Finite Group Databasebeta

← #407063

Subgroups — #407063

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1893C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#13880C211yesmaximal, characteristic, complemented
7#11378C212yesmaximal, complemented
8#11371C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#11049C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2212yes—
22C2×#386—22no—
23C2×#385—22no—
24C2×#384C2212yes—
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1893—24no—
28#1874—24no—
29#1853—24no—
30C2×#340—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1444—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8D414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121Q814yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4D412yes—
82C22×C42C2312yes—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×#102—24no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8D414yes—
96#390—24no—
97#390—42no—
98#386—48no—
99#385—48no—
100#384—24no—
101#384—42no—
102#380—22no—
103#380—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—21no—
109C2×#95D412yes—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#336—42no—
117#336—42no—
118C25—21no—
119C25C2×D411yescharacteristic
120C22×D4—24no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4C2×Q812yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C2×D414yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C22×C8—28no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
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—44no—
177C22×C8—44no—
178C22×C8—48no—
179C22×C8—48no—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D414yes—
183#121—28no—
184#121—28no—
185#121—44no—
186#121—44no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—84no—
223C2×C4⋊C4C2×Q812yes—
224C2×C4⋊C4#4114yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—84no—
236C2×C22⋊C4SD1618yes—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243#106—48no—
244#102—48no—
245#102—44no—
246#102—44no—
247C22⋊C8—28no—
248#95—24no—
249#95—48no—
250#95—42no—
251#95—48no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24C22⋊Q812yes—
266C24#12312yes—
267C24#12712yes—
268C24C2×SD1614yes—
269C24AGL(1,ℤ/8)14yes—
270C24C22×D411yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—48no—
276C2×D4—416no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—88no—
292C2×D4—84no—
293C2×D4—84no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—816no—
342C22×C4—816no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q812yes—
350C22×C4#12314yes—
351C22×C4#12314yes—
352C22×C4#12312yes—
353C22×C4#12712yes—
354C22×C4C2×SD1618yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)12yes—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—416no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—816no—
404C22⋊C4—216no—
405C22⋊C4—48no—
406C22⋊C4—816no—
407C22⋊C4—84no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—28no—
419C23—28no—
420C23—22no—
421C23—22no—
422C23—28no—
423C23—28no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—82no—
429C23—82no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—82no—
435C23—84no—
436C23—84no—
437C23—81no—
438C23—81no—
439C23—84no—
440C23—84no—
441C23#39914yes—
442C23#47618yes—
443C23#48118yes—
444C23#49214yes—
445C23#49614yes—
446C23C2×C22⋊Q811yescharacteristic
447C23C2×#12311yescharacteristic
448C23C2×#12711yescharacteristic
449C23#54612yes—
450C23#54612yes—
451C23C22×SD1612yes—
452C23C2×AGL(1,ℤ/8)12yes—
453C23#57612yes—
454C23#57812yes—
455D4—42no—
456D4—42no—
457D4—44no—
458D4—816no—
459D4—81no—
460D4—81no—
461D4—82no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—816no—
486C2×C4—816no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4#39914yes—
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4#52511yescharacteristic
511C2×C4#52912yes—
512C2×C4#52912yes—
513C2×C4#53411yescharacteristic
514C2×C4#53411yescharacteristic
515C2×C4#53812yes—
516C2×C4#53812yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4#55312yes—
520C2×C4#55312yes—
521C2×C4#55412yes—
522C2×C4#55412yes—
523C8—42no—
524C8—42no—
525C8—44no—
526C22—216no—
527C22—216no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—88no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—164no—
539C22—161no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22#209418yes—
544C22C2×#39911yescharacteristic
545C22#244412yes—
546C22#245012yes—
547C22#253911yescharacteristic
548C22#253911yescharacteristic
549C22C2×#47612yes—
550C22#308312yes—
551C22C2×#48112yes—
552C22#309812yes—
553C22C2×#49211yescharacteristic
554C22C2×#49611yescharacteristic
555C22#315511yescharacteristic
556C22#315611yescharacteristic
557C22#329514yes—
558C22#329814yes—
559C22#330814yes—
560C22#331814yes—
561C4—24no—
562C4—48no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—161no—
567C4—161no—
568C4#244412yes—
569C4#245012yes—
570C4#257911yescharacteristic
571C4#257911yescharacteristic
572C4#262211yescharacteristic
573C4#285311yescharacteristic
574C2—28no—
575C2—162no—
576C2#1494611yescharacteristic
577C2C2×#209411yescharacteristic
578C2#2373711yescharacteristic
579C2#2620214yes—
580C2#2641014yes—
581C2#2833211yescharacteristic
582C2#2833611yescharacteristic
583C2#2834111yescharacteristic
584C2#2834611yescharacteristic
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
3. C2×#1853C2semidirect
4. C22×#340C2semidirect
6. #13880C2semidirect
7. #11378C2semidirect
8. #11371C2semidirect
9. #11121C2semidirect
10. #11049C2semidirect
11. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18932
C2×#18742
C2×#18532
C22×#3402
#141372
#138802
#113782
#113712
#111212
#110492
#107142