Finite Group Databasebeta

← #360050

Subgroups — #360050

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11024C212yesmaximal
8#10932C212yesmaximal
9#10907C212yesmaximal
10#10842C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×C4×Q16—24no—
16C2×C4×Q16—24no—
17C2×C4×Q16C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2356—24no—
26#2356—24no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#387—22no—
30C2×#383—21no—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×#376—21no—
36#1805—24no—
37#1805—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47#1434—21no—
48C23×Q8C2311yescharacteristic
49C23×Q8C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16C2×C414yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8C2×C414yes—
69C2×C4×Q8C2×C414yes—
70C2×C4×Q8D414yes—
71C2×C4×Q8D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84C4×Q16—232no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107C2×C414yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103C2×C414yes—
102C2×#103C2×C414yes—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8C2×C414yes—
106#391—24no—
107#391—24no—
108#391—24no—
109#391—42no—
110#387—216no—
111#383—28no—
112#382—22no—
113#382—22no—
114#382—22no—
115#382D414yes—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120#376—28no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—42no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8C22×C412yes—
163C22×Q8C22×C412yes—
164C22×Q8C2×D412yes—
165C22×Q8C2×D412yes—
166C22×Q8#4112yes—
167C22×Q8#4112yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C2×Q16—28no—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—28no—
183C2×Q16—28no—
184C2×Q16—28no—
185C2×Q16—28no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8#4112yes—
206C22×C8#4112yes—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×Q8C22⋊C4116yes—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269#107—28no—
270C4⋊C8—44no—
271#103—28no—
272#103—28no—
273#103—48no—
274#103—48no—
275C4×C8—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—42no—
288#95—44no—
289C24C22×D411yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8C2×C22⋊C414yes—
345C2×Q8#11714yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4#12112yes—
406C22×C4#12112yes—
407C22×C4#13514yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C42−1+411yescharacteristic
421C22×C42−1+411yescharacteristic
422Q16—232no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8#11714yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—216no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—48no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4C2×C22⋊C414yes—
463C42—216no—
464C42—416no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—48no—
470C42C2×C22⋊C414yes—
471C42C4×D414yes—
472C42#12114yes—
473C42#12314yes—
474C23C2×C4×D411yescharacteristic
475C23C2×#12011yescharacteristic
476C23C2×#12111yescharacteristic
477C23#54611yescharacteristic
478C23#54711yescharacteristic
479C23#54811yescharacteristic
480C23#55011yescharacteristic
481C23C2×#13512yes—
482C23#57512yes—
483C23#57812yes—
484C23#57812yes—
485Q8—24no—
486Q8—24no—
487Q8—24no—
488Q8—24no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4#38614yes—
562C2×C4#38614yes—
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×#11711yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4#52011yescharacteristic
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#52511yescharacteristic
575C2×C4#53511yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53711yescharacteristic
578C2×C4#53711yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4#55011yescharacteristic
583C2×C4#55211yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55311yescharacteristic
586C2×C4#55311yescharacteristic
587C2×C4#55511yescharacteristic
588C2×C4#55511yescharacteristic
589C2×C4#55611yescharacteristic
590C2×C4#55611yescharacteristic
591C8—24no—
592C8—42no—
593C8—82no—
594C22C2×#38611yescharacteristic
595C22#235211yescharacteristic
596C22#235411yescharacteristic
597C22#240711yescharacteristic
598C22#240711yescharacteristic
599C22#241411yescharacteristic
600C22#241511yescharacteristic
601C22C4×#13511yescharacteristic
602C22#294311yescharacteristic
603C22#295011yescharacteristic
604C22#295011yescharacteristic
605C22#300212yes—
606C22#301112yes—
607C22#303512yes—
608C22#304412yes—
609C22#330012yes—
610C22#332712yes—
611C22#333612yes—
612C22#335212yes—
613C22#336412yes—
614C22#337712yes—
615C22#339312yes—
616C22#340212yes—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4—162no—
627C4#228011yescharacteristic
628C4#231111yescharacteristic
629C4#235211yescharacteristic
630C4#235411yescharacteristic
631C4#241411yescharacteristic
632C4#241511yescharacteristic
633C4#248511yescharacteristic
634C4#248511yescharacteristic
635C2#1387111yescharacteristic
636C2#2092411yescharacteristic
637C2#2093511yescharacteristic
638C2#2276111yescharacteristic
639C2#2276911yescharacteristic
640C2#2280911yescharacteristic
641C2#2281511yescharacteristic
642C2#2412712yes—
643C2#2419512yes—
644C2#2463712yes—
645C2#2468712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
17. C2×C4×Q16C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#23562
C2×#18052
#139422
#111902
#110242
#109322
#109072
#108422
#108342