Finite Group Databasebeta

← #214741

Subgroups — #214741

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17368C211yesmaximal, characteristic, complemented
4#9474C212yesmaximal, complemented
5#9386C212yesmaximal, complemented
6#9204C212yesmaximal, complemented
7#9027C212yesmaximal, complemented
8#8761C212yesmaximal, complemented
9#8376C212yesmaximal, complemented
10C2×#547—21no—
11C2×#547C2211yescharacteristic
12C2×#520C2211yescharacteristic
13#2973C2212yes—
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#1467—21no—
44#1465—21no—
45#1464—21no—
46#1463—21no—
47#1459C2212yes—
48C2×#324C2212yes—
49#1294C2214yes—
50#1289C2214yes—
51C22×#41—21no—
52C22×#41D411yescharacteristic
53#547—44no—
54#547—48no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#123—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#123—21no—
62C2×#123—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121C2×C411yescharacteristic
69C2×#121C2311yescharacteristic
70#520—44no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×#117—22no—
83C2×#117C2×C411yescharacteristic
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93#435—44no—
94#432—22no—
95#410—44no—
96C2×#99—22no—
97C2×#99—22no—
98#407—22no—
99C2×C22⋊C8C2314yes—
100#398—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398C2×C411yescharacteristic
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#397—21no—
115#397C2×C411yescharacteristic
116#397C2×C411yescharacteristic
117#397C2×C411yescharacteristic
118#396—21no—
119#396—21no—
120#396—21no—
121#396—21no—
122#396—21no—
123#394—21no—
124#394C2×C411yescharacteristic
125#394C2×C411yescharacteristic
126#394C2×C411yescharacteristic
127#392C2×C411yescharacteristic
128#391—21no—
129#391—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—44no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#386—21no—
146#385—21no—
147#385—21no—
148#382C2×C411yescharacteristic
149#381—21no—
150#381—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—22no—
155#380C2×C411yescharacteristic
156#380C2314yes—
157#380C2311yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C22⋊C4—21no—
160C2×#95—21no—
161C2×#95—21no—
162C2×#95C2×C411yescharacteristic
163C2×#95C2311yescharacteristic
164#342—22no—
165#342—22no—
166#340—44no—
167#327—44no—
168#324—22no—
169#324—22no—
170#324—22no—
171#323—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—24no—
175C2×#41—42no—
176C2×#41—44no—
177C2×#41—44no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8—21no—
181C22×Q8C22⋊C411yescharacteristic
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C22⋊C411yescharacteristic
186C22×D4C22×C411yescharacteristic
187C22×D4C2×D411yescharacteristic
188C22×D4C2×D411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2411yescharacteristic
203C2×M16—42no—
204C22×C8—24no—
205C4⋊Q8—42no—
206C4⋊Q8—44no—
207#123—44no—
208#123—44no—
209#123—44no—
210#123—44no—
211#123—44no—
212#123—44no—
213C22⋊Q8—44no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216#121—24no—
217#121—42no—
218#121—44no—
219#121—44no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—44no—
230C4×D4—44no—
231#117—42no—
232#117—42no—
233#117—42no—
234#117—44no—
235#117—44no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
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—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
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—24no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22×C411yescharacteristic
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D411yescharacteristic
297C2×C42—21no—
298C2×C42—42no—
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C4⋊C8—42no—
306#99—44no—
307#99—44no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—42no—
311#97—42no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
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—24no—
328#95—42no—
329#95C22×C411yescharacteristic
330#95C22×C411yescharacteristic
331#95C22×C411yescharacteristic
332#95C22×C411yescharacteristic
333C24—21no—
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C22×D411yescharacteristic
337#41—48no—
338#41—48no—
339#41—88no—
340#41—88no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4AGL(1,ℤ/8)12yes—
369C2×D4#13712yes—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
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—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4AGL(1,ℤ/8)14yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4H(5,𝔽2)14yes—
428M16—162no—
429C2×C8—44no—
430C2×C8—84no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—84no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—84no—
462C4⋊C4—162no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C42—22no—
487C42—22no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42—84no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23#41712yes—
513C23C22×C22⋊C411yescharacteristic
514C23#53412yes—
515C23C2×AGL(1,ℤ/8)12yes—
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23C2×#13711yescharacteristic
518Q8—44no—
519Q8—84no—
520D4—24no—
521D4—24no—
522D4—44no—
523D4—42no—
524D4—42no—
525D4—82no—
526D4—84no—
527D4—84no—
528D4—82no—
529D4—82no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
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—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
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—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—82no—
580C2×C4—82no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—161no—
592C2×C4—161no—
593C2×C4#41712yes—
594C2×C4#41812yes—
595C2×C4#41812yes—
596C2×C4#51812yes—
597C2×C4#53412yes—
598C8—84no—
599C22—21no—
600C22—22no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#230811yescharacteristic
624C22C2×#41811yescharacteristic
625C22C2×#41711yescharacteristic
626C22#319212yes—
627C22#319312yes—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—164no—
648C2—21no—
649C2—22no—
650C2—21no—
651C2—22no—
652C2—42no—
653C2—81no—
654C2—161no—
655C2#2176411yescharacteristic
656C2#2177711yescharacteristic
657C2#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. #17368C2semidirect
4. #9474C2semidirect
5. #9386C2semidirect
6. #9204C2semidirect
7. #9027C2semidirect
8. #8761C2semidirect
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
#173682
#94742
#93862
#92042
#90272
#87612
#83762