Finite Group Databasebeta

← #416479

Subgroups — #416479

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic, complemented
3C2×#1983C211yesmaximal, characteristic
4C2×#1939C211yesmaximal, characteristic, complemented
5#14160C211yesmaximal, characteristic, complemented
6#14153C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11189C212yesmaximal
9#11077C212yesmaximal, complemented
10#11076C212yesmaximal
11C4×#340C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C2×#488—24no—
15C2×#488—24no—
16C2×#488—24no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1989—24no—
32#1983—24no—
33#1939—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—21no—
40#1441—21no—
41#1439—21no—
42#1439—21no—
43#1437—22no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125D414yes—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#488—232no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8—42no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—42no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8D414yes—
108#391—24no—
109#391—42no—
110#390—24no—
111#390—42no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D412yes—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#336—24no—
132C25—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×D412yes—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4C2×D412yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180#125—28no—
181#125—28no—
182#125—44no—
183#125—44no—
184#124—28no—
185#124—28no—
186#124—44no—
187#124—44no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4114yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—81no—
243C2×C42—81no—
244C2×C42—81no—
245C2×C42—81no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C4⋊C8—48no—
253C4⋊C8—44no—
254#103—28no—
255#103—28no—
256#103—48no—
257#102—28no—
258#102—28no—
259#102—48no—
260C4×C8—28no—
261#95—24no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—82no—
269#95—82no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#13514yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+412yes—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—28no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—84no—
431C42—84no—
432C42#12014yes—
433C42#12314yes—
434C42#12314yes—
435C42#12414yes—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—82no—
458C23—82no—
459C23C2×#12111yescharacteristic
460C23C2×#12111yescharacteristic
461C23C2×#12111yescharacteristic
462C23#54612yes—
463C23#54812yes—
464C23C2×#13512yes—
465C23#57512yes—
466C23#57611yescharacteristic
467C23#57711yescharacteristic
468C23#57711yescharacteristic
469C23#57811yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—44no—
473Q8—81no—
474Q8—81no—
475Q8—82no—
476D4—42no—
477D4—42no—
478D4—44no—
479D4—81no—
480D4—81no—
481D4—82no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4#39414yes—
538C2×C4#39414yes—
539C2×C4C2×#12011yescharacteristic
540C2×C4C2×#12311yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4#52912yes—
544C2×C4#52912yes—
545C2×C4#53211yescharacteristic
546C2×C4#53211yescharacteristic
547C2×C4#53511yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#54212yes—
550C2×C4#54212yes—
551C2×C4#54612yes—
552C2×C4#54712yes—
553C2×C4#54812yes—
554C2×C4#55012yes—
555C2×C4#55612yes—
556C2×C4#55912yes—
557C8—42no—
558C8—42no—
559C8—82no—
560C22—82no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—161no—
566C22—161no—
567C22—162no—
568C22—162no—
569C22—161no—
570C22—161no—
571C22C2×#39411yescharacteristic
572C22#242612yes—
573C22#242712yes—
574C22#254411yescharacteristic
575C22#254511yescharacteristic
576C22#303512yes—
577C22#303512yes—
578C22#304511yescharacteristic
579C22#304511yescharacteristic
580C22#304611yescharacteristic
581C22#304611yescharacteristic
582C22#305011yescharacteristic
583C22#305411yescharacteristic
584C22#306011yescharacteristic
585C22#306111yescharacteristic
586C22#329712yes—
587C22#330012yes—
588C22#332312yes—
589C22#332612yes—
590C22#336012yes—
591C22#336512yes—
592C22#337012yes—
593C22#337712yes—
594C4—24no—
595C4—24no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4—84no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4#243111yescharacteristic
604C4#243211yescharacteristic
605C4#244712yes—
606C4#244812yes—
607C4#264011yescharacteristic
608C4#286111yescharacteristic
609C2—162no—
610C2#1522611yescharacteristic
611C2#2353911yescharacteristic
612C2#2354311yescharacteristic
613C2#2528012yes—
614C2#2528312yes—
615C2#2532712yes—
616C2#2532812yes—
617C2#2860111yescharacteristic
618C2#2860411yescharacteristic
619C2#2864211yescharacteristic
620C2#2864311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1989C2semidirect
4. C2×#1939C2semidirect
5. #14160C2semidirect
7. #11190C2semidirect
9. #11077C2semidirect
11. C4×#340C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4882
C2×#19892
C2×#19832
C2×#19392
#141602
#141532
#111902
#111892
#110772
#110762
C4×#3402