Finite Group Databasebeta

← #428420

Subgroups — #428420

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2094C211yesmaximal, characteristic, complemented
3C2×#2034C212yesmaximal
4C2×#1853C212yesmaximal, complemented
5C2×#1844C211yesmaximal, characteristic, complemented
6#15195C211yesmaximal, characteristic, complemented
7#11373C211yesmaximal, characteristic, complemented
8#11323C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#10925C211yesmaximal, characteristic, complemented
11#10921C211yesmaximal, characteristic, complemented
12C22×#123—22no—
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#106C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2501—24no—
23#2501—24no—
24C2×#399—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2094—24no—
33#2034—28no—
34#1853—28no—
35#1844—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1442—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#123—216no—
52C2×#123—48no—
53C2×#123—48no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—24no—
57C2×#121—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#106—42no—
86C2×#106—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8D418yes—
96C2×C4×C8—24no—
97C2×C4×C8—22no—
98#399—216no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390D414yes—
103#384—22no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384D414yes—
109#380—24no—
110#380—24no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—24no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—24no—
120#336—24no—
121#336—22no—
122C25—21no—
123C25C2×D411yescharacteristic
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—44no—
134C22×D4—42no—
135C22×D4—44no—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2411yescharacteristic
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—22no—
168C22×C8—22no—
169C22×C8C2×D414yes—
170C22×C8C2×D414yes—
171C22×C8#4114yes—
172C22×C8#4114yes—
173#123—464no—
174#121—48no—
175#121—48no—
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—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—84no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—84no—
226C2×C22⋊C4SD1618yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235#106—48no—
236#106—48no—
237#106—48no—
238#102—48no—
239#102—48no—
240C22⋊C8—216no—
241C4×C8—28no—
242#95—24no—
243#95—44no—
244#95—44no—
245#95—84no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—24no—
252C24—22no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—44no—
260C24#12712yes—
261C24C2×SD1614yes—
262C24C22×D411yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—48no—
268C2×D4—416no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—84no—
282C2×D4—88no—
283C2×D4—84no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—48no—
314C22×C4—416no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
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—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—84no—
338C22×C4—88no—
339C22×C4—84no—
340C22×C4#12014yes—
341C22×C4#12712yes—
342C22×C4C2×SD1614yes—
343C22×C4C2×SD1614yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8#12318yescomplemented
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—48no—
371C4⋊C4—416no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—416no—
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—88no—
385C22⋊C4—216no—
386C22⋊C4—48no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—88no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—41no—
394C42—41no—
395C42—42no—
396C23—28no—
397C23—28no—
398C23—22no—
399C23—24no—
400C23—22no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—44no—
405C23—44no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—84no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—81no—
421C23—81no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23#49214yes—
428C23C2×#12011yescharacteristic
429C23C2×#12711yescharacteristic
430C23#53511yescharacteristic
431C23#54712yes—
432C23#54712yes—
433C23C22×SD1612yes—
434C23C2×AGL(1,ℤ/8)12yes—
435C23#57714yes—
436D4—44no—
437D4—44no—
438D4—816no—
439D4—82no—
440D4—82no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—816no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—82no—
485C2×C4#45018yes—
486C2×C4C2×#12312yes—
487C2×C4#52511yescharacteristic
488C2×C4#53211yescharacteristic
489C2×C4#53611yescharacteristic
490C2×C4#53612yes—
491C2×C4#53611yescharacteristic
492C2×C4#53912yes—
493C2×C4#54212yes—
494C2×C4D4212yes—
495C2×C4D4212yes—
496C2×C4#54712yes—
497C2×C4#54712yes—
498C2×C4#54712yes—
499C2×C4#54812yes—
500C2×C4#55312yes—
501C2×C4#56312yes—
502C8—28no—
503C8—44no—
504C22—28no—
505C22—24no—
506C22—42no—
507C22—44no—
508C22—48no—
509C22—44no—
510C22—44no—
511C22—42no—
512C22—82no—
513C22—84no—
514C22—84no—
515C22—82no—
516C22—82no—
517C22—162no—
518C22—164no—
519C22—162no—
520C22#240611yescharacteristic
521C22#243211yescharacteristic
522C22#243311yescharacteristic
523C22#250112yes—
524C22#271812yes—
525C22C2×#45012yes—
526C22#301012yes—
527C22C2×#49211yescharacteristic
528C22#315111yescharacteristic
529C22#315412yes—
530C22#320812yes—
531C22#322112yes—
532C22#333514yes—
533C22#333514yes—
534C22#334514yes—
535C22#335414yes—
536C4—28no—
537C4—42no—
538C4—44no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—162no—
543C4—162no—
544C4#242212yes—
545C4#244312yes—
546C4#267412yes—
547C4#268012yes—
548C2—44no—
549C2—162no—
550C2#1538411yescharacteristic
551C2#2408912yes—
552C2#2569912yes—
553C2#2571711yescharacteristic
554C2#2573311yescharacteristic
555C2#2792412yes—
556C2#2794112yes—
557C2#3228214yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#2094C2semidirect
4. C2×#1853C2semidirect
5. C2×#1844C2semidirect
6. #15195C2semidirect
7. #11373C2semidirect
8. #11323C2semidirect
9. #11121C2semidirect
10. #10925C2semidirect
11. #10921C2semidirect
366. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#20942
C2×#20342
C2×#18532
C2×#18442
#151952
#113732
#113232
#111212
#109252
#109212