Finite Group Databasebeta

← #407064

Subgroups — #407064

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1893C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#13881C211yesmaximal, characteristic, complemented
7#11377C212yesmaximal, complemented
8#11372C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#11051C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2212yes—
22C2×#386—22no—
23C2×#382C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1893—24no—
27#1874—24no—
28#1852—24no—
29C2×#340—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1442—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
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—42no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4D412yes—
81C22×C42C2312yes—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#102—24no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8D414yes—
95#390—24no—
96#390—42no—
97#386—48no—
98#382—24no—
99#382—42no—
100#380—22no—
101#380—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95—24no—
105C2×#95—24no—
106C2×#95—21no—
107C2×#95—24no—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#336—42no—
116#336—42no—
117C25—21no—
118C25C2×D411yescharacteristic
119C22×D4—24no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4C2×Q812yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D414yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×Q812yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C22×C8—28no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
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—44no—
176C22×C8—44no—
177C22×C8—48no—
178C22×C8—48no—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D414yes—
182#121—28no—
183#121—28no—
184#121—44no—
185#121—44no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4#4114yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—84no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239#107—48no—
240#102—48no—
241#102—44no—
242#102—44no—
243C22⋊C8—28no—
244#95—24no—
245#95—48no—
246#95—48no—
247#95—42no—
248#95—84no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—41no—
255C24—41no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24C22⋊Q812yes—
263C24#12312yes—
264C24#12712yes—
265C24#13514yes—
266C24AGL(1,ℤ/8)14yes—
267C24C22×D411yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—48no—
273C2×D4—416no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—88no—
289C2×D4—84no—
290C2×D4—84no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
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—24no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—816no—
340C22×C4—816no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4C22⋊Q814yes—
346C22×C4C22⋊Q814yes—
347C22×C4C22⋊Q812yes—
348C22×C4#12314yes—
349C22×C4#12314yes—
350C22×C4#12312yes—
351C22×C4#12712yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)12yes—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—416no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—816no—
403C22⋊C4—84no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—22no—
417C23—22no—
418C23—28no—
419C23—28no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—44no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—82no—
431C23—84no—
432C23—84no—
433C23—81no—
434C23—81no—
435C23—84no—
436C23—84no—
437C23#39914yes—
438C23#47818yes—
439C23#48218yes—
440C23#49514yes—
441C23#49614yes—
442C23C2×C22⋊Q811yescharacteristic
443C23C2×#12311yescharacteristic
444C23C2×#12711yescharacteristic
445C23#54612yes—
446C23#54612yes—
447C23C2×#13512yes—
448C23C2×AGL(1,ℤ/8)12yes—
449C23#57612yes—
450C23#57812yes—
451D4—42no—
452D4—42no—
453D4—44no—
454D4—816no—
455D4—81no—
456D4—81no—
457D4—82no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—816no—
482C2×C4—816no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4#39914yes—
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4C2×#12311yescharacteristic
504C2×C4C2×#12311yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4#52511yescharacteristic
507C2×C4#52912yes—
508C2×C4#52912yes—
509C2×C4#53411yescharacteristic
510C2×C4#53411yescharacteristic
511C2×C4#53812yes—
512C2×C4#53812yes—
513C2×C4#54612yes—
514C2×C4#54612yes—
515C2×C4#55312yes—
516C2×C4#55312yes—
517C2×C4#55412yes—
518C2×C4#55412yes—
519C8—42no—
520C8—42no—
521C8—44no—
522C22—216no—
523C22—216no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—88no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—164no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22#209218yes—
540C22C2×#39911yescharacteristic
541C22#244412yes—
542C22#245012yes—
543C22#253911yescharacteristic
544C22#253911yescharacteristic
545C22C2×#47812yes—
546C22#308312yes—
547C22C2×#48212yes—
548C22#309812yes—
549C22C2×#49511yescharacteristic
550C22C2×#49611yescharacteristic
551C22#315511yescharacteristic
552C22#315611yescharacteristic
553C22#329714yes—
554C22#330014yes—
555C22#330814yes—
556C22#331814yes—
557C4—24no—
558C4—84no—
559C4—84no—
560C4—82no—
561C4—84no—
562C4—161no—
563C4—161no—
564C4#244412yes—
565C4#245012yes—
566C4#257911yescharacteristic
567C4#257911yescharacteristic
568C4#262211yescharacteristic
569C4#285311yescharacteristic
570C2—28no—
571C2—162no—
572C2#1494611yescharacteristic
573C2C2×#209211yescharacteristic
574C2#2373711yescharacteristic
575C2#2620814yes—
576C2#2642014yes—
577C2#2833111yescharacteristic
578C2#2833511yescharacteristic
579C2#2834111yescharacteristic
580C2#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×#1852C2semidirect
4. C22×#340C2semidirect
6. #13881C2semidirect
7. #11377C2semidirect
8. #11372C2semidirect
9. #11123C2semidirect
10. #11051C2semidirect
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×#18522
C22×#3402
#141372
#138812
#113772
#113722
#111232
#110512
#107142