Finite Group Databasebeta

← #160519

Subgroups — #160519

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34323C211yesmaximal, characteristic
2#22785C211yesmaximal, characteristic, complemented
3#20555C211yesmaximal, characteristic, complemented
4#7586C211yesmaximal, characteristic, complemented
5#7585C211yesmaximal, characteristic, complemented
6#7582C211yesmaximal, characteristic, complemented
7#7580C211yesmaximal, characteristic, complemented
8#7579C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7568C211yesmaximal, characteristic, complemented
11#7565C211yesmaximal, characteristic, complemented
12#7565C211yesmaximal, characteristic, complemented
13#7564C211yesmaximal, characteristic, complemented
14#7560C211yesmaximal, characteristic, complemented
15C2×#1271C211yesmaximal, characteristic, complemented
16C2×#553C411yescharacteristic
17C2×C8×D4—21no—
18#2373—21no—
19#2370C411yescharacteristic
20#2363C411yescharacteristic
21#2361—21no—
22#2361C411yescharacteristic
23#2359—21no—
24#2352—21no—
25#2352C411yescharacteristic
26#2349C411yescharacteristic
27#2346C411yescharacteristic
28#2345C411yescharacteristic
29#2335—21no—
30#2334—21no—
31#2325—21no—
32C4×#125C2211yescharacteristic
33C4×#121C2211yescharacteristic
34C4×#121C2211yescharacteristic
35#2296C2211yescharacteristic
36#2293C2211yescharacteristic
37C2×#389C2211yescharacteristic
38#2275—21no—
39C42×D4C2211yescharacteristic
40#1923—21no—
41#1916—21no—
42#1869—21no—
43#1839C2211yescharacteristic
44#1817C2211yescharacteristic
45#1797C2211yescharacteristic
46C8×C4⋊C4—21no—
47C4×C4⋊C8C2211yescharacteristic
48C8×C22⋊C4—21no—
49C8×C22⋊C4—21no—
50C8×C22⋊C4—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1744C2211yescharacteristic
55#1744C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C23×C8—21no—
77C23×C8—21no—
78#553—24no—
79#553C818yescomplemented
80C2×#126—21no—
81C2×#125C2×C411yescharacteristic
82C2×#123—21no—
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4C2×C411yescharacteristic
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4D411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×#117—21no—
97C2×#117C2×C411yescharacteristic
98C2×#117C2×C411yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C8×D4—28no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118#390—21no—
119#390C2×C411yescharacteristic
120#390C2311yescharacteristic
121#389—21no—
122#389—21no—
123#389—21no—
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2311yescharacteristic
130#389C2311yescharacteristic
131#389C2311yescharacteristic
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#386—21no—
149#386—21no—
150#386—21no—
151#386—21no—
152#386—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#382—21no—
159#382C2×C411yescharacteristic
160#380—21no—
161#380—21no—
162#380C2×C411yescharacteristic
163#380C2×C411yescharacteristic
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189#376—21no—
190C2×#95—21no—
191C2×#95—21no—
192C43C2311yescharacteristic
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C22⋊C411yescharacteristic
212C22×D4C22×C411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8—81no—
237C22×C8—81no—
238C22×C8—81no—
239C22×C8—81no—
240C22×C8—81no—
241#126—28no—
242#125—22no—
243#125C2×C814yescomplemented
244#123—28no—
245#121—28no—
246#121—22no—
247#121—22no—
248#121C2×C814yescomplemented
249#121C2×C814yescomplemented
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—22no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—44no—
258C4×D4—81no—
259C4×D4—81no—
260C4×D4—81no—
261C4×D4—81no—
262C4×D4C2×C814yes—
263C4×D4C2×C814yescomplemented
264C4×D4M1614yes—
265#117—22no—
266#117—81no—
267#117—81no—
268#117—81no—
269#117C2×C814yescomplemented
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—81no—
284C2×C4⋊C4—81no—
285C2×C4⋊C4—81no—
286C2×C4⋊C4C22⋊C411yescharacteristic
287C2×C4⋊C4C22⋊C411yescharacteristic
288C2×C4⋊C4C22⋊C411yescharacteristic
289C2×C4⋊C4C2×C814yescomplemented
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C22⋊C4C22⋊C411yescharacteristic
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—81no—
324C2×C42—81no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C4⋊C8—24no—
339C4⋊C8—81no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—81no—
345C4×C8—24no—
346C4×C8—42no—
347C4×C8—81no—
348C4×C8—81no—
349C4×C8—81no—
350C4×C8—81no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371#95C22⋊C411yescharacteristic
372#95C22⋊C411yescharacteristic
373#95C22⋊C411yescharacteristic
374C24—21no—
375C24C2×C22⋊C411yescharacteristic
376C24C2×C22⋊C411yescharacteristic
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4C22⋊C814yes—
389C2×D4C22×C812yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C22×C812yes—
441C22×C4C22×C812yes—
442C22×C4C2×M1612yes—
443C22×C4C2×M1612yes—
444C22×C4#13512yes—
445C22×C4#13512yes—
446C22×C4C23×C411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—84no—
474C2×C8—84no—
475C2×C8—84no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4C22⋊C814yes—
501C4⋊C4C22×C812yes—
502C4⋊C4C22×C812yes—
503C4⋊C4C22×C812yes—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—42no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4C22⋊C814yes—
523C22⋊C4C22⋊C814yes—
524C22⋊C4#10414yes—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—24no—
530C42—22no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—42no—
541C42—42no—
542C42—81no—
543C42—81no—
544C42—81no—
545C42—81no—
546C42C22×C812yes—
547C42C2×M1612yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×C22⋊C812yes—
566C23C2×C22⋊C812yes—
567C23#41612yes—
568C23C2×#10412yes—
569C23C22×C22⋊C411yescharacteristic
570D4—28no—
571D4—42no—
572D4—42no—
573D4—48no—
574D4—84no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—162no—
640C2×C4—162no—
641C2×C4—162no—
642C2×C4C2×C22⋊C812yes—
643C2×C4C2×C22⋊C812yes—
644C2×C4C2×C22⋊C812yes—
645C2×C4C2×C22⋊C812yes—
646C2×C4#41612yes—
647C2×C4C2×#10412yes—
648C2×C4C2×#10412yes—
649C2×C4C23×C811yescharacteristic
650C2×C4C22×M1611yescharacteristic
651C2×C4C2×#13511yescharacteristic
652C2×C4C2×#13511yescharacteristic
653C2×C4#57611yescharacteristic
654C2×C4#57711yescharacteristic
655C8—42no—
656C8—42no—
657C8—42no—
658C8—42no—
659C8—42no—
660C8—81no—
661C8—81no—
662C8—81no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22#147614yes—
684C22C22×C22⋊C811yescharacteristic
685C22#288311yescharacteristic
686C22#288911yescharacteristic
687C22C2×#41611yescharacteristic
688C22#289711yescharacteristic
689C22C22×#10411yescharacteristic
690C22#290111yescharacteristic
691C4—22no—
692C4—22no—
693C4—21no—
694C4—21no—
695C4—22no—
696C4—21no—
697C4—21no—
698C4—22no—
699C4—22no—
700C4—21no—
701C4—21no—
702C4—22no—
703C4—22no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—81no—
714C2—24no—
715C2—42no—
716C2—82no—
717C2#1735111yescharacteristic
718C2C2×#147611yescharacteristic
719C2#1743811yescharacteristic
720C2#1754711yescharacteristic
721C2#1761511yescharacteristic
722C2#1765311yescharacteristic
723C2#1772711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#553C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22785C2semidirect
3. #20555C2semidirect
4. #7586C2semidirect
5. #7585C2semidirect
6. #7582C2semidirect
7. #7580C2semidirect
8. #7579C2semidirect
9. #7576C2semidirect
10. #7568C2semidirect
11. #7565C2semidirect
12. #7565C2semidirect
13. #7564C2semidirect
14. #7560C2semidirect
15. C2×#1271C2semidirect
79. #553C8semidirect
243. #125C2×C8semidirect
248. #121C2×C8semidirect
249. #121C2×C8semidirect
263. C4×D4C2×C8semidirect
269. #117C2×C8semidirect
289. C2×C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#343232
#227852
#205552
#75862
#75852
#75822
#75802
#75792
#75762
#75682
#75652
#75642
#75602
C2×#12712