Finite Group Databasebeta

← #380075

Subgroups — #380075

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11189C212yesmaximal
9#11189C212yesmaximal
10#10850C211yesmaximal, characteristic, complemented
11#10832C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117—21no—
15C2×#478—24no—
16C2×#478—24no—
17C2×#478C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25#2293—24no—
26#2293—24no—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#388—22no—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33C2×C43C2211yescharacteristic
34#1989—24no—
35#1804—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—21no—
44#1436—22no—
45#1436—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125C2×C414yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C414yes—
63C2×C4×D4D414yes—
64C2×C4×D4D414yes—
65C2×#117—28no—
66C2×#117—48no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42D411yescharacteristic
80C22×C42D412yes—
81C22×C42D411yescharacteristic
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#478—232no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107C2×C414yes—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106C2×C414yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8C2×C414yes—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#102C2×C414yes—
105C2×#102C2×C414yes—
106#390—24no—
107#390—24no—
108#389—24no—
109#389—24no—
110#388—216no—
111#382—22no—
112#382—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C4×C22⋊C4—28no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C43—22no—
125C43—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—42no—
130#340—42no—
131#336—42no—
132#336—42no—
133C25—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C22×C412yes—
142C22×D4C2×D412yes—
143C22×D4C2×D412yes—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
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×C8C22×C412yes—
175C22×C8C22×C412yes—
176#125—28no—
177#125—28no—
178#125—44no—
179#125—44no—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4C22⋊C4116yes—
184#117—432no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254#107—28no—
255#106—28no—
256C4⋊C8—28no—
257C4⋊C8—44no—
258#102—28no—
259#102—28no—
260#102—48no—
261#95—28no—
262#95—24no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24C22×D411yescharacteristic
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4C2×C22⋊C414yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—216no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—48no—
321C22×C4—416no—
322C22×C4—48no—
323C22×C4—416no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
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×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#13514yes—
354C22×C4AGL(1,ℤ/8)14yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C22×C42−1+412yes—
364C22×C42−1+412yes—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—416no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4C2×C22⋊C414yes—
405C22⋊C4—216no—
406C22⋊C4—416no—
407C22⋊C4—416no—
408C42—28no—
409C42—48no—
410C42—44no—
411C42—48no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42C2×C22⋊C414yes—
417C42C4×D414yes—
418C42#12114yes—
419C42#12314yes—
420C23—24no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23C2×C4×D411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×#12111yescharacteristic
444C23#53612yes—
445C23#53712yes—
446C23C2×#13512yes—
447C23C2×AGL(1,ℤ/8)12yes—
448C23#57512yes—
449C23#57512yes—
450D4—24no—
451D4—24no—
452D4—44no—
453D4—42no—
454D4—42no—
455D4—44no—
456C2×C4—22no—
457C2×C4—22no—
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—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—432no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#38614yes—
509C2×C4#38614yes—
510C2×C4C22×C22⋊C411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4#51912yes—
513C2×C4#52012yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4#53511yescharacteristic
517C2×C4#53611yescharacteristic
518C2×C4#53611yescharacteristic
519C2×C4#53612yes—
520C2×C4#53611yescharacteristic
521C2×C4#53612yes—
522C2×C4#53712yes—
523C2×C4#53712yes—
524C2×C4#54112yes—
525C2×C4#54112yes—
526C2×C4#54212yes—
527C2×C4#54212yes—
528C8—42no—
529C8—42no—
530C8—82no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22C2×#38611yescharacteristic
541C22#232312yes—
542C22#261612yes—
543C22#262011yescharacteristic
544C22#262111yescharacteristic
545C22C4×#13511yescharacteristic
546C22#294311yescharacteristic
547C22C4×AGL(1,ℤ/8)12yes—
548C22#300212yes—
549C22#300311yescharacteristic
550C22#300511yescharacteristic
551C22#303512yes—
552C22#303612yes—
553C22#322812yes—
554C22#323212yes—
555C22#323412yes—
556C22#324012yes—
557C22#324712yes—
558C22#325012yes—
559C22#325112yes—
560C22#325612yes—
561C4—24no—
562C4—24no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—82no—
568C4—84no—
569C4—162no—
570C4#231011yescharacteristic
571C4#231111yescharacteristic
572C4#232312yes—
573C4#262011yescharacteristic
574C4#262111yescharacteristic
575C4#262512yes—
576C2—84no—
577C2#1425811yescharacteristic
578C2#2092411yescharacteristic
579C2#2092511yescharacteristic
580C2#2211412yes—
581C2#2212012yes—
582C2#2969312yes—
583C2#2969412yes—
584C2#2983212yes—
585C2#2987012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1989C2semidirect
5. #14709C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
10. #10850C2semidirect
11. #10832C2semidirect
17. C2×#478C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#22932
C2×#19892
C2×#18042
#147092
#112202
#112192
#111892
#108502
#108322