Finite Group Databasebeta

← #431011

Subgroups — #431011

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2059C212yesmaximal, complemented
4C2×#1854C212yesmaximal
5C2×#1845C211yesmaximal, characteristic, complemented
6#15135C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11096C212yesmaximal
10#10921C211yesmaximal, characteristic
11#10880C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2441—24no—
20#2441—24no—
21C2×#399—21no—
22C2×#399—21no—
23C2×#396—21no—
24C2×#389—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2084—24no—
35#2059—28no—
36#1854—28no—
37#1845—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2211yescharacteristic
45#1445—21no—
46#1444—21no—
47#1439—21no—
48C4⋊#95C2211yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—24no—
55C2×#125—42no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#107—42no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8Q818yes—
95C2×C4×C8—24no—
96C2×C4×C8—22no—
97#399—28no—
98#399—44no—
99#396—28no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389—42no—
106#389D414yes—
107#387—28no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384D414yes—
112#380—22no—
113#380—22no—
114#380—24no—
115#380—42no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340D418yes—
132#336—24no—
133#336—22no—
134C25C2×D411yescharacteristic
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q814yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×D414yes—
166C22×C8C2×Q814yes—
167C22×C8C2×Q814yes—
168C22×C8#4114yes—
169#125—48no—
170#125—44no—
171#125—44no—
172#117—48no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248#107—48no—
249#107—48no—
250#106—48no—
251C22⋊C8—216no—
252C4×C8—28no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—22no—
271C24—22no—
272C24C4⋊Q812yes—
273C24C22×D411yescharacteristic
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
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#12012yes—
343C22×C4#12012yes—
344C22×C4C4⋊Q812yes—
345C22×C4#13512yes—
346C22×C4#13512yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4#13712yes—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q812yes—
353C22×C4C2×#4112yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356C22×C42−1+412yes—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8C22⋊Q818yes—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—88no—
398C4⋊C4—84no—
399C4⋊C4#12714yes—
400C22⋊C4—48no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—84no—
404C22⋊C4—88no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—48no—
409C42—41no—
410C42—41no—
411C42—42no—
412C23—22no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C2×#12011yescharacteristic
436C23C2×C4⋊Q811yescharacteristic
437C23#53611yescharacteristic
438C23D4212yes—
439C23#54712yes—
440C23C2×#13511yescharacteristic
441C23C2×#13511yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×#13711yescharacteristic
444C23#57612yes—
445C23#57812yes—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
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—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#44914yes—
508C2×C4#45214yes—
509C2×C4C2×C22⋊Q812yes—
510C2×C4#52511yescharacteristic
511C2×C4C2×#12711yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53612yes—
515C2×C4#53912yes—
516C2×C4#54412yes—
517C2×C4D4212yes—
518C2×C4#54712yes—
519C2×C4#54712yes—
520C2×C4D4×Q812yes—
521C2×C4D4×Q812yes—
522C2×C4D4×Q812yes—
523C2×C4#55412yes—
524C2×C4#56312yes—
525C8—28no—
526C8—44no—
527C22—42no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—81no—
533C22—81no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22#240611yescharacteristic
539C22#243311yescharacteristic
540C22#243411yescharacteristic
541C22#244012yes—
542C22#268012yes—
543C22C2×#44911yescharacteristic
544C22C2×#45211yescharacteristic
545C22#300811yescharacteristic
546C22#301111yescharacteristic
547C22#316011yescharacteristic
548C22#316412yes—
549C22#316511yescharacteristic
550C22#322811yescharacteristic
551C22#323111yescharacteristic
552C22#324312yes—
553C22#327412yes—
554C22#327612yes—
555C22#328312yes—
556C22#328712yes—
557C22#333612yes—
558C22#333912yes—
559C22#334312yes—
560C22#334612yes—
561C4—24no—
562C4—24no—
563C4—42no—
564C4—44no—
565C4—82no—
566C4—84no—
567C4—81no—
568C4—81no—
569C4—82no—
570C4—82no—
571C4—162no—
572C4#244912yes—
573C4#254212yes—
574C4#267712yes—
575C4#272012yes—
576C2—82no—
577C2#1543011yescharacteristic
578C2#2409211yescharacteristic
579C2#2409411yescharacteristic
580C2#2575811yescharacteristic
581C2#2575911yescharacteristic
582C2#2581211yescharacteristic
583C2#2582111yescharacteristic
584C2#2606612yes—
585C2#2607712yes—
586C2#3108312yes—
587C2#3108412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
2. C2×#2084C2semidirect
3. C2×#2059C2semidirect
5. C2×#1845C2semidirect
7. #11357C2semidirect
8. #11328C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#20842
C2×#20592
C2×#18542
C2×#18452
#151352
#113572
#113282
#110962
#109212
#108802