Finite Group Databasebeta

← #475054

Subgroups — #475054

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1933C211yesmaximal, characteristic, complemented
3#15181C211yesmaximal, characteristic
4#15177C211yesmaximal, characteristic
5#11228C211yesmaximal, characteristic, complemented
6#11200C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal, complemented
10#10970C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26#1954—24no—
27#1933—24no—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35#1444—21no—
36#1442—21no—
37#1439—22no—
38#1439—22no—
39#1438—21no—
40#1438—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—42no—
52C2×C22⋊Q8—42no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—42no—
58C2×C4×D4—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D412yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—42no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C4×C8—42no—
90C2×C4×C8D414yes—
91#391—22no—
92#391—22no—
93#389—22no—
94#389—22no—
95#380—24no—
96#380—24no—
97#380—42no—
98#380—42no—
99#380—42no—
100#380—42no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118C25C2×D412yes—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8#4112yes—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4Q1614yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—48no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
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—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4#4114yes—
214C2×C4⋊C4#4112yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C2×D412yes—
233C2×C42#4112yes—
234C4⋊C8—44no—
235#103—28no—
236#103—44no—
237#103—48no—
238C22⋊C8—48no—
239C22⋊C8—48no—
240C4×C8—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—81no—
250#95—81no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24C2×Q1614yes—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—28no—
286C2×D4—48no—
287C2×D4—44no—
288C2×D4—48no—
289C2×D4—44no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
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—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—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
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—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4C2×Q1612yes—
368C22×C4#13512yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)11yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—44no—
401C4⋊C4—416no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C42—22no—
423C42—22no—
424C42—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23C2×#12111yescharacteristic
461C23#53412yes—
462C23#54611yescharacteristic
463C23#54612yes—
464C23#54611yescharacteristic
465C23C22×Q1611yescharacteristic
466C23C2×#13511yescharacteristic
467C23#57512yes—
468C23#57711yescharacteristic
469C23#57712yes—
470C23#57711yescharacteristic
471Q8—42no—
472Q8—42no—
473Q8—44no—
474Q8—81no—
475Q8—81no—
476Q8—82no—
477D4—416no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4#46714yes—
537C2×C4#52511yescharacteristic
538C2×C4#53211yescharacteristic
539C2×C4#53411yescharacteristic
540C2×C4#53511yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53511yescharacteristic
543C2×C4#53812yes—
544C2×C4#53812yes—
545C2×C4#53812yes—
546C2×C4#54012yes—
547C2×C4#54012yes—
548C2×C4#55112yes—
549C2×C4#55311yescharacteristic
550C2×C4#55311yescharacteristic
551C2×C4#55512yes—
552C2×C4#55912yes—
553C2×C4#56012yes—
554C2×C4#56012yes—
555C8—42no—
556C8—42no—
557C8—82no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22#253511yescharacteristic
574C22#254411yescharacteristic
575C22#261711yescharacteristic
576C22#269712yes—
577C22#269912yes—
578C22C2×#46711yescharacteristic
579C22#305411yescharacteristic
580C22#305711yescharacteristic
581C22#306011yescharacteristic
582C22#318912yes—
583C22#319112yes—
584C22#320012yes—
585C22#320412yes—
586C22#329911yescharacteristic
587C22#329912yes—
588C22#329911yescharacteristic
589C22#330312yes—
590C22#330411yescharacteristic
591C22#330411yescharacteristic
592C22#332312yes—
593C22#332312yes—
594C22#332612yes—
595C22#332612yes—
596C4—24no—
597C4—42no—
598C4—82no—
599C4—82no—
600C4—84no—
601C4—82no—
602C4—161no—
603C4—161no—
604C4—162no—
605C4#261011yescharacteristic
606C4#263511yescharacteristic
607C4#266111yescharacteristic
608C4#279712yes—
609C4#280312yes—
610C4#281811yescharacteristic
611C2—84no—
612C2#1702211yescharacteristic
613C2#2814211yescharacteristic
614C2#2814711yescharacteristic
615C2#2859811yescharacteristic
616C2#2860411yescharacteristic
617C2#2973111yescharacteristic
618C2#2974111yescharacteristic
619C2#3166912yes—
620C2#3167112yes—
621C2#3172112yes—
622C2#3172212yes—
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×#1933C2semidirect
5. #11228C2semidirect
6. #11200C2semidirect
7. #11200C2semidirect
8. #11077C2semidirect
9. #10970C2semidirect
10. #10970C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#19332
#151812
#151772
#112282
#112002
#110772
#109702
#108202