Finite Group Databasebeta

← #196489

Subgroups — #196489

Order 512.

1900 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic, complemented
4C2×#1525C212yesmaximal, complemented
5#9120C214yesmaximal, complemented
6#9119C214yesmaximal, complemented
7#8028C212yesmaximal, complemented
8C23×M16—21no—
9C2×#532C2211yescharacteristic
10C2×#525C412yes—
11C23×C42C2211yescharacteristic
12C2×#407—21no—
13C2×C4×M16—21no—
14C2×C4×M16—21no—
15C2×C4×M16—28no—
16C2×C4×M16C2212yes—
17C2×C4×M16C2212yes—
18C2×C4×M16C2212yes—
19C22×#97C2212yes—
20C22×C4×C8—21no—
21#2857C411yescharacteristic
22#2857C411yescharacteristic
23#2856C411yescharacteristic
24#2856C411yescharacteristic
25#2854—21no—
26#2854—21no—
27#2850C2212yes—
28#2850C2212yes—
29#2847—21no—
30#2847—21no—
31C4×#121—21no—
32C4×#121—21no—
33C4×#121—21no—
34C4×#121—21no—
35#2297C412yes—
36#2285C2211yescharacteristic
37#1749C2212yes—
38#1749C2212yes—
39C4×C22⋊C8—21no—
40C4×C22⋊C8—21no—
41#1728—21no—
42#1525—28no—
43C2×#329C2218yescomplemented
44#1295C2218yescomplemented
45C22×#41C2×C412yes—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—28no—
50C22×M16—28no—
51C22×M16—28no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C23×C8—22no—
56#532—24no—
57#532—24no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#127C2×C411yescharacteristic
60C2×#124C2312yes—
61#525—48no—
62C2×#123—22no—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121C2×C412yes—
72C2×#121C2×C412yes—
73C2×#120—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117C2×C412yes—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#407—28no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C4×M16—28no—
97C4×M16—28no—
98C4×M16—24no—
99C4×M16—28no—
100C4×M16—24no—
101C4×M16—24no—
102C4×M16D418yescomplemented
103C2×#97—22no—
104C2×#97—28no—
105C2×#97C2314yes—
106C2×#97C2314yes—
107C2×C4×C8—28no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111#391C2×C412yes—
112#391C2×C412yes—
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390C2×C412yes—
118#390C2×C412yes—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#384C2×C411yescharacteristic
134#384C2×C411yescharacteristic
135#382C2312yes—
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146#336—22no—
147#336—22no—
148#329—216no—
149C25C22⋊C411yescharacteristic
150C2×#41—28no—
151C2×#41—44no—
152C2×#41—44no—
153C22×Q8C22×C412yes—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C412yes—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C2×M16—264no—
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—28no—
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8—28no—
200C22×C8—28no—
201C22×C8—28no—
202C22×C8—24no—
203C22×C8—24no—
204C4⋊Q8—24no—
205#127—24no—
206#124—24no—
207#124—24no—
208#123—48no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211#121—44no—
212#121—44no—
213#121—44no—
214#121—44no—
215#121—44no—
216#121—42no—
217#121—44no—
218#121—42no—
219#121—44no—
220#120—48no—
221C4×D4—48no—
222C4×D4—48no—
223C4×D4—42no—
224C4×D4—44no—
225C4×D4—42no—
226#117—44no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D414yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42#4114yes—
282C2×C42C2411yescharacteristic
283C22⋊C8—28no—
284C22⋊C8—28no—
285#97—28no—
286#97—24no—
287#97—24no—
288#97—24no—
289#97C2×D418yes—
290C4×C8—28no—
291C4×C8—28no—
292C4×C8—44no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—41no—
306C24—41no—
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310#41—816no—
311C2×Q8—24no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×D4—24no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C22×C4—216no—
333C22×C4—216no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
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—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—416no—
387C22×C4—416no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4#10014yes—
392C22×C4#10414yes—
393C22×C4#10414yes—
394C22×C4#10414yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C4×D414yes—
405C22×C4#12012yes—
406C22×C4#12114yes—
407C22×C4#12312yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—232no—
412M16—232no—
413M16—416no—
414C2×C8—232no—
415C2×C8—28no—
416C2×C8—28no—
417C2×C8—28no—
418C2×C8—28no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C42—22no—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—44no—
458C42C4×D414yes—
459C42#12012yes—
460C42#12114yes—
461C42#12312yes—
462C42C22×D412yes—
463C42C2×#4112yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—28no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23#38612yes—
495C23C2×#10012yes—
496C23C2×#10412yes—
497C23C2×#10412yes—
498C23C2×#10412yes—
499C23C22×C22⋊C411yescharacteristic
500Q8—84no—
501D4—84no—
502D4—88no—
503D4—88no—
504D4—88no—
505D4—84no—
506D4—84no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—88no—
558C2×C4#38612yes—
559C2×C4#38612yes—
560C2×C4#38612yes—
561C2×C4C2×#10012yes—
562C2×C4C2×#10012yes—
563C2×C4#41312yes—
564C2×C4C2×#10412yes—
565C2×C4C2×#10412yes—
566C2×C4C2×#10412yes—
567C2×C4C2×#10412yes—
568C2×C4C2×#10412yes—
569C2×C4C2×#10412yes—
570C2×C4#42112yes—
571C2×C4C2×C4×D412yes—
572C2×C4C2×#12011yescharacteristic
573C2×C4C2×#12112yes—
574C2×C4C2×#12311yescharacteristic
575C8—216no—
576C8—48no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—416no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22#152514yescomplemented
606C22C2×#38611yescharacteristic
607C22C22×#10011yescharacteristic
608C22C2×#41311yescharacteristic
609C22C22×#10411yescharacteristic
610C22C22×#10411yescharacteristic
611C22C22×#10411yescharacteristic
612C22C2×#42111yescharacteristic
613C4—22no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4#187114yes—
627C4#189614yes—
628C4#190014yes—
629C2—24no—
630C2—42no—
631C2—42no—
632C2—162no—
633C2C2×#152511yescharacteristic
634C2#1763811yescharacteristic
635C2#1764711yescharacteristic
636C2#1764911yescharacteristic
637C2C2×#187111yescharacteristic
638C2C2×#189611yescharacteristic
639C2C2×#190011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×M16C2semidirect
3. C4×#407C2semidirect
4. C2×#1525C2semidirect
5. #9120C2semidirect
6. #9119C2semidirect
7. #8028C2semidirect
43. C2×#329C22semidirect
44. #1295C22semidirect
102. C4×M16D4semidirect
605. C22#1525semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
#366242
C4×#4072
C2×#15252
#91202
#91192
#80282