Finite Group Databasebeta

← #6388657

Subgroups — #6388657

Order 512.

1924 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
1C2×#2985C211yesmaximal, characteristic, complemented
2C2×#2920C211yesmaximal, characteristic
3#36625C211yesmaximal, characteristic
4#23053C214yesmaximal, complemented
5#22955C214yesmaximal, complemented
6#22605C214yesmaximal, complemented
7#22559C214yesmaximal
8#22102C212yesmaximal, complemented
9#22100C212yesmaximal, complemented
10#21954C211yesmaximal, characteristic, complemented
11#21821C211yesmaximal, characteristic, complemented
12#21598C212yesmaximal, complemented
13#21596C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15#20537C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2985—28no—
19C2×#446C2214yescomplemented
20C2×#446C2212yescomplemented
21C2×C8×Q8C2212yescomplemented
22#2920—28no—
23C2×#433C2214yes—
24C2×#432C2212yes—
25C2×#432C2211yescharacteristic
26C2×#432C2212yes—
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#423C2212yes—
33C2×#407C2212yes—
34C2×C4×M16C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2857C411yescharacteristic
37#2857C411yescharacteristic
38#2857C411yescharacteristic
39#2857C411yescharacteristic
40#2855C412yes—
41#2855C412yes—
42#2851C412yes—
43#2851C412yes—
44#2848C412yes—
45#2848C412yes—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50#2296C2212yes—
51#2285C2211yescharacteristic
52#2280C2212yes—
53#1985C2214yescomplemented
54#1984C2212yescomplemented
55#1984C2212yescomplemented
56#1940C2214yescomplemented
57#1940C2214yescomplemented
58#1939C2214yescomplemented
59#1939C2212yescomplemented
60#1939C2212yescomplemented
61#1924C2214yes—
62#1924C2214yes—
63#1923C2214yes—
64#1923C2214yes—
65#1917C2214yescomplemented
66#1917C2214yescomplemented
67#1917C2214yescomplemented
68#1916C2214yescomplemented
69#1851C2214yes—
70#1851C2214yes—
71#1818C2214yes—
72#1817C2212yes—
73#1798C2214yes—
74#1797C2212yes—
75#1797C2212yes—
76#1779C2212yes—
77#1771C2214yescomplemented
78#1771C2214yescomplemented
79#1770C2214yescomplemented
80#1770C2212yescomplemented
81C8×C4⋊C4C2212yescomplemented
82#1753C2214yes—
83#1753C2214yes—
84#1750C2214yes—
85#1749C2212yes—
86#1749C2212yes—
87C24×C4C2×C411yescharacteristic
88C24×C4C2×C411yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C22×M16C2312yes—
92#533—24no—
93#533—24no—
94C2×C4⋊Q8C2×C411yescharacteristic
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×#125C2×C412yes—
97C2×C22⋊Q8C2×C412yes—
98C2×C22⋊Q8C2×C412yes—
99C2×C22⋊Q8C2×C412yes—
100C2×C22⋊Q8C2×C412yes—
101C2×C22⋊Q8C2×C412yes—
102C2×C22⋊Q8C2×C412yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42C2×C411yescharacteristic
110C22×C42C2×C411yescharacteristic
111C22×C42C2×C411yescharacteristic
112C22×C42C2×C411yescharacteristic
113C22×C42C2×C411yescharacteristic
114C22×C42C2×C411yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#446—216no—
122#446—28no—
123C8×Q8—28no—
124#433—216no—
125#432—24no—
126#432—24no—
127#432—28no—
128#432—28no—
129#431—24no—
130#431—24no—
131#431—24no—
132#431D418yes—
133#423—28no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144#407—28no—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2312yes—
153C4×M16—24no—
154C4×M16—24no—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×#97C2314yes—
160C2×#97C2314yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2314yes—
164C2×C4×C8C2314yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#391C2×C412yes—
168#391C2×C412yes—
169#391C2×C412yes—
170#391C2×C412yes—
171#389C2×C412yes—
172#389C2×C412yes—
173#389C2×C412yes—
174#389C2×C412yes—
175#389C2×C412yes—
176#389C2×C412yes—
177#389C2×C412yes—
178#389C2×C412yes—
179#387C2×C414yes—
180#387C2×C414yes—
181#387C2×C414yes—
182#387C2×C414yes—
183#385C2×C414yes—
184#385C2×C414yes—
185#385C2×C414yes—
186#385C2×C414yes—
187#385C2×C414yes—
188#385C2×C414yes—
189#384C2×C411yescharacteristic
190#384C2×C411yescharacteristic
191#382C2×C412yes—
192#381C2×C414yes—
193#380C2×C412yes—
194#380C2×C412yes—
195#380C2×C412yes—
196#380C2×C412yes—
197#380C2×C412yes—
198#380C2×C412yes—
199#380C2×C412yes—
200#380C2×C412yes—
201#380C2×C412yes—
202#380C2×C412yes—
203#379C2×C414yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4C2314yes—
209C4×C22⋊C4C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25C22×C411yescharacteristic
221C22×Q8C22×C412yes—
222C22×Q8C22×C412yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C22×C411yescharacteristic
234C23×C4C22×C411yescharacteristic
235C23×C4C22×C411yescharacteristic
236C23×C4C22×C411yescharacteristic
237C23×C4C22×C411yescharacteristic
238C23×C4C22×C411yescharacteristic
239C23×C4C22×C411yescharacteristic
240C23×C4C22×C411yescharacteristic
241C23×C4C22×C411yescharacteristic
242C23×C4C22×C411yescharacteristic
243C23×C4C22×C411yescharacteristic
244C23×C4C22×C411yescharacteristic
245C23×C4C22×C411yescharacteristic
246C23×C4C22×C411yescharacteristic
247C23×C4C22×C411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265#125—28no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276#117—24no—
277#117—24no—
278#117—44no—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C414yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C22×C412yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C412yes—
290C2×C4⋊C4C22×C412yes—
291C2×C4⋊C4C22×C412yes—
292C2×C4⋊C4C22×C414yes—
293C2×C4⋊C4C22×C414yes—
294C2×C4⋊C4C22×C414yes—
295C2×C22⋊C4C22×C414yes—
296C2×C22⋊C4C22×C414yes—
297C2×C22⋊C4C22×C414yes—
298C2×C22⋊C4C22×C414yes—
299C2×C22⋊C4C22×C414yes—
300C2×C22⋊C4C22×C414yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C2×D414yes—
330C2×C42#4114yes—
331C2×C42C2414yes—
332C2×C42C2414yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C4⋊C8—24no—
338C4⋊C8—24no—
339C4⋊C8—28no—
340C4⋊C8—28no—
341C4⋊C8—24no—
342C4⋊C8—24no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345C4⋊C8—44no—
346C4⋊C8C2×D418yes—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C22⋊C8—28no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—44no—
354C22⋊C8C2×D418yes—
355#97—24no—
356#97—24no—
357#97—28no—
358#97—24no—
359#97—24no—
360#97—44no—
361C4×C8—28no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—44no—
365C4×C8C2×D414yes—
366C4×C8C2×D414yes—
367#95C22×C414yes—
368#95C22×C414yes—
369#95C22×C414yes—
370#95C22×C414yes—
371#95C22×C414yes—
372#95C22×C414yes—
373#95C22×C414yes—
374#95C22×C414yes—
375#95C22×C414yes—
376#95C22×C414yes—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24C23×C411yescharacteristic
385C24C23×C411yescharacteristic
386C24C23×C411yescharacteristic
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4C4×D414yes—
450C22×C4C4×D414yes—
451C22×C4C23×C411yescharacteristic
452C22×C4C23×C414yes—
453C22×C4C23×C414yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C411yescharacteristic
460C22×C4C23×C411yescharacteristic
461C22×C4C23×C411yescharacteristic
462C22×C4C23×C411yescharacteristic
463C22×C4C23×C411yescharacteristic
464C22×C4C23×C411yescharacteristic
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C2×#4112yes—
468C22×C42−1+414yes—
469C22×C4C2511yescharacteristic
470M16—48no—
471M16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8C22×D414yes—
489C2×C8C22×D414yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—28no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42C4×D414yes—
531C42C4×D414yes—
532C42C22×D412yes—
533C42C22×D412yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23C2×C4×D412yes—
562C23#51912yes—
563C23#56812yes—
564C23C24×C411yescharacteristic
565Q8—48no—
566Q8—84no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4C2×C4×D412yes—
625C2×C4C2×C4×D412yes—
626C2×C4C2×C4×D412yes—
627C2×C4C2×C4×D412yes—
628C2×C4C2×C4×D412yes—
629C2×C4#51912yes—
630C2×C4#56812yes—
631C2×C4#56812yes—
632C2×C4#56812yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×2−1+412yes—
636C2×C4#58512yes—
637C8—24no—
638C8—24no—
639C8—44no—
640C8—42no—
641C8—42no—
642C8—44no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—416no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22C22×C4×D411yescharacteristic
666C22C2×#51911yescharacteristic
667C22#342811yescharacteristic
668C22C2×#56811yescharacteristic
669C22C2×#56811yescharacteristic
670C22C2×#56811yescharacteristic
671C22C2×#56811yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—44no—
679C4—44no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—44no—
684C4—41no—
685C4—44no—
686C4#293014yes—
687C4#348912yes—
688C4#349512yes—
689C2—42no—
690C2—42no—
691C2—42no—
692C2#3420311yescharacteristic
693C2C2×#293011yescharacteristic
694C2#6014111yescharacteristic
695C2#6024911yescharacteristic
696C2#6025111yescharacteristic
697C2#6028511yescharacteristic
698C2#6029511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC25C22×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
1. C2×#2985C2semidirect
4. #23053C2semidirect
5. #22955C2semidirect
6. #22605C2semidirect
8. #22102C2semidirect
9. #22100C2semidirect
10. #21954C2semidirect
11. #21821C2semidirect
12. #21598C2semidirect
13. #21596C2semidirect
15. #20537C2semidirect
19. C2×#446C22semidirect
20. C2×#446C22semidirect
21. C2×C8×Q8C22semidirect
53. #1985C22semidirect
54. #1984C22semidirect
55. #1984C22semidirect
56. #1940C22semidirect
57. #1940C22semidirect
58. #1939C22semidirect
59. #1939C22semidirect
60. #1939C22semidirect
65. #1917C22semidirect
66. #1917C22semidirect
67. #1917C22semidirect
68. #1916C22semidirect
77. #1771C22semidirect
78. #1771C22semidirect
79. #1770C22semidirect
80. #1770C22semidirect
81. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29852
C2×#29202
#366252
#230532
#229552
#226052
#225592
#221022
#221002
#219542
#218212
#215982
#215962
#207152
#205372