Finite Group Databasebeta

← #469987

Subgroups — #469987

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4#15195C211yesmaximal, characteristic
5#11325C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11096C212yesmaximal
9#11095C212yesmaximal
10#11042C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2494—24no—
17#2494—24no—
18C2×#401—21no—
19C2×#397—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2084—24no—
32#1947—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1442—21no—
42#1442—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—42no—
70C2×#107—42no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#106—42no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79#401—28no—
80#397—28no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389D414yes—
85#388—28no—
86#387—28no—
87#384—22no—
88#384—22no—
89#382—22no—
90#382—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—42no—
96#380—42no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#336—24no—
122#336—22no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
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×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8#4114yes—
159C22×C8#4114yes—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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—24no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—82no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225#107—48no—
226#106—48no—
227C4⋊C8—28no—
228C22⋊C8—48no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—42no—
239#95—44no—
240#95—42no—
241#95—82no—
242#95—82no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24#13712yes—
254C24C22×D411yescharacteristic
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4#12012yes—
331C22×C4#12012yes—
332C22×C4#13512yes—
333C22×C4#13712yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+412yes—
342C22×C42−1+412yes—
343C22×C42−1+411yescharacteristic
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C22⋊C4—44no—
385C22⋊C4—48no—
386C22⋊C4—44no—
387C22⋊C4—84no—
388C22⋊C4—88no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—84no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23C2×#12011yescharacteristic
420C23#53511yescharacteristic
421C23#53711yescharacteristic
422C23#54712yes—
423C23#54712yes—
424C23C2×#13511yescharacteristic
425C23C2×#13711yescharacteristic
426C23C2×#13711yescharacteristic
427C23#57511yescharacteristic
428C23#57612yes—
429C23#57712yes—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#45214yes—
497C2×C4#52511yescharacteristic
498C2×C4#53211yescharacteristic
499C2×C4#53511yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53711yescharacteristic
504C2×C4#53711yescharacteristic
505C2×C4#53912yes—
506C2×C4#54112yes—
507C2×C4#54112yes—
508C2×C4#54212yes—
509C2×C4#54712yes—
510C2×C4#54812yes—
511C2×C4#55112yes—
512C2×C4#55312yes—
513C2×C4#56312yes—
514C2×C4#56312yes—
515C8—44no—
516C8—44no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—44no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—82no—
525C22—81no—
526C22—82no—
527C22—81no—
528C22#240611yescharacteristic
529C22#243211yescharacteristic
530C22#261511yescharacteristic
531C22#271812yes—
532C22#272212yes—
533C22C2×#45211yescharacteristic
534C22#300711yescharacteristic
535C22#300811yescharacteristic
536C22#301011yescharacteristic
537C22#321211yescharacteristic
538C22#321811yescharacteristic
539C22#322412yes—
540C22#324811yescharacteristic
541C22#325211yescharacteristic
542C22#325712yes—
543C22#333912yes—
544C22#334012yes—
545C22#334412yes—
546C22#334612yes—
547C22#334612yes—
548C22#334712yes—
549C22#335112yes—
550C22#335612yes—
551C4—24no—
552C4—42no—
553C4—82no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—162no—
561C4—162no—
562C4#260311yescharacteristic
563C4#263711yescharacteristic
564C4#266411yescharacteristic
565C4#272312yes—
566C4#272512yes—
567C4#282111yescharacteristic
568C2—82no—
569C2#1667311yescharacteristic
570C2#2409211yescharacteristic
571C2#2411111yescharacteristic
572C2#2570111yescharacteristic
573C2#2571311yescharacteristic
574C2#2966511yescharacteristic
575C2#2966911yescharacteristic
576C2#3229312yes—
577C2#3230212yes—
578C2#3241512yes—
579C2#3241612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#1947C2semidirect
5. #11325C2semidirect
6. #11292C2semidirect
7. #11227C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#20842
C2×#19472
#151952
#113252
#112922
#112272
#110962
#110952
#110422
#108072