Finite Group Databasebeta

← #6458872

Subgroups — #6458872

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#34189C211yesmaximal, characteristic
3#27873C214yesmaximal, complemented
4#27870C214yesmaximal, complemented
5#27866C214yesmaximal, complemented
6#27860C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2097C211yesmaximal, characteristic, complemented
9#23656C211yesmaximal, characteristic, complemented
10#23629C211yesmaximal, characteristic, complemented
11#21066C211yesmaximal, characteristic, complemented
12#21066C211yesmaximal, characteristic, complemented
13#21038C211yesmaximal, characteristic, complemented
14#21037C211yesmaximal, characteristic, complemented
15#20974C211yesmaximal, characteristic
16#20974C211yesmaximal, characteristic
17#20663C211yesmaximal, characteristic
18#20662C211yesmaximal, characteristic
19#20336C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C22×#131C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32#2855C2214yes—
33#2625C2214yes—
34#2447—22no—
35#2446—22no—
36#2419—24no—
37#2357—22no—
38#2356—22no—
39#2353—24no—
40#2316C2211yescharacteristic
41#2314C2214yes—
42#2311C2211yescharacteristic
43#2311C2211yescharacteristic
44#2311C2211yescharacteristic
45#2310C2211yescharacteristic
46#2296C2211yescharacteristic
47C2×#391C2211yescharacteristic
48#2285C2211yescharacteristic
49#2283C2214yes—
50C2×#382C2211yescharacteristic
51#2274C2211yescharacteristic
52#2274C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2097C2212yes—
56#2097C2212yes—
57#2097C2212yes—
58#2097C2212yes—
59#2085C2212yes—
60#2085C2212yes—
61#2084C2212yes—
62#2083C2212yes—
63#2074C2214yes—
64#2072C2214yes—
65#1928C2214yes—
66#1928C2214yes—
67#1927C2214yes—
68#1927C2214yes—
69#1921C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1918C2214yes—
73#1894C2214yes—
74#1894C2214yes—
75#1892C2214yes—
76#1892C2214yes—
77#1809C2214yes—
78#1809C2214yes—
79#1808C2214yes—
80#1807C2214yes—
81#1736C2211yescharacteristic
82#1736C2211yescharacteristic
83#1736C2211yescharacteristic
84#1736C2211yescharacteristic
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1735C2211yescharacteristic
88#1735C2211yescharacteristic
89#1729C2211yescharacteristic
90#1729C2211yescharacteristic
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94#1728C2212yes—
95C2×#336C2211yescharacteristic
96C2×#336C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—24no—
102C2×#131—24no—
103C2×#131—42no—
104C2×#131—42no—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C22×M16C2312yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C23×C8C2312yes—
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#536—24no—
114#536—24no—
115C2×C4⋊Q8—21no—
116C2×#124—21no—
117C2×#123—22no—
118C2×C22⋊Q8C2314yes—
119C4×#41—24no—
120C4×#41—24no—
121C2×C4×Q8D412yes—
122C2×C4×D4D414yes—
123C2×C4×D4D412yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#427—42no—
136#427—42no—
137C2×#107—24no—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106—24no—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147#419—42no—
148#419—42no—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157#400—24no—
158#398—24no—
159#397—24no—
160#397—24no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#389—22no—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#386—22no—
183#386—22no—
184#386—22no—
185#385—24no—
186#385—22no—
187#384C2312yes—
188#383—24no—
189#382—22no—
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#381—24no—
194#381—24no—
195#381—22no—
196#380—22no—
197#380—22no—
198#380C2314yes—
199#380C2314yes—
200#380C2314yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C22⋊C4—24no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210#376—24no—
211C2×#95—22no—
212C2×#95—22no—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2311yescharacteristic
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41#4114yes—
238C2×#41#4114yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D414yes—
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258#131—416no—
259#131—416no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—42no—
271C2×M16—42no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8—42no—
281C22×C8—42no—
282C22×C8#4114yes—
283C22×C8#4114yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2411yescharacteristic
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289C22×C8C2411yescharacteristic
290C4⋊Q8—44no—
291#124—44no—
292#123—48no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C4×Q8—28no—
296C4×D4—216no—
297C4×D4—28no—
298#117—24no—
299#117—24no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—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—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D414yes—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42C2×D414yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—24no—
360#107—24no—
361#107—42no—
362#107—42no—
363#106—24no—
364#106—24no—
365#106—42no—
366#106—42no—
367#103—24no—
368#103—24no—
369#103—24no—
370#103—24no—
371#103—44no—
372#103—44no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—28no—
392#41—28no—
393#41—28no—
394#41—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8#12314yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4#12318yes—
416C2×D4#12314yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458M16—48no—
459M16—48no—
460M16—44no—
461M16—44no—
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—48no—
479C2×C8—48no—
480C2×C8C2×#4112yes—
481C2×C8C2×#4112yes—
482C2×C8C2×#4112yes—
483C2×C8C2×#4112yes—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—84no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—48no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—28no—
520C42—24no—
521C42—21no—
522C42—21no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—84no—
532C42—84no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23C2×#12314yes—
540C23C2×#12312yes—
541C23#53212yes—
542C23#53712yes—
543C23C23×D411yescharacteristic
544Q8—28no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—84no—
550D4—216no—
551D4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4—162no—
592C2×C4—162no—
593C2×C4—162no—
594C2×C4C2×#12314yes—
595C2×C4C2×#12312yes—
596C2×C4#53212yes—
597C2×C4#53712yes—
598C2×C4#55114yes—
599C2×C4#55114yes—
600C2×C4#55214yes—
601C2×C4#55214yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C8—22no—
609C8—22no—
610C8—22no—
611C8—22no—
612C8—42no—
613C8—42no—
614C22—24no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—28no—
619C22—24no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22#249814yes—
624C22C22×#12311yescharacteristic
625C22C2×#53211yescharacteristic
626C22C2×#53711yescharacteristic
627C22#358412yes—
628C22#358612yes—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#249814yes—
642C4C2×#55111yescharacteristic
643C4C2×#55111yescharacteristic
644C4C2×#55211yescharacteristic
645C4C2×#55211yescharacteristic
646C2—24no—
647C2—44no—
648C2C2×#249811yescharacteristic
649C2#6052412yes—
650C2#6071311yescharacteristic
651C2#6072011yescharacteristic
652C2#6093212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27873C2semidirect
4. #27870C2semidirect
5. #27866C2semidirect
6. #27860C2semidirect
7. #23746C2semidirect
8. C2×#2097C2semidirect
9. #23656C2semidirect
10. #23629C2semidirect
11. #21066C2semidirect
12. #21066C2semidirect
13. #21038C2semidirect
14. #21037C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#341892
#278732
#278702
#278662
#278602
#237462
C2×#20972
#236562
#236292
#210662
#210382
#210372
#209742
#206632
#206622
#203362