Finite Group Databasebeta

← #6647804

Subgroups — #6647804

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic
3#33008C212yesmaximal
4#32660C214yesmaximal, complemented
5#32619C214yesmaximal, complemented
6#32582C214yesmaximal, complemented
7#32557C214yesmaximal, complemented
8#29913C212yesmaximal
9#27613C211yesmaximal, characteristic
10#27586C211yesmaximal, characteristic
11#24609C212yesmaximal
12#24508C212yesmaximal
13#24063C211yesmaximal, characteristic
14#23934C211yesmaximal, characteristic
15#21822C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#483C2212yes—
20C2×#451C2212yes—
21C2×#432C2212yes—
22#2855C2212yes—
23#2855C2212yes—
24#2627—21no—
25#2627—21no—
26#2582C2212yes—
27#2582C2212yes—
28#2581—22no—
29#2581—22no—
30#2581—22no—
31#2581C2214yes—
32#2581C2214yes—
33#2568C2212yes—
34#2567C2212yes—
35#2565—22no—
36C2×#400—21no—
37C2×#395—21no—
38#2317—21no—
39#2310—21no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#387—21no—
46#2285C2211yescharacteristic
47C2×#382C2211yescharacteristic
48#2101C2214yes—
49#2100C2212yes—
50#2097C2214yes—
51#2095C2212yes—
52#2088C2214yes—
53#2085C2214yes—
54#2084C2212yes—
55#2074C2214yes—
56#2072C2212yes—
57#2068C2214yes—
58#2066C2214yes—
59#2058C2212yes—
60#2056C2214yes—
61#2043C2212yes—
62#2024C2212yes—
63#2016C2212yes—
64#1985C2214yes—
65#1985C2214yes—
66#1971C2214yes—
67#1943C2214yes—
68#1935C2214yes—
69#1935C2214yes—
70#1928C2214yes—
71#1919C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1872C2212yes—
76#1872C2212yes—
77#1865C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1853C2212yes—
81#1852C2212yes—
82#1823C2212yes—
83#1817C2211yescharacteristic
84#1817C2211yescharacteristic
85#1817C2212yes—
86#1787C2212yes—
87C23×Q8D411yescharacteristic
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90#533—44no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#125—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—44no—
99C2×C4×Q8—44no—
100C2×C4×Q8D412yes—
101C2×C4×Q8D412yes—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
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⋊C4D411yescharacteristic
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—44no—
122#451—44no—
123#432—44no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2314yes—
137#400—22no—
138#400—22no—
139#400—24no—
140#400—24no—
141#398—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#395—24no—
146#395—24no—
147#395—22no—
148#395—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391—21no—
155#391—22no—
156#391C2312yes—
157#391C2314yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#384—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#382—22no—
189#382—22no—
190#382—21no—
191#382C2314yes—
192#381—22no—
193#381—22no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2312yes—
216#336C2314yes—
217#336C2312yes—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×D412yes—
227C22×Q8C2×D412yes—
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—44no—
243C2×Q16—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248#125—44no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—44no—
252C4×Q8—44no—
253C4×Q8—44no—
254C4×Q8—42no—
255C4×Q8—42no—
256C4×Q8—42no—
257C4×Q8—42no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8—48no—
263C4×Q8—44no—
264#117—44no—
265#117—44no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2412yes—
324#107—44no—
325#106—44no—
326#106—44no—
327C4⋊C8—44no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336#97—44no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—22no—
344#95—22no—
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#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—416no—
378C2×Q8—416no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q82−1+414yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
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—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4#13714yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4H(5,𝔽2)14yes—
435C22×C4C2511yescharacteristic
436Q16—88no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C42—24no—
480C42—22no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—48no—
490C42—44no—
491C42—44no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23#53412yes—
497C23C2×#13712yes—
498C23#57512yes—
499C23C23×D411yescharacteristic
500Q8—28no—
501Q8—48no—
502Q8—48no—
503Q8—48no—
504Q8—44no—
505Q8—44no—
506Q8—44no—
507Q8—84no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
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—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4#53412yes—
568C2×C4C2×#13712yes—
569C2×C4C2×#13712yes—
570C2×C4C2×H(5,𝔽2)12yes—
571C2×C4C2×2−1+412yes—
572C2×C4#58512yes—
573C8—44no—
574C8—44no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22#319614yes—
582C22C2×#53411yescharacteristic
583C22#348611yescharacteristic
584C22#348711yescharacteristic
585C22C22×#13711yescharacteristic
586C22C2×#57511yescharacteristic
587C22#358611yescharacteristic
588C22#358611yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—44no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4#353514yes—
607C2—24no—
608C2#3718611yescharacteristic
609C2C2×#319611yescharacteristic
610C2#6075411yescharacteristic
611C2#6182911yescharacteristic
612C2#6183211yescharacteristic
613C2#6188911yescharacteristic
614C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32660C2semidirect
5. #32619C2semidirect
6. #32582C2semidirect
7. #32557C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#25812
#330082
#326602
#326192
#325822
#325572
#299132
#276132
#275862
#246092
#245082
#240632
#239342
#218222