Finite Group Databasebeta

← #6502467

Subgroups — #6502467

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2451C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#28121C212yesmaximal
4#27611C211yesmaximal, characteristic
5#27583C211yesmaximal, characteristic
6#27441C214yesmaximal, complemented
7#27021C214yesmaximal, complemented
8#26911C214yesmaximal, complemented
9#26389C214yesmaximal, complemented
10#24301C212yesmaximal
11#24064C211yesmaximal, characteristic
12#23934C211yesmaximal, characteristic
13#22085C212yesmaximal
14#21916C211yesmaximal, characteristic
15#21455C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#475C2214yes—
21C2×#451C2212yes—
22C2×#441C2214yes—
23#2451—22no—
24#2451—22no—
25#2451C2214yes—
26#2451C2214yes—
27C2×#398—21no—
28C2×#395—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×#391—21no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46#2285C2211yescharacteristic
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2099C2212yes—
50#2096C2212yes—
51#2084C2212yes—
52#2072C2212yes—
53#2068C2214yes—
54#2058C2214yes—
55#2057C2212yes—
56#2056C2214yes—
57#2043C2212yes—
58#2024C2214yes—
59#2024C2212yes—
60#2016C2212yes—
61#1988C2214yes—
62#1943C2214yes—
63#1935C2214yes—
64#1928C2214yes—
65#1927C2214yes—
66#1920C2214yes—
67#1918C2214yes—
68#1917C2214yes—
69#1917C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1865C2214yes—
76#1865C2214yes—
77#1863C2214yes—
78#1853C2212yes—
79#1853C2212yes—
80#1824C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90#533—28no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×#125—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—21no—
100C2×C4×Q8—22no—
101C2×C4×Q8—44no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×#117—21no—
107C2×#117—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#484—28no—
122#475—48no—
123#451—28no—
124#441—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×#97C2314yes—
138#398—24no—
139#398—24no—
140#395—24no—
141#395—24no—
142#391—21no—
143#391—21no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—24no—
148#391—24no—
149#391C2312yes—
150#391C2314yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#387—24no—
167#387—24no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
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—24no—
224C22×Q8—42no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—82no—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×Q16—28no—
242C2×Q16—28no—
243C2×Q16—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—28no—
249C4⋊Q8—44no—
250C4⋊Q8—48no—
251#125—28no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8#4118yes—
255C4×Q8—28no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—28no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263#117—28no—
264#117—44no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
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—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326#107—28no—
327#106—28no—
328#106—44no—
329C4⋊C8—44no—
330#103—28no—
331#103—28no—
332#103—28no—
333#103—28no—
334#103—44no—
335#103—44no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338#97—44no—
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#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—48no—
372C2×Q8—416no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—48no—
376C2×Q8—81no—
377C2×Q8—81no—
378C2×Q8—84no—
379C2×Q8—816no—
380C2×Q8C2×#4114yes—
381C2×Q82−1+414yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4#13714yes—
421C22×C4#13714yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4114yes—
432C22×C4C2511yescharacteristic
433Q16—416no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—81no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C2×#4114yes—
472C4⋊C4H(5,𝔽2)14yes—
473C22⋊C4—28no—
474C22⋊C4—44no—
475C42—28no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—24no—
482C42—42no—
483C42—42no—
484C42—48no—
485C42—44no—
486C42—82no—
487C42—82no—
488C42—82no—
489C42—82no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#52512yes—
495C23C2×#13712yes—
496C23C2×#13712yes—
497C23C23×D411yescharacteristic
498Q8—28no—
499Q8—28no—
500Q8—48no—
501Q8—44no—
502Q8—44no—
503Q8—816no—
504Q8—84no—
505C2×C4—22no—
506C2×C4—28no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—48no—
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—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4#52512yes—
564C2×C4C2×#13712yes—
565C2×C4C2×#13712yes—
566C2×C4C2×#13714yes—
567C2×C4C22×#4112yes—
568C2×C4C2×H(5,𝔽2)12yes—
569C2×C4C2×2−1+412yes—
570C8—44no—
571C8—44no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22#311314yes—
579C22C2×#52511yescharacteristic
580C22#348411yescharacteristic
581C22#348611yescharacteristic
582C22C22×#13711yescharacteristic
583C22C22×#13711yescharacteristic
584C22#358511yescharacteristic
585C22#358611yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—24no—
590C4—44no—
591C4—44no—
592C4—42no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—44no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4#352014yes—
604C2—24no—
605C2#3535711yescharacteristic
606C2C2×#311311yescharacteristic
607C2#6056711yescharacteristic
608C2#6157011yescharacteristic
609C2#6158411yescharacteristic
610C2#6182911yescharacteristic
611C2#6183011yescharacteristic
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. #27441C2semidirect
7. #27021C2semidirect
8. #26911C2semidirect
9. #26389C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24512
#342132
#281212
#276112
#275832
#274412
#270212
#269112
#263892
#243012
#240642
#239342
#220852
#219162
#214552