Finite Group Databasebeta

← #358120

Subgroups — #358120

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3C2×C4×#99C212yesmaximal
4C4×#381C211yesmaximal, characteristic, complemented
5#11203C212yesmaximal, complemented
6#11197C212yesmaximal, complemented
7#11043C212yesmaximal
8#10966C212yesmaximal
9C4×#342C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×#99C2214yes—
14C4×#123—22no—
15C4×#123—22no—
16C4×#123—21no—
17C4×#123—21no—
18C4×#123—21no—
19C4×#123—21no—
20C2×#388C412yescomplemented
21C2×#387C412yescomplemented
22C2×#386C2211yescharacteristic
23C2×#385C2211yescharacteristic
24#2283—28no—
25C2×#383—21no—
26C2×#382—21no—
27C2×#381C2212yes—
28C2×#381C2212yes—
29C2×#381C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C412yescomplemented
33C2×C4×C22⋊C4C412yescomplemented
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×#376—21no—
38C4×#99—28no—
39C2×#342C2214yes—
40C2×#342C2214yes—
41C2×#342C2214yes—
42#1434—22no—
43C22⋊#95C2212yes—
44C4×#95—22no—
45C23×D4C2311yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C2×#123—24no—
51C2×#123—21no—
52C2×#123—21no—
53C2×#123—22no—
54C2×#123—24no—
55C2×C4×D4—24no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2311yescharacteristic
73C22×C42—24no—
74C22×C42—24no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42C2×C412yes—
80C22×C42C2×C412yes—
81C22×C42D412yes—
82C22×C42D412yes—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C2×#99—28no—
88C2×#99—28no—
89C2×#99—28no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—21no—
99#386—22no—
100#386—21no—
101#385—21no—
102#385—22no—
103#385—21no—
104#383—28no—
105#382—28no—
106#381—21no—
107#381—21no—
108#381—22no—
109#381—28no—
110#381—28no—
111#380—28no—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4D418yes—
129#376—28no—
130C2×#95—24no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—24no—
136C2×#95—22no—
137C2×#95C2×C412yes—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142#342—28no—
143#342—28no—
144#342—28no—
145C25C22×C411yescharacteristic
146C25C2411yescharacteristic
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4#4114yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D414yes—
189C23×C4#4114yes—
190#123—28no—
191#123—28no—
192#123—216no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—44no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4C2×D418yes—
242C2×C22⋊C4#4118yes—
243C2×C22⋊C4#4114yes—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—28no—
251C2×C42—28no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—28no—
262C2×C42—48no—
263C2×C42—44no—
264C2×C42—48no—
265C2×C42—44no—
266C2×C42C22⋊C412yes—
267C2×C42C22⋊C412yes—
268C2×C42C22⋊C412yes—
269C2×C42C22⋊C412yes—
270C2×C42C22⋊C412yes—
271C2×C42C22⋊C412yes—
272C2×C42C22⋊C412yes—
273C2×C42C22⋊C412yes—
274#99—232no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—216no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—28no—
290#95—48no—
291#95—44no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24#11712yes—
301C24C23×C411yescharacteristic
302C24C22×D411yescharacteristic
303C24C22×D411yescharacteristic
304C24C22×D412yes—
305C24C2×#4112yes—
306C24C2×#4112yes—
307C24C2×#4112yes—
308C24C2×#4114yes—
309C2×D4—28no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—88no—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4#11712yes—
403C22×C4#11714yes—
404C22×C4C4×D414yes—
405C22×C4C4×D414yes—
406C22×C4#12012yes—
407C22×C4#12012yes—
408C22×C4#12114yes—
409C22×C4#12114yes—
410C22×C4#12312yes—
411C22×C4#12312yes—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—416no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—44no—
428C22⋊C4—41no—
429C22⋊C4—42no—
430C22⋊C4—41no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—44no—
449C42—416no—
450C42—42no—
451C42—42no—
452C42—44no—
453C42—88no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—48no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—82no—
477C23—84no—
478C23—82no—
479C23C22×C22⋊C411yescharacteristic
480C23C2×#11711yescharacteristic
481C23C2×#11711yescharacteristic
482C23C2×C4×D412yes—
483C23C4×#4112yes—
484C23C2×#12011yescharacteristic
485C23C2×#12112yes—
486C23C2×#12311yescharacteristic
487C23#52512yes—
488C23#52512yes—
489C23#52512yes—
490C23#52514yes—
491C23#52914yes—
492C23#52914yes—
493C23#53212yes—
494C23#53214yes—
495D4—44no—
496D4—48no—
497D4—44no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—416no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—88no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4#38612yes—
566C2×C4#38612yes—
567C2×C4#38612yes—
568C2×C4#38612yes—
569C2×C4#41014yes—
570C2×C4#41014yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—24no—
578C22—21no—
579C22—22no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—84no—
593C22—84no—
594C22—82no—
595C22—81no—
596C22—81no—
597C22#228011yescharacteristic
598C22C2×#38611yescharacteristic
599C22#229012yes—
600C22#229312yes—
601C22#229711yescharacteristic
602C22#238812yes—
603C22#239214yes—
604C22#239512yes—
605C22C2×#41012yes—
606C22C2×#41012yes—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—24no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—88no—
617C4—88no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4#189914yes—
622C2—22no—
623C2—24no—
624C2—22no—
625C2—22no—
626C2—22no—
627C2—82no—
628C2#1383911yescharacteristic
629C2#2072312yes—
630C2C2×#189911yescharacteristic
631C2#2098311yescharacteristic
632C2#2142112yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC23C22×C22⋊C4
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
4. C4×#381C2semidirect
5. #11203C2semidirect
6. #11197C2semidirect
9. C4×#342C2semidirect
20. C2×#388C4semidirect
21. C2×#387C4semidirect
32. C2×C4×C4⋊C4C4semidirect
33. C2×C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#22832
C2×C4×#992
C4×#3812
#112032
#111972
#110432
#109662
C4×#3422