Finite Group Databasebeta

← #464808

Subgroups — #464808

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2489C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2059C212yesmaximal, complemented
4C2×#2036C212yesmaximal
5C2×#1947C211yesmaximal, characteristic, complemented
6#15135C211yesmaximal, characteristic
7#11095C212yesmaximal
8#11077C211yesmaximal, characteristic, complemented
9#11067C212yesmaximal, complemented
10#11042C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20#2489—24no—
21#2489—24no—
22C2×#400—21no—
23C2×#399—21no—
24C2×#397—21no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#389C2212yes—
28C2×#388—21no—
29C2×#384C2211yescharacteristic
30C2×#382—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#2096—24no—
36#2059—28no—
37#2036—28no—
38#1947—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45#1445—21no—
46#1444—21no—
47#1439—21no—
48C4⋊#95C2211yescharacteristic
49C23×Q8C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#125—42no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103—42no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—42no—
97#400—28no—
98#399—44no—
99#397—28no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391D414yes—
104#389—24no—
105#389—42no—
106#388—28no—
107#384—22no—
108#384—22no—
109#382—28no—
110#380—22no—
111#380—22no—
112#380—24no—
113#380—42no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4D414yes—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130#340—24no—
131#340—24no—
132#340—42no—
133#340D418yes—
134#336—24no—
135#336—22no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8C2×D412yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
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×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C2×D414yes—
182C22×C8#4114yes—
183#125—48no—
184#125—44no—
185#125—44no—
186C22⋊Q8—48no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254#107—48no—
255#106—48no—
256C4⋊C8—28no—
257#103—48no—
258#103—48no—
259C22⋊C8—48no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—42no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24#13712yes—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#13512yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C42−1+411yescharacteristic
388C22×C42−1+411yescharacteristic
389C22×C42−1+412yes—
390C22×C42−1+411yescharacteristic
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—84no—
426C4⋊C4—88no—
427C4⋊C4—84no—
428C4⋊C4#12714yes—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—88no—
434C42—22no—
435C42—22no—
436C42—24no—
437C42—41no—
438C42—41no—
439C42—42no—
440C42—48no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×#12011yescharacteristic
465C23#53611yescharacteristic
466C23#53711yescharacteristic
467C23D4212yes—
468C23#54712yes—
469C23C2×#13511yescharacteristic
470C23C2×AGL(1,ℤ/8)11yescharacteristic
471C23C2×#13711yescharacteristic
472C23#57511yescharacteristic
473C23#57712yes—
474C23#57812yes—
475Q8—44no—
476Q8—44no—
477Q8—81no—
478Q8—81no—
479Q8—81no—
480Q8—81no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#44914yes—
548C2×C4#52511yescharacteristic
549C2×C4C2×#12711yescharacteristic
550C2×C4#53511yescharacteristic
551C2×C4#53611yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#53612yes—
554C2×C4#53612yes—
555C2×C4#53611yescharacteristic
556C2×C4#53711yescharacteristic
557C2×C4#53711yescharacteristic
558C2×C4#54212yes—
559C2×C4#54212yes—
560C2×C4D4212yes—
561C2×C4#54612yes—
562C2×C4#54712yes—
563C2×C4#54812yes—
564C2×C4#56212yes—
565C2×C4#56312yes—
566C8—44no—
567C8—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—81no—
578C22—82no—
579C22#240611yescharacteristic
580C22#243311yescharacteristic
581C22#261611yescharacteristic
582C22#267212yes—
583C22#267412yes—
584C22C2×#44911yescharacteristic
585C22#300711yescharacteristic
586C22#301011yescharacteristic
587C22#301111yescharacteristic
588C22#323311yescharacteristic
589C22#323911yescharacteristic
590C22#324212yes—
591C22#324811yescharacteristic
592C22#325111yescharacteristic
593C22#325812yes—
594C22#326912yes—
595C22#327012yes—
596C22#327212yes—
597C22#328412yes—
598C22#333612yes—
599C22#334312yes—
600C22#334412yes—
601C22#335112yes—
602C4—24no—
603C4—42no—
604C4—44no—
605C4—81no—
606C4—82no—
607C4—81no—
608C4—81no—
609C4—82no—
610C4—84no—
611C4—82no—
612C4—162no—
613C4#254011yescharacteristic
614C4#263211yescharacteristic
615C4#265811yescharacteristic
616C4#267212yes—
617C4#267412yes—
618C4#282011yescharacteristic
619C2—82no—
620C2#1651011yescharacteristic
621C2#2409411yescharacteristic
622C2#2411111yescharacteristic
623C2#2575411yescharacteristic
624C2#2576711yescharacteristic
625C2#2968911yescharacteristic
626C2#2969911yescharacteristic
627C2#3066612yes—
628C2#3068712yes—
629C2#3078212yes—
630C2#3080312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2096C2semidirect
3. C2×#2059C2semidirect
5. C2×#1947C2semidirect
8. #11077C2semidirect
9. #11067C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24892
C2×#20962
C2×#20592
C2×#20362
C2×#19472
#151352
#110952
#110772
#110672
#110422
#108062