Finite Group Databasebeta

← #6508499

Subgroups — #6508499

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2453C211yesmaximal, characteristic
3#29111C211yesmaximal, characteristic
4#29082C214yesmaximal, complemented
5#28953C214yesmaximal, complemented
6#28313C211yesmaximal, characteristic
7#26847C214yesmaximal, complemented
8#26497C214yesmaximal, complemented
9#25912C211yesmaximal, characteristic
10#25819C211yesmaximal, characteristic
11#24601C212yesmaximal
12#24502C212yesmaximal
13#22084C212yesmaximal
14#21455C212yesmaximal
15#20536C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#501C2212yes—
19C2×#491C2212yes—
20C2×#483C2212yes—
21C2×#451C2212yes—
22C2×#441C2214yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yes—
32#2507C2212yes—
33#2453—22no—
34#2453—22no—
35#2453C2214yes—
36#2453C2214yes—
37#2451C2212yes—
38#2446C2212yes—
39C2×#400—21no—
40C2×#398—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2096C2212yes—
52#2088C2214yes—
53#2072C2212yes—
54#2068C2214yes—
55#2066C2214yes—
56#2059C2214yes—
57#2056C2214yes—
58#2026C2214yes—
59#2024C2212yes—
60#2016C2212yes—
61#1992C2212yes—
62#1992C2212yes—
63#1971C2214yes—
64#1969C2214yes—
65#1944C2214yes—
66#1941C2214yes—
67#1935C2214yes—
68#1935C2214yes—
69#1933C2214yes—
70#1933C2214yes—
71#1920C2214yes—
72#1878C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1865C2214yes—
76#1863C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1823C2212yes—
80#1787C2212yes—
81#1770C2214yes—
82#1770C2214yes—
83#1763C2214yes—
84#1749C2211yescharacteristic
85#1749C2212yes—
86#1749C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C22×Q16C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125—21no—
92C2×#125—21no—
93C2×#125—44no—
94C2×#125C2312yes—
95C2×#125C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—44no—
100C2×C4×Q8C2314yes—
101C2×C4×Q8C2312yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#501—44no—
117#491—44no—
118#483—28no—
119#451—28no—
120#441—48no—
121#431—28no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2314yes—
137C2×C4×C8C2314yes—
138#400—22no—
139#400—22no—
140#400—22no—
141#400—24no—
142#400—24no—
143#398—22no—
144#398—22no—
145#398—24no—
146#398—24no—
147#398—22no—
148#398—22no—
149#395—22no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2314yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#389—24no—
163#389—24no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#385—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—24no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×Q8C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×Q16—28no—
239C2×Q16—28no—
240C2×Q16—44no—
241C22×C8C2414yes—
242C22×C8C2414yes—
243C4⋊Q8—42no—
244C4⋊Q8—42no—
245#125—24no—
246#125—24no—
247#125—42no—
248#125—42no—
249#125—44no—
250#125—44no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—44no—
254C4×Q8—28no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—48no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2412yes—
318#107—44no—
319#106—28no—
320#106—28no—
321C4⋊C8—28no—
322#103—28no—
323#103—28no—
324#103—28no—
325#103—28no—
326#103—44no—
327#103—44no—
328C22⋊C8—24no—
329C22⋊C8#4118yes—
330#97—44no—
331#97—42no—
332#97—42no—
333C4×C8—28no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
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—22no—
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—416no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—81no—
374C2×Q8—81no—
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—22no—
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—41no—
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—42no—
415C22×C4#13714yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
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×C8C2×#4118yes—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4H(5,𝔽2)14yes—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—42no—
479C42—41no—
480C42—41no—
481C42—48no—
482C42—82no—
483C42—82no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#53212yes—
489C23C2×#13712yes—
490C23#57512yes—
491C23C23×D411yescharacteristic
492Q8—28no—
493Q8—48no—
494Q8—44no—
495Q8—44no—
496Q8—44no—
497Q8—162no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—42no—
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—42no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4#53212yes—
559C2×C4C2×#13714yes—
560C2×C4C22×#4112yes—
561C2×C4C2×H(5,𝔽2)12yes—
562C2×C4C2×2−1+412yes—
563C8—28no—
564C8—44no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22#318114yes—
572C22C2×#53211yescharacteristic
573C22#348411yescharacteristic
574C22#348611yescharacteristic
575C22C22×#13711yescharacteristic
576C22C2×#57511yescharacteristic
577C22#358611yescharacteristic
578C22#358611yescharacteristic
579C4—24no—
580C4—22no—
581C4—42no—
582C4—44no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—44no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4#352014yes—
597C2—24no—
598C2#3539711yescharacteristic
599C2C2×#318111yescharacteristic
600C2#6070511yescharacteristic
601C2#6158311yescharacteristic
602C2#6158611yescharacteristic
603C2#6182911yescharacteristic
604C2#6183211yescharacteristic
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
4. #29082C2semidirect
5. #28953C2semidirect
7. #26847C2semidirect
8. #26497C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24532
#291112
#290822
#289532
#283132
#268472
#264972
#259122
#258192
#246012
#245022
#220842
#214552
#205362