Finite Group Databasebeta

← #451779

Subgroups — #451779

Order 512.

1460 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
1#13872C211yesmaximal, characteristic
2#13872C211yesmaximal, characteristic
3#11316C212yesmaximal, complemented
4#11315C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10964C211yesmaximal, characteristic
8#10730C212yesmaximal, complemented
9#10730C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C2×#386—21no—
16C2×#386—21no—
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1441—21no—
32#1441—21no—
33#1439—21no—
34#1439—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78#386—44no—
79#386—44no—
80#380—24no—
81#380—24no—
82#380—24no—
83#380—24no—
84#380—42no—
85#380—42no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—41no—
103#336—41no—
104C25C2×D411yescharacteristic
105C25C2×D411yescharacteristic
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—41no—
111C22×D4—41no—
112C22×D4SD1614yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4#4114yes—
146C23×C4#4114yes—
147C23×C4C2411yescharacteristic
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—22no—
161C22×C8—22no—
162C4×D4—48no—
163C4×D4—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
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—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4#4114yes—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C22⋊C8—48no—
224C22⋊C8—48no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—42no—
230#95—44no—
231#95—82no—
232#95—82no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24#12312yes—
250C24#12312yes—
251C24C2×SD1612yes—
252C24C2×SD1612yes—
253C24AGL(1,ℤ/8)12yes—
254C24#13712yes—
255C24C22×D411yescharacteristic
256C2×D4—28no—
257C2×D4—28no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—48no—
317C22×C4—48no—
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—88no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4#12312yes—
353C22×C4#12312yes—
354C22×C4C2×SD1612yes—
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
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—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—42no—
403C42—42no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23#48114yes—
449C23#48314yes—
450C23C2×#12311yescharacteristic
451C23C2×#12311yescharacteristic
452C23#53411yescharacteristic
453C23#54612yes—
454C23#54612yes—
455C23C22×SD1611yescharacteristic
456C23C2×AGL(1,ℤ/8)11yescharacteristic
457C23C2×#13711yescharacteristic
458C23#57511yescharacteristic
459C23#57612yes—
460C23#57812yes—
461D4—416no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—816no—
499C2×C4—816no—
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—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#52912yes—
523C2×C4#52912yes—
524C2×C4#53812yes—
525C2×C4#53812yes—
526C2×C4#53812yes—
527C2×C4#53812yes—
528C2×C4#55111yescharacteristic
529C2×C4#55111yescharacteristic
530C2×C4#55112yes—
531C2×C4#55112yes—
532C2×C4#55312yes—
533C2×C4#55312yes—
534C2×C4#55312yes—
535C2×C4#55312yes—
536C2×C4#56011yescharacteristic
537C2×C4#56011yescharacteristic
538C8—82no—
539C8—82no—
540C22—28no—
541C22—28no—
542C22—24no—
543C22—24no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22#244412yes—
563C22#244412yes—
564C22#248311yescharacteristic
565C22#269711yescharacteristic
566C22#269711yescharacteristic
567C22C2×#48311yescharacteristic
568C22C2×#48111yescharacteristic
569C22#309411yescharacteristic
570C22#309511yescharacteristic
571C22#309812yes—
572C22#309812yes—
573C22#318711yescharacteristic
574C22#319311yescharacteristic
575C22#319911yescharacteristic
576C22#320211yescharacteristic
577C22#329512yes—
578C22#329812yes—
579C22#330812yes—
580C22#331212yes—
581C22#331412yes—
582C22#331812yes—
583C22#332212yes—
584C22#332712yes—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—84no—
589C4—84no—
590C4—81no—
591C4—162no—
592C4—162no—
593C4#257812yes—
594C4#257812yes—
595C4#277812yes—
596C4#277812yes—
597C2—44no—
598C2—44no—
599C2#1601411yescharacteristic
600C2#2620012yes—
601C2#2620212yes—
602C2#2621512yes—
603C2#2622112yes—
604C2#2760111yescharacteristic
605C2#2760811yescharacteristic
606C2#3166711yescharacteristic
607C2#3166811yescharacteristic
608C2#3168011yescharacteristic
609C2#3168311yescharacteristic
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
3. #11316C2semidirect
4. #11315C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
8. #10730C2semidirect
9. #10730C2semidirect

Maximal subgroups

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

subgroupindex
#138722
#113162
#113152
#112002
#111992
#109642
#107302