Finite Group Databasebeta

← #214709

Subgroups — #214709

Order 512.

1119 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#9470C212yesmaximal, complemented
5#9384C212yesmaximal, complemented
6#9204C212yesmaximal, complemented
7#9027C212yesmaximal, complemented
8#8760C212yesmaximal, complemented
9#8376C212yesmaximal, complemented
10C2×#547—21no—
11C2×#547C2211yescharacteristic
12C2×#520C2211yescharacteristic
13#2972C2212yes—
14C2×#410—21no—
15#2621C411yescharacteristic, complemented
16#2618C411yescharacteristic, complemented
17#2618C2211yescharacteristic
18#2546C411yescharacteristic, complemented
19#2543C411yescharacteristic, complemented
20#2543C2211yescharacteristic
21#2536C411yescharacteristic, complemented
22#2486C411yescharacteristic, complemented
23#2486C2211yescharacteristic
24#2429—21no—
25#2425—21no—
26#2425—21no—
27#2417—21no—
28#2413—21no—
29#2413—21no—
30#2392C411yescharacteristic, complemented
31C2×#398C411yescharacteristic, complemented
32C2×#398C2211yescharacteristic
33#2349C2211yescharacteristic
34#2339—21no—
35#2316C2212yes—
36#1823C2214yes—
37#1799—21no—
38#1799—21no—
39#1799C2212yes—
40#1799C2212yes—
41#1740—21no—
42#1519C2214yes—
43#1461—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#324—21no—
48C2×#324C2212yes—
49#1294C2214yes—
50#1289C2214yes—
51C22×#41—21no—
52C22×#41D411yescharacteristic
53#547—44no—
54#547—44no—
55#547—44no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#123—21no—
62C2×#123—21no—
63C2×#123—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2311yescharacteristic
71#520—44no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117—22no—
84C2×#117C2×C411yescharacteristic
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C8×D4—44no—
95#431—22no—
96#410—44no—
97C2×#99—22no—
98C2×#99—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101#398—21no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#398C2×C411yescharacteristic
107#398C2×C411yescharacteristic
108#398C2×C411yescharacteristic
109#398C2×C411yescharacteristic
110#398C2×C411yescharacteristic
111#398C2×C411yescharacteristic
112#398C2×C411yescharacteristic
113#398C2×C411yescharacteristic
114#398C2×C411yescharacteristic
115#397—21no—
116#397C2×C411yescharacteristic
117#397C2×C411yescharacteristic
118#397C2×C411yescharacteristic
119#396—21no—
120#396—21no—
121#396—21no—
122#396—21no—
123#396—21no—
124#394—21no—
125#394C2×C411yescharacteristic
126#394C2×C411yescharacteristic
127#394C2×C411yescharacteristic
128#392C2×C411yescharacteristic
129#391—21no—
130#391—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—44no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#386—21no—
146#386—21no—
147#385—21no—
148#385—21no—
149#382C2×C411yescharacteristic
150#381—21no—
151#381—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—22no—
156#380C2×C411yescharacteristic
157#380C2314yes—
158#380C2311yescharacteristic
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C22⋊C4—21no—
161C2×#95—21no—
162C2×#95—21no—
163C2×#95C2×C411yescharacteristic
164C2×#95C2311yescharacteristic
165#342—22no—
166#342—22no—
167#340—44no—
168#327—44no—
169#324—22no—
170#324—22no—
171#324—22no—
172#323—22no—
173C2×#41—42no—
174C2×#41—42no—
175C2×#41—44no—
176C2×#41—44no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41D814yes—
180C2×#41SD1614yes—
181C22×Q8—21no—
182C22×Q8C22⋊C411yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22×C411yescharacteristic
188C22×D4C2×D411yescharacteristic
189C22×D4C2×D411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2411yescharacteristic
204C22×C8—24no—
205C22×C8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—44no—
208#123—44no—
209#123—44no—
210#123—44no—
211#123—42no—
212#123—42no—
213#123—44no—
214#123—42no—
215#123—42no—
216C22⋊Q8—44no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219#121—24no—
220#121—42no—
221#121—44no—
222#121—42no—
223#121—42no—
224C4×Q8—42no—
225C4×Q8—42no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—44no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117—44no—
240#117—44no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22×C411yescharacteristic
300C2×C22⋊C4C2×D411yescharacteristic
301C2×C22⋊C4C2×D411yescharacteristic
302C2×C22⋊C4C2×D411yescharacteristic
303C2×C42—21no—
304C2×C42—42no—
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C4⋊C8—42no—
312#99—44no—
313#99—44no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—42no—
317C4×C8—42no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—24no—
334#95—42no—
335#95C22×C411yescharacteristic
336#95C22×C411yescharacteristic
337#95C22×C411yescharacteristic
338#95C22×C411yescharacteristic
339C24—21no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24C22×D411yescharacteristic
343#41—48no—
344#41—48no—
345#41—88no—
346#41—88no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8#10214yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4#10214yes—
372C2×D4C2×D812yes—
373C2×D4C2×D812yes—
374C2×D4C2×SD1612yes—
375C2×D4C2×SD1612yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4#10214yes—
419C22×C4#10214yes—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×D812yes—
431C22×C4C2×SD1612yes—
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4H(5,𝔽2)14yes—
435C2×C8—44no—
436C2×C8—84no—
437C2×C8—162no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—84no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C4⋊C4—162no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—42no—
501C42—84no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×#10212yes—
520C23C2×#10212yes—
521C23C22×C22⋊C411yescharacteristic
522C23#53412yes—
523C23C22×D811yescharacteristic
524C23C22×SD1611yescharacteristic
525C23#57512yes—
526Q8—44no—
527Q8—84no—
528D4—24no—
529D4—24no—
530D4—24no—
531D4—24no—
532D4—44no—
533D4—82no—
534D4—84no—
535D4—84no—
536D4—82no—
537D4—82no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—161no—
600C2×C4—161no—
601C2×C4C2×#10212yes—
602C2×C4C2×#10212yes—
603C2×C4C2×#10212yes—
604C2×C4C2×#10212yes—
605C2×C4#51812yes—
606C2×C4#53412yes—
607C8—84no—
608C22—21no—
609C22—22no—
610C22—21no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22#230811yescharacteristic
633C22C22×#10211yescharacteristic
634C22#289611yescharacteristic
635C22#318612yes—
636C22#318712yes—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—164no—
657C2—21no—
658C2—22no—
659C2—21no—
660C2—22no—
661C2—42no—
662C2—42no—
663C2—161no—
664C2#2175811yescharacteristic
665C2#2177711yescharacteristic
666C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36020C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17351C2semidirect
4. #9470C2semidirect
5. #9384C2semidirect
6. #9204C2semidirect
7. #9027C2semidirect
8. #8760C2semidirect
9. #8376C2semidirect
15. #2621C4semidirect
16. #2618C4semidirect
18. #2546C4semidirect
19. #2543C4semidirect
21. #2536C4semidirect
22. #2486C4semidirect
30. #2392C4semidirect
31. C2×#398C4semidirect

Maximal subgroups

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

subgroupindex
#360202
#206342
#173512
#94702
#93842
#92042
#90272
#87602
#83762