Finite Group Databasebeta

← #6502461

Subgroups — #6502461

Order 512.

1648 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2453C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#28124C212yesmaximal
4#27610C211yesmaximal, characteristic
5#27583C211yesmaximal, characteristic
6#27425C214yesmaximal, complemented
7#27005C214yesmaximal, complemented
8#26895C214yesmaximal, complemented
9#26371C214yesmaximal, complemented
10#24307C212yesmaximal
11#24063C211yesmaximal, characteristic
12#23934C211yesmaximal, characteristic
13#22078C212yesmaximal
14#21912C211yesmaximal, characteristic
15C4×#451C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#483C2212yes—
20C2×#479C2214yes—
21C2×#451C2212yes—
22C2×C4×Q16C2214yes—
23#2453—22no—
24#2453—22no—
25#2453C2214yes—
26#2453C2214yes—
27C2×#400—21no—
28C2×#398—21no—
29#2370—22no—
30#2369C2212yes—
31#2365C2212yes—
32#2355—22no—
33#2355C2212yes—
34Q8×C4⋊C4C2212yes—
35#2329—21no—
36#2329—21no—
37#2317—21no—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46#2285C2211yescharacteristic
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2099C2212yes—
50#2096C2212yes—
51#2083C2212yes—
52#2072C2212yes—
53#2066C2214yes—
54#2059C2214yes—
55#2058C2212yes—
56#2056C2214yes—
57#2043C2212yes—
58#2026C2214yes—
59#2024C2212yes—
60#2016C2212yes—
61#1989C2214yes—
62#1941C2214yes—
63#1933C2214yes—
64#1927C2214yes—
65#1927C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1916C2214yes—
69#1916C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1865C2214yes—
76#1865C2214yes—
77#1863C2214yes—
78#1853C2212yes—
79#1852C2212yes—
80#1823C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1787C2212yes—
85C4×#103C2214yes—
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C22×Q16C2314yes—
89#533—28no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—21no—
93C2×#125—22no—
94C2×#125—44no—
95C2×#125C2314yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8—22no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×#117—21no—
106C2×#117—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#483—28no—
122#479—48no—
123#451—28no—
124C4×Q16—48no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C4×C8C2314yes—
138#400—24no—
139#400—24no—
140#398—24no—
141#398—24no—
142#391—22no—
143#391—21no—
144#391—21no—
145#391—22no—
146#391—22no—
147#391C2312yes—
148#391C2314yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#389—22no—
153#389—22no—
154#389—24no—
155#389—24no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—24no—
173#387—24no—
174#385—22no—
175#385—22no—
176#384—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#382—21no—
181#380—21no—
182#380C2312yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4D412yes—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8C2412yes—
225C22×Q8C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×Q16—28no—
238C2×Q16—28no—
239C2×Q16—44no—
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244#125—28no—
245#125—28no—
246#125—44no—
247#125—48no—
248C22⋊Q8—28no—
249C22⋊Q8—24no—
250C22⋊Q8#4118yes—
251C4×Q8—28no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—28no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—44no—
258C4×Q8—48no—
259#117—28no—
260#117—44no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324#107—28no—
325#106—28no—
326#106—44no—
327C4⋊C8—44no—
328#103—28no—
329#103—28no—
330#103—28no—
331#103—28no—
332#103—44no—
333#103—44no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C4×C8—44no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24C22×D411yescharacteristic
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—416no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×Q8C2×#4114yes—
375C2×Q82−1+414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4#13514yes—
415C22×C4#13714yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C2×#4114yes—
426C22×C4C2511yescharacteristic
427Q16—416no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×#4114yes—
467C4⋊C4H(5,𝔽2)14yes—
468C22⋊C4—28no—
469C22⋊C4—44no—
470C42—28no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—24no—
477C42—42no—
478C42—42no—
479C42—48no—
480C42—44no—
481C42—82no—
482C42—82no—
483C42—82no—
484C42—82no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23#52512yes—
490C23C2×#13512yes—
491C23C2×#13712yes—
492C23C23×D411yescharacteristic
493Q8—28no—
494Q8—28no—
495Q8—48no—
496Q8—44no—
497Q8—44no—
498Q8—84no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4#52512yes—
558C2×C4C2×#13512yes—
559C2×C4C2×#13512yes—
560C2×C4C2×#13714yes—
561C2×C4C22×#4112yes—
562C2×C4C2×H(5,𝔽2)12yes—
563C2×C4C2×2−1+412yes—
564C8—44no—
565C8—44no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22#310814yes—
573C22C2×#52511yescharacteristic
574C22#348411yescharacteristic
575C22#348611yescharacteristic
576C22C22×#13511yescharacteristic
577C22C22×#13711yescharacteristic
578C22#358611yescharacteristic
579C22#358611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—24no—
584C4—44no—
585C4—44no—
586C4—42no—
587C4—42no—
588C4—44no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—44no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4#352014yes—
598C2—24no—
599C2#3535711yescharacteristic
600C2C2×#310811yescharacteristic
601C2#6056911yescharacteristic
602C2#6156511yescharacteristic
603C2#6157311yescharacteristic
604C2#6182211yescharacteristic
605C2#6182911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #27425C2semidirect
7. #27005C2semidirect
8. #26895C2semidirect
9. #26371C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24532
#342132
#281242
#276102
#275832
#274252
#270052
#268952
#263712
#243072
#240632
#239342
#220782
#219122
C4×#4512