Finite Group Databasebeta

← #6856698

Subgroups — #6856698

Order 512.

1535 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic
4#32084C214yesmaximal
5#30880C214yesmaximal
6#29799C212yesmaximal
7#29259C212yesmaximal
8#28748C212yesmaximal
9#28748C211yesmaximal, characteristic
10#28401C212yesmaximal
11#28284C211yesmaximal, characteristic
12#27769C212yesmaximal
13#25651C212yesmaximal
14#24191C212yesmaximal
15#22821C211yesmaximal, characteristic
16#22148C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#487C2212yes—
20C2×#487C2214yes—
21C2×#477C2212yes—
22C2×#462C2211yescharacteristic
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#462C2212yes—
26C2×C4×Q16C2212yes—
27#2868C2212yes—
28#2868C2211yescharacteristic
29#2586—22no—
30#2582C2211yescharacteristic
31#2567C2211yescharacteristic
32#2567C2211yescharacteristic
33#2552—21no—
34#2552—22no—
35#2552—22no—
36#2552—21no—
37#2551C2211yescharacteristic
38#2546C2214yes—
39#2546C2212yes—
40#2507—21no—
41#2507C2212yes—
42#2451—21no—
43#2451C2212yes—
44#2399—22no—
45Q8×C4⋊C4—21no—
46Q8×C4⋊C4—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52#2295—21no—
53#2295C2212yes—
54C42×Q8C2211yescharacteristic
55C42×Q8C2211yescharacteristic
56#2095C2214yes—
57#2068C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2214yes—
63#2024C2214yes—
64#2016C2214yes—
65#2016C2214yes—
66#1984C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2211yescharacteristic
72#1981C2212yes—
73#1981C2212yes—
74#1969C2214yes—
75#1956C2214yes—
76#1956C2212yes—
77#1947C2211yescharacteristic
78#1947C2212yes—
79#1933C2214yes—
80#1928C2212yes—
81#1916C2214yes—
82#1878C2214yes—
83#1878C2212yes—
84#1863C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97#557—44no—
98#557—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2314yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×#125—24no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121#487—44no—
122#487—48no—
123#477—44no—
124#462—24no—
125#462—28no—
126#462—44no—
127#462—44no—
128C4×Q16—44no—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#398—24no—
156#398—24no—
157#398—24no—
158#395—22no—
159#395—22no—
160#395—22no—
161#395—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2314yes—
169#391C2312yes—
170#391C2312yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—24no—
175#389—24no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#384—21no—
196#384—21no—
197#384—22no—
198#384—21no—
199#384—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227#376—22no—
228C43—21no—
229C43C2312yes—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2412yes—
242C22×Q8C2414yes—
243C22×Q8C2411yescharacteristic
244C2×Q16—24no—
245C2×Q16—28no—
246C2×Q16—42no—
247C2×Q16—42no—
248C2×Q16—44no—
249C2×Q16—44no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#125—416no—
271#125—48no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—41no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—48no—
285C4×Q8—84no—
286C4×Q8—82no—
287C4×Q8Q1618yescomplemented
288C4×Q8#4114yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—42no—
354#107—44no—
355C4⋊C8—22no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—42no—
360#103—28no—
361#103—24no—
362#103—28no—
363#103—24no—
364#103—44no—
365#103—42no—
366#103—42no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373C4×C8—24no—
374C4×C8—44no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×Q8C2×Q1614yes—
421C2×Q8C2×#4112yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442Q16—48no—
443Q16—88no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4C2×Q1618yes—
497C4⋊C4C2×Q1614yes—
498C4⋊C4#13714yes—
499C4⋊C4C2×#4112yes—
500C4⋊C4H(5,𝔽2)12yes—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—48no—
516C42—42no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—81no—
522C42C2×Q1618yes—
523C42C2×Q1614yes—
524C42C2×#4112yes—
525C42H(5,𝔽2)12yes—
526C42H(5,𝔽2)12yes—
527C422−1+412yes—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—42no—
535Q8—44no—
536Q8—44no—
537Q8—42no—
538Q8—42no—
539Q8—44no—
540Q8—42no—
541Q8—44no—
542Q8—88no—
543Q8—84no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4#53512yes—
590C2×C4#53512yes—
591C2×C4#54812yes—
592C2×C4#54812yes—
593C2×C4C22×Q1612yes—
594C2×C4C22×Q1614yes—
595C2×C4C22×Q1614yes—
596C2×C4C22×Q1612yes—
597C2×C4C22×Q1612yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4#57712yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58511yescharacteristic
607C8—42no—
608C8—44no—
609C8—42no—
610C22C2×#53511yescharacteristic
611C22C2×#54811yescharacteristic
612C22#348811yescharacteristic
613C22C23×Q1611yescharacteristic
614C22C22×#13711yescharacteristic
615C22C2×#57711yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—24no—
619C4—24no—
620C4—24no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—84no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4#320914yes—
635C4#336314yes—
636C4#350611yescharacteristic
637C4#350811yescharacteristic
638C4#354811yescharacteristic
639C4#355411yescharacteristic
640C2#3972511yescharacteristic
641C2C2×#320911yescharacteristic
642C2#6083611yescharacteristic
643C2C2×#336311yescharacteristic
644C2#6132211yescharacteristic
645C2#6195211yescharacteristic
646C2#6198011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
287. C4×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#365932
#349502
#331632
#320842
#308802
#297992
#292592
#287482
#284012
#282842
#277692
#256512
#241912
#228212
#221482
C4×#4622