Finite Group Databasebeta

← #415015

Subgroups — #415015

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic, complemented
3C2×#1842C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic, complemented
6C4×#390C211yesmaximal, characteristic
7#11232C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11076C212yesmaximal
10#11076C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#478—24no—
15C2×#478—24no—
16C2×#478—24no—
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2212yes—
26C2×#390C2212yes—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—22no—
30C2×C43C2211yescharacteristic
31#1989—24no—
32#1842—24no—
33#1804—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1437—21no—
40#1437—21no—
41#1436—21no—
42#1436—22no—
43#1436—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125D414yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D411yescharacteristic
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#478—232no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—42no—
97C2×C4⋊C8—42no—
98C2×C4⋊C8D414yes—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—24no—
106C2×#102—42no—
107C2×#102—42no—
108#390—24no—
109#390Q818yes—
110#382—22no—
111#382—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—48no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
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—
131C25—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4C2×Q814yes—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
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×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174#125—28no—
175#125—28no—
176#125—44no—
177#125—44no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—44no—
181C4×D4—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
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—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q814yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×Q814yes—
244C2×C42#4114yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248#107—28no—
249#106—28no—
250C4⋊C8—28no—
251C4⋊C8—48no—
252C4⋊C8—48no—
253C4⋊C8—44no—
254#102—28no—
255#102—28no—
256#102—48no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—44no—
264#95—44no—
265#95—44no—
266C24—22no—
267C24—24no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24C22×D411yescharacteristic
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4C4⋊Q814yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
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—24no—
312C22×C4—416no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—48no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4—88no—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4#13514yes—
351C22×C4AGL(1,ℤ/8)14yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q812yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C42−1+412yes—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C42—28no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—48no—
410C42—44no—
411C42—84no—
412C42#12014yes—
413C42#12314yes—
414C42#12314yes—
415C42#12414yes—
416C23—24no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—82no—
437C23C2×#12111yescharacteristic
438C23C2×#12111yescharacteristic
439C23C2×#12111yescharacteristic
440C23C2×C22⋊Q812yes—
441C23#53712yes—
442C23C2×#13512yes—
443C23C2×AGL(1,ℤ/8)12yes—
444C23#57512yes—
445C23#57512yes—
446D4—24no—
447D4—24no—
448D4—44no—
449D4—42no—
450D4—44no—
451D4—42no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#39414yes—
504C2×C4#39414yes—
505C2×C4#47818yescomplemented
506C2×C4C2×#12011yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4C2×#12411yescharacteristic
510C2×C4C2×C4⋊Q811yescharacteristic
511C2×C4#53211yescharacteristic
512C2×C4#53312yes—
513C2×C4#53511yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#54212yes—
516C2×C4#54612yes—
517C2×C4D4×Q812yes—
518C2×C4D4×Q812yes—
519C2×C4#55012yes—
520C2×C4#55212yes—
521C2×C4#55412yes—
522C2×C4#55412yes—
523C2×C4#55612yes—
524C8—42no—
525C8—42no—
526C8—82no—
527C22—44no—
528C22—48no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22C2×#39411yescharacteristic
537C22#241712yes—
538C22#253812yes—
539C22#262011yescharacteristic
540C22#285711yescharacteristic
541C22#303512yes—
542C22#303512yes—
543C22#303612yes—
544C22#303811yescharacteristic
545C22#304011yescharacteristic
546C22#305011yescharacteristic
547C22#305411yescharacteristic
548C22#305512yes—
549C22C2×#47812yes—
550C22#307612yes—
551C22#307712yes—
552C22#308012yes—
553C22#324712yes—
554C22#325012yes—
555C22#325112yes—
556C22#325612yes—
557C4—28no—
558C4—24no—
559C4—24no—
560C4—44no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—162no—
566C4#242312yes—
567C4#243111yescharacteristic
568C4#243411yescharacteristic
569C4#249212yes—
570C4#249911yescharacteristic
571C4#250211yescharacteristic
572C2—84no—
573C2#1518211yescharacteristic
574C2#2353911yescharacteristic
575C2#2354611yescharacteristic
576C2#2483912yes—
577C2#2484112yes—
578C2#2828512yes—
579C2#2828612yes—
580C2#2983212yes—
581C2#3319912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
2. C2×#1989C2semidirect
3. C2×#1842C2semidirect
5. #14709C2semidirect
7. #11232C2semidirect
8. #11231C2semidirect
11. #10850C2semidirect
505. C2×C4#478semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#19892
C2×#18422
C2×#18042
#147092
C4×#3902
#112322
#112312
#110762
#108502