Finite Group Databasebeta

← #219862

Subgroups — #219862

Order 512.

1411 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36126C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3C2×#1502C212yesmaximal, complemented
4#13821C211yesmaximal, characteristic, complemented
5#9712C211yesmaximal, characteristic, complemented
6#9680C211yesmaximal, characteristic, complemented
7#9148C212yesmaximal, complemented
8#9114C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic, complemented
10#8332C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C2×#532C2212yes—
14C2×#431C2212yes—
15#2856C2211yescharacteristic
16#2854C2212yes—
17#2551C411yescharacteristic, complemented
18#2541C411yescharacteristic, complemented
19#2491—21no—
20#2461—22no—
21#2452C412yescomplemented
22#2445C412yescomplemented
23#2430—22no—
24#2416—21no—
25#2366C411yescharacteristic, complemented
26#2361—21no—
27D4×C4⋊C4C411yescharacteristic, complemented
28C4×#121C2211yescharacteristic
29C2×#389—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1939—21no—
34#1916—22no—
35#1817C2212yes—
36#1797C2211yescharacteristic
37C8×C4⋊C4—21no—
38C4×C22⋊C8C2211yescharacteristic
39#1502—28no—
40C2×#327C2211yescharacteristic
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45C2×#327C2212yes—
46C2×#327C2211yescharacteristic
47#1436—22no—
48C4⋊#95C2211yescharacteristic
49#1434C2212yes—
50C4×#95C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54C22×#41—22no—
55#547—44no—
56#547—44no—
57#532—24no—
58#532—24no—
59#532—42no—
60#532—42no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8C2×C412yes—
63C2×#127C2×C412yes—
64C2×#127C2×C411yescharacteristic
65C2×#124—22no—
66C2×#123—22no—
67C2×#123—22no—
68C2×C22⋊Q8—22no—
69C2×#121—21no—
70C2×#121—22no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121C2312yes—
75C2×#121C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C412yes—
81C2×C4×D4C2×C411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42—22no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#431—24no—
92#431—42no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2311yescharacteristic
97C2×C22⋊C8C2311yescharacteristic
98C2×C4×C8—22no—
99C2×C4×C8—21no—
100#396—22no—
101#395—22no—
102#394—22no—
103#394—22no—
104#393—22no—
105#392—22no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390C2×C412yes—
110#390C2×C412yes—
111#390C2×C412yes—
112#390C2×C411yescharacteristic
113#389—28no—
114#389—21no—
115#389C2×C412yes—
116#389C2×C412yes—
117#389C2×C411yescharacteristic
118#388—22no—
119#388—22no—
120#388—21no—
121#386—22no—
122#386—21no—
123#386—22no—
124#386—22no—
125#385—22no—
126#385—22no—
127#384C2×C411yescharacteristic
128#380—21no—
129#380C2311yescharacteristic
130C4×C4⋊C4—24no—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D412yes—
139C4×C4⋊C4D411yescharacteristic
140C4×C22⋊C4—21no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95D412yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149#336—21no—
150#336—22no—
151#336—22no—
152#336—21no—
153#327—24no—
154#327—22no—
155#327—24no—
156#327—24no—
157#327—42no—
158#327—42no—
159#327D414yescomplemented
160C2×#41—44no—
161C2×#41—44no—
162C2×#41—44no—
163C2×#41—44no—
164C2×#41—82no—
165C2×#41—82no—
166C22×Q8—22no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—21no—
173C22×D4C22×C412yes—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C4⋊Q8—24no—
190C4⋊Q8—42no—
191C4⋊Q8—44no—
192#127—24no—
193#127—42no—
194#127—42no—
195#124—44no—
196#124—44no—
197#123—44no—
198#123—44no—
199#123—48no—
200C22⋊Q8—48no—
201#121—21no—
202#121—21no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121—44no—
208#121—44no—
209#121—42no—
210#121—42no—
211#121—44no—
212#121—44no—
213#121—81no—
214#121—81no—
215#121C2×D412yes—
216#121#4112yes—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—44no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C412yes—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4D814yes—
248C2×C4⋊C4SD1614yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—82no—
279C2×C42C22×C412yes—
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C412yes—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C4⋊C8—44no—
288C4⋊C8—42no—
289C22⋊C8—21no—
290C22⋊C8—22no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—22no—
294C22⋊C8C2×D412yes—
295C4×C8—44no—
296C4×C8—42no—
297#95—28no—
298#95—21no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95—21no—
303#95—21no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95—22no—
308#95—42no—
309#95—44no—
310#95—81no—
311#95C22⋊C412yes—
312#95C22⋊C412yes—
313#95C22⋊C412yes—
314#95C22⋊C412yes—
315#95C2×D412yes—
316C24—21no—
317C24—22no—
318C24C22×D411yescharacteristic
319#41—88no—
320#41—88no—
321#41—88no—
322#41—88no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—82no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4—82no—
354C2×D4—82no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4C4×D412yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×D814yes—
417C22×C4C2×D812yes—
418C22×C4C2×SD1612yes—
419C22×C4C2×SD1614yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C2×C8—22no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—41no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—44no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—48no—
444C4⋊C4—416no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—81no—
450C4⋊C4#10214yes—
451C4⋊C4#10214yes—
452C4⋊C4#10218yes—
453C4⋊C4#10414yes—
454C4⋊C4C4×D412yes—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—81no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—48no—
485C42—42no—
486C42—88no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C22×C22⋊C411yescharacteristic
506C23D4211yescharacteristic
507C23#54611yescharacteristic
508C23C22×D811yescharacteristic
509C23C22×SD1611yescharacteristic
510C23#57611yescharacteristic
511C23#57711yescharacteristic
512Q8—88no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—88no—
520D4—88no—
521D4—82no—
522D4—82no—
523D4—82no—
524D4—82no—
525D4—82no—
526D4—82no—
527D4—84no—
528D4—162no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—88no—
588C2×C4—88no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—84no—
600C2×C4#41012yes—
601C2×C4C2×#10212yes—
602C2×C4C2×#10214yes—
603C2×C4C2×#10214yes—
604C2×C4C2×#10212yes—
605C2×C4C2×#10414yes—
606C2×C4C2×#10412yes—
607C2×C4C2×C4×D411yescharacteristic
608C2×C4#51811yescharacteristic
609C2×C4D4211yescharacteristic
610C2×C4#54611yescharacteristic
611C8—42no—
612C8—82no—
613C8—81no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22D4×C22⋊C411yescharacteristic
641C22C2×#41011yescharacteristic
642C22#288411yescharacteristic
643C22C22×#10211yescharacteristic
644C22#289711yescharacteristic
645C22C22×#10411yescharacteristic
646C22#290111yescharacteristic
647C22D4×D811yescharacteristic
648C22D4×SD1611yescharacteristic
649C22#329411yescharacteristic
650C22#329511yescharacteristic
651C4—22no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—24no—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—84no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4#150214yescomplemented
670C2—21no—
671C2—21no—
672C2—21no—
673C2—21no—
674C2—82no—
675C2—162no—
676C2C2×#150211yescharacteristic
677C2#1755011yescharacteristic
678C2#1760711yescharacteristic
679C2#1761811yescharacteristic
680C2#2240611yescharacteristic
681C2D4×#10211yescharacteristic
682C2D4×#10411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36126C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22599C2semidirect
3. C2×#1502C2semidirect
4. #13821C2semidirect
5. #9712C2semidirect
6. #9680C2semidirect
7. #9148C2semidirect
8. #9114C2semidirect
9. #8571C2semidirect
10. #8332C2semidirect
11. C4×#327C2semidirect
17. #2551C4semidirect
18. #2541C4semidirect
21. #2452C4semidirect
22. #2445C4semidirect
25. #2366C4semidirect
27. D4×C4⋊C4C4semidirect
159. #327D4semidirect
669. C4#1502semidirect

Maximal subgroups

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

subgroupindex
#361262
#225992
C2×#15022
#138212
#97122
#96802
#91482
#91142
#85712
#83322
C4×#3272