Finite Group Databasebeta

← #216722

Subgroups — #216722

Order 512.

993 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36638C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C4×#382C211yesmaximal, characteristic, complemented
4#9886C212yesmaximal, complemented
5#9681C212yesmaximal
6#9148C212yesmaximal
7#9144C212yesmaximal, complemented
8#8576C212yesmaximal, complemented
9#8575C212yesmaximal
10#2866—21no—
11#2865—21no—
12#2864C411yescharacteristic, complemented
13#2863C411yescharacteristic, complemented
14#2861C411yescharacteristic, complemented
15#2860C411yescharacteristic, complemented
16#2855C2211yescharacteristic
17#2854C2211yescharacteristic
18#2853—21no—
19#2852—21no—
20#2851C2211yescharacteristic
21#2850C2211yescharacteristic
22C4×#125C411yescharacteristic, complemented
23C4×#126—21no—
24C4×#124C411yescharacteristic, complemented
25C4×C22⋊Q8C2211yescharacteristic
26C4×#123—21no—
27C4×#121C2211yescharacteristic
28#2298—21no—
29#2295C411yescharacteristic, complemented
30#2293C411yescharacteristic, complemented
31#2290—21no—
32C2×#382—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C43C2211yescharacteristic
35#1839—21no—
36#1831—21no—
37#1817C2212yes—
38#1797C2212yes—
39C4×C4⋊C8—21no—
40C4×C22⋊C8C2212yes—
41#1744—21no—
42C2×#328C2214yes—
43C2×#327C2214yes—
44C4⋊#95—21no—
45#1434C2212yes—
46#1434C2212yes—
47C4×#95—21no—
48C4×#95—21no—
49#1294C2214yes—
50#1294C2214yes—
51C2×#126—21no—
52C2×#125C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#123—21no—
55C2×C22⋊Q8C2311yescharacteristic
56C2×#121C2311yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×#117—21no—
61C22×C4⋊C4—22no—
62C22×C42—22no—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C22⋊C8C2314yes—
72C2×C4×C8—22no—
73#391C2×C411yescharacteristic
74#391C2×C411yescharacteristic
75#391C2×C411yescharacteristic
76#391C2×C411yescharacteristic
77#390C2×C411yescharacteristic
78#390C2×C411yescharacteristic
79#390C2×C411yescharacteristic
80#390C2×C411yescharacteristic
81#389C2×C411yescharacteristic
82#389C2×C411yescharacteristic
83#389C2×C411yescharacteristic
84#389C2×C411yescharacteristic
85#389C2×C411yescharacteristic
86#389C2×C411yescharacteristic
87#389C2×C411yescharacteristic
88#389C2×C411yescharacteristic
89#388—21no—
90#388—21no—
91#388—21no—
92#388—21no—
93#388—21no—
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#387—21no—
106#387—21no—
107#387—21no—
108#387—21no—
109#387—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#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#385—21no—
126#385—21no—
127#385—21no—
128#385—21no—
129#383—21no—
130#382—28no—
131#382C2×C411yescharacteristic
132#382C2×C411yescharacteristic
133#381—21no—
134#380C2311yescharacteristic
135#380C2311yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2×C411yescharacteristic
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148#376—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95C2314yes—
154C43C2×C411yescharacteristic
155C43C2×C411yescharacteristic
156C43C2×C411yescharacteristic
157C43C2×C411yescharacteristic
158C43D411yescharacteristic
159C43D411yescharacteristic
160C43D411yescharacteristic
161C43D411yescharacteristic
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#328—44no—
167#327—44no—
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—24no—
172C23×C4—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C22×C8—24no—
181#126—44no—
182#125—42no—
183#124—42no—
184#123—44no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187#121—22no—
188#121—22no—
189C4×Q8—42no—
190C4×D4—42no—
191C4×D4—44no—
192#117—44no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—21no—
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—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
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×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C4⋊C8—44no—
272C22⋊C8—24no—
273C22⋊C8—24no—
274C4×C8—44no—
275#95—28no—
276#95—28no—
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—
300#95—21no—
301#95—24no—
302#95—24no—
303#95—44no—
304#95—44no—
305C24—21no—
306C24C22×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—42no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
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—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4#13514yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+412yes—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—416no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—88no—
441C42#10414yes—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23C22×C22⋊C411yescharacteristic
454C23#53511yescharacteristic
455C23#53611yescharacteristic
456C23C2×#13512yes—
457C23#57512yes—
458Q8—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
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—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4#41012yes—
534C2×C4#41012yes—
535C2×C4#41612yes—
536C2×C4#41612yes—
537C2×C4C2×#10412yes—
538C2×C4C2×#10412yes—
539C2×C4C2×#10412yes—
540C2×C4#42112yes—
541C2×C4#52012yes—
542C2×C4#53712yes—
543C8—84no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—22no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#231111yescharacteristic
560C22C2×#41011yescharacteristic
561C22C2×#41011yescharacteristic
562C22C2×#41611yescharacteristic
563C22#289611yescharacteristic
564C22C22×#10411yescharacteristic
565C22C2×#42111yescharacteristic
566C22#321112yes—
567C22#323212yes—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—88no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C2—22no—
591C2—22no—
592C2—161no—
593C2#1753311yescharacteristic
594C2#1754311yescharacteristic
595C2#1757611yescharacteristic
596C2#1758011yescharacteristic
597C2#2186111yescharacteristic
598C2#2187511yescharacteristic
599C2#2189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. C4×#382C2semidirect
4. #9886C2semidirect
7. #9144C2semidirect
8. #8576C2semidirect
12. #2864C4semidirect
13. #2863C4semidirect
14. #2861C4semidirect
15. #2860C4semidirect
22. C4×#125C4semidirect
24. C4×#124C4semidirect
29. #2295C4semidirect
30. #2293C4semidirect

Maximal subgroups

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

subgroupindex
#366382
#207772
C4×#3822
#98862
#96812
#91482
#91442
#85762
#85752