Finite Group Databasebeta

← #412582

Subgroups — #412582

Order 512.

1186 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
1C2×#2072C212yesmaximal
2C2×#1803C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11267C212yesmaximal
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×#389—21no—
20C2×#389—21no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2072—28no—
27#1803—24no—
28C4×#103—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444—22no—
39#1441—22no—
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—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103D414yes—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#391—24no—
88#391—42no—
89#389—44no—
90#389—44no—
91#384—24no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—42no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—42no—
116#336—24no—
117#336—24no—
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×Q8—21no—
128C22×Q8—21no—
129C22×Q8C2×D412yes—
130C22×Q8C2×D412yes—
131C22×Q8#4112yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
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—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C2×D412yes—
163C22×C8#4112yes—
164C4×Q8—28no—
165C4×Q8—28no—
166C4×Q8—44no—
167C4×Q8—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
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—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
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—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4Q1614yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4#4112yes—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C4⋊C8—44no—
244#103—28no—
245#103—28no—
246#103—48no—
247C4×C8—44no—
248#95—24no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—42no—
253#95—44no—
254#95—82no—
255#95—82no—
256C24C22×D411yescharacteristic
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—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8#12114yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
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—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
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—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
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#12112yes—
344C22×C4#12112yes—
345C22×C4#12312yes—
346C22×C4#12312yes—
347C22×C4#12412yes—
348C22×C4#12412yes—
349C22×C4C2×SD1614yes—
350C22×C4C2×SD1612yes—
351C22×C4C2×Q1614yes—
352C22×C4C2×Q1612yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4#13712yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+411yescharacteristic
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—88no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4—88no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—48no—
424C42—84no—
425C42#12314yes—
426C42#12314yes—
427C42#12614yes—
428C42#12614yes—
429C23C2×#12111yescharacteristic
430C23C2×#12311yescharacteristic
431C23C2×#12411yescharacteristic
432C23D4211yescharacteristic
433C23#54711yescharacteristic
434C23#54811yescharacteristic
435C23#54811yescharacteristic
436C23C22×SD1611yescharacteristic
437C23C22×Q1611yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23C2×#13711yescharacteristic
440C23#57611yescharacteristic
441C23#57711yescharacteristic
442C23#57711yescharacteristic
443C23#57811yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
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—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4#39714yes—
515C2×C4#39714yes—
516C2×C4#46014yes—
517C2×C4#46214yes—
518C2×C4#48314yes—
519C2×C4#48414yes—
520C2×C4#48714yes—
521C2×C4#48914yes—
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12311yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C2×C4#52511yescharacteristic
526C2×C4C2×#12611yescharacteristic
527C2×C4C2×#12611yescharacteristic
528C2×C4#52911yescharacteristic
529C2×C4#52911yescharacteristic
530C2×C4#53211yescharacteristic
531C2×C4#53211yescharacteristic
532C2×C4#53611yescharacteristic
533C2×C4#53711yescharacteristic
534C2×C4#54111yescharacteristic
535C2×C4#54111yescharacteristic
536C2×C4#54211yescharacteristic
537C2×C4#54211yescharacteristic
538C2×C4#54711yescharacteristic
539C2×C4#54711yescharacteristic
540C2×C4#54711yescharacteristic
541C2×C4#55211yescharacteristic
542C2×C4#55211yescharacteristic
543C2×C4#55211yescharacteristic
544C2×C4#55211yescharacteristic
545C2×C4#55311yescharacteristic
546C2×C4#55311yescharacteristic
547C2×C4#55511yescharacteristic
548C2×C4#55611yescharacteristic
549C2×C4#55611yescharacteristic
550C8—42no—
551C8—42no—
552C8—82no—
553C22C2×#39711yescharacteristic
554C22#241511yescharacteristic
555C22#241611yescharacteristic
556C22#242111yescharacteristic
557C22#242211yescharacteristic
558C22#248911yescharacteristic
559C22#249111yescharacteristic
560C22C2×#46011yescharacteristic
561C22C2×#46211yescharacteristic
562C22#304511yescharacteristic
563C22#304611yescharacteristic
564C22C2×#48311yescharacteristic
565C22C2×#48411yescharacteristic
566C22#309811yescharacteristic
567C22#309911yescharacteristic
568C22C2×#48711yescharacteristic
569C22C2×#48911yescharacteristic
570C22#313611yescharacteristic
571C22#313711yescharacteristic
572C22#327411yescharacteristic
573C22#327811yescharacteristic
574C22D4×SD1611yescharacteristic
575C22D4×Q1611yescharacteristic
576C22#333511yescharacteristic
577C22#333811yescharacteristic
578C22#335311yescharacteristic
579C22#335611yescharacteristic
580C22#335911yescharacteristic
581C22#336211yescharacteristic
582C22#336311yescharacteristic
583C22#336311yescharacteristic
584C22#336811yescharacteristic
585C22#337311yescharacteristic
586C22#337411yescharacteristic
587C22#337511yescharacteristic
588C4—24no—
589C4—44no—
590C4—44no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—162no—
596C4—162no—
597C4#207218yescomplemented
598C4#248511yescharacteristic
599C4#248611yescharacteristic
600C4#249411yescharacteristic
601C4#249511yescharacteristic
602C4#249611yescharacteristic
603C4#249611yescharacteristic
604C4#262511yescharacteristic
605C4#285411yescharacteristic
606C2#1513211yescharacteristic
607C2C2×#207211yescharacteristic
608C2#2363411yescharacteristic
609C2#2467811yescharacteristic
610C2#2468211yescharacteristic
611C2#2472811yescharacteristic
612C2#2473211yescharacteristic
613C2#2499411yescharacteristic
614C2#2499611yescharacteristic
615C2#2501411yescharacteristic
616C2#2501611yescharacteristic
617C2#2772911yescharacteristic
618C2#2773711yescharacteristic
619C2#2776911yescharacteristic
620C2#2777811yescharacteristic
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
597. C4#2072semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#18032
C2×C4×#1032
#139262
#139102
#113162
#112672
#111902
#108492
#108412
#108342