Finite Group Databasebeta

← #6434366

Subgroups — #6434366

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic
3#30088C214yesmaximal, complemented
4#30087C214yesmaximal, complemented
5#30081C214yesmaximal, complemented
6#30080C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23629C211yesmaximal, characteristic, complemented
12#23627C211yesmaximal, characteristic
13#23512C211yesmaximal, characteristic
14#23512C211yesmaximal, characteristic
15#20882C211yesmaximal, characteristic
16#20881C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20877C211yesmaximal, characteristic, complemented
19#20774C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#519C2211yescharacteristic
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#2864C2214yes—
36#2647C2214yes—
37#2570—22no—
38#2570—22no—
39#2569—24no—
40#2362—22no—
41#2360—22no—
42#2359—24no—
43#2327C2212yes—
44#2326C2212yes—
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#124C2212yes—
52#2297C2211yescharacteristic
53C2×#391C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yes—
58#2097C2214yes—
59#2097C2212yes—
60#2097C2214yes—
61#2092C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2085C2214yes—
65#2084C2214yes—
66#2083C2214yes—
67#2074C2212yes—
68#2074C2214yes—
69#2073C2212yes—
70#2072C2212yes—
71#2072C2214yes—
72#2071C2212yes—
73#2043C2214yes—
74#2043C2214yes—
75#2042C2214yes—
76#2042C2214yes—
77#1865C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1864C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1862C2214yes—
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#536—24no—
104#536—24no—
105C2×C4⋊Q8C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109#520—42no—
110#520—42no—
111#519—42no—
112#519—42no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117—22no—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
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×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162#400—24no—
163#400—24no—
164#398—24no—
165#398—24no—
166#391—22no—
167#391C2311yescharacteristic
168#391C2312yes—
169#391C2312yes—
170#391C2311yescharacteristic
171#390C2311yescharacteristic
172#390C2311yescharacteristic
173#389—22no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#388—24no—
182#388—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#382—21no—
200#382—22no—
201#382C2312yes—
202#382C2312yes—
203#382C2312yes—
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D412yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D414yes—
215#376C2312yes—
216C2×#95—22no—
217C2×#95—22no—
218C43C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#336C2312yes—
226#336C2312yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41#4114yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8#4114yes—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276#124—24no—
277#124—24no—
278#123—216no—
279C22⋊Q8—216no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—48no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—48no—
302#117—44no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—44no—
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—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
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—22no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—21no—
392C24C22×D411yescharacteristic
393#41—28no—
394#41—28no—
395#41—44no—
396#41—44no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—28no—
401C2×Q8—28no—
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—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8C2×#4114yes—
415C2×Q8C2×#4112yes—
416C2×D4—24no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×#4112yes—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464M16—44no—
465M16—44no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
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—44no—
479C2×C8—44no—
480C4⋊C4—216no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C22⋊C4—216no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—82no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—44no—
530C42—44no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—42no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23#52512yes—
545C23#53512yes—
546C23#53612yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—22no—
551Q8—22no—
552Q8—48no—
553Q8—42no—
554Q8—42no—
555Q8—41no—
556Q8—41no—
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4—164no—
601C2×C4—162no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4#52512yes—
605C2×C4#53512yes—
606C2×C4#53612yes—
607C2×C4#53914yes—
608C2×C4#53914yes—
609C2×C4#53914yes—
610C2×C4#53914yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—42no—
618C8—42no—
619C8—42no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—44no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22#263314yes—
631C22C2×#52511yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53611yescharacteristic
634C22#358412yes—
635C22#358612yes—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4#263314yes—
651C4C2×#53911yescharacteristic
652C4C2×#53911yescharacteristic
653C4C2×#53911yescharacteristic
654C4C2×#53911yescharacteristic
655C2—24no—
656C2—82no—
657C2C2×#263311yescharacteristic
658C2#6059812yes—
659C2#6081911yescharacteristic
660C2#6082811yescharacteristic
661C2#6087612yes—
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
1. #35339C2semidirect
3. #30088C2semidirect
4. #30087C2semidirect
5. #30081C2semidirect
6. #30080C2semidirect
8. C2×#2097C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
11. #23629C2semidirect
17. #20877C2semidirect
18. #20877C2semidirect
19. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#353392
#342762
#300882
#300872
#300812
#300802
#237332
C2×#20972
#236612
#236602
#236292
#236272
#235122
#208822
#208812
#208772
#207742