Finite Group Databasebeta

← #470289

Subgroups — #470289

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1919C211yesmaximal, characteristic, complemented
4#15129C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11200C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19#2547—24no—
20#2547—24no—
21C2×#397—22no—
22C2×#394—21no—
23C2×#391—21no—
24C2×#389C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2057—24no—
31#1919—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—21no—
43#1440—22no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×#124—28no—
56C2×#124—48no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—42no—
79C2×#107—42no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89#397—216no—
90#394—44no—
91#391—28no—
92#389—22no—
93#389—22no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—42no—
100#380—42no—
101#380—42no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—22no—
121#336—22no—
122C25C2×D412yes—
123C22×Q8—24no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8#4114yes—
173C22×C8#4112yes—
174C4⋊Q8—48no—
175C4⋊Q8—44no—
176C4⋊Q8—44no—
177#124—432no—
178C4×D4—48no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4112yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—82no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—44no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241#107—48no—
242#106—28no—
243C22⋊C8—48no—
244C22⋊C8—48no—
245#95—28no—
246#95—28no—
247#95—24no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—42no—
252#95—81no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24AGL(1,ℤ/8)14yes—
268C24C22×D411yescharacteristic
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×D4—28no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—44no—
278C2×D4—44no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
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—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—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4#13512yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+412yes—
369C22×C42−1+411yescharacteristic
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—84no—
412C22⋊C4—48no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C42—24no—
418C42—22no—
419C42—22no—
420C42—48no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C2×#12011yescharacteristic
457C23#53412yes—
458C23#54611yescharacteristic
459C23#54711yescharacteristic
460C23#54712yes—
461C23C2×#13511yescharacteristic
462C23C2×AGL(1,ℤ/8)12yes—
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23#57711yescharacteristic
465C23#57812yes—
466C23#57811yescharacteristic
467Q8—816no—
468D4—416no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
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—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#44914yes—
530C2×C4#52511yescharacteristic
531C2×C4#53211yescharacteristic
532C2×C4#53412yes—
533C2×C4#53511yescharacteristic
534C2×C4#53511yescharacteristic
535C2×C4#53611yescharacteristic
536C2×C4#53711yescharacteristic
537C2×C4#53812yes—
538C2×C4#54212yes—
539C2×C4#54212yes—
540C2×C4#54212yes—
541C2×C4#54812yes—
542C2×C4#54811yescharacteristic
543C2×C4#55011yescharacteristic
544C2×C4#55112yes—
545C2×C4#55512yes—
546C2×C4#56212yes—
547C2×C4#56312yes—
548C8—42no—
549C8—44no—
550C8—81no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22#240711yescharacteristic
567C22#243211yescharacteristic
568C22#261311yescharacteristic
569C22#268612yes—
570C22#271712yes—
571C22C2×#44911yescharacteristic
572C22#300311yescharacteristic
573C22#301111yescharacteristic
574C22#301311yescharacteristic
575C22#318812yes—
576C22#319212yes—
577C22#320112yes—
578C22#320212yes—
579C22#330211yescharacteristic
580C22#330812yes—
581C22#332411yescharacteristic
582C22#334011yescharacteristic
583C22#334212yes—
584C22#334312yes—
585C22#334312yes—
586C22#334512yes—
587C22#335311yescharacteristic
588C22#335312yes—
589C4—24no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—84no—
596C4—162no—
597C4—161no—
598C4#261011yescharacteristic
599C4#263211yescharacteristic
600C4#266211yescharacteristic
601C4#273012yes—
602C4#273112yes—
603C4#282011yescharacteristic
604C2—84no—
605C2#1668111yescharacteristic
606C2#2411911yescharacteristic
607C2#2413211yescharacteristic
608C2#2570511yescharacteristic
609C2#2570911yescharacteristic
610C2#2959411yescharacteristic
611C2#2960511yescharacteristic
612C2#3133312yes—
613C2#3134012yes—
614C2#3226712yes—
615C2#3227112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2057C2semidirect
3. C2×#1919C2semidirect
5. #11384C2semidirect
6. #11200C2semidirect
7. #11096C2semidirect
8. #11096C2semidirect
9. #11095C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#20572
C2×#19192
#151292
#113842
#112002
#110962
#110952
#109642
#108072