Finite Group Databasebeta

← #163894

Subgroups — #163894

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34281C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic
3#22589C211yesmaximal, characteristic
4#7605C211yesmaximal, characteristic, complemented
5#7600C211yesmaximal, characteristic
6#7596C211yesmaximal, characteristic, complemented
7#7589C211yesmaximal, characteristic
8#7585C211yesmaximal, characteristic, complemented
9#7585C211yesmaximal, characteristic, complemented
10#7583C211yesmaximal, characteristic
11#7581C211yesmaximal, characteristic, complemented
12#7577C211yesmaximal, characteristic, complemented
13#7571C211yesmaximal, characteristic, complemented
14#7566C211yesmaximal, characteristic
15#7562C211yesmaximal, characteristic, complemented
16C2×#541C411yescharacteristic
17#2373—21no—
18#2373—21no—
19#2373—21no—
20#2373—21no—
21#2371—21no—
22#2371—21no—
23#2370—21no—
24#2370—21no—
25#2330C411yescharacteristic
26#2327C411yescharacteristic
27#2323C411yescharacteristic
28#2323C411yescharacteristic
29#2314C411yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31C4×#125C2211yescharacteristic
32C4×C22⋊Q8C2211yescharacteristic
33C4×C22⋊Q8C2211yescharacteristic
34#2295C2211yescharacteristic
35#2295C2211yescharacteristic
36#2276C411yescharacteristic
37#2276C411yescharacteristic
38C4×#117C2211yescharacteristic
39#1939—21no—
40#1923—21no—
41#1923—21no—
42#1916—21no—
43#1916—21no—
44#1850—21no—
45#1839C2211yescharacteristic
46#1839C2211yescharacteristic
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51C8×C4⋊C4—21no—
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C8×C22⋊C4—21no—
56C4×C22⋊C8C2211yescharacteristic
57#1744C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C23×C8—21no—
76#541—24no—
77#541C818yescomplemented
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#126—21no—
80C2×#126—21no—
81C2×#125—21no—
82C2×#125—21no—
83C2×#125C2×C411yescharacteristic
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C2×#117C2×C411yescharacteristic
94C22×C42C2311yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#391C2311yescharacteristic
111#391C2311yescharacteristic
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2311yescharacteristic
121#389C2311yescharacteristic
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#385—21no—
147#385—21no—
148#384C2×C411yescharacteristic
149#382C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#380C2×C411yescharacteristic
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4C2×C411yescharacteristic
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4D411yescharacteristic
179C4×C22⋊C4D411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182#376—21no—
183#376—21no—
184#376—21no—
185#376—21no—
186#376C2×C411yescharacteristic
187#376C2×C411yescharacteristic
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C22×Q8C22×C411yescharacteristic
209C22×Q8C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—81no—
224C4⋊Q8—22no—
225C4⋊Q8C2×C814yescomplemented
226#126—28no—
227#126—28no—
228#125—28no—
229#125—28no—
230#125—22no—
231#125C2×C814yescomplemented
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8C2×C814yes—
235C22⋊Q8C2×C814yes—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×Q8—42no—
241C4×Q8—42no—
242C4×Q8C2×C814yescomplemented
243C4×Q8C2×C814yescomplemented
244#117—42no—
245#117C2×C814yes—
246#117M1614yes—
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—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
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⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—42no—
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2411yescharacteristic
312C4⋊C8—42no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316C4⋊C8—81no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—81no—
320C4×C8—42no—
321C4×C8—42no—
322C4×C8—81no—
323C4×C8—81no—
324C4×C8—81no—
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—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355C24C2×C22⋊C411yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8C22×C812yes—
363C2×Q8C22×C812yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C22×C812yes—
394C22×C4C2×M1612yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—84no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4C22⋊C814yes—
452C4⋊C4C22⋊C814yes—
453C4⋊C4C22×C812yes—
454C4⋊C4C22×C812yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—41no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4C22⋊C814yes—
462C22⋊C4C22⋊C814yes—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—24no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—81no—
488C42—81no—
489C42—81no—
490C42—81no—
491C42—81no—
492C42—81no—
493C42C22×C812yes—
494C42C22×C812yes—
495C42C2×M1612yes—
496C42C2×M1612yes—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23C2×C22⋊C812yes—
504C23C22×C22⋊C411yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
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—81no—
570C2×C4—81no—
571C2×C4C2×C22⋊C812yes—
572C2×C4C2×C22⋊C812yes—
573C2×C4C2×C22⋊C812yes—
574C2×C4C2×C22⋊C812yes—
575C2×C4C2×C22⋊C812yes—
576C2×C4#41012yes—
577C2×C4#41312yes—
578C2×C4C23×C811yescharacteristic
579C2×C4C22×M1611yescharacteristic
580C2×C4#57711yescharacteristic
581C2×C4#57711yescharacteristic
582C2×C4#57811yescharacteristic
583C2×C4#57811yescharacteristic
584C8—42no—
585C8—42no—
586C8—81no—
587C8—81no—
588C8—81no—
589C8—81no—
590C8—81no—
591C8—81no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22C22×C22⋊C811yescharacteristic
600C22C2×#41011yescharacteristic
601C22C2×#41311yescharacteristic
602C22#289811yescharacteristic
603C22#289811yescharacteristic
604C22#290111yescharacteristic
605C22#290111yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—21no—
612C4—21no—
613C4—22no—
614C4—21no—
615C4—21no—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C2—42no—
632C2#1734911yescharacteristic
633C2#1743911yescharacteristic
634C2#1743911yescharacteristic
635C2#1760911yescharacteristic
636C2#1761211yescharacteristic
637C2#1768711yescharacteristic
638C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7605C2semidirect
6. #7596C2semidirect
8. #7585C2semidirect
9. #7585C2semidirect
11. #7581C2semidirect
12. #7577C2semidirect
13. #7571C2semidirect
15. #7562C2semidirect
77. #541C8semidirect
225. C4⋊Q8C2×C8semidirect
231. #125C2×C8semidirect
242. C4×Q8C2×C8semidirect
243. C4×Q8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#342812
#226392
#225892
#76052
#76002
#75962
#75892
#75852
#75832
#75812
#75772
#75712
#75662
#75622