Finite Group Databasebeta

← #196506

Subgroups — #196506

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic, complemented
4C2×#1527C212yesmaximal, complemented
5#9123C214yesmaximal, complemented
6#9119C214yesmaximal, complemented
7#8033C212yesmaximal, complemented
8C22×#124C2211yescharacteristic
9C22×C22⋊Q8C411yescharacteristic
10C22×#121C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C2×#432—21no—
14C2×#432—28no—
15C2×#432C2212yes—
16C2×#432C2212yescomplemented
17C22×C4⋊C8—21no—
18C2×#407—21no—
19C22×C22⋊C8—21no—
20C2×C4×M16C2212yes—
21C22×#97C2212yescomplemented
22#2850—21no—
23#2850—21no—
24#2850—21no—
25#2850—21no—
26#2850C412yes—
27#2850C412yes—
28#2850C2212yes—
29#2850C2212yes—
30C2×#391C411yescharacteristic
31C2×#390C411yescharacteristic
32C2×#388—21no—
33C2×#388—21no—
34C2×#386—21no—
35C2×#386—21no—
36C2×#382C2211yescharacteristic
37#1749C2212yescomplemented
38#1749C2212yescomplemented
39C4×C22⋊C8—21no—
40C4×C22⋊C8—21no—
41#1728—21no—
42#1527—28no—
43C2×#329C2218yescomplemented
44#1295C2218yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—22no—
52C23×C8—22no—
53C2×#124—21no—
54C2×#124—21no—
55C2×#124—24no—
56C2×#124—24no—
57C2×#124C2×C412yes—
58C2×#124C2312yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—44no—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—44no—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93#432—28no—
94#432—24no—
95#432—28no—
96#432D418yescomplemented
97C2×C4⋊C8—28no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100#407—28no—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C4×M16—24no—
107C4×M16—24no—
108C2×#97—22no—
109C2×#97—28no—
110C2×#97C2314yes—
111C2×#97C2314yes—
112C2×C4×C8—22no—
113#391—21no—
114#391—21no—
115#391C2×C412yes—
116#391C2×C412yes—
117#390—21no—
118#390—21no—
119#390C2×C412yes—
120#390C2×C412yes—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#386—22no—
137#382—21no—
138#382—21no—
139#382C2×C412yes—
140#382C2312yes—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145#336—22no—
146#336—22no—
147#329—216no—
148C25—21no—
149C25C22⋊C411yescharacteristic
150C22×Q8—21no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8C22×C412yes—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—24no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4C22×C412yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×M16—28no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8—28no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—44no—
203C22×C8—44no—
204#124—28no—
205#124—24no—
206#124—24no—
207#124—42no—
208#124—44no—
209#124—42no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214#121—44no—
215#121—44no—
216#121—44no—
217#121—44no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D414yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4114yes—
279C2×C42C2411yescharacteristic
280C4⋊C8—28no—
281C4⋊C8—28no—
282C22⋊C8—28no—
283C22⋊C8—28no—
284C22⋊C8—28no—
285C22⋊C8—28no—
286#97—28no—
287#97—24no—
288#97—24no—
289#97—24no—
290#97C2×D418yescomplemented
291C4×C8—44no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—41no—
306C24—41no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24#10414yes—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C2×Q8—24no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—88no—
325C2×D4—24no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—88no—
338C2×D4—816no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
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—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
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—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4#10014yes—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C4×D414yes—
418C22×C4#12012yes—
419C22×C4#12114yes—
420C22×C4#12312yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424M16—416no—
425C2×C8—28no—
426C2×C8—28no—
427C2×C8—416no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42C4×D414yes—
466C42#12012yes—
467C42#12114yes—
468C42#12312yes—
469C42C22×D412yes—
470C42C2×#4112yes—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—28no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—28no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23#38612yes—
507C23C2×#10012yes—
508C23C2×#10412yes—
509C23C2×#10412yes—
510C23C2×#10412yes—
511C23#42112yes—
512C23#42112yes—
513C23C22×C22⋊C411yescharacteristic
514Q8—84no—
515Q8—84no—
516D4—84no—
517D4—88no—
518D4—88no—
519D4—84no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4#38612yes—
580C2×C4#38612yes—
581C2×C4#38612yes—
582C2×C4C2×#10012yes—
583C2×C4C2×#10012yes—
584C2×C4#41312yes—
585C2×C4#42112yes—
586C2×C4C2×C4×D412yes—
587C2×C4C2×#12011yescharacteristic
588C2×C4C2×#12112yes—
589C2×C4C2×#12311yescharacteristic
590C8—48no—
591C8—48no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—168no—
620C22#152714yescomplemented
621C22#187114yes—
622C22C2×#38611yescharacteristic
623C22C22×#10011yescharacteristic
624C22C2×#41311yescharacteristic
625C22C22×#10411yescharacteristic
626C22C2×#42111yescharacteristic
627C22C2×#42111yescharacteristic
628C22C2×#42111yescharacteristic
629C4—22no—
630C4—22no—
631C4—21no—
632C4—21no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#190014yes—
645C2—24no—
646C2—24no—
647C2—42no—
648C2—162no—
649C2C2×#152711yescharacteristic
650C2#1765011yescharacteristic
651C2#1766811yescharacteristic
652C2#1767111yescharacteristic
653C2C2×#187111yescharacteristic
654C2#2097911yescharacteristic
655C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#432C2semidirect
3. C4×#407C2semidirect
4. C2×#1527C2semidirect
5. #9123C2semidirect
6. #9119C2semidirect
7. #8033C2semidirect
16. C2×#432C22semidirect
21. C22×#97C22semidirect
37. #1749C22semidirect
38. #1749C22semidirect
43. C2×#329C22semidirect
44. #1295C22semidirect
96. #432D4semidirect
290. #97C2×D4semidirect
620. C22#1527semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#28502
C4×#4072
C2×#15272
#91232
#91192
#80332