Finite Group Databasebeta

← #452006

Subgroups — #452006

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#11323C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11200C211yesmaximal, characteristic
9#11200C211yesmaximal, characteristic
10#11147C211yesmaximal, characteristic
11#10908C212yesmaximal, complemented
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2497—24no—
18#2497—24no—
19C2×#400—21no—
20C2×#396—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
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#1853—24no—
34#1853—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—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84#400—28no—
85#396—28no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#388—28no—
91#387—28no—
92#384—22no—
93#384—22no—
94#381—44no—
95#381—44no—
96#380—22no—
97#380—22no—
98#380—42no—
99#380D414yes—
100#380D414yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—24no—
125C25C2×D411yescharacteristic
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
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—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8#4114yes—
166C22×C8#4114yes—
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—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4#4114yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4SD1614yes—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—82no—
232#106—48no—
233#106—48no—
234C22⋊C8—28no—
235C22⋊C8—28no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—42no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24#12012yes—
257C24C2×SD1612yes—
258C24#13712yes—
259C24C22×D411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
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—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—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4#12012yes—
339C22×C4C2×SD1612yes—
340C22×C4C2×SD1612yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4C2×#4112yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349C22×C42−1+412yes—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C22⋊C4—28no—
388C22⋊C4—44no—
389C22⋊C4—84no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—42no—
398C42—84no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23#44814yes—
422C23C2×#12011yescharacteristic
423C23#52511yescharacteristic
424C23#53511yescharacteristic
425C23#54712yes—
426C23#54712yes—
427C23C22×SD1611yescharacteristic
428C23C2×AGL(1,ℤ/8)11yescharacteristic
429C23C2×#13711yescharacteristic
430C23#57511yescharacteristic
431C23#57712yes—
432C23#57712yes—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—816no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#52511yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52912yes—
500C2×C4#53611yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53912yes—
503C2×C4#54112yes—
504C2×C4#54212yes—
505C2×C4#54612yes—
506C2×C4#54612yes—
507C2×C4#54712yes—
508C2×C4#54712yes—
509C2×C4#55212yes—
510C2×C4#55312yes—
511C2×C4#55612yes—
512C2×C4#56312yes—
513C8—44no—
514C8—44no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22#240411yescharacteristic
527C22#240611yescharacteristic
528C22#240611yescharacteristic
529C22#247812yes—
530C22#271812yes—
531C22C2×#44811yescharacteristic
532C22#300611yescharacteristic
533C22#300911yescharacteristic
534C22#300911yescharacteristic
535C22#310111yescharacteristic
536C22#311011yescharacteristic
537C22#312612yes—
538C22#320811yescharacteristic
539C22#321511yescharacteristic
540C22#322112yes—
541C22#333512yes—
542C22#333512yes—
543C22#334412yes—
544C22#334412yes—
545C22#334512yes—
546C22#334812yes—
547C22#335112yes—
548C22#335412yes—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—81no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4—82no—
559C4—162no—
560C4#246612yes—
561C4#246812yes—
562C4#268912yes—
563C4#269812yes—
564C2—44no—
565C2#1602411yescharacteristic
566C2#2393311yescharacteristic
567C2#2394011yescharacteristic
568C2#2409711yescharacteristic
569C2#2409711yescharacteristic
570C2#2410111yescharacteristic
571C2#2410411yescharacteristic
572C2#2745612yes—
573C2#2747712yes—
574C2#3228212yes—
575C2#3229612yes—
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. #11323C2semidirect
6. #11292C2semidirect
7. #11220C2semidirect
11. #10908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#18532
#139042
#113232
#112922
#112202
#112002
#111472
#109082