Finite Group Databasebeta

← #412546

Subgroups — #412546

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal
2C2×#1939C211yesmaximal, characteristic
3C2×#1844C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#14151C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic
8#11231C212yesmaximal
9#11220C212yesmaximal
10#11122C212yesmaximal
11#10849C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2066—28no—
29#1939—24no—
30#1844—24no—
31#1803—24no—
32C4×#103—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—22no—
40#1441—22no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#106—42no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103D414yes—
93C2×#103Q814yes—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8D414yes—
97#391—24no—
98#391—42no—
99#389—24no—
100#389—42no—
101#389—44no—
102#384—22no—
103#384—22no—
104#384—22no—
105#384D414yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D412yes—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—42no—
124#336—24no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8C2×D412yes—
137C22×Q8C2×Q812yes—
138C22×Q8C2×Q812yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×Q812yes—
169C22×C8#4112yes—
170C22×C8#4112yes—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—44no—
174C4×Q8—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4SD1614yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#106—48no—
248#106—48no—
249C4⋊C8—48no—
250C4⋊C8—44no—
251#103—28no—
252#103—28no—
253#103—48no—
254C4×C8—28no—
255#95—24no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—44no—
261#95—82no—
262#95—82no—
263#95—82no—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8C4⋊Q814yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
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—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
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—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C2×SD1614yes—
359C22×C4C2×SD1612yes—
360C22×C4#13512yes—
361C22×C4#13712yes—
362C22×C4#13712yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8#12114yes—
397C2×C8C22⋊Q814yes—
398C4⋊C4—28no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—48no—
431C42—84no—
432C42#12114yes—
433C42C22⋊Q814yes—
434C42C22⋊Q814yes—
435C42#12614yes—
436C23C2×#12111yescharacteristic
437C23C2×#12111yescharacteristic
438C23C2×C22⋊Q811yescharacteristic
439C23D4211yescharacteristic
440C23#54711yescharacteristic
441C23D4×Q811yescharacteristic
442C23#55011yescharacteristic
443C23C22×SD1611yescharacteristic
444C23C2×#13511yescharacteristic
445C23C2×#13711yescharacteristic
446C23C2×#13711yescharacteristic
447C23#57711yescharacteristic
448C23#57711yescharacteristic
449C23#57711yescharacteristic
450C23#57711yescharacteristic
451Q8—24no—
452Q8—24no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
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—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#39614yes—
522C2×C4#39614yes—
523C2×C4#46114yes—
524C2×C4#46414yes—
525C2×C4#46514yes—
526C2×C4#47014yes—
527C2×C4#47514yes—
528C2×C4#47914yes—
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4C2×#12611yescharacteristic
535C2×C4C2×C4⋊Q811yescharacteristic
536C2×C4#53211yescharacteristic
537C2×C4#53311yescharacteristic
538C2×C4#53311yescharacteristic
539C2×C4#53711yescharacteristic
540C2×C4#53711yescharacteristic
541C2×C4#54011yescharacteristic
542C2×C4#54111yescharacteristic
543C2×C4#54411yescharacteristic
544C2×C4#54411yescharacteristic
545C2×C4D4211yescharacteristic
546C2×C4#54611yescharacteristic
547C2×C4#54711yescharacteristic
548C2×C4#54711yescharacteristic
549C2×C4#54811yescharacteristic
550C2×C4D4×Q811yescharacteristic
551C2×C4#55211yescharacteristic
552C2×C4#55311yescharacteristic
553C2×C4#55411yescharacteristic
554C2×C4#55411yescharacteristic
555C2×C4#55511yescharacteristic
556C2×C4#55611yescharacteristic
557C8—24no—
558C8—24no—
559C8—82no—
560C22C2×#39611yescharacteristic
561C22#242811yescharacteristic
562C22#242911yescharacteristic
563C22#244011yescharacteristic
564C22#244111yescharacteristic
565C22#253711yescharacteristic
566C22#253811yescharacteristic
567C22C2×#46111yescharacteristic
568C22C2×#46411yescharacteristic
569C22#304211yescharacteristic
570C22#304311yescharacteristic
571C22C2×#46511yescharacteristic
572C22C2×#47011yescharacteristic
573C22#306011yescharacteristic
574C22#306011yescharacteristic
575C22C2×#47511yescharacteristic
576C22C2×#47911yescharacteristic
577C22#308411yescharacteristic
578C22#308411yescharacteristic
579C22#327811yescharacteristic
580C22#327811yescharacteristic
581C22D4×SD1611yescharacteristic
582C22#328411yescharacteristic
583C22#333511yescharacteristic
584C22#334011yescharacteristic
585C22#335611yescharacteristic
586C22#335611yescharacteristic
587C22Q8×SD1611yescharacteristic
588C22#338111yescharacteristic
589C22#338711yescharacteristic
590C22#338711yescharacteristic
591C22#339211yescharacteristic
592C22#339411yescharacteristic
593C22#340011yescharacteristic
594C22#340011yescharacteristic
595C4—24no—
596C4—44no—
597C4—44no—
598C4—84no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—162no—
604C4#206618yescomplemented
605C4#241611yescharacteristic
606C4#241711yescharacteristic
607C4#243911yescharacteristic
608C4#244011yescharacteristic
609C4#249611yescharacteristic
610C4#249711yescharacteristic
611C4#262911yescharacteristic
612C4#285711yescharacteristic
613C2#1513111yescharacteristic
614C2C2×#206611yescharacteristic
615C2#2361011yescharacteristic
616C2#2536811yescharacteristic
617C2#2536911yescharacteristic
618C2#2545311yescharacteristic
619C2#2545411yescharacteristic
620C2#2604211yescharacteristic
621C2#2604311yescharacteristic
622C2#2608811yescharacteristic
623C2#2608911yescharacteristic
624C2#2822311yescharacteristic
625C2#2823211yescharacteristic
626C2#2826411yescharacteristic
627C2#2827411yescharacteristic
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
604. C4#2066semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#19392
C2×#18442
C2×#18032
C2×C4×#1032
#141512
#139262
#112312
#112202
#111222
#108492