Finite Group Databasebeta

← #431042

Subgroups — #431042

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#2084C211yesmaximal, characteristic, complemented
4C2×#2059C212yesmaximal, complemented
5C2×#1864C212yesmaximal
6#13910C211yesmaximal, characteristic, complemented
7#11346C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11066C212yesmaximal
10#10931C211yesmaximal, characteristic
11#10885C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2454—24no—
19#2454—24no—
20C2×#399—21no—
21C2×#396—21no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#385—21no—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2093—24no—
35#2084—24no—
36#2059—28no—
37#1864—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46#1444—21no—
47#1444—21no—
48C4⋊#95—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#125—42no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—42no—
96C2×#102Q818yes—
97#399—28no—
98#396—28no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390D414yes—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—24no—
107#389—24no—
108#389—42no—
109#389—42no—
110#389D414yes—
111#385—28no—
112#384—22no—
113#384—22no—
114#384—44no—
115#380—28no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—24no—
128#340—24no—
129#340D418yes—
130#336—24no—
131#336—24no—
132#336—22no—
133#336—22no—
134C25—21no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C2×D412yes—
142C22×D4C2×Q814yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—82no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
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×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×D414yes—
173C22×C8C2×Q814yes—
174#125—48no—
175#125—44no—
176#125—44no—
177C4×D4—48no—
178C4×D4—44no—
179C4×D4—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
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—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×Q814yes—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255#107—48no—
256#106—48no—
257#102—216no—
258#102—48no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—44no—
265#95—44no—
266C24—22no—
267C24—24no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24C22×D411yescharacteristic
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
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—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4#13512yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4#13712yes—
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q812yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+412yes—
367C22×C42−1+412yes—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
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—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4—84no—
406C4⋊C4C22⋊Q818yes—
407C4⋊C4#12714yes—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C42—24no—
414C42—24no—
415C42—48no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—42no—
422C23—24no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—81no—
436C23—81no—
437C23—82no—
438C23—81no—
439C23—81no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23C2×#12011yescharacteristic
446C23#52511yescharacteristic
447C23#53611yescharacteristic
448C23D4212yes—
449C23D4×Q812yes—
450C23C2×#13511yescharacteristic
451C23C2×AGL(1,ℤ/8)11yescharacteristic
452C23C2×#13711yescharacteristic
453C23#57511yescharacteristic
454C23#57612yes—
455C23#57712yes—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—816no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#44914yes—
522C2×C4C2×C22⋊Q812yes—
523C2×C4C2×#12711yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4#53511yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#53612yes—
528C2×C4#53912yes—
529C2×C4#54412yes—
530C2×C4D4212yes—
531C2×C4#54712yes—
532C2×C4#54712yes—
533C2×C4#54712yes—
534C2×C4D4×Q812yes—
535C2×C4D4×Q812yes—
536C2×C4#55412yes—
537C2×C4#56312yes—
538C8—44no—
539C8—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—84no—
546C22—84no—
547C22—82no—
548C22—82no—
549C22—84no—
550C22—84no—
551C22#240611yescharacteristic
552C22#243311yescharacteristic
553C22#243411yescharacteristic
554C22#244912yes—
555C22#267712yes—
556C22C2×#44911yescharacteristic
557C22#300711yescharacteristic
558C22#300811yescharacteristic
559C22#300911yescharacteristic
560C22#310811yescharacteristic
561C22#311411yescharacteristic
562C22#311912yes—
563C22#323211yescharacteristic
564C22#323811yescharacteristic
565C22#324112yes—
566C22#326612yes—
567C22#326712yes—
568C22#327412yes—
569C22#328712yes—
570C22#338112yes—
571C22#338412yes—
572C22#338512yes—
573C22#338912yes—
574C4—24no—
575C4—42no—
576C4—48no—
577C4—42no—
578C4—44no—
579C4—82no—
580C4—81no—
581C4—81no—
582C4—82no—
583C4—81no—
584C4—162no—
585C4#244012yes—
586C4#254212yes—
587C4#268012yes—
588C4#272012yes—
589C2—84no—
590C2#1543011yescharacteristic
591C2#2406611yescharacteristic
592C2#2406911yescharacteristic
593C2#2575911yescharacteristic
594C2#2576911yescharacteristic
595C2#2583511yescharacteristic
596C2#2585011yescharacteristic
597C2#2637912yes—
598C2#2639012yes—
599C2#3095912yes—
600C2#3097012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2084C2semidirect
4. C2×#2059C2semidirect
6. #13910C2semidirect
7. #11346C2semidirect
8. #11328C2semidirect
11. #10885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#20932
C2×#20842
C2×#20592
C2×#18642
#139102
#113462
#113282
#110662
#109312
#108852