Finite Group Databasebeta

← #6407489

Subgroups — #6407489

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34809C211yesmaximal, characteristic
2#23324C212yesmaximal, complemented
3#23323C212yesmaximal, complemented
4#23321C214yesmaximal, complemented
5#23318C214yesmaximal, complemented
6#23316C212yesmaximal, complemented
7#23315C212yesmaximal, complemented
8C2×#1955C211yesmaximal, characteristic
9#21147C212yesmaximal, complemented
10#21053C212yesmaximal, complemented
11#21050C212yesmaximal, complemented
12#21021C212yesmaximal, complemented
13#20898C211yesmaximal, characteristic
14#20836C212yesmaximal, complemented
15#20628C212yesmaximal, complemented
16C23×M16C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#424C2212yes—
20C2×#424C2212yes—
21C2×#424C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C2×#407C2212yes—
25C22×C22⋊C8C2212yes—
26C22×C22⋊C8C2212yes—
27C2×#405C2212yes—
28C22×#97C2211yescharacteristic
29#2487C411yescharacteristic
30#2487C411yescharacteristic
31#2486C411yescharacteristic
32#2486C411yescharacteristic
33#2485C412yes—
34#2485C412yes—
35#2484C412yes—
36#2483C412yes—
37#2482C412yes—
38#2405C412yes—
39#2395C2212yes—
40#2395C2212yes—
41C4×#123C2212yes—
42C4×#123C2212yes—
43#2296C2212yes—
44#2280C2212yes—
45#2280C2211yescharacteristic
46#1955C2211yescharacteristic
47#1955C2211yescharacteristic
48#1955C2212yes—
49#1955C2212yes—
50#1955C2211yescharacteristic
51#1955C2211yescharacteristic
52#1940C2214yescomplemented
53#1940C2214yescomplemented
54#1939C2214yescomplemented
55#1939C2214yescomplemented
56#1924C2214yescomplemented
57#1924C2214yescomplemented
58#1924C2214yescomplemented
59#1924C2214yescomplemented
60#1923C2214yescomplemented
61#1923C2214yescomplemented
62#1923C2214yescomplemented
63#1923C2214yescomplemented
64#1917C2214yescomplemented
65#1917C2214yescomplemented
66#1917C2214yescomplemented
67#1917C2214yescomplemented
68#1870C2212yes—
69#1870C2212yes—
70#1869C2212yes—
71#1869C2212yes—
72#1851C2214yescomplemented
73#1851C2214yescomplemented
74#1850C2214yescomplemented
75#1850C2214yescomplemented
76#1798C2214yescomplemented
77#1798C2214yescomplemented
78#1797C2214yescomplemented
79#1797C2214yescomplemented
80#1729C2212yes—
81#1729C2212yes—
82#1729C2212yes—
83#1729C2212yes—
84#1728C2212yes—
85C23×D4C2×C411yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C22×M16—44no—
89C22×M16—42no—
90C22×M16C2312yes—
91C22×M16C2312yes—
92C22×M16C2312yes—
93C23×C8C2312yes—
94C23×C8C2312yes—
95C23×C8C2312yes—
96C2×#123C2×C412yes—
97C2×#123C2×C412yes—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4C2×C411yescharacteristic
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4C2×C411yescharacteristic
103C2×C4×D4C2×C411yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2311yescharacteristic
108C2×#117—42no—
109C2×#117—42no—
110C2×#117C2×C412yes—
111C2×#117C2×C412yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C22×C4⋊C4C2×C411yescharacteristic
116C22×C4⋊C4C2×C411yescharacteristic
117C22×C4⋊C4C2×C411yescharacteristic
118C22×C22⋊C4C2×C412yes—
119C22×C22⋊C4C2×C411yescharacteristic
120C22×C22⋊C4C2×C411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#424—44no—
124#424—44no—
125#424—44no—
126C2×C4⋊C8—42no—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136#407—44no—
137#407—42no—
138C2×C22⋊C8—44no—
139C2×C22⋊C8—44no—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152#405—44no—
153C2×#97—42no—
154C2×#97C2311yescharacteristic
155C2×#97C2311yescharacteristic
156C2×#97C2312yes—
157C2×#97C2314yes—
158C2×C4×C8C2314yes—
159#400C2×C412yes—
160#400C2×C412yes—
161#399C2×C412yes—
162#399C2×C412yes—
163#398C2×C412yes—
164#398C2×C412yes—
165#397C2×C412yes—
166#397C2×C412yes—
167#397C2×C414yes—
168#397C2×C414yes—
169#394C2×C414yes—
170#394C2×C414yes—
171#389C2×C412yes—
172#389C2×C412yes—
173#389C2×C412yes—
174#389C2×C412yes—
175#388C2×C414yes—
176#388C2×C414yes—
177#387C2×C414yes—
178#387C2×C414yes—
179#386C2×C412yes—
180#386C2×C412yes—
181#385C2×C414yes—
182#385C2×C414yes—
183#385C2×C412yes—
184#385C2×C412yes—
185#381C2×C412yes—
186#381C2×C414yes—
187#381C2×C412yes—
188#381C2×C412yes—
189#380C2×C412yes—
190#380C2×C412yes—
191#380C2×C412yes—
192#380C2×C412yes—
193#380C2×C412yes—
194#380C2×C411yescharacteristic
195#380C2×C411yescharacteristic
196#379C2×C414yes—
197#379C2×C412yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2314yes—
200C4×C22⋊C4C2314yes—
201C4×C22⋊C4C2314yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204#336C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25C22×C411yescharacteristic
215C25C22×C411yescharacteristic
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4C22×C411yescharacteristic
220C22×D4C22×C411yescharacteristic
221C22×D4C22×C411yescharacteristic
222C22×D4C22×C411yescharacteristic
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4C22×C411yescharacteristic
232C23×C4C22×C411yescharacteristic
233C23×C4C22×C411yescharacteristic
234C23×C4C22×C411yescharacteristic
235C23×C4C22×C411yescharacteristic
236C23×C4C22×C411yescharacteristic
237C23×C4C22×C411yescharacteristic
238C23×C4C22×C412yes—
239C23×C4C22×C411yescharacteristic
240C23×C4C22×C412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—44no—
253C2×M16—42no—
254C2×M16—416no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8—44no—
262C22×C8—42no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269#123—44no—
270#123—44no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—41no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—42no—
288#117—42no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4C22×C412yes—
295C2×C4⋊C4C22×C412yes—
296C2×C4⋊C4C22×C412yes—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C414yes—
305C2×C4⋊C4C22×C414yes—
306C2×C4⋊C4C22×C414yes—
307C2×C4⋊C4C22×C414yes—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4C22×C414yes—
312C2×C22⋊C4C22×C414yes—
313C2×C22⋊C4C22×C412yes—
314C2×C22⋊C4C22×C414yes—
315C2×C22⋊C4C22×C414yes—
316C2×C22⋊C4C22×C411yescharacteristic
317C2×C22⋊C4C22×C411yescharacteristic
318C2×C22⋊C4C22×C411yescharacteristic
319C2×C22⋊C4C22×C411yescharacteristic
320C2×C22⋊C4C22×C411yescharacteristic
321C2×C22⋊C4C22×C411yescharacteristic
322C2×C22⋊C4C22×C411yescharacteristic
323C2×C22⋊C4C22×C411yescharacteristic
324C2×C22⋊C4C22×C412yes—
325C2×C22⋊C4C22×C414yes—
326C2×C22⋊C4C22×C414yes—
327C2×C22⋊C4C22×C414yes—
328C2×C22⋊C4C22×C414yes—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C22×C411yescharacteristic
334C2×C42C22×C411yescharacteristic
335C2×C42C22×C414yes—
336C2×C42C2414yes—
337C2×C42C2414yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C4⋊C8—41no—
341C4⋊C8—42no—
342C4⋊C8—41no—
343C4⋊C8—44no—
344C4⋊C8—44no—
345C4⋊C8—44no—
346C4⋊C8—44no—
347C4⋊C8—44no—
348C4⋊C8—44no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—44no—
358C22⋊C8—44no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361#97—41no—
362#97—41no—
363#97—42no—
364#97—44no—
365C4×C8—44no—
366#95C22×C414yes—
367#95C22×C414yes—
368#95C22×C414yes—
369#95C22×C414yes—
370#95C22×C414yes—
371#95C22×C412yes—
372#95C22×C412yes—
373#95C22×C414yes—
374#95C22×C414yes—
375#95C22×C414yes—
376#95C22×C414yes—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—41no—
384C24—41no—
385C24—42no—
386C24—42no—
387C24C23×C411yescharacteristic
388C24C23×C411yescharacteristic
389C24C23×C411yescharacteristic
390C24C23×C411yescharacteristic
391C24C23×C411yescharacteristic
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—416no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4C23×C411yescharacteristic
450C22×C4C23×C411yescharacteristic
451C22×C4C23×C414yes—
452C22×C4C23×C414yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C414yes—
457C22×C4C23×C414yes—
458C22×C4C23×C411yescharacteristic
459C22×C4C23×C411yescharacteristic
460C22×C4C23×C411yescharacteristic
461C22×C4C23×C411yescharacteristic
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C22×C4C2511yescharacteristic
467M16—48no—
468M16—48no—
469M16—48no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—48no—
489C2×C8—48no—
490C2×C8—48no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—28no—
509C22⋊C4—44no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—24no—
531C42—24no—
532C42—24no—
533C42—44no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—44no—
556C23—44no—
557C23—48no—
558C23—48no—
559C23—44no—
560C23—44no—
561C23—82no—
562C23—82no—
563C23—84no—
564C23—84no—
565C23#51812yes—
566C23#52014yes—
567C23#56812yes—
568C23#56812yes—
569C23C24×C411yescharacteristic
570D4—44no—
571D4—44no—
572D4—44no—
573D4—44no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4#51812yes—
627C2×C4#52014yes—
628C2×C4#56014yes—
629C2×C4#56218yes—
630C2×C4#56314yes—
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+412yes—
635C8—42no—
636C8—42no—
637C8—44no—
638C8—42no—
639C8—42no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—24no—
644C22—24no—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—88no—
666C22—88no—
667C22#237614yes—
668C22C2×#51811yescharacteristic
669C22C2×#52012yes—
670C22C2×#56811yescharacteristic
671C22C2×#56811yescharacteristic
672C22#357312yes—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—44no—
680C4—41no—
681C4—41no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4#237614yes—
686C4C2×#56011yescharacteristic
687C4C2×#56212yes—
688C4C2×#56311yescharacteristic
689C2—24no—
690C2—42no—
691C2—42no—
692C2—81no—
693C2—81no—
694C2C2×#237611yescharacteristic
695C2#6023411yescharacteristic
696C2#6023511yescharacteristic
697C2#6027212yes—
698C2#6027212yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23324C2semidirect
3. #23323C2semidirect
4. #23321C2semidirect
5. #23318C2semidirect
6. #23316C2semidirect
7. #23315C2semidirect
9. #21147C2semidirect
10. #21053C2semidirect
11. #21050C2semidirect
12. #21021C2semidirect
14. #20836C2semidirect
15. #20628C2semidirect
52. #1940C22semidirect
53. #1940C22semidirect
54. #1939C22semidirect
55. #1939C22semidirect
56. #1924C22semidirect
57. #1924C22semidirect
58. #1924C22semidirect
59. #1924C22semidirect
60. #1923C22semidirect
61. #1923C22semidirect
62. #1923C22semidirect
63. #1923C22semidirect
64. #1917C22semidirect
65. #1917C22semidirect
66. #1917C22semidirect
67. #1917C22semidirect
72. #1851C22semidirect
73. #1851C22semidirect
74. #1850C22semidirect
75. #1850C22semidirect
76. #1798C22semidirect
77. #1798C22semidirect
78. #1797C22semidirect
79. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#348092
#233242
#233232
#233212
#233182
#233162
#233152
C2×#19552
#211472
#210532
#210502
#210212
#208982
#208362
#206282