Finite Group Databasebeta

← #412583

Subgroups — #412583

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C212yesmaximal
2C2×#1803C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#11315C212yesmaximal
7#11268C212yesmaximal
8#11190C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#391C2212yes—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2074—28no—
25#1803—24no—
26C4×#103—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1442—22no—
37#1441—22no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8D414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103D414yes—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85#391—24no—
86#391—42no—
87#384—22no—
88#384—22no—
89#384—22no—
90#384—22no—
91#382—24no—
92#382—42no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—42no—
114#336—24no—
115#336—24no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8C2×D412yes—
128C22×Q8C2×D412yes—
129C22×Q8#4112yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
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×C8C2×D412yes—
161C22×C8#4112yes—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—44no—
165C4×Q8—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4#4112yes—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C2×D412yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C4⋊C8—44no—
236#103—28no—
237#103—28no—
238#103—48no—
239C4×C8—44no—
240#95—24no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250C24C22×D411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8#12114yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
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#12112yes—
340C22×C4#12112yes—
341C22×C4#12312yes—
342C22×C4#12312yes—
343C22×C4#12412yes—
344C22×C4#12412yes—
345C22×C4#13512yes—
346C22×C4#13512yes—
347C22×C4#13712yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C42−1+411yescharacteristic
361C22×C42−1+411yescharacteristic
362C22×C42−1+411yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—48no—
416C42—84no—
417C42#12314yes—
418C42#12314yes—
419C42#12614yes—
420C42#12614yes—
421C23C2×#12111yescharacteristic
422C23C2×#12311yescharacteristic
423C23C2×#12411yescharacteristic
424C23D4211yescharacteristic
425C23#54711yescharacteristic
426C23#54811yescharacteristic
427C23#54811yescharacteristic
428C23C2×#13511yescharacteristic
429C23C2×#13511yescharacteristic
430C23C2×#13711yescharacteristic
431C23C2×#13711yescharacteristic
432C23#57611yescharacteristic
433C23#57711yescharacteristic
434C23#57711yescharacteristic
435C23#57811yescharacteristic
436Q8—24no—
437Q8—24no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
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—82no—
506C2×C4#39714yes—
507C2×C4#39714yes—
508C2×C4#46414yes—
509C2×C4#46414yes—
510C2×C4#48514yes—
511C2×C4#48514yes—
512C2×C4#48814yes—
513C2×C4#49114yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C2×C4#52511yescharacteristic
518C2×C4C2×#12611yescharacteristic
519C2×C4C2×#12611yescharacteristic
520C2×C4#52911yescharacteristic
521C2×C4#52911yescharacteristic
522C2×C4#53211yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53611yescharacteristic
525C2×C4#53711yescharacteristic
526C2×C4#54111yescharacteristic
527C2×C4#54111yescharacteristic
528C2×C4#54211yescharacteristic
529C2×C4#54211yescharacteristic
530C2×C4#54711yescharacteristic
531C2×C4#54711yescharacteristic
532C2×C4#54711yescharacteristic
533C2×C4#55211yescharacteristic
534C2×C4#55211yescharacteristic
535C2×C4#55211yescharacteristic
536C2×C4#55211yescharacteristic
537C2×C4#55311yescharacteristic
538C2×C4#55311yescharacteristic
539C2×C4#55511yescharacteristic
540C2×C4#55611yescharacteristic
541C2×C4#55611yescharacteristic
542C8—42no—
543C8—42no—
544C8—82no—
545C22C2×#39711yescharacteristic
546C22#241511yescharacteristic
547C22#241611yescharacteristic
548C22#242111yescharacteristic
549C22#242211yescharacteristic
550C22#248911yescharacteristic
551C22#249111yescharacteristic
552C22C2×#46411yescharacteristic
553C22C2×#46411yescharacteristic
554C22#304511yescharacteristic
555C22#304611yescharacteristic
556C22C2×#48511yescharacteristic
557C22C2×#48511yescharacteristic
558C22#309811yescharacteristic
559C22#309911yescharacteristic
560C22C2×#48811yescharacteristic
561C22C2×#49111yescharacteristic
562C22#313611yescharacteristic
563C22#313711yescharacteristic
564C22#327611yescharacteristic
565C22#327811yescharacteristic
566C22#328411yescharacteristic
567C22#328711yescharacteristic
568C22#334011yescharacteristic
569C22#334211yescharacteristic
570C22#335611yescharacteristic
571C22#335711yescharacteristic
572C22#336111yescharacteristic
573C22#336411yescharacteristic
574C22#336511yescharacteristic
575C22#336511yescharacteristic
576C22#336911yescharacteristic
577C22#337211yescharacteristic
578C22#337411yescharacteristic
579C22#337511yescharacteristic
580C4—24no—
581C4—44no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—162no—
588C4—162no—
589C4#207418yescomplemented
590C4#248511yescharacteristic
591C4#248611yescharacteristic
592C4#249411yescharacteristic
593C4#249511yescharacteristic
594C4#249611yescharacteristic
595C4#249611yescharacteristic
596C4#262511yescharacteristic
597C4#285411yescharacteristic
598C2#1513211yescharacteristic
599C2C2×#207411yescharacteristic
600C2#2363411yescharacteristic
601C2#2467911yescharacteristic
602C2#2468311yescharacteristic
603C2#2472911yescharacteristic
604C2#2473311yescharacteristic
605C2#2499511yescharacteristic
606C2#2499711yescharacteristic
607C2#2501511yescharacteristic
608C2#2501711yescharacteristic
609C2#2773011yescharacteristic
610C2#2773811yescharacteristic
611C2#2777011yescharacteristic
612C2#2777711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
589. C4#2074semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
C2×#18032
C2×C4×#1032
#139262
#139112
#113152
#112682
#111902
#108492
#108412
#108342