Finite Group Databasebeta

← #460088

Subgroups — #460088

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2024C212yesmaximal, complemented
2C2×#1894C212yesmaximal
3#15178C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11315C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10970C212yesmaximal
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#393—21no—
20C2×#391C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#2024—28no—
24#1894—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1446—21no—
32#1444—21no—
33#1444—21no—
34#1442—21no—
35#1439—21no—
36#1439—21no—
37#1439—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×Q8C2312yes—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—24no—
46C22×M16—42no—
47C22×M16—41no—
48C22×M16—41no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—41no—
54C2×C4⋊Q8—41no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8—41no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—41no—
93#393—44no—
94#391—24no—
95#391—42no—
96#380—24no—
97#380—42no—
98C4×C22⋊C4—41no—
99C4×C22⋊C4—41no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—24no—
112#340—24no—
113#340—42no—
114#340D418yes—
115#336—24no—
116#336—42no—
117C25C2×D411yescharacteristic
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—82no—
136C22×Q8#4114yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C2×M16—44no—
158C2×M16—48no—
159C2×M16—48no—
160C2×M16—44no—
161C2×M16—44no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×D414yes—
176C4⋊Q8—44no—
177C4⋊Q8—48no—
178C4⋊Q8—44no—
179C4⋊Q8—48no—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C22⋊Q8—216no—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C2×C4⋊C4—22no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—82no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233#103—48no—
234#103—48no—
235#103—48no—
236#103—48no—
237C22⋊C8—44no—
238#95—41no—
239#95—41no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24#12412yes—
260C24C22×D411yescharacteristic
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—48no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—88no—
291C2×Q8—84no—
292C2×Q8—88no—
293C2×Q8—84no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
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—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12412yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4#13712yes—
364C22×C4#13712yes—
365C22×C4C22×D412yes—
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+411yescharacteristic
375M16—416no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—416no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—416no—
409C22⋊C4—88no—
410C42—42no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23C2×#12411yescharacteristic
440C23#53411yescharacteristic
441C23#53711yescharacteristic
442C23D4212yes—
443C23#54612yes—
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×#13711yescharacteristic
446C23C2×#13711yescharacteristic
447C23C2×#13711yescharacteristic
448C23#57712yes—
449C23#57812yes—
450Q8—44no—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—816no—
455Q8—82no—
456Q8—82no—
457Q8—82no—
458Q8—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
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—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—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#52512yes—
519C2×C4#53512yes—
520C2×C4#53512yes—
521C2×C4#53812yes—
522C2×C4#53811yescharacteristic
523C2×C4#53812yes—
524C2×C4#53812yes—
525C2×C4#53811yescharacteristic
526C2×C4#53911yescharacteristic
527C2×C4#53911yescharacteristic
528C2×C4D4211yescharacteristic
529C2×C4#54612yes—
530C2×C4#54812yes—
531C2×C4#55112yes—
532C2×C4#55111yescharacteristic
533C2×C4#56011yescharacteristic
534C2×C4H(3,𝔽4)11yescharacteristic
535C2×C4#56212yes—
536C8—44no—
537C8—82no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—82no—
548C22—82no—
549C22#247212yes—
550C22#263911yescharacteristic
551C22#266911yescharacteristic
552C22#267612yes—
553C22#268411yescharacteristic
554C22#313712yes—
555C22#313911yescharacteristic
556C22#314111yescharacteristic
557C22#319311yescharacteristic
558C22#319511yescharacteristic
559C22#320011yescharacteristic
560C22#320211yescharacteristic
561C22#325411yescharacteristic
562C22#325511yescharacteristic
563C22#325812yes—
564C22#326612yes—
565C22#327012yes—
566C22#327212yes—
567C22#327612yes—
568C22#330812yes—
569C22#331612yes—
570C22#332812yes—
571C22#332912yes—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—84no—
578C4—84no—
579C4—84no—
580C4—162no—
581C4#246311yescharacteristic
582C4#263112yes—
583C4#267011yescharacteristic
584C4#269612yes—
585C4#276011yescharacteristic
586C4#282411yescharacteristic
587C2—82no—
588C2#1635711yescharacteristic
589C2#2727712yes—
590C2#2727812yes—
591C2#3019311yescharacteristic
592C2#3019911yescharacteristic
593C2#3051011yescharacteristic
594C2#3051711yescharacteristic
595C2#3092912yes—
596C2#3093712yes—
597C2#3127211yescharacteristic
598C2#3127311yescharacteristic
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
1. C2×#2024C2semidirect
5. #11315C2semidirect
6. #11190C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20242
C2×#18942
#151782
#151252
#113152
#111902
#110102
#109702
#107152