Finite Group Databasebeta

← #470035

Subgroups — #470035

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic
5#11328C212yesmaximal, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11208C212yesmaximal, complemented
8#11199C212yesmaximal
9#11096C212yesmaximal
10#11042C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2494—24no—
16#2494—24no—
17C2×#401—21no—
18C2×#399—21no—
19C2×#397—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2100—24no—
33#1947—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—42no—
71C2×#107—42no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77#401—28no—
78#399—44no—
79#397—28no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—24no—
84#389—42no—
85#389D414yes—
86#388—28no—
87#387—28no—
88#384—22no—
89#384—22no—
90#382—22no—
91#382—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—24no—
97#380—42no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
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—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#340—42no—
122#336—22no—
123#336—42no—
124C25C2×D411yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
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—22no—
159C22×C8—22no—
160C22×C8#4114yes—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
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—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
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—24no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4#4114yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—82no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230#107—48no—
231C4⋊C8—28no—
232C22⋊C8—48no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95—22no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24AGL(1,ℤ/8)12yes—
257C24C22×D411yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
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—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4#12012yes—
334C22×C4#12012yes—
335C22×C4#13512yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+412yes—
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—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C22⋊C4—44no—
389C22⋊C4—48no—
390C22⋊C4—44no—
391C22⋊C4—84no—
392C22⋊C4—88no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—41no—
397C42—41no—
398C42—42no—
399C42—84no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23C2×#12011yescharacteristic
424C23#53611yescharacteristic
425C23#53711yescharacteristic
426C23#54612yes—
427C23#54712yes—
428C23C2×#13511yescharacteristic
429C23C2×AGL(1,ℤ/8)11yescharacteristic
430C23C2×#13711yescharacteristic
431C23#57511yescharacteristic
432C23#57612yes—
433C23#57812yes—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#45214yes—
501C2×C4#52511yescharacteristic
502C2×C4#53211yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53611yescharacteristic
507C2×C4#53711yescharacteristic
508C2×C4#53711yescharacteristic
509C2×C4#53812yes—
510C2×C4#53912yes—
511C2×C4#53912yes—
512C2×C4#54112yes—
513C2×C4#54712yes—
514C2×C4#55012yes—
515C2×C4#55212yes—
516C2×C4#55612yes—
517C2×C4#56312yes—
518C2×C4#56312yes—
519C8—44no—
520C8—82no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—82no—
530C22—82no—
531C22—81no—
532C22#240611yescharacteristic
533C22#243111yescharacteristic
534C22#261611yescharacteristic
535C22#268912yes—
536C22#269512yes—
537C22C2×#45211yescharacteristic
538C22#300611yescharacteristic
539C22#300811yescharacteristic
540C22#301111yescharacteristic
541C22#323311yescharacteristic
542C22#323811yescharacteristic
543C22#324312yes—
544C22#324711yescharacteristic
545C22#325211yescharacteristic
546C22#326212yes—
547C22#330212yes—
548C22#330912yes—
549C22#331812yes—
550C22#332812yes—
551C22#333912yes—
552C22#334312yes—
553C22#334612yes—
554C22#335212yes—
555C4—24no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—81no—
560C4—84no—
561C4—81no—
562C4—81no—
563C4—82no—
564C4—82no—
565C4—162no—
566C4#260411yescharacteristic
567C4#263611yescharacteristic
568C4#266411yescharacteristic
569C4#272112yes—
570C4#272712yes—
571C4#282111yescharacteristic
572C2—82no—
573C2#1667411yescharacteristic
574C2#2409211yescharacteristic
575C2#2411011yescharacteristic
576C2#2557011yescharacteristic
577C2#2558111yescharacteristic
578C2#2969211yescharacteristic
579C2#2969611yescharacteristic
580C2#3143712yes—
581C2#3144612yes—
582C2#3161712yes—
583C2#3161912yes—
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×#2100C2semidirect
3. C2×#1947C2semidirect
5. #11328C2semidirect
6. #11227C2semidirect
7. #11208C2semidirect
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×#21002
C2×#19472
#151352
#113282
#112272
#112082
#111992
#110962
#110422
#108072