Finite Group Databasebeta

← #416097

Subgroups — #416097

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4C4×#389C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11327C212yesmaximal
7#11220C212yesmaximal
8#11219C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10849C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2859—24no—
18#2859—24no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1948—24no—
33#1947—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—28no—
50C2×#125—44no—
51C2×#125—44no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—42no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8Q814yes—
86C2×C4⋊C8Q814yes—
87#389—28no—
88#389—28no—
89#389—24no—
90#389—24no—
91#389—24no—
92#389—24no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#384—22no—
98#384—22no—
99#384D414yes—
100#384D414yes—
101#382—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—44no—
106C4×C4⋊C4Q814yes—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C43—22no—
118C43—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—42no—
126#336—42no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
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×C8C2×Q814yes—
153C22×C8C2×Q814yes—
154#125—432no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4#4112yes—
212C2×C4⋊C4#4112yes—
213C2×C4⋊C4#4112yes—
214C2×C4⋊C4#4112yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×Q812yes—
248C2×C42C2×Q812yes—
249C2×C42C2×Q812yes—
250C2×C42#4112yes—
251#107—48no—
252#106—48no—
253C4⋊C8—28no—
254C4⋊C8—28no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265C24C22×D411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
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—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4#12012yes—
337C22×C4#12012yes—
338C22×C4C22⋊Q812yes—
339C22×C4C22⋊Q812yes—
340C22×C4C4⋊Q812yes—
341C22×C4C4⋊Q812yes—
342C22×C4#13512yes—
343C22×C4#13512yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4#13712yes—
348C22×C4#13712yes—
349C22×C4#13712yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4#12514yescomplemented
410C42—28no—
411C42—28no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—44no—
421C42—44no—
422C42—48no—
423C42—84no—
424C42C22⋊Q814yes—
425C42C22⋊Q814yes—
426C42C4⋊Q814yes—
427C42C4⋊Q814yes—
428C23C2×#12011yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#52511yescharacteristic
431C23#52511yescharacteristic
432C23C2×C4⋊Q811yescharacteristic
433C23#53711yescharacteristic
434C23#53711yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×AGL(1,ℤ/8)11yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23C2×AGL(1,ℤ/8)11yescharacteristic
440C23C2×#13711yescharacteristic
441C23C2×#13711yescharacteristic
442C23C2×#13711yescharacteristic
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
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—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
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—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4#39514yes—
520C2×C4#39514yes—
521C2×C4#44914yes—
522C2×C4#45214yes—
523C2×C4#47814yes—
524C2×C4#47914yes—
525C2×C4#49914yes—
526C2×C4#50114yes—
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×#12511yescharacteristic
530C2×C4#52911yescharacteristic
531C2×C4#52911yescharacteristic
532C2×C4C2×C4⋊Q811yescharacteristic
533C2×C4C2×C4⋊Q811yescharacteristic
534C2×C4#53311yescharacteristic
535C2×C4#54011yescharacteristic
536C2×C4#54111yescharacteristic
537C2×C4#54411yescharacteristic
538C2×C4#54411yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54611yescharacteristic
541C2×C4#54711yescharacteristic
542C2×C4#54711yescharacteristic
543C2×C4#54811yescharacteristic
544C2×C4#54811yescharacteristic
545C2×C4#55011yescharacteristic
546C2×C4#55011yescharacteristic
547C2×C4#55411yescharacteristic
548C2×C4#55411yescharacteristic
549C2×C4#55411yescharacteristic
550C2×C4#55411yescharacteristic
551C2×C4#55711yescharacteristic
552C2×C4#55711yescharacteristic
553C2×C4#55711yescharacteristic
554C2×C4#55711yescharacteristic
555C8—44no—
556C8—44no—
557C22C2×#39511yescharacteristic
558C22#240711yescharacteristic
559C22#240711yescharacteristic
560C22#258411yescharacteristic
561C22#258411yescharacteristic
562C22#262911yescharacteristic
563C22#285111yescharacteristic
564C22C2×#44911yescharacteristic
565C22C2×#45211yescharacteristic
566C22#300311yescharacteristic
567C22#300411yescharacteristic
568C22C2×#47811yescharacteristic
569C22C2×#47911yescharacteristic
570C22#307911yescharacteristic
571C22#307911yescharacteristic
572C22#310611yescharacteristic
573C22#310611yescharacteristic
574C22#310711yescharacteristic
575C22#310711yescharacteristic
576C22#310911yescharacteristic
577C22#311011yescharacteristic
578C22#311211yescharacteristic
579C22#311311yescharacteristic
580C22C2×#49911yescharacteristic
581C22C2×#50111yescharacteristic
582C22#316511yescharacteristic
583C22#316511yescharacteristic
584C22#324711yescharacteristic
585C22#324711yescharacteristic
586C22#324811yescharacteristic
587C22#324811yescharacteristic
588C22#325311yescharacteristic
589C22#325411yescharacteristic
590C22#325411yescharacteristic
591C22#325511yescharacteristic
592C4—24no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—82no—
600C4—84no—
601C4—82no—
602C4—84no—
603C4#205918yes—
604C4#245311yescharacteristic
605C4#245311yescharacteristic
606C4#245411yescharacteristic
607C4#245411yescharacteristic
608C4#255011yescharacteristic
609C4#255011yescharacteristic
610C4#255211yescharacteristic
611C4#255211yescharacteristic
612C2#1521811yescharacteristic
613C2C2×#205911yescharacteristic
614C2#2355211yescharacteristic
615C2#2415211yescharacteristic
616C2#2415311yescharacteristic
617C2#2415611yescharacteristic
618C2#2415711yescharacteristic
619C2#2931411yescharacteristic
620C2#2931411yescharacteristic
621C2#2931911yescharacteristic
622C2#2932011yescharacteristic
623C2#2999511yescharacteristic
624C2#2999711yescharacteristic
625C2#3307211yescharacteristic
626C2#3307311yescharacteristic
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
409. C4⋊C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#19482
C2×#19472
C4×#3892
#113282
#113272
#112202
#112192
#108492
#108322