16—28no—
143C4×SD16—44no—
144C2×#107C2312yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#97C2314yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166#400—24no—
167#398—24no—
168#398—24no—
169#398—24no—
170#397—24no—
171#397—24no—
172#397—24no—
173#391—22no—
174#391—22no—
175#391D412yes—
176#391C2312yes—
177#391C2314yes—
178#391C2314yes—
179#391C2314yes—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—21no—
191#389—21no—
192#389D412yes—
193#389D412yes—
194#389C2314yes—
195#387—24no—
196#387—24no—
197#385—24no—
198#385—24no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384D411yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#382—22no—
207#381—24no—
208#380—22no—
209#380—22no—
210#380—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C43—21no—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2312yes—
232#336C2312yes—
233C22×Q8C2×D412yes—
234C22×Q8C2412yes—
235C22×Q8C2414yes—
236C22×Q8C2411yescharacteristic
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2412yes—
240C23×C4—22no—
241C23×C4—24no—
242C2×Q16—24no—
243C2×Q16—28no—
244C2×Q16—24no—
245C2×Q16—42no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—42no—
256C2×SD16—42no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×SD16C2×D418yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275#127—22no—
276#127C2×D414yes—
277#125—48no—
278#125—416no—
279#124—28no—
280#124—44no—
281#124—48no—
282#123—416no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8—41no—
292C4×Q8—41no—
293C4×Q8—44no—
294C4×Q8—48no—
295C4×Q8C2×D414yes—
296C4×D4—28no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—44no—
302C4×D4—82no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2414yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#106—42no—
359C4⋊C8—28no—
360C4⋊C8—22no—
361C4⋊C8—22no—
362C4⋊C8C2×D414yes—
363C4⋊C8#4114yes—
364#103—24no—
365#103—28no—
366#103—28no—
367#103—24no—
368#103—24no—
369#103—42no—
370#103—42no—
371#103—44no—
372#103—44no—
373#102—42no—
374#102—42no—
375#102C2×D418yes—
376#97—28no—
377C4×C8—24no—
378C4×C8—24no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8C22×D412yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4C22×D414yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D414yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450Q16—48no—
451Q16—88no—
452Q16—88no—
453SD16—48no—
454SD16—416no—
455SD16—88no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8C22×D414yes—
467C2×C8C2×#4114yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—84no—
502C4⋊C4—81no—
503C4⋊C4C22×D412yes—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—21no—
516C42—21no—
517C42—41no—
518C42—48no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42#13714yes—
526C42C22×D412yes—
527C42C2×#4112yes—
528C42H(5,𝔽2)12yes—
529C42H(5,𝔽2)12yes—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×D411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—48no—
542Q8—41no—
543Q8—44no—
544Q8—42no—
545Q8—44no—
546Q8—42no—
547Q8—44no—
548Q8—42no—
549Q8—42no—
550Q8—44no—
551D4—24no—
552D4—24no—
553D4—42no—
554D4—48no—
555D4—42no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—88no—
597C2×C4—168no—
598C2×C4D4212yes—
599C2×C4D4212yes—
600C2×C4#55012yes—
601C2×C4#55012yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4#57712yes—
606C2×C4#57812yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4#58511yescharacteristic
612C2×C4#58511yescharacteristic
613C8—24no—
614C8—24no—
615C8—42no—
616C8—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C2×D4211yescharacteristic
625C22C2×#55011yescharacteristic
626C22#348711yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57711yescharacteristic
629C22C2×#57811yescharacteristic
630C22#358611yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—44no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—82no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—81no—
644C4—81no—
645C4#327614yes—
646C4#340014yescomplemented
647C4D4×#4111yescharacteristic
648C4#349611yescharacteristic
649C4#351711yescharacteristic
650C4#354611yescharacteristic
651C2—84no—
652C2#5905111yescharacteristic
653C2C2×#327611yescharacteristic
654C2#6109411yescharacteristic
655C2C2×#340011yescharacteristic
656C2#6142211yescharacteristic
657C2#6189311yescharacteristic
658C2#6193411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #36560C2semidirect
4. #30757C2semidirect
5. #28471C2semidirect
6. #28440C2semidirect
8. #26604C2semidirect
10. #25365C2semidirect
11. #25302C2semidirect
12. #24730C2semidirect
13. #24191C2semidirect
14. #23153C2semidirect
15. #23002C2semidirect
16. C4×#462C2semidirect
646. C4#3400semidirect

Maximal subgroups

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

subgroupindex
C2×#34002
#598272
#365602
#307572
#284712
#284402
#284382
#266042
#256292
#253652
#253022
#247302
#241912
#231532
#230022
C4×#4622
C4×#4602