Finite Group Databasebeta

← #6434769

Subgroups — #6434769

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#30070C214yesmaximal, complemented
4#30068C214yesmaximal, complemented
5#30067C214yesmaximal, complemented
6#30065C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23556C211yesmaximal, characteristic, complemented
10#23555C211yesmaximal, characteristic, complemented
11#23510C211yesmaximal, characteristic
12#23510C211yesmaximal, characteristic
13#23455C211yesmaximal, characteristic, complemented
14#23451C211yesmaximal, characteristic
15#21068C211yesmaximal, characteristic
16#21067C211yesmaximal, characteristic
17#21064C211yesmaximal, characteristic, complemented
18#21064C211yesmaximal, characteristic, complemented
19#20774C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35#2862C2212yes—
36#2861C2212yes—
37#2642C2212yes—
38#2640C2212yes—
39#2492—22no—
40#2489—22no—
41#2488—24no—
42#2368—24no—
43#2355—22no—
44#2355—22no—
45#2331C2214yes—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2311C2211yescharacteristic
49#2310C2211yescharacteristic
50C4×#125C2214yes—
51#2297C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#382C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2214yes—
58#2097C2212yes—
59#2097C2214yes—
60#2097C2212yes—
61#2092C2212yes—
62#2092C2212yes—
63#2059C2214yes—
64#2059C2214yes—
65#2058C2214yes—
66#2057C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2026C2214yes—
72#2026C2212yes—
73#2025C2212yes—
74#2024C2214yes—
75#2024C2212yes—
76#2023C2212yes—
77#1928C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1927C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1840C2214yes—
86#1839C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1729C2211yescharacteristic
92#1729C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#544—42no—
104#544—42no—
105#536—42no—
106#536—42no—
107#533—24no—
108#533—24no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—22no—
122C2×C4×Q8—22no—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#428—42no—
133#428—42no—
134C2×#107—42no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#106—42no—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142#419—24no—
143#419—24no—
144#417—42no—
145#417—42no—
146#416—24no—
147#416—24no—
148C2×#103—42no—
149C2×#103—42no—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162#400—24no—
163#400—24no—
164#397—24no—
165#393—24no—
166#391—22no—
167#391—24no—
168#391—22no—
169#391—24no—
170#391—24no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2311yescharacteristic
176#391C2312yes—
177#391C2311yescharacteristic
178#390C2312yes—
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389C2314yes—
182#389C2314yes—
183#388—24no—
184#388—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#382—21no—
196#382—21no—
197#382—24no—
198#382—22no—
199#382C2312yes—
200#382C2312yes—
201#381—22no—
202#380—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—24no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D414yes—
215#376C2312yes—
216C2×#95—22no—
217C2×#95—22no—
218C43C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4—21no—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—42no—
280C4⋊Q8—42no—
281C4⋊Q8—44no—
282#125—24no—
283#125—24no—
284#125—42no—
285#125—42no—
286#125—44no—
287#124—42no—
288#124—42no—
289#123—48no—
290C22⋊Q8—216no—
291C22⋊Q8—48no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294C4×Q8—216no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117#4114yes—
304#117#4114yes—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42C2×D412yes—
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#107—42no—
360#107—42no—
361#106—44no—
362#106—42no—
363#106—42no—
364#103—24no—
365#103—24no—
366#103—28no—
367#103—28no—
368#103—44no—
369#103—44no—
370#103—42no—
371#103—42no—
372#102—24no—
373#102—24no—
374#102—42no—
375#102—42no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—21no—
392C24C22×D411yescharacteristic
393#41—44no—
394#41—44no—
395#41—44no—
396#41—44no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—84no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467M16—44no—
468M16—44no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—44no—
482C2×C8—44no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4C2×#4114yes—
512C4⋊C4C2×#4114yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C42—24no—
524C42—24no—
525C42—24no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—84no—
538C42—84no—
539C23—22no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23#52512yes—
547C23#53512yes—
548C23#53612yes—
549C23C23×D411yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—88no—
559Q8—84no—
560Q8—164no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—84no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—162no—
605C2×C4—162no—
606C2×C4—162no—
607C2×C4#52512yes—
608C2×C4#53512yes—
609C2×C4#53612yes—
610C2×C4#54214yes—
611C2×C4#54214yes—
612C2×C4#54214yes—
613C2×C4#54214yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C8—42no—
621C8—42no—
622C8—42no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—24no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—82no—
633C22#263214yes—
634C22C2×#52511yescharacteristic
635C22C2×#53511yescharacteristic
636C22C2×#53611yescharacteristic
637C22#358412yes—
638C22#358612yes—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—81no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4#263214yes—
654C4C2×#54211yescharacteristic
655C4C2×#54211yescharacteristic
656C4C2×#54211yescharacteristic
657C4C2×#54211yescharacteristic
658C2—24no—
659C2—161no—
660C2C2×#263211yescharacteristic
661C2#6058612yes—
662C2#6081911yescharacteristic
663C2#6082811yescharacteristic
664C2#6089112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34215C2semidirect
3. #30070C2semidirect
4. #30068C2semidirect
5. #30067C2semidirect
6. #30065C2semidirect
8. C2×#2097C2semidirect
9. #23556C2semidirect
10. #23555C2semidirect
13. #23455C2semidirect
17. #21064C2semidirect
18. #21064C2semidirect
19. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#342152
#300702
#300682
#300672
#300652
#237332
C2×#20972
#235562
#235552
#235102
#234552
#234512
#210682
#210672
#210642
#207742