Finite Group Databasebeta

← #468057

Subgroups — #468057

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1880C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11208C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#11051C212yesmaximal
10#11023C212yesmaximal
11#10789C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19#2568—24no—
20#2568—24no—
21C2×#400—22no—
22C2×#399—21no—
23C2×#391—21no—
24C2×#391—21no—
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#382C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2023—24no—
32#1880—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—42no—
51C22×SD16—42no—
52C22×SD16Q814yescomplemented
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8D414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90#400—216no—
91#399—44no—
92#391—28no—
93#391—28no—
94#391—24no—
95#391—24no—
96#390—24no—
97#390—22no—
98#390—22no—
99#390—42no—
100#389—24no—
101#389—42no—
102#382—24no—
103#382—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—22no—
115#340—24no—
116#340—42no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—22no—
121#336—42no—
122#336—42no—
123C25—21no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8C2×D412yes—
137C22×Q8C2×Q812yes—
138C22×Q8#4112yes—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×Q812yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×SD16—28no—
160C2×SD16—48no—
161C2×SD16—44no—
162C2×SD16—44no—
163C2×SD16—48no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8C2×Q812yes—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—48no—
187C2×C4⋊C4—22no—
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—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242#103—48no—
243#103—44no—
244#102—44no—
245#102—48no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—44no—
252#95—42no—
253#95—81no—
254C24—22no—
255C24—41no—
256C24—42no—
257C24—42no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—42no—
262C24—41no—
263C24—42no—
264C24—41no—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—48no—
286C2×Q8—88no—
287C2×Q8—88no—
288C2×Q8C22⋊Q814yes—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
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—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4#13512yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C42−1+411yescharacteristic
369C22×C42−1+411yescharacteristic
370SD16—416no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C22⋊C4—84no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C22⋊C4—84no—
411C42—22no—
412C42—22no—
413C42—24no—
414C42—48no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—44no—
424C23—81no—
425C23—81no—
426C23—82no—
427C23—81no—
428C23—81no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23C2×#12011yescharacteristic
443C23#53512yes—
444C23#54612yes—
445C23#54811yescharacteristic
446C23D4×Q811yescharacteristic
447C23C2×#13511yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449C23#57512yes—
450C23#57711yescharacteristic
451C23#57712yes—
452C23#57811yescharacteristic
453Q8—24no—
454Q8—24no—
455Q8—42no—
456Q8—42no—
457Q8—44no—
458Q8—44no—
459Q8—816no—
460D4—42no—
461D4—42no—
462D4—44no—
463D4—81no—
464D4—81no—
465D4—82no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
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—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#44914yes—
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4#53211yescharacteristic
521C2×C4#53411yescharacteristic
522C2×C4#53411yescharacteristic
523C2×C4#53412yes—
524C2×C4#53511yescharacteristic
525C2×C4#53611yescharacteristic
526C2×C4#53812yes—
527C2×C4#53912yes—
528C2×C4#54212yes—
529C2×C4#54312yes—
530C2×C4#54811yescharacteristic
531C2×C4D4×Q811yescharacteristic
532C2×C4#55012yes—
533C2×C4#55112yes—
534C2×C4#55412yes—
535C2×C4#56212yes—
536C2×C4#56212yes—
537C8—42no—
538C8—81no—
539C8—82no—
540C22—82no—
541C22—82no—
542C22—84no—
543C22—84no—
544C22—82no—
545C22—162no—
546C22—162no—
547C22—161no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22#240911yescharacteristic
552C22#243111yescharacteristic
553C22#261311yescharacteristic
554C22#269212yes—
555C22#269612yes—
556C22C2×#44911yescharacteristic
557C22#300511yescharacteristic
558C22#301011yescharacteristic
559C22#301311yescharacteristic
560C22#321112yes—
561C22#321312yes—
562C22#322112yes—
563C22#322512yes—
564C22#330412yes—
565C22#331112yes—
566C22#331312yes—
567C22#332412yes—
568C22#336411yescharacteristic
569C22#336811yescharacteristic
570C22#337112yes—
571C22#338111yescharacteristic
572C22#338611yescharacteristic
573C22#338812yes—
574C4—24no—
575C4—42no—
576C4—44no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—161no—
583C4#256411yescharacteristic
584C4#262611yescharacteristic
585C4#266311yescharacteristic
586C4#273012yes—
587C4#273212yes—
588C4#281611yescharacteristic
589C2—162no—
590C2#1660011yescharacteristic
591C2#2432811yescharacteristic
592C2#2434011yescharacteristic
593C2#2557411yescharacteristic
594C2#2558011yescharacteristic
595C2#2962311yescharacteristic
596C2#2963111yescharacteristic
597C2#3153112yes—
598C2#3154312yes—
599C2#3164712yes—
600C2#3166112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
1. C2×#2568C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11208C2semidirect
8. #11067C2semidirect
11. #10789C2semidirect
52. C22×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#20232
C2×#18802
#151482
#113842
#113462
#112082
#110672
#110512
#110232
#107892