Finite Group Databasebeta

← #404866

Subgroups — #404866

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#1880C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11067C212yesmaximal, complemented
8#11023C212yesmaximal
9#10931C212yesmaximal
10#10884C212yesmaximal, complemented
11#10789C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C2×#399—21no—
19#2355—24no—
20#2355—24no—
21C2×#391C2212yes—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#387—22no—
26C2×#384C2211yescharacteristic
27C2×#382—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2071—24no—
32#1880—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C22×SD16—22no—
48C22×SD16—22no—
49C22×SD16—24no—
50C22×SD16—22no—
51C22×SD16—42no—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—22no—
58C2×C4×Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×#103—22no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—42no—
87#399—44no—
88#391—24no—
89#391—24no—
90#390—24no—
91#390—22no—
92#390—22no—
93#390—42no—
94#389—24no—
95#389—42no—
96#387—216no—
97#384—22no—
98#384D414yes—
99#382—28no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—24no—
116#340—42no—
117#336—22no—
118#336—22no—
119#336—42no—
120#336—42no—
121C25—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8C2×D412yes—
131C22×Q8#4114yes—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4#4112yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C2×SD16—28no—
153C2×SD16—48no—
154C2×SD16—48no—
155C2×SD16—44no—
156C2×SD16—44no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8#4112yes—
174C4×Q8—216no—
175C4×Q8—28no—
176C4×Q8—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4SD1614yescomplemented
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4#4114yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—28no—
233C2×C42—82no—
234C2×C42—82no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#103—48no—
238#103—44no—
239#102—48no—
240#102—44no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—24no—
245#95—44no—
246#95—42no—
247#95—81no—
248C24—22no—
249C24—41no—
250C24—42no—
251C24—42no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8#12314yes—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
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—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4C2×SD1614yes—
347C22×C4C2×SD1612yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C42−1+411yescharacteristic
359SD16—416no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—216no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4#12314yes—
397C22⋊C4—88no—
398C22⋊C4—84no—
399C22⋊C4—88no—
400C22⋊C4—84no—
401C42—216no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—88no—
410C23—42no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—82no—
416C23—82no—
417C23—81no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—81no—
422C23—81no—
423C23—82no—
424C23—81no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C2×#12011yescharacteristic
434C23#52512yes—
435C23#54612yes—
436C23#54811yescharacteristic
437C23#55011yescharacteristic
438C23C22×SD1611yescharacteristic
439C23C2×#13711yescharacteristic
440C23#57512yes—
441C23#57612yes—
442C23#57711yescharacteristic
443C23#57811yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—42no—
447Q8—42no—
448Q8—44no—
449Q8—44no—
450D4—42no—
451D4—44no—
452D4—42no—
453D4—81no—
454D4—82no—
455D4—81no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4#44814yes—
509C2×C4C2×#12311yescharacteristic
510C2×C4C2×#12311yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4#52512yes—
513C2×C4#52912yes—
514C2×C4#52912yes—
515C2×C4#53211yescharacteristic
516C2×C4#53511yescharacteristic
517C2×C4#53711yescharacteristic
518C2×C4#54012yes—
519C2×C4#54212yes—
520C2×C4#54611yescharacteristic
521C2×C4#54711yescharacteristic
522C2×C4#55012yes—
523C2×C4#55112yes—
524C2×C4#55112yes—
525C2×C4#55112yes—
526C2×C4#55212yes—
527C8—42no—
528C8—81no—
529C8—82no—
530C22—84no—
531C22—84no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—162no—
536C22—162no—
537C22—161no—
538C22—161no—
539C22—161no—
540C22—161no—
541C22#240711yescharacteristic
542C22#243111yescharacteristic
543C22#247112yes—
544C22#247712yes—
545C22#284911yescharacteristic
546C22C2×#44811yescharacteristic
547C22#300511yescharacteristic
548C22#300811yescharacteristic
549C22#301311yescharacteristic
550C22#310312yes—
551C22#311712yes—
552C22#311912yes—
553C22#312412yes—
554C22#329512yes—
555C22#331112yes—
556C22#331312yes—
557C22#333312yes—
558C22#335911yescharacteristic
559C22#337012yes—
560C22#337211yescharacteristic
561C22#339211yescharacteristic
562C22#339811yescharacteristic
563C22#340312yes—
564C4—24no—
565C4—44no—
566C4—44no—
567C4—42no—
568C4—44no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—161no—
573C4#239511yescharacteristic
574C4#247112yes—
575C4#247912yes—
576C4#248211yescharacteristic
577C4#249811yescharacteristic
578C4#262411yescharacteristic
579C2—162no—
580C2#1490511yescharacteristic
581C2#2418711yescharacteristic
582C2#2419811yescharacteristic
583C2#2556911yescharacteristic
584C2#2558011yescharacteristic
585C2#2719412yes—
586C2#2721212yes—
587C2#2742612yes—
588C2#2744212yes—
589C2#3300511yescharacteristic
590C2#3301211yescharacteristic
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×#2355C2semidirect
5. #11371C2semidirect
6. #11346C2semidirect
7. #11067C2semidirect
10. #10884C2semidirect
11. #10789C2semidirect
206. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#20712
C2×#18802
#151482
#113712
#113462
#110672
#110232
#109312
#108842
#107892