Finite Group Databasebeta

← #416422

Subgroups — #416422

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic, complemented
3#13905C211yesmaximal, characteristic
4#13905C211yesmaximal, characteristic
5#11200C212yesmaximal, complemented
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10850C211yesmaximal, characteristic, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1954—24no—
26#1835—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1445—21no—
38#1445—21no—
39#1438—21no—
40#1438—21no—
41#1437—22no—
42#1437—22no—
43C23×D4D411yescharacteristic
44C24×C4C2312yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D412yes—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8D414yes—
81#382—22no—
82#382—22no—
83#382—22no—
84#382—22no—
85#380—24no—
86#380—24no—
87#380—24no—
88#380—24no—
89#380—42no—
90#380—42no—
91#380—42no—
92#380—42no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117C25C2×D412yes—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—42no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—44no—
167C4×D4—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
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—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4#4114yes—
198C2×C4⋊C4#4114yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—81no—
217C2×C42—81no—
218C2×C42—81no—
219C2×C42—81no—
220C2×C42C2×D412yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C4⋊C8—44no—
227C22⋊C8—48no—
228C22⋊C8—48no—
229C4×C8—28no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24#13514yes—
255C24C22×D411yescharacteristic
256C2×D4—28no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—44no—
260C2×D4—44no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12312yes—
339C22×C4#12312yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C22⋊C4—48no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C42—84no—
396C42—84no—
397C42#12314yes—
398C42#12414yes—
399C42#12614yes—
400C42#12614yes—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23C2×#12111yescharacteristic
437C23C2×#12311yescharacteristic
438C23C2×#12311yescharacteristic
439C23#54612yes—
440C23#54612yes—
441C23C2×#13512yes—
442C23#57512yes—
443C23#57611yescharacteristic
444C23#57711yescharacteristic
445C23#57711yescharacteristic
446C23#57811yescharacteristic
447D4—416no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#39714yes—
508C2×C4#39714yes—
509C2×C4C2×#12311yescharacteristic
510C2×C4C2×#12411yescharacteristic
511C2×C4C2×#12611yescharacteristic
512C2×C4C2×#12611yescharacteristic
513C2×C4#52912yes—
514C2×C4#52912yes—
515C2×C4#53211yescharacteristic
516C2×C4#53211yescharacteristic
517C2×C4#53511yescharacteristic
518C2×C4#53511yescharacteristic
519C2×C4#54012yes—
520C2×C4#54012yes—
521C2×C4#55312yes—
522C2×C4#55312yes—
523C2×C4#55512yes—
524C2×C4#55512yes—
525C2×C4#55912yes—
526C2×C4#55912yes—
527C8—42no—
528C8—42no—
529C8—82no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×#39711yescharacteristic
546C22#242012yes—
547C22#242012yes—
548C22#254411yescharacteristic
549C22#254411yescharacteristic
550C22#305011yescharacteristic
551C22#305411yescharacteristic
552C22#306011yescharacteristic
553C22#306111yescharacteristic
554C22#309112yes—
555C22#309112yes—
556C22#309611yescharacteristic
557C22#309611yescharacteristic
558C22#309711yescharacteristic
559C22#309711yescharacteristic
560C22#329612yes—
561C22#329712yes—
562C22#330012yes—
563C22#330112yes—
564C22#332212yes—
565C22#332312yes—
566C22#332612yes—
567C22#332712yes—
568C4—24no—
569C4—24no—
570C4—82no—
571C4—84no—
572C4—84no—
573C4—161no—
574C4—161no—
575C4—161no—
576C4—161no—
577C4#250011yescharacteristic
578C4#250011yescharacteristic
579C4#251312yes—
580C4#251312yes—
581C4#264111yescharacteristic
582C4#286111yescharacteristic
583C2—84no—
584C2#1522411yescharacteristic
585C2#2363811yescharacteristic
586C2#2364111yescharacteristic
587C2#2497312yes—
588C2#2497412yes—
589C2#2497512yes—
590C2#2497612yes—
591C2#2860111yescharacteristic
592C2#2860211yescharacteristic
593C2#2860311yescharacteristic
594C2#2860411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1835C2semidirect
5. #11200C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #11199C2semidirect
9. #10850C2semidirect
10. #10850C2semidirect
11. C4×#340C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18352
#139052
#112002
#111992
#108502
C4×#3402