Finite Group Databasebeta

← #402302

Subgroups — #402302

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2094C211yesmaximal, characteristic, complemented
2C2×#2077C212yesmaximal, complemented
3C2×#2034C212yesmaximal
4C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6#14686C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#11323C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic, complemented
10#10925C212yesmaximal, complemented
11#10861C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#106C2212yes—
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2212yes—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#386—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380—22no—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28#2094—24no—
29#2077—28no—
30#2034—28no—
31C4×C22⋊C8—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39C4⋊#95—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—24no—
52C2×#121—42no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—24no—
68C22×C42—42no—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8D418yes—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—42no—
90#390—22no—
91#390—22no—
92#390—24no—
93#386—48no—
94#384—22no—
95#384—22no—
96#384—42no—
97#380—24no—
98#380—24no—
99#380—24no—
100#380—48no—
101C2×#95—22no—
102C2×#95—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—22no—
111#336D414yescomplemented
112C25—21no—
113C25C2×D411yescharacteristic
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—42no—
125C22×D4—44no—
126C22×D4—42no—
127C22×D4—44no—
128C22×D4#4112yes—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4SD1618yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×D414yes—
166C22×C8C2×D414yes—
167C22×C8C2×D414yes—
168C22×C8#4114yes—
169#121—48no—
170#121—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4#4112yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—48no—
222C2×C42—48no—
223C2×C42—42no—
224C2×C42C2×D412yes—
225C2×C42#4112yes—
226C2×C42#4114yes—
227#106—48no—
228#106—48no—
229#102—48no—
230#102—48no—
231C22⋊C8—216no—
232C4×C8—48no—
233#95—44no—
234#95—42no—
235#95—44no—
236#95—44no—
237#95—48no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24#12712yes—
259C24C2×SD1614yes—
260C24C22×D411yescharacteristic
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—48no—
266C2×D4—416no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—84no—
280C2×D4—88no—
281C2×D4—84no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—216no—
303C22×C4—216no—
304C22×C4—22no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4#12114yes—
345C22×C4#12314yes—
346C22×C4#12712yes—
347C22×C4C2×SD1614yes—
348C22×C4C2×SD1614yes—
349C22×C4#13514yes—
350C22×C4AGL(1,ℤ/8)14yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C2×#4112yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8#12118yes—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—88no—
390C4⋊C4—88no—
391C22⋊C4—48no—
392C22⋊C4—88no—
393C22⋊C4—84no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—48no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—82no—
429C23—82no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—82no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23#49214yes—
447C23C2×#12111yescharacteristic
448C23C2×#12311yescharacteristic
449C23#52511yescharacteristic
450C23C2×#12711yescharacteristic
451C23#53211yescharacteristic
452C23#53411yescharacteristic
453C23#53611yescharacteristic
454C23C22×SD1612yes—
455C23C2×#13512yes—
456C23C2×AGL(1,ℤ/8)12yes—
457C23#57512yes—
458D4—44no—
459D4—44no—
460D4—816no—
461D4—82no—
462D4—82no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—432no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4#46518yes—
513C2×C4#48118yes—
514C2×C4C2×#12112yes—
515C2×C4#52912yes—
516C2×C4#52912yes—
517C2×C4#53212yes—
518C2×C4#53912yes—
519C2×C4#54212yes—
520C2×C4#54612yes—
521C2×C4#54612yes—
522C2×C4#54612yes—
523C2×C4#54711yescharacteristic
524C2×C4#54712yes—
525C2×C4#54712yes—
526C2×C4#54811yescharacteristic
527C2×C4#55112yes—
528C2×C4#55211yescharacteristic
529C2×C4#55611yescharacteristic
530C8—28no—
531C8—44no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22—44no—
542C22—42no—
543C22—84no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—82no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22—162no—
552C22—161no—
553C22—162no—
554C22#239511yescharacteristic
555C22#241211yescharacteristic
556C22#241511yescharacteristic
557C22#250111yescharacteristic
558C22#253911yescharacteristic
559C22#266311yescharacteristic
560C22#285611yescharacteristic
561C22C2×#46512yes—
562C22#305512yes—
563C22C2×#48112yes—
564C22#309112yes—
565C22#310112yes—
566C22#310612yes—
567C22C2×#49211yescharacteristic
568C22#314711yescharacteristic
569C22#315111yescharacteristic
570C22#315311yescharacteristic
571C22#316712yes—
572C22#317112yes—
573C22#318712yes—
574C22#318812yes—
575C22#322712yes—
576C22#323312yes—
577C4—28no—
578C4—28no—
579C4—42no—
580C4—82no—
581C4—161no—
582C4—161no—
583C4—162no—
584C4—162no—
585C4#242512yes—
586C4#242712yes—
587C4#252112yes—
588C4#255312yes—
589C2—44no—
590C2—162no—
591C2#1483911yescharacteristic
592C2#2375512yes—
593C2#2455312yes—
594C2#2470112yes—
595C2#2790611yescharacteristic
596C2#2791111yescharacteristic
597C2#2830211yescharacteristic
598C2#2830711yescharacteristic
599C2#3035712yes—
600C2#3317011yescharacteristic
601C2#3318511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2094C2semidirect
2. C2×#2077C2semidirect
4. C2×C4×C22⋊C8C2semidirect
6. #14686C2semidirect
7. #11371C2semidirect
8. #11323C2semidirect
9. #11049C2semidirect
10. #10925C2semidirect
11. #10861C2semidirect
111. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20942
C2×#20772
C2×#20342
C2×C4×C22⋊C82
#146982
#146862
#113712
#113232
#110492
#109252
#108612