Finite Group Databasebeta

← #360071

Subgroups — #360071

Order 512.

2096 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3C2×#1806C212yesmaximal, complemented
4#20626C211yesmaximal, characteristic, complemented
5#11201C212yesmaximal, complemented
6#10971C212yesmaximal
7#10923C212yesmaximal, complemented
8#10878C212yesmaximal, complemented
9#10852C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×#117—21no—
13C23×C42C2211yescharacteristic
14C2×#423C2212yes—
15C22×#104C2214yes—
16C2×#407C2212yes—
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C412yescomplemented
19C4×#123—21no—
20C4×#123—21no—
21C4×#121C412yescomplemented
22C4×#121C412yescomplemented
23C4×#120—21no—
24C4×#120—21no—
25#2297C2211yescharacteristic
26#2296C2211yescharacteristic
27#2290—22no—
28#2280C2211yescharacteristic
29C42×D4—22no—
30C4×#117—28no—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2212yes—
36C2×#376—21no—
37C2×C43C2212yes—
38#1806—28no—
39#1797—22no—
40#1797—22no—
41C2×#339C2214yes—
42C2×#339C2214yes—
43C2×#339C2214yes—
44#1728C2212yes—
45C22×#41C2311yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16C2314yes—
50#525—24no—
51#525—24no—
52C2×#123—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×#121C2×C412yes—
55C2×#120—21no—
56C4×#41—24no—
57C4×#41—24no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×#117—216no—
68C2×#117—28no—
69C2×#117—21no—
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—22no—
76C22×C42—24no—
77C22×C42—24no—
78C22×C42—22no—
79C22×C42—21no—
80C22×C42C2×C412yes—
81C22×C42C2×C412yes—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D412yes—
85C22×C42D412yes—
86C22×C42C2314yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#423—24no—
90#423—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×#104—28no—
94C2×#104—28no—
95C2×#104—44no—
96#407—44no—
97C2×C22⋊C8—24no—
98#391C2×C412yes—
99#390C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—22no—
106#386—22no—
107#386—22no—
108#386—21no—
109#386—21no—
110#385—22no—
111#385—22no—
112#385—21no—
113#385—21no—
114#381—21no—
115#380C2×C412yes—
116#380C2×C412yes—
117#379—21no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D418yescomplemented
126C4×C22⋊C4—28no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—22no—
136#376—28no—
137#376—21no—
138C2×#95—22no—
139C2×#95—22no—
140C43—28no—
141C43—21no—
142#339—28no—
143#339—28no—
144#339—44no—
145#336—24no—
146C25C2×D411yescharacteristic
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41#4114yes—
153C22×Q8C22×C411yescharacteristic
154C22×D4—22no—
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—28no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—44no—
173C23×C4C22⋊C412yes—
174C23×C4C22⋊C412yes—
175C23×C4C22⋊C412yes—
176C23×C4C22⋊C412yes—
177C23×C4C22×C412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C22×C8—24no—
192#123—28no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195#121—24no—
196#121—24no—
197#120—28no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×D4—216no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209#117—264no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—44no—
216#117#4114yes—
217C2×C4⋊C4—216no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C42—28no—
242C2×C42—216no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—28no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—28no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—48no—
267C2×C42—44no—
268C2×C42C22⋊C412yes—
269C2×C42C22⋊C412yes—
270C2×C42C22⋊C412yes—
271C2×C42C22⋊C412yes—
272C2×C42C22×C412yes—
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D418yes—
276C2×C42C2×D412yes—
277C2×C42C2×D414yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42#4114yes—
281C4⋊C8—44no—
282C4⋊C8—44no—
283#104—416no—
284C22⋊C8—48no—
285#95—28no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—28no—
295C24—21no—
296C24—21no—
297C24—24no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—42no—
302C24C2×C22⋊C412yes—
303C24C22×D411yescharacteristic
304#41—24no—
305#41—24no—
306#41—24no—
307#41—24no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312C2×Q8—21no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8#11712yescomplemented
316C2×D4—28no—
317C2×D4—24no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—44no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4#11712yescomplemented
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—28no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4#10418yescomplemented
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4#12312yes—
399C22×C4#12312yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408M16—44no—
409M16—44no—
410M16—44no—
411M16—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—232no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4#11714yescomplemented
426C22⋊C4—232no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—24no—
430C22⋊C4—44no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C42—232no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—28no—
448C42—28no—
449C42—416no—
450C42—44no—
451C42—42no—
452C42—41no—
453C42—41no—
454C42—44no—
455C42—44no—
456C42—42no—
457C42—42no—
458C42—44no—
459C42C4×D414yes—
460C42C4×D414yes—
461C42#12114yes—
462C42#12114yes—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—48no—
475C23—48no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23C2×#10414yes—
485C23C22×C22⋊C411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×#12311yescharacteristic
488C23#52512yes—
489C23#53712yes—
490Q8—24no—
491Q8—42no—
492D4—24no—
493D4—416no—
494D4—44no—
495D4—44no—
496D4—42no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—216no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—216no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4#38612yes—
548C2×C4#38612yes—
549C2×C4#38612yes—
550C2×C4#38612yes—
551C2×C4C2×#10414yes—
552C2×C4C2×#10414yes—
553C2×C4#42114yes—
554C2×C4C2×#11711yescharacteristic
555C2×C4C2×#11711yescharacteristic
556C2×C4C2×C4×D412yes—
557C2×C4#52012yes—
558C2×C4C2×#12112yes—
559C2×C4#52512yes—
560C2×C4#53511yescharacteristic
561C2×C4#53611yescharacteristic
562C2×C4#54612yes—
563C2×C4#54712yes—
564C2×C4#54812yes—
565C2×C4#55012yes—
566C2×C4#55212yes—
567C2×C4#55312yes—
568C2×C4#55512yes—
569C2×C4#55612yes—
570C8—84no—
571C22—24no—
572C22—24no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—48no—
581C22—48no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22#228011yescharacteristic
592C22C2×#38611yescharacteristic
593C22#231111yescharacteristic
594C22#240712yes—
595C22#248512yes—
596C22C22×#10412yes—
597C22C2×#42112yes—
598C4—22no—
599C4—24no—
600C4—24no—
601C4—22no—
602C4—22no—
603C4—416no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—81no—
610C4#189614yes—
611C4#235212yes—
612C4#235412yes—
613C4#241412yes—
614C4#241512yes—
615C2—22no—
616C2—22no—
617C2—44no—
618C2—82no—
619C2#1387111yescharacteristic
620C2#2074712yes—
621C2C2×#189611yescharacteristic
622C2#2097911yescharacteristic
623C2#2185612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#117C2semidirect
3. C2×#1806C2semidirect
4. #20626C2semidirect
5. #11201C2semidirect
7. #10923C2semidirect
8. #10878C2semidirect
9. #10852C2semidirect
17. C4×C22⋊Q8C4semidirect
18. C4×C22⋊Q8C4semidirect
21. C4×#121C4semidirect
22. C4×#121C4semidirect
125. C4×C4⋊C4D4semidirect
315. C2×Q8#117semidirect
326. C2×D4#117semidirect
388. C22×C4#104semidirect
425. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×C4×#1172
C2×#18062
#206262
#112012
#109712
#109232
#108782
#108522