Finite Group Databasebeta

← #482409

Subgroups — #482409

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11883C212yesmaximal, complemented
5#11879C212yesmaximal, complemented
6#11805C212yesmaximal, complemented
7#11803C212yesmaximal, complemented
8#11783C212yesmaximal, complemented
9#11780C212yesmaximal, complemented
10C2×#547—21no—
11C2×#547C411yescharacteristic
12C2×#520C2211yescharacteristic
13#2975—21no—
14#2921—21no—
15#2621C411yescharacteristic
16#2618C411yescharacteristic
17#2618C2211yescharacteristic
18#2546C2211yescharacteristic
19#2543C411yescharacteristic
20#2543C2211yescharacteristic
21#2536C2211yescharacteristic
22#2486C411yescharacteristic
23#2486C2211yescharacteristic
24#2429—21no—
25#2425—21no—
26#2425—21no—
27#2417—21no—
28#2413—21no—
29#2413—21no—
30#2392C411yescharacteristic
31C2×#398C411yescharacteristic
32C2×#398C2211yescharacteristic
33#2349C411yescharacteristic
34#2339—21no—
35#1821C2212yes—
36#1801C2212yes—
37#1800—21no—
38#1785—21no—
39C4×#101—21no—
40C4×#100C2212yes—
41#1465—21no—
42#1465C2212yes—
43#1463—21no—
44#1463C2212yes—
45#1461C2212yes—
46#1458—21no—
47#1346C2214yes—
48#1346C2214yes—
49#1346C2214yes—
50#1345C2214yes—
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×#121C2×C411yescharacteristic
70#520—44no—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117D411yescharacteristic
84C2×#117D411yescharacteristic
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92#436—44no—
93#433—44no—
94#431—22no—
95#423—22no—
96C2×#101—22no—
97C2×#100—22no—
98#407—22no—
99#407—22no—
100#407—22no—
101C4×M16—22no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#398—21no—
107#398C2×C411yescharacteristic
108#398C2×C411yescharacteristic
109#398C2×C411yescharacteristic
110#398C2×C411yescharacteristic
111#398C2×C411yescharacteristic
112#398C2×C411yescharacteristic
113#398C2311yescharacteristic
114#398C2311yescharacteristic
115#398C2311yescharacteristic
116#397—21no—
117#397C2×C411yescharacteristic
118#397C2×C411yescharacteristic
119#397C2×C411yescharacteristic
120#396—21no—
121#396—21no—
122#396—21no—
123#396—21no—
124#396—21no—
125#394—21no—
126#394C2×C411yescharacteristic
127#394C2×C411yescharacteristic
128#394C2311yescharacteristic
129#392C2×C411yescharacteristic
130#391—21no—
131#391—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
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#380C2×C411yescharacteristic
156#380C2×C411yescharacteristic
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C22⋊C4—21no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95C2×C411yescharacteristic
162C2×#95C2×C411yescharacteristic
163#343—22no—
164#343C2314yes—
165#324—22no—
166#324—22no—
167#324C2314yes—
168#323—22no—
169C2×#41—24no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—44no—
173C2×#41—44no—
174C2×#41—44no—
175C22×Q8—21no—
176C22×Q8C2×D411yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4C22⋊C411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C22×D4C2×D411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C2×M16—24no—
199C2×M16—42no—
200C2×M16—42no—
201C4⋊Q8—42no—
202C4⋊Q8—44no—
203#123—44no—
204#123—44no—
205#123—44no—
206#123—44no—
207#123—44no—
208#123—44no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212#121—42no—
213#121—44no—
214#121—44no—
215#121—42no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—44no—
226C4×D4—44no—
227#117—24no—
228#117—42no—
229#117—42no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
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—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2411yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22×C411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C42—21no—
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C4⋊C8—44no—
295C4⋊C8—42no—
296#101—44no—
297#100—44no—
298C22⋊C8—24no—
299C22⋊C8—44no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302#97—42no—
303C4×C8—42no—
304C4×C8—42no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
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#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323#95C22×C411yescharacteristic
324C24—21no—
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C411yescharacteristic
327C24C2×C22⋊C411yescharacteristic
328#41—416no—
329#41—816no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8AGL(1,ℤ/8)14yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—81no—
355C2×D4—81no—
356C22×C4—21no—
357C22×C4—22no—
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—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—44no—
412M16—44no—
413M16—162no—
414M16—162no—
415C2×C8—44no—
416C2×C8—44no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4#9914yes—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—41no—
472C42—41no—
473C42—42no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×#9912yes—
495C23#42112yes—
496C23C22×C22⋊C411yescharacteristic
497Q8—44no—
498Q8—84no—
499D4—28no—
500D4—44no—
501D4—44no—
502D4—84no—
503D4—82no—
504D4—84no—
505D4—84no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—161no—
563C2×C4—161no—
564C2×C4C2×#9912yes—
565C2×C4C2×#9912yes—
566C2×C4#41812yes—
567C2×C4#41812yes—
568C2×C4C2×AGL(1,ℤ/8)12yes—
569C8—84no—
570C8—84no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C22×#9911yescharacteristic
596C22C2×#41811yescharacteristic
597C22C2×#42111yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—81no—
617C2—21no—
618C2—21no—
619C2—42no—
620C2—42no—
621C2—81no—
622C2—161no—
623C2#1757711yescharacteristic
624C2#3335011yescharacteristic
625C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#36020C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20731C2semidirect
3. #17370C2semidirect
4. #11883C2semidirect
5. #11879C2semidirect
6. #11805C2semidirect
7. #11803C2semidirect
8. #11783C2semidirect
9. #11780C2semidirect

Maximal subgroups

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

subgroupindex
#360202
#207312
#173702
#118832
#118792
#118052
#118032
#117832
#117802