Finite Group Databasebeta

← #431007

Subgroups — #431007

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2059C212yesmaximal, complemented
4C2×#1853C212yesmaximal
5C2×#1847C211yesmaximal, characteristic, complemented
6#15135C211yesmaximal, characteristic
7#11358C211yesmaximal, characteristic, complemented
8#11327C212yesmaximal, complemented
9#11096C212yesmaximal
10#10922C211yesmaximal, characteristic
11#10881C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19#2442—24no—
20#2442—24no—
21C2×#400—21no—
22C2×#399—21no—
23C2×#398—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2211yescharacteristic
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2084—24no—
35#2059—28no—
36#1853—28no—
37#1847—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×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#125—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—42no—
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—22no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—42no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8Q818yes—
93C2×#97—24no—
94C2×#97—22no—
95#400—28no—
96#399—44no—
97#398—28no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—24no—
102#389—42no—
103#389D414yes—
104#385—28no—
105#384—22no—
106#384—22no—
107#382—22no—
108#382D414yes—
109#380—28no—
110#380—22no—
111#380—22no—
112#380—24no—
113#380—42no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—21no—
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—22no—
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×Q814yes—
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×C8C2×D414yes—
171C22×C8C2×Q814yes—
172C22×C8C2×Q814yes—
173C22×C8#4114yes—
174#125—48no—
175#125—44no—
176#125—44no—
177C2×C4⋊C4—21no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246#107—48no—
247#106—48no—
248#106—48no—
249C22⋊C8—216no—
250#97—28no—
251#95—24no—
252#95—24no—
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—22no—
269C24—22no—
270C24C4⋊Q812yes—
271C24AGL(1,ℤ/8)12yes—
272C24#13712yes—
273C24C22×D411yescharacteristic
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—48no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
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×C4—42no—
348C22×C4—84no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4C4⋊Q812yes—
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q812yes—
358C22×C4C2×#4112yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361C22×C42−1+412yes—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8C22⋊Q818yes—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C4⋊C4#12714yes—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—48no—
407C22⋊C4—88no—
408C42—22no—
409C42—22no—
410C42—24no—
411C42—41no—
412C42—41no—
413C42—42no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23#50114yes—
437C23C2×#12011yescharacteristic
438C23C2×C4⋊Q811yescharacteristic
439C23#53611yescharacteristic
440C23D4212yes—
441C23#54712yes—
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×#13711yescharacteristic
444C23#57511yescharacteristic
445C23#57511yescharacteristic
446C23#57712yes—
447C23#57712yes—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—816no—
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—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4C2×C22⋊Q812yes—
513C2×C4#52511yescharacteristic
514C2×C4C2×#12711yescharacteristic
515C2×C4#53511yescharacteristic
516C2×C4#53611yescharacteristic
517C2×C4#53612yes—
518C2×C4#53912yes—
519C2×C4#54412yes—
520C2×C4D4212yes—
521C2×C4#54712yes—
522C2×C4#54712yes—
523C2×C4D4×Q812yes—
524C2×C4D4×Q812yes—
525C2×C4D4×Q812yes—
526C2×C4#55412yes—
527C2×C4#56312yes—
528C8—28no—
529C8—44no—
530C22—28no—
531C22—24no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22#240611yescharacteristic
542C22#243311yescharacteristic
543C22#243411yescharacteristic
544C22#244012yes—
545C22#268012yes—
546C22#300611yescharacteristic
547C22#300711yescharacteristic
548C22#300911yescharacteristic
549C22#301011yescharacteristic
550C22C2×#50111yescharacteristic
551C22#316312yes—
552C22#316611yescharacteristic
553C22#323411yescharacteristic
554C22#323611yescharacteristic
555C22#324412yes—
556C22#326512yes—
557C22#326612yes—
558C22#326712yes—
559C22#327112yes—
560C22#334412yes—
561C22#334512yes—
562C22#335112yes—
563C22#335112yes—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—82no—
569C4—84no—
570C4—81no—
571C4—82no—
572C4—81no—
573C4—82no—
574C4—162no—
575C4#244912yes—
576C4#254212yes—
577C4#267712yes—
578C4#272012yes—
579C2—44no—
580C2#1543011yescharacteristic
581C2#2410111yescharacteristic
582C2#2411111yescharacteristic
583C2#2576811yescharacteristic
584C2#2576911yescharacteristic
585C2#2581911yescharacteristic
586C2#2582011yescharacteristic
587C2#2606812yes—
588C2#2607512yes—
589C2#3109412yes—
590C2#3109712yes—
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×#1847C2semidirect
7. #11358C2semidirect
8. #11327C2semidirect
11. #10881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#20842
C2×#20592
C2×#18532
C2×#18472
#151352
#113582
#113272
#110962
#109222
#108812