Finite Group Databasebeta

← #402430

Subgroups — #402430

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
3#14698C211yesmaximal, characteristic
4#14682C211yesmaximal, characteristic, complemented
5#11358C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11051C211yesmaximal, characteristic, complemented
9#10969C212yesmaximal
10#10925C212yesmaximal, complemented
11#10820C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2212yes—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#386—22no—
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25#2073—24no—
26C4×C22⋊C8—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1437—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—22no—
40C22⋊#95—22no—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121—24no—
50C2×#121—42no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—24no—
64C22×C42—42no—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C4×C8—24no—
78C2×C4×C8—24no—
79C2×C4×C8—42no—
80#390—22no—
81#390—22no—
82#390—24no—
83#386—48no—
84#382—22no—
85#382—22no—
86#382—42no—
87#380—24no—
88#380—42no—
89#380—42no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—22no—
105#336D414yescomplemented
106C25—21no—
107C25C2×D411yescharacteristic
108C22×D4—24no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—42no—
119C22×D4—44no—
120C22×D4—42no—
121C22×D4—44no—
122C22×D4#4112yes—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
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×C8C2×D414yes—
162C22×C8#4114yes—
163#121—48no—
164#121—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
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—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4#4112yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—48no—
211C2×C42—48no—
212C2×C42—42no—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215C2×C42#4114yes—
216#102—48no—
217#102—48no—
218C22⋊C8—216no—
219C4×C8—48no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—44no—
226#95—44no—
227#95—44no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24#12412yes—
249C24#13514yes—
250C24C22×D411yescharacteristic
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—48no—
256C2×D4—416no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—84no—
270C2×D4—88no—
271C2×D4—84no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—216no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4#12114yes—
341C22×C4#12314yes—
342C22×C4#12412yes—
343C22×C4#13514yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)11yescharacteristic
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
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—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C22⋊C4—48no—
384C22⋊C4—88no—
385C22⋊C4—84no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—48no—
393C23—22no—
394C23—24no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—82no—
421C23—82no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—82no—
428C23—82no—
429C23—81no—
430C23—81no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—81no—
436C23—82no—
437C23—82no—
438C23#48814yes—
439C23C2×#12111yescharacteristic
440C23C2×#12311yescharacteristic
441C23#52511yescharacteristic
442C23C2×#12411yescharacteristic
443C23#53211yescharacteristic
444C23#53411yescharacteristic
445C23#53511yescharacteristic
446C23C2×#13512yes—
447C23C2×#13512yes—
448C23C2×AGL(1,ℤ/8)12yes—
449C23#57512yes—
450D4—44no—
451D4—44no—
452D4—816no—
453D4—82no—
454D4—82no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—432no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#48218yes—
510C2×C4#52512yes—
511C2×C4#52912yes—
512C2×C4#52912yes—
513C2×C4#52912yes—
514C2×C4#53812yes—
515C2×C4#54012yes—
516C2×C4#54611yescharacteristic
517C2×C4#54612yes—
518C2×C4#55011yescharacteristic
519C2×C4#55112yes—
520C2×C4#55112yes—
521C2×C4#55112yes—
522C2×C4#55311yescharacteristic
523C2×C4#55312yes—
524C2×C4#55312yes—
525C2×C4#55511yescharacteristic
526C8—44no—
527C8—44no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—44no—
534C22—44no—
535C22—42no—
536C22—44no—
537C22—44no—
538C22—42no—
539C22—84no—
540C22—82no—
541C22—82no—
542C22—84no—
543C22—82no—
544C22—161no—
545C22—161no—
546C22—161no—
547C22—161no—
548C22—162no—
549C22—162no—
550C22#239511yescharacteristic
551C22#241211yescharacteristic
552C22#241411yescharacteristic
553C22#250411yescharacteristic
554C22#256111yescharacteristic
555C22#266111yescharacteristic
556C22#285011yescharacteristic
557C22#305012yes—
558C22#305512yes—
559C22C2×#48212yes—
560C22#309112yes—
561C22#310212yes—
562C22#310512yes—
563C22C2×#48811yescharacteristic
564C22#313111yescharacteristic
565C22#313811yescharacteristic
566C22#314211yescharacteristic
567C22#316812yes—
568C22#317112yes—
569C22#318812yes—
570C22#319112yes—
571C22#320712yes—
572C22#321112yes—
573C4—28no—
574C4—44no—
575C4—42no—
576C4—82no—
577C4—161no—
578C4—161no—
579C4—162no—
580C4—162no—
581C4#245712yes—
582C4#246712yes—
583C4#253112yes—
584C4#257812yes—
585C2—44no—
586C2—162no—
587C2#1484211yescharacteristic
588C2#2375812yes—
589C2#2455512yes—
590C2#2465412yes—
591C2#2797411yescharacteristic
592C2#2797711yescharacteristic
593C2#2910411yescharacteristic
594C2#2910811yescharacteristic
595C2#3032012yes—
596C2#3304411yescharacteristic
597C2#3305811yescharacteristic
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×#2073C2semidirect
2. C2×C4×C22⋊C8C2semidirect
4. #14682C2semidirect
5. #11358C2semidirect
6. #11268C2semidirect
7. #11200C2semidirect
8. #11051C2semidirect
10. #10925C2semidirect
11. #10820C2semidirect
105. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
C2×C4×C22⋊C82
#146982
#146822
#113582
#112682
#112002
#110512
#109692
#109252
#108202