Finite Group Databasebeta

← #396570

Subgroups — #396570

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1877C211yesmaximal, characteristic
2C2×#1877C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4#15185C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11255C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11077C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#384C2211yescharacteristic
23C22×#95—21no—
24#1877—28no—
25#1877—24no—
26#1802—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1445—22no—
36#1442—21no—
37#1442—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C22⋊#95—22no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—22no—
47C24×C4—21no—
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—24no—
51C22×SD16—24no—
52C22×SD16D414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C2×C4×D4D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88#391—24no—
89#391—22no—
90#391—22no—
91#391—42no—
92#384—22no—
93#384—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—44no—
105C2×#95D411yescharacteristic
106C2×#95D412yes—
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—22no—
111#340—24no—
112#340—42no—
113#336—24no—
114#336—22no—
115#336—42no—
116C25—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8C2×D412yes—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×D412yes—
133C22×D4C2×D412yes—
134C22×D4#4112yes—
135C22×D4#4114yes—
136C22×D4#4114yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—28no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×SD16—28no—
160C2×SD16—216no—
161C2×SD16—44no—
162C2×SD16—44no—
163C2×SD16—48no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8C2×D412yes—
181C4×D4—28no—
182C4×D4—44no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—48no—
187C4×D4—48no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42C2×D412yes—
226C2×C42#4112yes—
227C4⋊C8—44no—
228#103—48no—
229#103—44no—
230#102—48no—
231#102—44no—
232#95—22no—
233#95—42no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—44no—
239#95—42no—
240#95—82no—
241#95—81no—
242#95—82no—
243#95—82no—
244#95SD1614yescomplemented
245C24—22no—
246C24—24no—
247C24—22no—
248C24—24no—
249C24—22no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—42no—
254C24—41no—
255C24—41no—
256C24C22×D411yescharacteristic
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4#12114yes—
289C2×D4#12314yes—
290C2×D4#12318yes—
291C2×D4#12414yes—
292C2×D4#12618yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—48no—
316C22×C4—416no—
317C22×C4—416no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—84no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4C2×SD1614yes—
356C22×C4C2×SD1612yes—
357C22×C4#13712yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C42−1+411yescharacteristic
368SD16—416no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—44no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23C2×#12111yescharacteristic
435C23#52512yes—
436C23D4211yescharacteristic
437C23#54612yes—
438C23#54811yescharacteristic
439C23C22×SD1611yescharacteristic
440C23C2×#13711yescharacteristic
441C23#57512yes—
442C23#57612yes—
443C23#57711yescharacteristic
444C23#57811yescharacteristic
445Q8—42no—
446Q8—44no—
447Q8—42no—
448Q8—81no—
449Q8—81no—
450Q8—82no—
451D4—24no—
452D4—28no—
453D4—24no—
454D4—28no—
455D4—24no—
456D4#39718yes—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—816no—
489C2×C4—832no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4#46114yes—
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12312yes—
514C2×C4C2×#12311yescharacteristic
515C2×C4C2×#12411yescharacteristic
516C2×C4C2×#12612yes—
517C2×C4#52912yes—
518C2×C4#53211yescharacteristic
519C2×C4#53411yescharacteristic
520C2×C4#53711yescharacteristic
521C2×C4#53812yes—
522C2×C4#53912yes—
523C2×C4#54612yes—
524C2×C4#54711yescharacteristic
525C2×C4#55112yes—
526C2×C4#55112yes—
527C2×C4#55212yes—
528C2×C4#55311yescharacteristic
529C2×C4#55312yes—
530C8—42no—
531C8—81no—
532C8—82no—
533C22—28no—
534C22—44no—
535C22—44no—
536C22—48no—
537C22—48no—
538C22—44no—
539C22—48no—
540C22—44no—
541C22—48no—
542C22—44no—
543C22—44no—
544C22#241211yescharacteristic
545C22#241611yescharacteristic
546C22#241812yes—
547C22#248112yes—
548C22#285011yescharacteristic
549C22C2×#46111yescharacteristic
550C22#304011yescharacteristic
551C22#304111yescharacteristic
552C22#304611yescharacteristic
553C22#310112yes—
554C22#311612yes—
555C22#311812yes—
556C22#312412yes—
557C22#326812yes—
558C22#327611yescharacteristic
559C22D4×SD1611yescharacteristic
560C22#329512yes—
561C22#331412yes—
562C22#331512yes—
563C22#332312yes—
564C22#335911yescharacteristic
565C22#337012yes—
566C22#337211yescharacteristic
567C4—24no—
568C4—44no—
569C4—42no—
570C4—84no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—161no—
575C4—162no—
576C4C2×#39711yescharacteristic
577C4#241912yes—
578C4#249811yescharacteristic
579C4#251012yes—
580C4#254311yescharacteristic
581C4#263911yescharacteristic
582C2—48no—
583C2#1473511yescharacteristic
584C2#2453011yescharacteristic
585C2#2454511yescharacteristic
586C2#2473011yescharacteristic
587C2#2474911yescharacteristic
588C2#2490112yes—
589C2#2491912yes—
590C2#2755212yes—
591C2#2757612yes—
592C2#3302611yescharacteristic
593C2#3303611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15185C2semidirect
6. #11371C2semidirect
7. #11315C2semidirect
8. #11255C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect
244. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#18772
C2×#18022
#151852
D4×#952
#113712
#113152
#112552
#111902
#110772
#107882