Finite Group Databasebeta

← #450874

Subgroups — #450874

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#11267C212yesmaximal, complemented
6#11220C212yesmaximal, complemented
7#11200C211yesmaximal, characteristic
8#11147C211yesmaximal, characteristic
9#11122C212yesmaximal, complemented
10#10970C211yesmaximal, characteristic
11#10789C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2423—24no—
21#2423—24no—
22C2×#399—21no—
23C2×#398—21no—
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#387—21no—
28C2×#384C2211yescharacteristic
29C2×#381—21no—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1872—24no—
35#1853—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8D414yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—24no—
59C2×C4×Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—42no—
96#399—28no—
97#398—28no—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#388—28no—
103#387—28no—
104#384—22no—
105#384—22no—
106#381—44no—
107#381—44no—
108#380—22no—
109#380—42no—
110#380D414yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#340—42no—
129#336—24no—
130#336—42no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D412yes—
139C22×Q8#4114yes—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
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×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8#4114yes—
181C22⋊Q8—28no—
182C22⋊Q8—44no—
183C22⋊Q8—48no—
184C4×Q8—48no—
185C4×Q8—44no—
186C4×Q8—44no—
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—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
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—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4SD1614yes—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—44no—
250#106—48no—
251#103—48no—
252#103—48no—
253C22⋊C8—28no—
254C22⋊C8—44no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—42no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24#12012yes—
275C24C2×SD1612yes—
276C24#13712yes—
277C24C22×D411yescharacteristic
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8#12314yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
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—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4#12012yes—
373C22×C4C2×SD1612yes—
374C22×C4C2×SD1612yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C22⋊C4—28no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C42—48no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—42no—
431C42—42no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23#44814yes—
455C23C2×#12011yescharacteristic
456C23#52511yescharacteristic
457C23#53511yescharacteristic
458C23#54612yes—
459C23#54712yes—
460C23C22×SD1611yescharacteristic
461C23C2×AGL(1,ℤ/8)11yescharacteristic
462C23C2×#13711yescharacteristic
463C23#57511yescharacteristic
464C23#57712yes—
465C23#57812yes—
466Q8—28no—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—816no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4C2×#12311yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#52912yes—
537C2×C4#53411yescharacteristic
538C2×C4#53611yescharacteristic
539C2×C4#53912yes—
540C2×C4#54012yes—
541C2×C4#54212yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54612yes—
545C2×C4#54712yes—
546C2×C4#55112yes—
547C2×C4#55212yes—
548C2×C4#55312yes—
549C2×C4#56212yes—
550C8—44no—
551C8—82no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22#240411yescharacteristic
564C22#240511yescharacteristic
565C22#240611yescharacteristic
566C22#247012yes—
567C22#270612yes—
568C22C2×#44811yescharacteristic
569C22#300611yescharacteristic
570C22#300911yescharacteristic
571C22#301111yescharacteristic
572C22#310111yescharacteristic
573C22#311011yescharacteristic
574C22#312112yes—
575C22#320811yescharacteristic
576C22#321511yescharacteristic
577C22#322512yes—
578C22#329812yes—
579C22#331212yes—
580C22#331712yes—
581C22#333212yes—
582C22#333512yes—
583C22#334312yes—
584C22#334412yes—
585C22#335212yes—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—44no—
590C4—42no—
591C4—84no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4—82no—
596C4—162no—
597C4#241912yes—
598C4#245712yes—
599C4#268712yes—
600C4#270112yes—
601C2—44no—
602C2#1598211yescharacteristic
603C2#2393311yescharacteristic
604C2#2394611yescharacteristic
605C2#2404111yescharacteristic
606C2#2404511yescharacteristic
607C2#2409711yescharacteristic
608C2#2411011yescharacteristic
609C2#2716012yes—
610C2#2718112yes—
611C2#3193212yes—
612C2#3195112yes—
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
5. #11267C2semidirect
6. #11220C2semidirect
9. #11122C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#18722
C2×#18532
#139042
#112672
#112202
#112002
#111472
#111222
#109702
#107892