Finite Group Databasebeta

← #493867

Subgroups — #493867

Order 512.

1911 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34258C211yesmaximal, characteristic
2#20585C212yesmaximal
3#12859C212yesmaximal, complemented
4#12858C212yesmaximal, complemented
5#12857C212yesmaximal, complemented
6#12856C212yesmaximal, complemented
7#12582C212yesmaximal, complemented
8C2×#1393C212yesmaximal, complemented
9C22×H(5,𝔽2)C2211yescharacteristic
10C2×#520C411yescharacteristic, complemented
11C2×C4×#41C411yescharacteristic, complemented
12C22×#99—22no—
13C22×#99C2214yes—
14#2341—22no—
15D4×C22⋊C4—22no—
16D4×C22⋊C4—22no—
17D4×C22⋊C4—22no—
18#2311C411yescharacteristic, complemented
19#2311C411yescharacteristic, complemented
20#2311C2211yescharacteristic
21#2309C411yescharacteristic, complemented
22#2309C2211yescharacteristic
23#2309C2211yescharacteristic
24#2308—22no—
25#2297C411yescharacteristic, complemented
26#2297C411yescharacteristic, complemented
27#2297C2211yescharacteristic
28C2×#390C411yescharacteristic, complemented
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31#1782—22no—
32#1782C2214yes—
33#1781—22no—
34#1781C2214yes—
35#1393C2214yescomplemented
36#1393C2214yescomplemented
37#1393C2214yescomplemented
38#1393C2214yescomplemented
39C2×H(5,𝔽2)—24no—
40C2×H(5,𝔽2)—24no—
41C2×H(5,𝔽2)—44no—
42C2×H(5,𝔽2)—44no—
43C22×#41C2×C411yescharacteristic
44C22×#41C2×C411yescharacteristic
45C22×#41C2×C411yescharacteristic
46C22×#41D412yes—
47C22×#41C2311yescharacteristic
48C23×D4C2×C411yescharacteristic
49C23×D4D412yes—
50C23×D4D412yes—
51C23×D4D412yes—
52C23×D4C2311yescharacteristic
53C23×D4C2311yescharacteristic
54#520—44no—
55C4×#41—44no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic, complemented
67C2×#117C2×C411yescharacteristic, complemented
68C2×#117C2×C411yescharacteristic, complemented
69C2×#117C2×C411yescharacteristic, complemented
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic, complemented
72C22×C4⋊C4C2×C411yescharacteristic, complemented
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—24no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic, complemented
79C22×C42C2×C411yescharacteristic, complemented
80C2×#99—28no—
81C2×#99—28no—
82C2×#99—24no—
83C2×#99—24no—
84C2×#99—24no—
85C2×#99—48no—
86C2×#99C2318yes—
87#391C2×C411yescharacteristic, complemented
88#391C2×C411yescharacteristic, complemented
89#391C2×C411yescharacteristic, complemented
90#391C2×C411yescharacteristic, complemented
91#390—22no—
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390—22no—
97#390C2×C411yescharacteristic
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#390C2×C411yescharacteristic, complemented
102#390C2×C411yescharacteristic, complemented
103#390C2×C411yescharacteristic, complemented
104#390C2×C411yescharacteristic, complemented
105#390C2311yescharacteristic
106#390C2311yescharacteristic
107#390C2311yescharacteristic
108#390C2311yescharacteristic
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—48no—
116#381—24no—
117#380—22no—
118#380—22no—
119#380—22no—
120#379—22no—
121#379—22no—
122#379—22no—
123#379—24no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C25C22⋊C412yes—
128C25C2×D412yes—
129C25C2×D412yes—
130H(5,𝔽2)—416no—
131H(5,𝔽2)—432no—
132H(5,𝔽2)—416no—
133C2×#41—24no—
134C2×#41—24no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—28no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—44no—
151C2×#41—44no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22×C411yescharacteristic
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—28no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4C22⋊C411yescharacteristic
174C22×D4C22⋊C412yes—
175C22×D4C22⋊C412yes—
176C22×D4C22⋊C412yes—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C22×D4C22×C411yescharacteristic
181C22×D4C2×D412yes—
182C22×D4C2×D412yes—
183C22×D4C2×D412yes—
184C22×D4#4114yes—
185C22×D4#4114yes—
186C22×D4C2411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—44no—
194C23×C4C22⋊C412yes—
195C23×C4C22⋊C412yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D412yes—
200C4×Q8—44no—
201C4×Q8—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—84no—
252C2×C22⋊C4—84no—
253C2×C22⋊C4—84no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267#99—28no—
268#99—28no—
269#99—48no—
270#99—48no—
271#99—48no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—48no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—42no—
284C24—42no—
285C24—82no—
286C24—82no—
287C24—82no—
288C24—82no—
289C24—82no—
290C24—82no—
291C24C2×C22⋊C412yes—
292C24C2×C22⋊C412yes—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24#12318yes—
297C24C22×D412yes—
298#41—216no—
299#41—216no—
300#41—48no—
301#41—48no—
302#41—48no—
303#41—48no—
304#41—48no—
305#41—48no—
306#41—432no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—432no—
358C2×D4#9914yes—
359C2×D4#9914yes—
360C2×D4#11712yes—
361C2×D4#11712yes—
362C2×D4#12318yes—
363C2×D4C2×#4112yes—
364C2×D4C2×#4112yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—48no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—88no—
409C22×C4—164no—
410C22×C4—164no—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4C23×C411yescharacteristic
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—88no—
449C22⋊C4—164no—
450C22⋊C4—164no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—28no—
468C23—28no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—48no—
479C23—48no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—88no—
493C23—88no—
494C23—88no—
495C23—162no—
496C23—162no—
497C23—164no—
498C23—164no—
499C23—164no—
500C23#38114yes—
501C23#38112yes—
502C23C2×#9912yes—
503C23C2×#9912yes—
504C23C2×#9912yes—
505C23C2×#9912yes—
506C23C22×C22⋊C411yescharacteristic
507C23C2×#12314yes—
508Q8—28no—
509Q8—42no—
510Q8—42no—
511D4—28no—
512D4—28no—
513D4—28no—
514D4—42no—
515D4—42no—
516D4—48no—
517D4—42no—
518D4—42no—
519D4—48no—
520D4—48no—
521D4—42no—
522D4—42no—
523D4—48no—
524D4—48no—
525D4—48no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4—162no—
573C2×C4—162no—
574C2×C4—162no—
575C2×C4—1616no—
576C2×C4#38112yes—
577C2×C4C2×#9912yes—
578C2×C4C2×#9912yes—
579C2×C4C2×#11711yescharacteristic
580C2×C4C2×#11711yescharacteristic
581C22—21no—
582C22—22no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—24no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—88no—
610C22—82no—
611C22—82no—
612C22—88no—
613C22—82no—
614C22—168no—
615C22—168no—
616C22#181914yes—
617C22C2×#38111yescharacteristic
618C22C22×#9911yescharacteristic
619C22C22×#9911yescharacteristic
620C4—21no—
621C4—21no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—168no—
635C2—21no—
636C2—21no—
637C2—24no—
638C2—44no—
639C2—42no—
640C2—42no—
641C2—44no—
642C2—82no—
643C2C2×#181911yescharacteristic
644C2#3335911yescharacteristic
645C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12859C2semidirect
4. #12858C2semidirect
5. #12857C2semidirect
6. #12856C2semidirect
7. #12582C2semidirect
8. C2×#1393C2semidirect
10. C2×#520C4semidirect
11. C2×C4×#41C4semidirect
18. #2311C4semidirect
19. #2311C4semidirect
21. #2309C4semidirect
25. #2297C4semidirect
26. #2297C4semidirect
28. C2×#390C4semidirect
35. #1393C22semidirect
36. #1393C22semidirect
37. #1393C22semidirect
38. #1393C22semidirect
66. C2×#117C2×C4semidirect
67. C2×#117C2×C4semidirect
68. C2×#117C2×C4semidirect
69. C2×#117C2×C4semidirect
71. C22×C4⋊C4C2×C4semidirect
72. C22×C4⋊C4C2×C4semidirect
78. C22×C42C2×C4semidirect
79. C22×C42C2×C4semidirect
87. #391C2×C4semidirect
88. #391C2×C4semidirect
89. #391C2×C4semidirect
90. #391C2×C4semidirect
101. #390C2×C4semidirect
102. #390C2×C4semidirect
103. #390C2×C4semidirect
104. #390C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342582
#205852
#128592
#128582
#128572
#128562
#125822
C2×#13932