Finite Group Databasebeta

← #207459

Subgroups — #207459

Order 512.

973 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36638C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
4#9866C212yesmaximal, complemented
5#9675C212yesmaximal, complemented
6#8939C212yesmaximal
7#8937C212yesmaximal, complemented
8#8183C212yesmaximal, complemented
9#8182C212yesmaximal, complemented
10C22×C4⋊C8C2212yes—
11C22×C4×C8—21no—
12#2866—21no—
13#2865—21no—
14#2864C411yescharacteristic
15#2863C411yescharacteristic
16#2861C411yescharacteristic
17#2860C411yescharacteristic
18#2855C411yescharacteristic
19#2854C411yescharacteristic
20#2853—21no—
21#2852—21no—
22#2851C411yescharacteristic
23#2850C411yescharacteristic
24C4×#125C2211yescharacteristic
25C4×#126—21no—
26C4×#124C2211yescharacteristic
27C4×C22⋊Q8C2211yescharacteristic
28C4×#123—21no—
29C4×#121C2211yescharacteristic
30#2298—21no—
31#2295C2211yescharacteristic
32#2293C2211yescharacteristic
33#2290—21no—
34C2×C43C2211yescharacteristic
35#1839—21no—
36#1831—21no—
37#1817—21no—
38#1797—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8—21no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43C4×C4⋊C8C2212yes—
44C4×C22⋊C8C2212yes—
45#1744C2212yes—
46C2×#336—21no—
47C2×Q8⋊C8C2214yes—
48C2×D4⋊C8C2214yes—
49#1279C2214yes—
50#1279C2214yes—
51C23×C8—22no—
52C2×#126—21no—
53C2×#125C2×C411yescharacteristic
54C2×#124C2×C411yescharacteristic
55C2×#123—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×#121C2×C411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4—21no—
60C2×C4×D4C2311yescharacteristic
61C2×#117—21no—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—28no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2314yes—
73C2×C22⋊C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8—44no—
77C2×C4×C8C2314yes—
78#391C2×C411yescharacteristic
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#391C2×C411yescharacteristic
82#390C2×C411yescharacteristic
83#390C2×C411yescharacteristic
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#389C2×C411yescharacteristic
87#389C2×C411yescharacteristic
88#389C2×C411yescharacteristic
89#389C2×C411yescharacteristic
90#389C2×C411yescharacteristic
91#389C2×C411yescharacteristic
92#389C2×C411yescharacteristic
93#389C2×C411yescharacteristic
94#388—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#383—21no—
135#382C2×C411yescharacteristic
136#382C2×C411yescharacteristic
137#381—21no—
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151#376—21no—
152C43C2×C411yescharacteristic
153C43C2×C411yescharacteristic
154C43D411yescharacteristic
155C43D411yescharacteristic
156C43D411yescharacteristic
157C43D411yescharacteristic
158C43C2311yescharacteristic
159C43C2311yescharacteristic
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166Q8⋊C8—28no—
167D4⋊C8—28no—
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C22×C8—28no—
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—44no—
183C22×C8—44no—
184#126—44no—
185#125—24no—
186#124—24no—
187#123—44no—
188C22⋊Q8—24no—
189#121—24no—
190C4×Q8—24no—
191C4×D4—24no—
192C4×D4—44no—
193#117—42no—
194#117—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2411yescharacteristic
267C4⋊C8—28no—
268C4⋊C8—24no—
269C4⋊C8—44no—
270C4⋊C8#4118yes—
271C22⋊C8—44no—
272C4×C8—24no—
273C4×C8—24no—
274C4×C8—44no—
275C4×C8—44no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24C2×C22⋊C411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4#13114yes—
354C22×C4#13514yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C2×C8—28no—
359C2×C8—416no—
360C2×C8—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊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—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—21no—
407C42—21no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—42no—
425C42—42no—
426C42—41no—
427C42—41no—
428C42#13114yes—
429C42AGL(1,ℤ/8)12yes—
430C42#13712yes—
431C42C2×#4114yes—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23#40812yes—
444C23#41612yes—
445C23#42112yes—
446C23C22×C22⋊C411yescharacteristic
447Q8—44no—
448D4—44no—
449D4—84no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#40812yes—
510C2×C4#40812yes—
511C2×C4#40812yes—
512C2×C4#41612yes—
513C2×C4#41712yes—
514C2×C4#41712yes—
515C2×C4#42112yes—
516C2×C4C4×#4112yes—
517C2×C4#52512yes—
518C2×C4#53212yes—
519C2×C4C2×#13112yes—
520C2×C4C2×#13112yes—
521C2×C4C2×#13512yes—
522C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
523C2×C4C2×#13711yescharacteristic
524C8—48no—
525C8—84no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22#148214yes—
540C22#229711yescharacteristic
541C22C2×#40811yescharacteristic
542C22C2×#40811yescharacteristic
543C22C2×#41611yescharacteristic
544C22C2×#41711yescharacteristic
545C22C2×#42111yescharacteristic
546C22C2×#42111yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4#296614yes—
567C4#310512yes—
568C4#310812yes—
569C2—24no—
570C2—161no—
571C2C2×#148211yescharacteristic
572C2#1745811yescharacteristic
573C2#1746611yescharacteristic
574C2#1746611yescharacteristic
575C2#2137511yescharacteristic
576C2#2138311yescharacteristic
577C2#2138811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. C2×C4×C4⋊C8C2semidirect
4. #9866C2semidirect
5. #9675C2semidirect
7. #8937C2semidirect
8. #8183C2semidirect
9. #8182C2semidirect

Maximal subgroups

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

subgroupindex
#366382
#207772
C2×C4×C4⋊C82
#98662
#96752
#89392
#89372
#81832
#81822