Finite Group Databasebeta

← #412217

Subgroups — #412217

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#2059C212yesmaximal
3C2×#1948C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5C4×#389C211yesmaximal, characteristic
6#11385C212yesmaximal
7#11325C212yesmaximal
8#11208C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10849C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2859—24no—
18#2859—24no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#2059—28no—
33#1948—24no—
34#1947—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C4×#95—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—42no—
55C2×#125—42no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8Q814yes—
90C2×C4⋊C8Q814yes—
91#389—28no—
92#389—28no—
93#389—28no—
94#389—28no—
95#389—22no—
96#389—22no—
97#389D414yes—
98#389Q814yes—
99#384—22no—
100#384—22no—
101#384D414yes—
102#384D414yes—
103#382—24no—
104#382—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4Q814yes—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C43—22no—
121C43—22no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—24no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C2×Q814yes—
156C22×C8C2×Q814yes—
157#125—48no—
158#125—48no—
159#125—48no—
160#125—48no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
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—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×Q812yes—
251C2×C42C2×Q812yes—
252C2×C42C2×Q812yes—
253C2×C42C2×Q812yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257#107—48no—
258#106—48no—
259C4⋊C8—28no—
260C4⋊C8—28no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271C24C22×D411yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
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—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4#12012yes—
343C22×C4#12012yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4C4⋊Q812yes—
351C22×C4C4⋊Q812yes—
352C22×C4#13512yes—
353C22×C4#13512yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4#13712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×Q811yescharacteristic
360C22×C4C22×Q811yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369C22×C42−1+411yescharacteristic
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—42no—
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—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—416no—
402C4⋊C4—416no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4C4⋊Q814yes—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—44no—
425C42—44no—
426C42—44no—
427C42—44no—
428C42—48no—
429C42—84no—
430C42C22⋊Q814yes—
431C42C22⋊Q814yes—
432C42C4⋊Q814yes—
433C42C4⋊Q814yes—
434C23C2×#12011yescharacteristic
435C23C2×#12111yescharacteristic
436C23C2×C22⋊Q811yescharacteristic
437C23C2×C22⋊Q811yescharacteristic
438C23C2×C4⋊Q811yescharacteristic
439C23#53511yescharacteristic
440C23#53611yescharacteristic
441C23C2×#13511yescharacteristic
442C23C2×#13511yescharacteristic
443C23C2×AGL(1,ℤ/8)11yescharacteristic
444C23C2×#13711yescharacteristic
445C23#57511yescharacteristic
446C23#57511yescharacteristic
447C23#57511yescharacteristic
448C23#57511yescharacteristic
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#39514yes—
526C2×C4#39514yes—
527C2×C4#44914yes—
528C2×C4#45214yes—
529C2×C4#47814yes—
530C2×C4#47914yes—
531C2×C4#49914yes—
532C2×C4#50114yes—
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×C4⋊Q811yescharacteristic
536C2×C4C2×C4⋊Q811yescharacteristic
537C2×C4C2×C4⋊Q811yescharacteristic
538C2×C4#53211yescharacteristic
539C2×C4#53311yescharacteristic
540C2×C4#53311yescharacteristic
541C2×C4#54211yescharacteristic
542C2×C4#54211yescharacteristic
543C2×C4#54411yescharacteristic
544C2×C4#54411yescharacteristic
545C2×C4D4211yescharacteristic
546C2×C4D4211yescharacteristic
547C2×C4#54611yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54811yescharacteristic
550C2×C4#54811yescharacteristic
551C2×C4D4×Q811yescharacteristic
552C2×C4D4×Q811yescharacteristic
553C2×C4D4×Q811yescharacteristic
554C2×C4D4×Q811yescharacteristic
555C2×C4#55411yescharacteristic
556C2×C4#55411yescharacteristic
557C2×C4#55711yescharacteristic
558C2×C4#55711yescharacteristic
559C2×C4Q8211yescharacteristic
560C2×C4Q8211yescharacteristic
561C8—44no—
562C8—44no—
563C22C2×#39511yescharacteristic
564C22#240811yescharacteristic
565C22#240911yescharacteristic
566C22#254611yescharacteristic
567C22#256711yescharacteristic
568C22#262811yescharacteristic
569C22#285711yescharacteristic
570C22C2×#44911yescharacteristic
571C22C2×#45211yescharacteristic
572C22#300511yescharacteristic
573C22#300511yescharacteristic
574C22#303511yescharacteristic
575C22#303511yescharacteristic
576C22#303611yescharacteristic
577C22#303711yescharacteristic
578C22C2×#47811yescharacteristic
579C22C2×#47911yescharacteristic
580C22#307611yescharacteristic
581C22#307611yescharacteristic
582C22#307711yescharacteristic
583C22#307811yescharacteristic
584C22#308011yescharacteristic
585C22#308011yescharacteristic
586C22C2×#49911yescharacteristic
587C22C2×#50111yescharacteristic
588C22#316611yescharacteristic
589C22#316611yescharacteristic
590C22#321211yescharacteristic
591C22#321211yescharacteristic
592C22#321411yescharacteristic
593C22#321611yescharacteristic
594C22#323311yescharacteristic
595C22#323311yescharacteristic
596C22#323511yescharacteristic
597C22#323711yescharacteristic
598C4—24no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—84no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4#205918yescomplemented
610C4#245111yescharacteristic
611C4#245111yescharacteristic
612C4#245211yescharacteristic
613C4#245211yescharacteristic
614C4#255011yescharacteristic
615C4#255011yescharacteristic
616C4#255111yescharacteristic
617C4#255111yescharacteristic
618C2#1512811yescharacteristic
619C2C2×#205911yescharacteristic
620C2#2355311yescharacteristic
621C2#2421011yescharacteristic
622C2#2421311yescharacteristic
623C2#2430011yescharacteristic
624C2#2430311yescharacteristic
625C2#2869111yescharacteristic
626C2#2869311yescharacteristic
627C2#2921011yescharacteristic
628C2#2921211yescharacteristic
629C2#2998611yescharacteristic
630C2#2998811yescharacteristic
631C2#3319911yescharacteristic
632C2#3320011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
609. C4#2059semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#20592
C2×#19482
C2×#19472
C4×#3892
#113852
#113252
#112082
#108492
#108322