Finite Group Databasebeta

← #452002

Subgroups — #452002

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#11324C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11199C211yesmaximal, characteristic
9#11199C211yesmaximal, characteristic
10#11147C211yesmaximal, characteristic
11#10907C212yesmaximal, complemented
12C22×#126—21no—
13C23×C4⋊C4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2496—24no—
19#2496—24no—
20C2×#400—21no—
21C2×#397—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#381—21no—
27C2×#381—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1852—24no—
34#1852—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#126—28no—
52C2×#126—44no—
53C2×#126—44no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#107—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87#400—28no—
88#397—28no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#387—28no—
94#384—22no—
95#384—22no—
96#381—44no—
97#381—44no—
98#380—22no—
99#380—22no—
100#380—42no—
101#380D414yes—
102#380D414yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—24no—
126#336—24no—
127C25C2×D411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
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×C8—22no—
167C22×C8#4114yes—
168C22×C8#4114yes—
169#126—432no—
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—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
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—22no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4D814yes—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—42no—
240C2×C42—42no—
241#107—48no—
242#107—48no—
243C22⋊C8—28no—
244C22⋊C8—28no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24#12012yes—
266C24C2×D812yes—
267C24AGL(1,ℤ/8)12yes—
268C24C22×D411yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
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—84no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4#12012yes—
352C22×C4C2×D812yes—
353C22×C4C2×D812yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C42−1+412yes—
362C22×C42−1+412yes—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
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—24no—
382C4⋊C4—24no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C22⋊C4—28no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—84no—
405C42—48no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—42no—
411C42—42no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#44714yes—
435C23C2×#12011yescharacteristic
436C23#52511yescharacteristic
437C23#53511yescharacteristic
438C23#54712yes—
439C23#54712yes—
440C23C22×D811yescharacteristic
441C23C2×AGL(1,ℤ/8)11yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23#57511yescharacteristic
444C23#57812yes—
445C23#57812yes—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—816no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
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#52511yescharacteristic
515C2×C4#52511yescharacteristic
516C2×C4#52912yes—
517C2×C4#53611yescharacteristic
518C2×C4#53611yescharacteristic
519C2×C4#53912yes—
520C2×C4#54112yes—
521C2×C4#54212yes—
522C2×C4#54612yes—
523C2×C4#54612yes—
524C2×C4#54712yes—
525C2×C4#54712yes—
526C2×C4#55212yes—
527C2×C4#55312yes—
528C2×C4#55612yes—
529C2×C4#56312yes—
530C8—44no—
531C8—44no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—44no—
536C22—42no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22#240411yescharacteristic
544C22#240611yescharacteristic
545C22#240611yescharacteristic
546C22#247812yes—
547C22#271812yes—
548C22C2×#44711yescharacteristic
549C22#300611yescharacteristic
550C22#301111yescharacteristic
551C22#301111yescharacteristic
552C22#310011yescharacteristic
553C22#310911yescharacteristic
554C22#312712yes—
555C22#320511yescharacteristic
556C22#321311yescharacteristic
557C22#322312yes—
558C22#333412yes—
559C22#333412yes—
560C22#334312yes—
561C22#334312yes—
562C22#334312yes—
563C22#334912yes—
564C22#335212yes—
565C22#335312yes—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—82no—
571C4—82no—
572C4—81no—
573C4—82no—
574C4—82no—
575C4—84no—
576C4—162no—
577C4#246612yes—
578C4#246812yes—
579C4#268912yes—
580C4#269812yes—
581C2—44no—
582C2#1602411yescharacteristic
583C2#2393011yescharacteristic
584C2#2394311yescharacteristic
585C2#2408811yescharacteristic
586C2#2408811yescharacteristic
587C2#2410711yescharacteristic
588C2#2411011yescharacteristic
589C2#2745412yes—
590C2#2747512yes—
591C2#3227812yes—
592C2#3230012yes—
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. #11324C2semidirect
6. #11292C2semidirect
7. #11219C2semidirect
11. #10907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#18522
#139042
#113242
#112922
#112192
#111992
#111472
#109072