Finite Group Databasebeta

← #6508275

Subgroups — #6508275

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic
3#29329C211yesmaximal, characteristic
4#29297C211yesmaximal, characteristic
5#29038C214yesmaximal, complemented
6#28904C214yesmaximal, complemented
7#26838C214yesmaximal, complemented
8#26265C214yesmaximal, complemented
9#25856C211yesmaximal, characteristic
10#25722C211yesmaximal, characteristic
11#24508C212yesmaximal
12#24508C212yesmaximal
13#22050C212yesmaximal
14#21455C212yesmaximal
15#20534C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#494C2212yes—
19C2×#487C2212yes—
20C2×#451C2212yes—
21C2×#451C2212yes—
22C2×#441C2214yes—
23C2×#432C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yes—
32#2507C2212yes—
33#2451C2212yes—
34#2446—22no—
35#2446—22no—
36#2446C2214yes—
37#2446C2214yes—
38#2446C2212yes—
39C2×#400—21no—
40C2×#398—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2212yes—
52#2084C2212yes—
53#2066C2214yes—
54#2066C2214yes—
55#2058C2212yes—
56#2056C2214yes—
57#2056C2214yes—
58#2043C2214yes—
59#2043C2212yes—
60#2016C2214yes—
61#1971C2214yes—
62#1971C2214yes—
63#1971C2214yes—
64#1956C2212yes—
65#1956C2212yes—
66#1941C2214yes—
67#1935C2214yes—
68#1935C2214yes—
69#1935C2214yes—
70#1933C2214yes—
71#1927C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1872C2212yes—
75#1872C2212yes—
76#1865C2214yes—
77#1853C2212yes—
78#1853C2212yes—
79#1787C2212yes—
80#1787C2212yes—
81#1771C2214yes—
82#1771C2214yes—
83#1763C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C22×Q16C2314yes—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—44no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#494—44no—
117#487—44no—
118#451—28no—
119#451—28no—
120#441—48no—
121#432—28no—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#400—22no—
141#400—22no—
142#400—24no—
143#400—24no—
144#398—22no—
145#398—22no—
146#398—24no—
147#398—24no—
148#398—22no—
149#398—22no—
150#395—22no—
151#391—21no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#391C2314yes—
160#391C2312yes—
161#391C2312yes—
162#391C2314yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—24no—
187#387—24no—
188#385—22no—
189#385—22no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#381—22no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
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—21no—
224C22×Q8—24no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8C2412yes—
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—28no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×Q16—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262C4×Q8—28no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—48no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
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—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
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—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2412yes—
322#106—44no—
323#106—44no—
324C4⋊C8—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—44no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333C22⋊C8—24no—
334C22⋊C8#4118yes—
335#97—28no—
336#97—44no—
337C4×C8—42no—
338C4×C8—42no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
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#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—416no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—88no—
388C2×Q8H(5,𝔽2)14yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
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—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
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—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4#13714yes—
428C22×C4#13714yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4114yes—
439C22×C4C2511yescharacteristic
440Q16—416no—
441Q16—88no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8C2×#4118yes—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—42no—
490C42—41no—
491C42—41no—
492C42—48no—
493C42—82no—
494C42—82no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#53212yes—
500C23C2×#13712yes—
501C23C2×#13712yes—
502C23C23×D411yescharacteristic
503Q8—28no—
504Q8—48no—
505Q8—44no—
506Q8—44no—
507Q8—44no—
508Q8—44no—
509Q8—44no—
510Q8—84no—
511Q8—162no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—44no—
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—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4#53212yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13712yes—
572C2×C4C2×#13712yes—
573C2×C4C2×#13712yes—
574C2×C4C22×#4112yes—
575C2×C4C2×H(5,𝔽2)12yes—
576C2×C4#58512yes—
577C8—28no—
578C8—44no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22#318514yes—
586C22C2×#53211yescharacteristic
587C22#348411yescharacteristic
588C22#348711yescharacteristic
589C22C22×#13711yescharacteristic
590C22C22×#13711yescharacteristic
591C22#358611yescharacteristic
592C22#358611yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4#351314yes—
611C2—24no—
612C2#3539611yescharacteristic
613C2C2×#318511yescharacteristic
614C2#6070511yescharacteristic
615C2#6157311yescharacteristic
616C2#6157411yescharacteristic
617C2#6188911yescharacteristic
618C2#6189311yescharacteristic
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
5. #29038C2semidirect
6. #28904C2semidirect
7. #26838C2semidirect
8. #26265C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24462
#293292
#292972
#290382
#289042
#268382
#262652
#258562
#257222
#245082
#220502
#214552
#205342