Finite Group Databasebeta

← #430471

Subgroups — #430471

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1935C212yesmaximal, complemented
2C2×#1872C212yesmaximal
3#15178C211yesmaximal, characteristic
4#14130C211yesmaximal, characteristic
5#11315C212yesmaximal, complemented
6#11067C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10970C212yesmaximal
9#10922C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#385—21no—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1935—28no—
25#1872—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1444—21no—
33#1444—21no—
34#1444—21no—
35#1442—21no—
36#1439—21no—
37#1439—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C22⋊#95—21no—
41C23×Q8C2312yes—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8—42no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—42no—
89C2×#97—24no—
90C2×#97—41no—
91#391—24no—
92#391—42no—
93#389—24no—
94#389—42no—
95#385—44no—
96#380—24no—
97C4×C22⋊C4—41no—
98C4×C22⋊C4—41no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109#340—24no—
110#340—24no—
111#340—42no—
112#340D418yes—
113#336—42no—
114#336—41no—
115C25C2×D411yescharacteristic
116C22×Q8—22no—
117C22×Q8—22no—
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#4114yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4114yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C4⋊Q8—44no—
168C4⋊Q8—48no—
169C4⋊Q8—44no—
170C22⋊Q8—216no—
171C22⋊Q8—48no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227#103—216no—
228#103—48no—
229#103—48no—
230#103—48no—
231C22⋊C8—48no—
232#97—44no—
233#95—24no—
234#95—41no—
235#95—41no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—44no—
241#95—42no—
242#95—82no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—21no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24#12412yes—
252C24#13712yes—
253C24#13712yes—
254C24C22×D411yescharacteristic
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
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—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—48no—
326C22×C4—42no—
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—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—88no—
351C22×C4—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4#12412yes—
355C22×C4AGL(1,ℤ/8)12yes—
356C22×C4#13712yes—
357C22×C4C22×D412yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)11yescharacteristic
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—416no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—48no—
399C22⋊C4—416no—
400C42—42no—
401C42—42no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23#49114yes—
429C23#52511yescharacteristic
430C23C2×#12411yescharacteristic
431C23#53411yescharacteristic
432C23D4212yes—
433C23#54612yes—
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23C2×#13711yescharacteristic
436C23C2×#13711yescharacteristic
437C23C2×#13711yescharacteristic
438C23#57712yes—
439C23#57812yes—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—82no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—816no—
490C2×C4—82no—
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—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#52512yes—
507C2×C4#52512yes—
508C2×C4#53512yes—
509C2×C4#53812yes—
510C2×C4#53812yes—
511C2×C4#53812yes—
512C2×C4D4211yescharacteristic
513C2×C4#54612yes—
514C2×C4#55012yes—
515C2×C4#55111yescharacteristic
516C2×C4#55112yes—
517C2×C4#55312yes—
518C2×C4#55312yes—
519C2×C4#55312yes—
520C2×C4#56011yescharacteristic
521C2×C4#56011yescharacteristic
522C8—44no—
523C8—82no—
524C22—28no—
525C22—24no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22#245612yes—
536C22#247212yes—
537C22#259111yescharacteristic
538C22#266911yescharacteristic
539C22#269711yescharacteristic
540C22#311011yescharacteristic
541C22#311311yescharacteristic
542C22#312012yes—
543C22C2×#49111yescharacteristic
544C22#313712yes—
545C22#314111yescharacteristic
546C22#319311yescharacteristic
547C22#319511yescharacteristic
548C22#320011yescharacteristic
549C22#320211yescharacteristic
550C22#326612yes—
551C22#327012yes—
552C22#327212yes—
553C22#327612yes—
554C22#330812yes—
555C22#331212yes—
556C22#331612yes—
557C22#332012yes—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—84no—
562C4—81no—
563C4—84no—
564C4—81no—
565C4—84no—
566C4—84no—
567C4#245612yes—
568C4#259812yes—
569C4#271412yes—
570C4#277812yes—
571C2—44no—
572C2#1542011yescharacteristic
573C2#2663612yes—
574C2#2664112yes—
575C2#2726812yes—
576C2#2727812yes—
577C2#2933511yescharacteristic
578C2#2934411yescharacteristic
579C2#3051011yescharacteristic
580C2#3051711yescharacteristic
581C2#3168211yescharacteristic
582C2#3168311yescharacteristic
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×#1935C2semidirect
5. #11315C2semidirect
6. #11067C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19352
C2×#18722
#151782
#141302
#113152
#110672
#110102
#109702
#109222