Finite Group Databasebeta

← #360072

Subgroups — #360072

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3C2×#1806C212yesmaximal, complemented
4#20626C211yesmaximal, characteristic, complemented
5#11201C212yesmaximal, complemented
6#10971C212yesmaximal
7#10924C212yesmaximal, complemented
8#10879C212yesmaximal, complemented
9#10852C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#423C2212yes—
14C22×#104C2214yes—
15C2×#407C2212yes—
16C4×C22⋊Q8C412yescomplemented
17C4×C22⋊Q8C412yescomplemented
18C4×#123—21no—
19C4×#123—21no—
20C4×#121C412yescomplemented
21C4×#121C412yescomplemented
22C4×#120—21no—
23C4×#120—21no—
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#2290—22no—
27#2283—28no—
28C2×#383—21no—
29C2×#382—21no—
30#2280C2211yescharacteristic
31C2×#381—21no—
32C2×#380—21no—
33C42×D4—22no—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×#376C2212yes—
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—21no—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—24no—
74C22×C42—24no—
75C22×C42—21no—
76C22×C42C2×C412yes—
77C22×C42C2×C412yes—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D412yes—
81C22×C42D412yes—
82C22×C42C2314yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#423—24no—
86#423—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×#104—28no—
90C2×#104—28no—
91C2×#104—44no—
92#407—44no—
93C2×C22⋊C8—24no—
94#391C2×C412yes—
95#390C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#386—22no—
104#386—21no—
105#386—21no—
106#385—22no—
107#385—22no—
108#385—21no—
109#385—21no—
110#383—28no—
111#382—28no—
112#381—28no—
113#381—21no—
114#380—28no—
115#380C2×C412yes—
116#380C2×C412yes—
117#379—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D418yescomplemented
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4—22no—
134#376—28no—
135#376—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C43—21no—
141#339—28no—
142#339—28no—
143#339—44no—
144#336—24no—
145C25C2×D411yescharacteristic
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41#4114yes—
152C22×Q8C22×C411yescharacteristic
153C22×D4—22no—
154C22×D4—21no—
155C22×D4C22×C411yescharacteristic
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4C22⋊C412yes—
173C23×C4C22⋊C412yes—
174C23×C4C22⋊C412yes—
175C23×C4C22⋊C412yes—
176C23×C4C22×C412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C22×C8—24no—
191#123—28no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194#121—24no—
195#121—24no—
196#120—28no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×D4—216no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—44no—
214#117#4114yes—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—48no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—44no—
262C2×C42C22⋊C412yes—
263C2×C42C22⋊C412yes—
264C2×C42C22⋊C412yes—
265C2×C42C22⋊C412yes—
266C2×C42C22×C412yes—
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D418yes—
270C2×C42C2×D412yes—
271C2×C42C2×D414yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4114yes—
275C4⋊C8—44no—
276C4⋊C8—44no—
277#104—416no—
278C22⋊C8—48no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—28no—
291C24—21no—
292C24—21no—
293C24—24no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—42no—
298C24C2×C22⋊C412yes—
299C24C22×D411yescharacteristic
300#41—24no—
301#41—24no—
302#41—24no—
303#41—24no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—42no—
308C2×Q8—21no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8#11712yes—
312C2×D4—28no—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—44no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4#11712yes—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—28no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12312yes—
395C22×C4#12312yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404M16—44no—
405M16—44no—
406M16—44no—
407M16—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C4⋊C4—28no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—416no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4#11714yes—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—416no—
428C22⋊C4—44no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—28no—
443C42—28no—
444C42—44no—
445C42—42no—
446C42—41no—
447C42—41no—
448C42—44no—
449C42—44no—
450C42—42no—
451C42—42no—
452C42—44no—
453C42C4×D414yes—
454C42C4×D414yes—
455C42#12114yes—
456C42#12114yes—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—48no—
469C23—48no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23#42114yes—
479C23C22×C22⋊C411yescharacteristic
480C23C2×#12011yescharacteristic
481C23C2×#12311yescharacteristic
482C23#52512yes—
483C23#53712yes—
484Q8—24no—
485Q8—42no—
486D4—24no—
487D4—416no—
488D4—44no—
489D4—44no—
490D4—42no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—48no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—88no—
541C2×C4—88no—
542C2×C4—84no—
543C2×C4#38612yes—
544C2×C4#38612yes—
545C2×C4#38612yes—
546C2×C4#38612yes—
547C2×C4#42114yes—
548C2×C4C2×#11711yescharacteristic
549C2×C4C2×#11711yescharacteristic
550C2×C4C2×C4×D412yes—
551C2×C4#52012yes—
552C2×C4C2×#12112yes—
553C2×C4#52512yes—
554C2×C4#53511yescharacteristic
555C2×C4#53611yescharacteristic
556C2×C4#54612yes—
557C2×C4#54712yes—
558C2×C4#54812yes—
559C2×C4#55012yes—
560C2×C4#55212yes—
561C2×C4#55312yes—
562C2×C4#55512yes—
563C2×C4#55612yes—
564C8—84no—
565C22—24no—
566C22—24no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—48no—
575C22—48no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—82no—
583C22—81no—
584C22—81no—
585C22#228011yescharacteristic
586C22C2×#38611yescharacteristic
587C22#231111yescharacteristic
588C22#240712yes—
589C22#248512yes—
590C22C2×#42112yes—
591C22C2×#42112yes—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—88no—
601C4—82no—
602C4—81no—
603C4—81no—
604C4#189714yes—
605C4#235212yes—
606C4#235412yes—
607C4#241412yes—
608C4#241512yes—
609C2—22no—
610C2—22no—
611C2—44no—
612C2—82no—
613C2#1387111yescharacteristic
614C2#2074812yes—
615C2C2×#189711yescharacteristic
616C2#2097911yescharacteristic
617C2#2185712yes—
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×#2283C2semidirect
3. C2×#1806C2semidirect
4. #20626C2semidirect
5. #11201C2semidirect
7. #10924C2semidirect
8. #10879C2semidirect
9. #10852C2semidirect
16. C4×C22⋊Q8C4semidirect
17. C4×C22⋊Q8C4semidirect
20. C4×#121C4semidirect
21. C4×#121C4semidirect
124. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#22832
C2×#18062
#206262
#112012
#109712
#109242
#108792
#108522