Finite Group Databasebeta

← #6434770

Subgroups — #6434770

Order 512.

1693 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35264C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#30069C214yesmaximal, complemented
4#30068C214yesmaximal, complemented
5#30066C214yesmaximal, complemented
6#30065C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23556C211yesmaximal, characteristic, complemented
10#23555C211yesmaximal, characteristic, complemented
11#23509C211yesmaximal, characteristic
12#23509C211yesmaximal, characteristic
13#23454C211yesmaximal, characteristic, complemented
14#23451C211yesmaximal, characteristic, complemented
15#21068C211yesmaximal, characteristic, complemented
16#21067C211yesmaximal, characteristic, complemented
17#21063C211yesmaximal, characteristic, complemented
18#21063C211yesmaximal, characteristic, complemented
19#20774C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35#2861C2212yes—
36#2858C2212yes—
37#2641C2212yes—
38#2640C2212yes—
39#2491—22no—
40#2490—22no—
41#2472—24no—
42#2368—24no—
43#2355—22no—
44#2355—22no—
45#2331C2214yescomplemented
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2311C2211yescharacteristic
49#2309C2211yescharacteristic
50C4×#125C2214yescomplemented
51#2297C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#382C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2092C2214yes—
60#2092C2214yes—
61#2092C2212yescomplemented
62#2092C2212yescomplemented
63#2059C2214yescomplemented
64#2059C2214yescomplemented
65#2058C2214yescomplemented
66#2057C2214yescomplemented
67#2042C2214yes—
68#2042C2214yes—
69#2041C2214yes—
70#2041C2214yes—
71#2026C2212yescomplemented
72#2025C2214yes—
73#2025C2212yescomplemented
74#2024C2212yescomplemented
75#2023C2214yes—
76#2023C2212yescomplemented
77#1928C2214yescomplemented
78#1927C2214yescomplemented
79#1926C2214yes—
80#1926C2214yes—
81#1926C2214yescomplemented
82#1925C2214yes—
83#1925C2214yes—
84#1925C2214yescomplemented
85#1840C2214yescomplemented
86#1839C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1729C2211yescharacteristic
92#1729C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#544—42no—
104#544—42no—
105#535—42no—
106#535—42no—
107#533—24no—
108#533—24no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#127C2312yes—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#124—24no—
117C2×#124C2312yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#120—22no—
123C2×C4×Q8—22no—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134#428—42no—
135#428—42no—
136C2×#107—42no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106—42no—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144#418—24no—
145#418—24no—
146#417—42no—
147#417—42no—
148#416—24no—
149#416—24no—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#102—42no—
155C2×#102—42no—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164#397—24no—
165#394—24no—
166#394—24no—
167#393—24no—
168#391—24no—
169#391—24no—
170#391C2312yes—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#390—22no—
174#390—24no—
175#390—22no—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2311yescharacteristic
180#390C2312yes—
181#390C2312yes—
182#390C2311yescharacteristic
183#389C2314yes—
184#389C2314yes—
185#388—24no—
186#388—24no—
187#387—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384—21no—
196#384—21no—
197#382—21no—
198#382—21no—
199#382—22no—
200#382C2312yes—
201#382C2312yes—
202#380—22no—
203#379—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D414yes—
216#376C2312yes—
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#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8C2411yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2412yes—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—42no—
280C4⋊Q8—42no—
281C4⋊Q8—44no—
282#127—42no—
283#127—42no—
284#125—24no—
285#125—24no—
286#125—42no—
287#125—42no—
288#125—44no—
289#124—42no—
290#124—42no—
291#124—88no—
292C22⋊Q8—216no—
293C22⋊Q8—48no—
294#121—44no—
295#121—44no—
296#120—48no—
297C4×Q8—216no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117#4114yes—
307#117#4114yes—
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—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42C2×D412yes—
358C2×C42C2×D414yes—
359C2×C42C2×D414yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#107—42no—
366#107—42no—
367#106—44no—
368#106—42no—
369#106—42no—
370#103—24no—
371#103—24no—
372#103—42no—
373#103—42no—
374#102—24no—
375#102—24no—
376#102—28no—
377#102—28no—
378#102—44no—
379#102—44no—
380#102—42no—
381#102—42no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—22no—
394C24—21no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—82no—
399C24—82no—
400C24C22×D411yescharacteristic
401#41—44no—
402#41—44no—
403#41—44no—
404#41—44no—
405C2×Q8—28no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—44no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—84no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—84no—
441C2×D4—84no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475M16—44no—
476M16—44no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—44no—
490C2×C8—44no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4C2×#4114yes—
518C4⋊C4C2×#4114yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—24no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42—84no—
548C42—84no—
549C23—22no—
550C23—22no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—88no—
562C23—84no—
563C23—162no—
564C23—162no—
565C23—164no—
566C23#52512yes—
567C23#53512yes—
568C23#53612yes—
569C23C23×D411yescharacteristic
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—88no—
575Q8—164no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—84no—
585D4—84no—
586D4—164no—
587D4—164no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—162no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4#52512yes—
626C2×C4#53512yes—
627C2×C4#53612yes—
628C2×C4#54214yes—
629C2×C4#54214yes—
630C2×C4#54214yes—
631C2×C4#54214yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—22no—
643C22—22no—
644C22—24no—
645C22—24no—
646C22—81no—
647C22—82no—
648C22—81no—
649C22—82no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—82no—
655C22—84no—
656C22—168no—
657C22#263214yes—
658C22C2×#52511yescharacteristic
659C22C2×#53511yescharacteristic
660C22C2×#53611yescharacteristic
661C22#358412yes—
662C22#358512yes—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—81no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—81no—
676C4#263214yes—
677C4C2×#54211yescharacteristic
678C4C2×#54211yescharacteristic
679C4C2×#54211yescharacteristic
680C4C2×#54211yescharacteristic
681C2—24no—
682C2—161no—
683C2—162no—
684C2C2×#263211yescharacteristic
685C2#6058712yes—
686C2#6081911yescharacteristic
687C2#6082711yescharacteristic
688C2#6089012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #30069C2semidirect
4. #30068C2semidirect
5. #30066C2semidirect
6. #30065C2semidirect
7. #23733C2semidirect
8. C2×#2092C2semidirect
9. #23556C2semidirect
10. #23555C2semidirect
13. #23454C2semidirect
14. #23451C2semidirect
15. #21068C2semidirect
16. #21067C2semidirect
17. #21063C2semidirect
18. #21063C2semidirect
19. #20774C2semidirect
45. #2331C22semidirect
50. C4×#125C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
61. #2092C22semidirect
62. #2092C22semidirect
63. #2059C22semidirect
64. #2059C22semidirect
65. #2058C22semidirect
66. #2057C22semidirect
71. #2026C22semidirect
73. #2025C22semidirect
74. #2024C22semidirect
76. #2023C22semidirect
77. #1928C22semidirect
78. #1927C22semidirect
81. #1926C22semidirect
84. #1925C22semidirect
85. #1840C22semidirect
86. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#342152
#300692
#300682
#300662
#300652
#237332
C2×#20922
#235562
#235552
#235092
#234542
#234512
#210682
#210672
#210632
#207742