Finite Group Databasebeta

← #6342980

Subgroups — #6342980

Order 512.

3103 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×#3460C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic, complemented
3#60276C211yesmaximal, characteristic, complemented
4#17483C2116yesmaximal, complemented
5#17480C214yesmaximal, complemented
6#17478C214yesmaximal, complemented
7#17437C212yesmaximal, complemented
8C2×#1478C212yesmaximal, complemented
9#3460—24no—
10#3460—24no—
11#3460C418yes—
12#3460C2218yes—
13C2×#544C411yescharacteristic
14C2×#542—21no—
15C2×#542C412yes—
16C2×#540—21no—
17C2×#540C412yes—
18C2×#538—22no—
19C2×#537C411yescharacteristic
20C2×#535C411yescharacteristic
21C2×#535C411yescharacteristic
22C2×#532—21no—
23C2×#532C2212yes—
24C2×#529—21no—
25C2×#529C2212yes—
26C2×#525—22no—
27C2×C4×#41C2211yescharacteristic
28C2×C4×#41C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30#2989C2218yescomplemented
31#2988—24no—
32#2983—21no—
33#2982—21no—
34#2975—21no—
35#2973—22no—
36#2972—22no—
37#2972—21no—
38#2965—24no—
39C8×#41C2218yescomplemented
40#2920C2214yescomplemented
41C2×#431C2212yes—
42C2×#431C2212yes—
43C2×#431C2214yescomplemented
44#1496C22116yescomplemented
45#1495C22116yescomplemented
46#1494C22116yescomplemented
47#1492C22116yescomplemented
48#1479C2214yescomplemented
49#1478C2218yes—
50#1478C2218yescomplemented
51#1478C2214yescomplemented
52#1477C2218yescomplemented
53#1476C2218yescomplemented
54C2×D4⋊C8C2218yescomplemented
55C22×#41—22no—
56C22×#41C2×C411yescharacteristic
57C22×#41C2×C411yescharacteristic
58C23×D4C2×C411yescharacteristic
59C24×C4—21no—
60#544—24no—
61#544C2×C418yes—
62#542—28no—
63#542—28no—
64#542—28no—
65#542C2×C4116yes—
66#540—28no—
67#540—28no—
68#540—28no—
69#540C2×C4116yes—
70#538—216no—
71#538—216no—
72#537—24no—
73#537C2×C418yes—
74#535—24no—
75#535—24no—
76#535C2×C418yes—
77#535C2×C418yes—
78#532—28no—
79#532—28no—
80#532—24no—
81#532—24no—
82#532C2×C418yes—
83#532C2318yes—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2×C412yes—
86C2×#127—21no—
87C2×#127C2×C412yes—
88#529—28no—
89#529—28no—
90#529—24no—
91#529—24no—
92#529C2×C418yes—
93#529C2318yes—
94C2×#126—22no—
95C2×#126—22no—
96C2×#125—21no—
97C2×#125C2×C412yes—
98C2×#124—21no—
99C2×#124—22no—
100C2×#124—22no—
101C2×#124C2×C412yes—
102#525—216no—
103#525—216no—
104C2×#123—22no—
105C2×#123—22no—
106C2×#123—22no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2×C412yes—
112C2×#121—22no—
113C2×#121—21no—
114C2×#121—22no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—22no—
118C2×#121—22no—
119C2×#121C2×C412yes—
120C2×#121C2×C412yes—
121C2×#121C2×C412yes—
122C2×#120—22no—
123C2×#120—22no—
124C4×#41—22no—
125C4×#41—22no—
126C4×#41C2×C414yes—
127C4×#41C2×C414yes—
128C4×#41D414yes—
129C4×#41D414yes—
130C4×#41C2314yes—
131C4×#41C2314yes—
132C2×C4×Q8C2×C411yescharacteristic
133C2×C4×Q8D411yescharacteristic
134C2×C4×Q8D411yescharacteristic
135C2×C4×Q8C2311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—22no—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4—22no—
142C2×C4×D4—22no—
143C2×C4×D4—22no—
144C2×C4×D4C2×C412yes—
145C2×C4×D4C2×C414yes—
146C2×C4×D4C2×C411yescharacteristic
147C2×C4×D4D412yes—
148C2×C4×D4D411yescharacteristic
149C2×C4×D4D412yes—
150C2×C4×D4D411yescharacteristic
151C2×C4×D4C2312yes—
152C2×C4×D4C2314yes—
153C2×C4×D4C2311yescharacteristic
154C2×#117—21no—
155C2×#117—22no—
156C2×#117C2×C411yescharacteristic
157C2×#117C2×C411yescharacteristic
158C2×#117C2×C412yes—
159C22×C4⋊C4C2×C411yescharacteristic
160C22×C22⋊C4—21no—
161C22×C42—21no—
162C22×C42C2311yescharacteristic
163C22×C42C2312yes—
164#446—28no—
165C8×Q8—28no—
166#436—28no—
167#435—216no—
168C8×D4—28no—
169C8×D4—216no—
170#432—24no—
171#432C2318yescomplemented
172#431—22no—
173#431—22no—
174#431—22no—
175#431—22no—
176#431—24no—
177#431—24no—
178#431—24no—
179#431C2314yes—
180#431C2314yes—
181#431C2314yes—
182#431C2314yes—
183#431C2318yescomplemented
184#423C2318yescomplemented
185C2×C4⋊C8—22no—
186C2×C4⋊C8—24no—
187C2×C4⋊C8C2314yes—
188#407—24no—
189C2×C22⋊C8—22no—
190C2×C22⋊C8—22no—
191C2×C22⋊C8—24no—
192C4×M16—24no—
193C2×C4×C8—22no—
194C2×C4×C8C2314yes—
195C2×C4×C8C2318yescomplemented
196Q8⋊C8C23116yescomplemented
197D4⋊C8C23132yes—
198D4⋊C8C23116yescomplemented
199C25—21no—
200C2×#41—22no—
201C2×#41—28no—
202C2×#41—28no—
203C2×#41—28no—
204C2×#41—28no—
205C2×#41—88no—
206C2×#41C22⋊C414yes—
207C2×#41C22×C414yes—
208C2×#41C22×C414yes—
209C22×Q8—22no—
210C22×Q8C22⋊C411yescharacteristic
211C22×Q8C22×C411yescharacteristic
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—28no—
221C22×D4C22⋊C412yes—
222C22×D4C22⋊C411yescharacteristic
223C22×D4C22×C412yes—
224C22×D4C22×C411yescharacteristic
225C22×D4C22×C414yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—84no—
235C23×C4C22×C412yes—
236C23×C4C22×C411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C2×M16—28no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—28no—
245C4⋊Q8—28no—
246C4⋊Q8—24no—
247C4⋊Q8C22×C418yes—
248#127—28no—
249#127—24no—
250#127C22×C418yes—
251#126—216no—
252#126—28no—
253#126—28no—
254#125—28no—
255#125—24no—
256#125C22×C418yes—
257#124—28no—
258#124—216no—
259#124—28no—
260#124—24no—
261#124—28no—
262#124C22×C418yes—
263#123—28no—
264#123—28no—
265#123—216no—
266#123—216no—
267#123—216no—
268C22⋊Q8—28no—
269C22⋊Q8—24no—
270C22⋊Q8—216no—
271C22⋊Q8—216no—
272C22⋊Q8C22×C418yes—
273#121—216no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—28no—
280#121—28no—
281#121—28no—
282#121—216no—
283#121—216no—
284#121C22×C418yes—
285#121C22×C418yes—
286#121C22×C418yes—
287#120—216no—
288#120—216no—
289C4×Q8—22no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8C22⋊C414yes—
293C4×Q8C22⋊C414yes—
294C4×Q8C22×C414yes—
295C4×Q8C22×C412yes—
296C4×Q8C2×D414yes—
297C4×Q8C2×D414yes—
298C4×Q8C2412yes—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—28no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—28no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—28no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—28no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4C22⋊C418yes—
318C4×D4C22⋊C414yes—
319C4×D4C22⋊C418yes—
320C4×D4C22⋊C414yes—
321C4×D4C22×C414yes—
322C4×D4C22×C418yes—
323C4×D4C22×C414yes—
324C4×D4C22×C412yes—
325C4×D4C2×D418yes—
326C4×D4C2×D414yes—
327C4×D4C2×D418yes—
328C4×D4C2×D414yes—
329C4×D4C2414yes—
330C4×D4C2412yes—
331#117—22no—
332#117—24no—
333#117—28no—
334#117—24no—
335#117—28no—
336#117C22⋊C414yes—
337#117C22⋊C418yes—
338#117C22×C414yes—
339#117C22×C414yes—
340#117C22×C418yes—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊C411yescharacteristic
350C2×C4⋊C4C22⋊C411yescharacteristic
351C2×C4⋊C4C22⋊C412yes—
352C2×C4⋊C4C22⋊C412yes—
353C2×C4⋊C4C22×C412yes—
354C2×C4⋊C4C22×C414yes—
355C2×C4⋊C4C22×C411yescharacteristic
356C2×C4⋊C4C22×C411yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4C22⋊C412yes—
371C2×C22⋊C4C22⋊C412yes—
372C2×C22⋊C4C22⋊C412yes—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42C22×C411yescharacteristic
379C2×C42C22×C412yes—
380C2×C42C22×C412yes—
381C2×C42C22×C414yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2×D414yes—
389C2×C42C2×D414yes—
390C2×C42C2411yescharacteristic
391C2×C42C2412yes—
392C2×C42C2414yes—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—28no—
396C4⋊C8—24no—
397C4⋊C8—28no—
398C4⋊C8C2418yes—
399C22⋊C8—24no—
400C22⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C22⋊C8—28no—
404C22⋊C8—28no—
405#97—28no—
406C4×C8—24no—
407C4×C8—24no—
408C4×C8—24no—
409C4×C8—28no—
410C4×C8C2418yes—
411C24—22no—
412C24—21no—
413C24—24no—
414C24—44no—
415C24—82no—
416C24—82no—
417C24C2×C22⋊C412yes—
418C24C2×C22⋊C411yescharacteristic
419#41—216no—
420#41—216no—
421#41—216no—
422#41—48no—
423#41—816no—
424#41—816no—
425C2×Q8—21no—
426C2×Q8—28no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—84no—
430C2×Q8C2×C22⋊C414yes—
431C2×Q8C23×C412yes—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—28no—
435C2×D4—28no—
436C2×D4—28no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—216no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—48no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—84no—
452C2×D4—84no—
453C2×D4—84no—
454C2×D4C2×C22⋊C418yes—
455C2×D4C2×C22⋊C414yes—
456C2×D4C23×C414yes—
457C2×D4C23×C412yes—
458C22×C4—28no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—88no—
477C22×C4—81no—
478C22×C4—82no—
479C22×C4—88no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—82no—
483C22×C4C2×C22⋊C412yes—
484C22×C4C2×C22⋊C414yes—
485C22×C4C2×C22⋊C411yescharacteristic
486C22×C4C2×C22⋊C412yes—
487C22×C4C2×C22⋊C412yes—
488C22×C4C2×C22⋊C411yescharacteristic
489C22×C4C2×C22⋊C412yes—
490C22×C4C2×C22⋊C412yes—
491C22×C4C2×C22⋊C414yes—
492C22×C4C2×C22⋊C414yes—
493C22×C4C2×C22⋊C411yescharacteristic
494C22×C4C2×C22⋊C412yes—
495C22×C4#13118yes—
496C22×C4C23×C414yes—
497C22×C4C23×C411yescharacteristic
498C22×C4C23×C412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D414yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D414yes—
505M16—88no—
506C2×C8—28no—
507C2×C8—28no—
508C2×C8—84no—
509C2×C8—84no—
510C2×C8—88no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4C2×C22⋊C414yes—
523C4⋊C4C2×C22⋊C414yes—
524C4⋊C4C2×C22⋊C418yes—
525C4⋊C4C2×C22⋊C418yes—
526C4⋊C4C23×C414yes—
527C4⋊C4C23×C412yes—
528C4⋊C4C23×C412yes—
529C22⋊C4—24no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—22no—
542C22⋊C4C2×C22⋊C418yes—
543C22⋊C4C2×C22⋊C418yes—
544C22⋊C4C2×C22⋊C418yes—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42C2×C22⋊C412yes—
550C42C2×C22⋊C414yes—
551C42C2×C22⋊C412yes—
552C42C2×C22⋊C412yes—
553C42C2×C22⋊C414yes—
554C42C2×C22⋊C414yes—
555C42C23×C411yescharacteristic
556C42C23×C412yes—
557C42C23×C414yes—
558C42C22×D412yes—
559C42C22×D414yes—
560C42C22×D411yescharacteristic
561C42C22×D411yescharacteristic
562C42C22×D412yes—
563C42C22×D414yes—
564C42C2511yescharacteristic
565C23—24no—
566C23—24no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—48no—
579C23—84no—
580C23—84no—
581C23—816no—
582C23#40814yes—
583C23C22×C22⋊C414yes—
584C23C22×C22⋊C412yes—
585C23C22×C22⋊C411yescharacteristic
586Q8—28no—
587Q8—24no—
588Q8—42no—
589Q8—88no—
590D4—216no—
591D4—28no—
592D4—28no—
593D4—24no—
594D4—48no—
595D4—44no—
596D4—48no—
597D4—42no—
598D4—88no—
599D4—88no—
600D4—88no—
601D4—88no—
602D4—88no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4#40814yes—
637C2×C4#40818yes—
638C2×C4C22×C22⋊C411yescharacteristic
639C2×C4C22×C22⋊C412yes—
640C2×C4C22×C22⋊C411yescharacteristic
641C2×C4C22×C22⋊C414yes—
642C2×C4C22×C22⋊C412yes—
643C2×C4C22×C22⋊C414yes—
644C2×C4C22×C22⋊C414yes—
645C2×C4C22×C22⋊C412yes—
646C2×C4C22×C22⋊C411yescharacteristic
647C2×C4C2×#13114yes—
648C2×C4C2×#13118yes—
649C2×C4C24×C411yescharacteristic
650C2×C4C23×D411yescharacteristic
651C2×C4C23×D411yescharacteristic
652C8—48no—
653C8—48no—
654C22—21no—
655C22—22no—
656C22—21no—
657C22—22no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—44no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—84no—
669C22—84no—
670C22—88no—
671C22—816no—
672C22C2×#40812yes—
673C22C2×#40814yes—
674C22C23×C22⋊C411yescharacteristic
675C4—24no—
676C4—21no—
677C4—22no—
678C4—24no—
679C4—21no—
680C4—21no—
681C4—22no—
682C4—24no—
683C4—42no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4C2×#40812yes—
688C4C2×#40814yes—
689C4C22×#13112yes—
690C4#358512yes—
691C2—22no—
692C2—24no—
693C2—44no—
694C2—82no—
695C2—84no—
696C2C22×#40811yescharacteristic
697C2#6008711yescharacteristic
698C2#6009311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radicalC2×#3460C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60277C2semidirect
3. #60276C2semidirect
4. #17483C2semidirect
5. #17480C2semidirect
6. #17478C2semidirect
7. #17437C2semidirect
8. C2×#1478C2semidirect
30. #2989C22semidirect
39. C8×#41C22semidirect
40. #2920C22semidirect
43. C2×#431C22semidirect
44. #1496C22semidirect
45. #1495C22semidirect
46. #1494C22semidirect
47. #1492C22semidirect
48. #1479C22semidirect
50. #1478C22semidirect
51. #1478C22semidirect
52. #1477C22semidirect
53. #1476C22semidirect
54. C2×D4⋊C8C22semidirect
171. #432C23semidirect
183. #431C23semidirect
184. #423C23semidirect
195. C2×C4×C8C23semidirect
196. Q8⋊C8C23semidirect
198. D4⋊C8C23semidirect

Maximal subgroups

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

subgroupindex
C2×#34602
#602772
#602762
#174832
#174802
#174782
#174372
C2×#14782