Finite Group Databasebeta

← #470295

Subgroups — #470295

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1921C211yesmaximal, characteristic, complemented
4#15181C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19#2547—24no—
20#2547—24no—
21C2×#397—22no—
22C2×#391—21no—
23C2×#389C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2058—24no—
30#1921—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1442—21no—
42#1439—22no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
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⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
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—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#106—42no—
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#391—28no—
91#389—22no—
92#389—22no—
93#380—24no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—42no—
100#380—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111#340—24no—
112#340—24no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—42no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—22no—
121C25C2×D412yes—
122C22×Q8—24no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—42no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
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—21no—
151C23×C4—22no—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8#4114yes—
172C22×C8#4112yes—
173C4⋊Q8—48no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#124—432no—
177C4×D4—48no—
178C4×D4—44no—
179C4×D4—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
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—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4#4112yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—44no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239#107—28no—
240#106—48no—
241C22⋊C8—48no—
242C22⋊C8—48no—
243#95—28no—
244#95—28no—
245#95—24no—
246#95—44no—
247#95—44no—
248#95—42no—
249#95—42no—
250#95—81no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24AGL(1,ℤ/8)14yes—
266C24C22×D411yescharacteristic
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—48no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×D4—28no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—44no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
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—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—41no—
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—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#13512yes—
355C22×C4#13712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+412yes—
365C22×C42−1+411yescharacteristic
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—416no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C42—24no—
414C42—22no—
415C42—22no—
416C42—48no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
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—
452C23C2×#12011yescharacteristic
453C23#53412yes—
454C23#54611yescharacteristic
455C23#54712yes—
456C23#54711yescharacteristic
457C23C2×#13511yescharacteristic
458C23C2×AGL(1,ℤ/8)12yes—
459C23C2×#13711yescharacteristic
460C23#57712yes—
461C23#57711yescharacteristic
462C23#57811yescharacteristic
463Q8—816no—
464D4—416no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
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—81no—
501C2×C4—81no—
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—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#45214yes—
524C2×C4#52511yescharacteristic
525C2×C4#53211yescharacteristic
526C2×C4#53412yes—
527C2×C4#53511yescharacteristic
528C2×C4#53511yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#53711yescharacteristic
531C2×C4#53812yes—
532C2×C4#54212yes—
533C2×C4#54212yes—
534C2×C4#54212yes—
535C2×C4#54812yes—
536C2×C4#54811yescharacteristic
537C2×C4#55011yescharacteristic
538C2×C4#55112yes—
539C2×C4#55512yes—
540C2×C4#56212yes—
541C2×C4#56312yes—
542C8—42no—
543C8—44no—
544C8—81no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22#240711yescharacteristic
561C22#243211yescharacteristic
562C22#261311yescharacteristic
563C22#268612yes—
564C22#271712yes—
565C22C2×#45211yescharacteristic
566C22#300311yescharacteristic
567C22#301011yescharacteristic
568C22#301311yescharacteristic
569C22#318812yes—
570C22#319312yes—
571C22#320012yes—
572C22#320112yes—
573C22#330411yescharacteristic
574C22#330612yes—
575C22#332811yescharacteristic
576C22#334012yes—
577C22#334211yescharacteristic
578C22#334312yes—
579C22#334512yes—
580C22#334512yes—
581C22#335611yescharacteristic
582C22#335612yes—
583C4—24no—
584C4—42no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—84no—
590C4—162no—
591C4—161no—
592C4#261011yescharacteristic
593C4#263211yescharacteristic
594C4#266211yescharacteristic
595C4#273012yes—
596C4#273112yes—
597C4#282011yescharacteristic
598C2—84no—
599C2#1668111yescharacteristic
600C2#2412311yescharacteristic
601C2#2412811yescharacteristic
602C2#2570111yescharacteristic
603C2#2570911yescharacteristic
604C2#2959811yescharacteristic
605C2#2960311yescharacteristic
606C2#3133112yes—
607C2#3133812yes—
608C2#3226712yes—
609C2#3226912yes—
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×#2058C2semidirect
3. C2×#1921C2semidirect
5. #11384C2semidirect
6. #11199C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
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×#20582
C2×#19212
#151812
#113842
#111992
#110962
#110952
#109642
#108072