Finite Group Databasebeta

← #6492616

Subgroups — #6492616

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34155C211yesmaximal, characteristic
3#33199C212yesmaximal, complemented
4#33066C212yesmaximal, complemented
5#29998C212yesmaximal, complemented
6#29986C212yesmaximal, complemented
7#26533C212yesmaximal, complemented
8#26511C212yesmaximal, complemented
9#26463C212yesmaximal, complemented
10#26441C212yesmaximal, complemented
11#26439C214yesmaximal, complemented
12#26363C214yesmaximal, complemented
13#21974C211yesmaximal, characteristic, complemented
14#21893C211yesmaximal, characteristic
15#21859C211yesmaximal, characteristic
16#21362C211yesmaximal, characteristic, complemented
17#21171C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#478C2214yes—
20C2×#431C2211yescharacteristic
21C2×#428C2212yes—
22C2×#423C2211yescharacteristic
23#2859C2212yes—
24#2859C2212yes—
25#2647—22no—
26#2571—21no—
27#2571C2212yes—
28#2552—21no—
29#2552C2212yes—
30#2442—22no—
31#2442C2214yes—
32#2372—22no—
33#2369C2211yescharacteristic
34#2369C2211yescharacteristic
35#2366C2211yescharacteristic
36#2366C2211yescharacteristic
37#2363—21no—
38#2363—21no—
39#2363C2212yes—
40#2363C2212yes—
41#2317—21no—
42#2316—21no—
43#2297—22no—
44#2293—21no—
45#2293C2212yes—
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C42×D4—21no—
49C42×D4C2212yes—
50C4×#117C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2092C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2065C2214yes—
57#2065C2214yes—
58#2059C2214yes—
59#2025C2214yes—
60#1990C2212yes—
61#1989C2214yes—
62#1987C2212yes—
63#1982C2212yes—
64#1957C2212yes—
65#1949C2214yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1948C2212yes—
69#1948C2211yescharacteristic
70#1947C2211yescharacteristic
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1939C2212yes—
76#1918C2214yes—
77#1918C2214yes—
78#1895C2214yes—
79#1874C2214yes—
80#1862C2214yes—
81#1862C2214yes—
82#1854C2212yes—
83#1854C2212yes—
84#1851C2214yes—
85#1842C2212yes—
86#1831C2212yes—
87#1804C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1736C2212yes—
95C22×#41—21no—
96C22×M16C2312yes—
97#532—24no—
98#532Q818yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#125—22no—
102C2×#125C2314yes—
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—22no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C42—22no—
122C22×C42C2311yescharacteristic
123#478—48no—
124#431—24no—
125#428—44no—
126C2×#107C2314yes—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132#423—24no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#400—24no—
147#398—24no—
148#398—24no—
149#391—22no—
150#391—22no—
151#390—21no—
152#390—21no—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#385—24no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#382—22no—
188#382—22no—
189#382—21no—
190#382C2314yes—
191#380—21no—
192#380—24no—
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—21no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—21no—
219#376D412yes—
220C43—21no—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—28no—
235C2×#41—82no—
236C2×#41—82no—
237C22×Q8—21no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—24no—
250C2×M16—42no—
251C2×M16—42no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8C2×Q814yes—
257#127—22no—
258#127C2×Q814yes—
259#125—44no—
260#125—44no—
261#125—84no—
262#124—28no—
263#121—28no—
264#121—24no—
265#121C2×Q818yes—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4C2×Q818yes—
275#117—22no—
276#117—22no—
277#117—42no—
278#117—42no—
279#117—41no—
280#117—41no—
281#117—44no—
282#117—84no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—84no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×Q814yes—
345C2×C42#4114yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—28no—
351#107—24no—
352#107—42no—
353#106—24no—
354#106—42no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—44no—
361C4⋊C8—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366C22⋊C8—24no—
367C4×C8—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—22no—
387C24C22×D411yescharacteristic
388#41—816no—
389C2×Q8—24no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4C22×Q814yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4C22⋊Q814yes—
436C22×C4C22⋊Q814yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×Q812yes—
447C22×C4C2×#4112yes—
448C22×C4C2511yescharacteristic
449M16—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4C22×Q814yes—
493C4⋊C4H(5,𝔽2)12yes—
494C4⋊C42−1+412yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—82no—
524C42—82no—
525C42C22⋊Q814yes—
526C42C22⋊Q814yes—
527C42C22×Q812yes—
528C42C22×Q812yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C2×C22⋊Q812yes—
545C23C23×D411yescharacteristic
546Q8—84no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—42no—
551D4—42no—
552D4—84no—
553D4—88no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—416no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—164no—
606C2×C4C2×C22⋊Q812yes—
607C2×C4C2×C22⋊Q812yes—
608C2×C4C2×C22⋊Q812yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4#57512yes—
612C2×C4#57512yes—
613C2×C4#57612yes—
614C2×C4#57712yes—
615C2×C4C23×Q811yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58511yescharacteristic
620C2×C4#58511yescharacteristic
621C8—42no—
622C8—44no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C22×C22⋊Q811yescharacteristic
639C22#348811yescharacteristic
640C22#348911yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57611yescharacteristic
644C22C2×#57711yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4—82no—
657C4—84no—
658C4—81no—
659C4—82no—
660C4#308014yes—
661C4#308114yes—
662C4#346011yescharacteristic
663C4#346111yescharacteristic
664C4#350311yescharacteristic
665C4#350411yescharacteristic
666C2—42no—
667C2—84no—
668C2#3525711yescharacteristic
669C2C2×#308011yescharacteristic
670C2C2×#308111yescharacteristic
671C2#6198711yescharacteristic
672C2#6199511yescharacteristic
673C2#6213511yescharacteristic
674C2#6214311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35286C2semidirect
3. #33199C2semidirect
4. #33066C2semidirect
5. #29998C2semidirect
6. #29986C2semidirect
7. #26533C2semidirect
8. #26511C2semidirect
9. #26463C2semidirect
10. #26441C2semidirect
11. #26439C2semidirect
12. #26363C2semidirect
13. #21974C2semidirect
16. #21362C2semidirect
17. #21171C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#341552
#331992
#330662
#299982
#299862
#265332
#265112
#264632
#264412
#264392
#263632
#219742
#218932
#218592
#213622
#211712