Finite Group Databasebeta

← #354900

Subgroups — #354900

Order 512.

2056 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2275C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3C2×C4×#99C212yesmaximal
4#13778C212yesmaximal, complemented
5#10846C212yesmaximal, complemented
6#10801C212yesmaximal, complemented
7#10761C212yesmaximal
8C4×#342C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#99C2214yes—
13C2×#380C2214yes—
14#2275—28no—
15C2×C4×C4⋊C4C411yescharacteristic, complemented
16C2×C4×C4⋊C4C411yescharacteristic, complemented
17C2×C4×C4⋊C4C411yescharacteristic, complemented
18C2×C4×C22⋊C4C411yescharacteristic, complemented
19C2×C4×C22⋊C4C411yescharacteristic, complemented
20C2×C4×C22⋊C4C411yescharacteristic, complemented
21C2×C4×C22⋊C4C2214yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×#376C411yescharacteristic, complemented
26C22×#95C411yescharacteristic, complemented
27C22×#95C2211yescharacteristic
28#1799C4116yescomplemented
29C4×#99—28no—
30C2×#342C2214yes—
31C2×#342C2214yes—
32C2×#342C2214yes—
33#1436—28no—
34C22⋊#95C2214yes—
35C23×D4C2311yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4C2×C418yes—
50C2×C4×D4D414yes—
51C2×C4×D4Q814yes—
52C22×C4⋊C4—24no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2318yes—
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C42—24no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C2×#99—28no—
74C2×#99—28no—
75C2×#99C2×C4116yes—
76#380—28no—
77#380C2×C4116yescomplemented
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—28no—
88C4×C22⋊C4—28no—
89#376—24no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95D418yes—
106#342—28no—
107#342—28no—
108#342C2×C4116yescomplemented
109C25C22×C411yescharacteristic
110C25C2411yescharacteristic
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22×C414yes—
116C22×D4C2×D412yes—
117C22×D4C2×Q812yes—
118C22×D4#4114yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—28no—
129C23×C4—81no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4C4214yescomplemented
142C23×C4C4⋊C414yescomplemented
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C414yes—
157C23×C4C2×D414yes—
158C23×C4C2×D414yes—
159C23×C4C2×D414yes—
160C23×C4C2×D412yes—
161C23×C4C2×Q812yes—
162C23×C4#4114yes—
163C23×C4#4114yes—
164C4×D4—232no—
165C4×D4—216no—
166C4×D4—216no—
167C4×D4—216no—
168C4×D4—216no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—216no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4—84no—
179C2×C4⋊C4C4214yescomplemented
180C2×C4⋊C4C4⋊C414yescomplemented
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4C4214yescomplemented
200C2×C22⋊C4C4⋊C414yescomplemented
201C2×C22⋊C4C22×C4116yes—
202C2×C22⋊C4C22×C414yes—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×Q812yes—
205C2×C22⋊C4#4114yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—48no—
220C2×C42—84no—
221C2×C42—84no—
222C2×C42—81no—
223C2×C42—81no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42—81no—
227C2×C42—81no—
228C2×C42C4214yescomplemented
229C2×C42C4⋊C414yescomplemented
230#99—232no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—216no—
238#95—48no—
239#95—84no—
240#95—84no—
241#95C22⋊C414yes—
242#95C22⋊C414yes—
243#95C22⋊C414yes—
244#95C22⋊C414yes—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24C2×C4212yes—
258C24C2×C4⋊C412yes—
259C24#11712yes—
260C24C23×C411yescharacteristic
261C24C23×C412yes—
262C24C22×D412yes—
263C24C22×D411yescharacteristic
264C24C22×Q811yescharacteristic
265C24C2×#4112yes—
266C24C2×#4112yes—
267C24C2×#4112yes—
268C24C2×#4112yes—
269C2×D4—28no—
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—28no—
274C2×D4—28no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4C4×D418yes—
278C2×D4C4×Q818yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—84no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4C2×C4212yes—
360C22×C4C2×C4212yes—
361C22×C4C2×C4212yes—
362C22×C4C2×C4212yes—
363C22×C4C2×C4212yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C4⋊C412yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C2×C4⋊C412yes—
374C22×C4C2×C4⋊C412yes—
375C22×C4#11712yes—
376C22×C4#11712yes—
377C22×C4#11712yes—
378C22×C4#11712yes—
379C22×C4#11712yes—
380C22×C4#11712yes—
381C22×C4#11712yes—
382C22×C4C4×D418yes—
383C22×C4C4×D418yes—
384C22×C4#12118yes—
385C22×C4C22⋊Q818yes—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—816no—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—216no—
398C22⋊C4—216no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C42—28no—
402C42—28no—
403C42—28no—
404C42—82no—
405C42—82no—
406C42—82no—
407C42—82no—
408C42—82no—
409C42—82no—
410C42—816no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23C4×C4⋊C414yes—
456C23C22×C4211yescharacteristic
457C23C22×C22⋊C411yescharacteristic
458C23C22×C4⋊C411yescharacteristic
459C23C2×#11711yescharacteristic
460C23C2×#11711yescharacteristic
461C23C2×#11711yescharacteristic
462C23C2×#11711yescharacteristic
463C23C2×C4×D414yes—
464C23C2×C4×D412yes—
465C23C2×C4×Q812yes—
466C23C4×#4114yes—
467C23C2×#12112yes—
468C23C2×C22⋊Q812yes—
469C23#52514yes—
470C23#52914yes—
471C23#53212yes—
472C23#53312yes—
473D4—216no—
474D4—216no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
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—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—88no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4—88no—
552C2×C4C4×C22⋊C414yes—
553C2×C4C4×C22⋊C414yes—
554C2×C4C4×C4⋊C414yes—
555C2×C4#38014yes—
556C2×C4#38014yes—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—24no—
566C22—21no—
567C22—21no—
568C22—24no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—24no—
574C22—21no—
575C22—21no—
576C22—24no—
577C22—24no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22C2×C4×C22⋊C411yescharacteristic
595C22C2×C4×C4⋊C411yescharacteristic
596C22C4×#11711yescharacteristic
597C22C2×#38011yescharacteristic
598C22#228011yescharacteristic
599C22#228311yescharacteristic
600C22#228511yescharacteristic
601C22C4×#12114yes—
602C22C4×C22⋊Q814yes—
603C22#288311yescharacteristic
604C22#288311yescharacteristic
605C22#288311yescharacteristic
606C22#288411yescharacteristic
607C4—24no—
608C4—24no—
609C4—24no—
610C4—24no—
611C4—24no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—816no—
621C2—24no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—24no—
627C2—44no—
628C2C4×#38011yescharacteristic
629C2#2039211yescharacteristic
630C2#2039311yescharacteristic
631C2#2063311yescharacteristic
632C2#2063411yescharacteristic
633C2#2072411yescharacteristic
634C2#2072511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13778C2semidirect
5. #10846C2semidirect
6. #10801C2semidirect
8. C4×#342C2semidirect
15. C2×C4×C4⋊C4C4semidirect
16. C2×C4×C4⋊C4C4semidirect
17. C2×C4×C4⋊C4C4semidirect
18. C2×C4×C22⋊C4C4semidirect
19. C2×C4×C22⋊C4C4semidirect
20. C2×C4×C22⋊C4C4semidirect
25. C2×#376C4semidirect
26. C22×#95C4semidirect
28. #1799C4semidirect
77. #380C2×C4semidirect
108. #342C2×C4semidirect
141. C23×C4C42semidirect
142. C23×C4C4⋊C4semidirect
179. C2×C4⋊C4C42semidirect
180. C2×C4⋊C4C4⋊C4semidirect
199. C2×C22⋊C4C42semidirect
200. C2×C22⋊C4C4⋊C4semidirect
228. C2×C42C42semidirect
229. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22752
C2×#17992
C2×C4×#992
#137782
#108462
#108012
#107612
C4×#3422