Finite Group Databasebeta

← #6343112

Subgroups — #6343112

Order 512.

3093 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3475C211yesmaximal, characteristic
2#60276C211yesmaximal, characteristic, complemented
3#60125C211yesmaximal, characteristic, complemented
4#17508C2116yesmaximal, complemented
5#17487C214yesmaximal, complemented
6#17486C212yesmaximal, complemented
7#17443C214yesmaximal, complemented
8C2×#1486C212yesmaximal, complemented
9#3475—24no—
10#3475—24no—
11#3475C418yes—
12#3475C2218yes—
13C2×#556C411yescharacteristic
14C2×#555C411yescharacteristic
15C2×#554C411yescharacteristic
16C2×#553—21no—
17C2×#552—21no—
18C2×#552C412yes—
19C2×#551—22no—
20C2×#547C411yescharacteristic
21C2×#546C412yes—
22C2×#533C2211yescharacteristic
23C2×#529—22no—
24C2×#529C2212yes—
25C2×#525—21no—
26C2×C4×#41C2212yes—
27C2×C4×#41C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×#117—21no—
30#2988—24no—
31#2982—21no—
32#2972—22no—
33#2972—21no—
34C8×#41C2218yescomplemented
35#2925—24no—
36#2925C2218yescomplemented
37#2921—22no—
38#2920C2214yescomplemented
39C2×#431C2212yes—
40C2×#431C2214yescomplemented
41C2×#423C2212yes—
42#2879—21no—
43#2873—21no—
44#1494C22116yescomplemented
45#1493C2218yescomplemented
46#1488C22116yescomplemented
47#1487C2218yescomplemented
48#1486C2214yes—
49#1486C2214yes—
50#1486C2218yescomplemented
51#1482C22116yescomplemented
52#1481C2218yescomplemented
53#1476C22116yescomplemented
54C2×D4⋊C8C2218yescomplemented
55C22×#41C2×C412yes—
56C22×#41D411yescharacteristic
57C23×D4C2×C411yescharacteristic
58C24×C4—21no—
59C22×M16—22no—
60#556—24no—
61#556C2×C418yes—
62#555—24no—
63#555C2×C418yes—
64#554—24no—
65#554C2×C418yes—
66#553—28no—
67#553—28no—
68#552—28no—
69#552—28no—
70#552—28no—
71#552C2×C4116yes—
72#551—216no—
73#551—216no—
74#547—24no—
75#547C2×C418yes—
76#546—28no—
77#546C2×C4116yes—
78#533—22no—
79#533—22no—
80#533C2×C414yes—
81#533C2314yes—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C4⋊Q8C2×C411yescharacteristic
84#529—216no—
85#529—216no—
86#529—24no—
87#529—24no—
88#529C2×C418yes—
89#529C2318yes—
90C2×#126—22no—
91C2×#126—22no—
92C2×#126—22no—
93C2×#125—22no—
94C2×#125—21no—
95C2×#125C2×C412yes—
96C2×#124—22no—
97C2×#124C2×C412yes—
98#525—28no—
99#525—28no—
100C2×#123—22no—
101C2×#123—22no—
102C2×#123—22no—
103C2×#123—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2×C412yes—
109C2×C22⋊Q8C2×C412yes—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121C2×C412yes—
114C2×#120—22no—
115C4×#41—24no—
116C4×#41—24no—
117C4×#41C2×C418yes—
118C4×#41C2×C414yes—
119C4×#41D414yes—
120C4×#41D414yes—
121C4×#41C2318yes—
122C4×#41C2314yes—
123C2×C4×Q8—21no—
124C2×C4×Q8C2×C412yes—
125C2×C4×Q8C2×C411yescharacteristic
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4—22no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—22no—
134C2×C4×D4—22no—
135C2×C4×D4C2×C411yescharacteristic
136C2×C4×D4C2×C411yescharacteristic
137C2×C4×D4C2×C414yes—
138C2×C4×D4C2×C412yes—
139C2×C4×D4C2×C412yes—
140C2×C4×D4C2×C411yescharacteristic
141C2×C4×D4D412yes—
142C2×C4×D4D411yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C2×C4×D4C2311yescharacteristic
145C2×C4×D4C2314yes—
146C2×#117—21no—
147C2×#117—28no—
148C2×#117—28no—
149C2×#117—21no—
150C2×#117—21no—
151C2×#117—22no—
152C2×#117—22no—
153C2×#117—22no—
154C2×#117C2×C412yes—
155C2×#117D411yescharacteristic
156C2×#117D412yes—
157C22×C4⋊C4—21no—
158C22×C4⋊C4—21no—
159C22×C4⋊C4C2×C411yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—22no—
162C22×C42—22no—
163C22×C42C2311yescharacteristic
164C22×C42C2312yes—
165C8×Q8—28no—
166C8×D4—28no—
167C8×D4—216no—
168#433—216no—
169#432—24no—
170#432C2318yescomplemented
171#431—22no—
172#431—22no—
173#431—24no—
174#431—24no—
175#431—24no—
176#431C2314yes—
177#431C2314yes—
178#431C2318yescomplemented
179#431C2318yescomplemented
180#423—22no—
181#423—22no—
182#423C2314yes—
183#423C2314yes—
184C2×C4⋊C8—22no—
185C2×C4⋊C8—22no—
186C2×C4⋊C8—24no—
187C2×C4⋊C8C2314yes—
188#408—28no—
189#407—24no—
190C2×C22⋊C8—22no—
191C2×C22⋊C8—24no—
192#405—28no—
193C4×M16—24no—
194C2×C4×C8C2314yes—
195C2×C4×C8C2318yescomplemented
196Q8⋊C8C23132yescomplemented
197D4⋊C8C23116yes—
198D4⋊C8C23116yes—
199C25—21no—
200C2×#41—24no—
201C2×#41—28no—
202C2×#41—48no—
203C2×#41C22⋊C414yes—
204C2×#41C22×C418yes—
205C2×#41C2×D414yes—
206C22×Q8—21no—
207C22×Q8C22⋊C411yescharacteristic
208C22×Q8C22×C412yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—44no—
214C22×D4C22⋊C412yes—
215C22×D4C22⋊C411yescharacteristic
216C22×D4C22×C411yescharacteristic
217C22×D4C22×C414yes—
218C22×D4C22×C411yescharacteristic
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—48no—
229C23×C4C22×C411yescharacteristic
230C23×C4C22×C412yes—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C2×M16—24no—
236C2×M16—24no—
237C2×M16—48no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—28no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8C22×C414yes—
244C4⋊Q8C22×C414yes—
245#126—216no—
246#126—216no—
247#126—28no—
248#126—28no—
249#125—216no—
250#125—24no—
251#125—24no—
252#125—24no—
253#125C22×C418yes—
254#124—216no—
255#124—24no—
256#124C22×C418yes—
257#123—216no—
258#123—28no—
259#123—28no—
260#123—216no—
261#123—216no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—216no—
269C22⋊Q8C22×C418yes—
270C22⋊Q8C22×C418yes—
271#121—28no—
272#121—28no—
273#121—24no—
274#121—216no—
275#121C22×C418yes—
276#120—216no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8C22⋊C414yes—
283C4×Q8C22⋊C414yes—
284C4×Q8C22×C418yes—
285C4×Q8C22×C414yes—
286C4×Q8C22×C414yes—
287C4×Q8C2×D414yes—
288C4×Q8C2414yes—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4C22⋊C418yes—
307C4×D4C22⋊C414yes—
308C4×D4C22⋊C418yes—
309C4×D4C22⋊C414yes—
310C4×D4C22×C412yes—
311C4×D4C22×C412yes—
312C4×D4C22×C414yes—
313C4×D4C22×C414yes—
314C4×D4C22×C418yes—
315C4×D4C22×C418yes—
316C4×D4C22×C414yes—
317C4×D4C2×D418yes—
318C4×D4C2×D414yes—
319C4×D4C2412yes—
320C4×D4C2412yes—
321#117—24no—
322#117—24no—
323#117—24no—
324#117—28no—
325#117—28no—
326#117—216no—
327#117—28no—
328#117—28no—
329#117—28no—
330#117—28no—
331#117C22⋊C414yes—
332#117C22⋊C418yes—
333#117C22×C418yes—
334#117C2×D414yes—
335#117C2×D418yes—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4C22⋊C411yescharacteristic
350C2×C4⋊C4C22⋊C411yescharacteristic
351C2×C4⋊C4C22⋊C412yes—
352C2×C4⋊C4C22⋊C412yes—
353C2×C4⋊C4C22×C411yescharacteristic
354C2×C4⋊C4C22×C411yescharacteristic
355C2×C4⋊C4C22×C414yes—
356C2×C4⋊C4C22×C412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—216no—
371C2×C22⋊C4C22⋊C412yes—
372C2×C22⋊C4C22⋊C412yes—
373C2×C22⋊C4C22⋊C412yes—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—28no—
378C2×C42—28no—
379C2×C42C22×C411yescharacteristic
380C2×C42C22×C412yes—
381C2×C42C22×C412yes—
382C2×C42C22×C414yes—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D412yes—
388C2×C42C2×D412yes—
389C2×C42C2×D414yes—
390C2×C42C2×D414yes—
391C2×C42C2411yescharacteristic
392C2×C42C2412yes—
393C2×C42C2414yes—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—28no—
399C4⋊C8—24no—
400C4⋊C8C2418yes—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C22⋊C8—28no—
404C22⋊C8—28no—
405C22⋊C8—28no—
406#97—28no—
407C4×C8—24no—
408C4×C8—28no—
409C4×C8—28no—
410C4×C8C2418yes—
411C24—21no—
412C24—21no—
413C24—24no—
414C24—44no—
415C24—44no—
416C24—82no—
417C24C2×C22⋊C412yes—
418C24C2×C22⋊C411yescharacteristic
419#41—28no—
420#41—28no—
421#41—216no—
422#41—216no—
423#41—48no—
424#41—416no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—44no—
429C2×Q8—82no—
430C2×Q8C2×C22⋊C414yes—
431C2×Q8C23×C414yes—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—28no—
435C2×D4—28no—
436C2×D4—28no—
437C2×D4—21no—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—24no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—48no—
444C2×D4—44no—
445C2×D4—48no—
446C2×D4—84no—
447C2×D4C2×C22⋊C418yes—
448C2×D4C2×C22⋊C414yes—
449C2×D4C23×C412yes—
450C2×D4C23×C412yes—
451C22×C4—28no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—416no—
476C22×C4—82no—
477C22×C4—81no—
478C22×C4—84no—
479C22×C4C2×C22⋊C412yes—
480C22×C4C2×C22⋊C414yes—
481C22×C4C2×C22⋊C412yes—
482C22×C4C2×C22⋊C412yes—
483C22×C4C2×C22⋊C411yescharacteristic
484C22×C4C2×C22⋊C412yes—
485C22×C4C2×C22⋊C414yes—
486C22×C4C2×C22⋊C411yescharacteristic
487C22×C4C2×C22⋊C412yes—
488C22×C4C2×C22⋊C414yes—
489C22×C4C2×C22⋊C411yescharacteristic
490C22×C4C2×C22⋊C412yes—
491C22×C4#13518yes—
492C22×C4C23×C411yescharacteristic
493C22×C4C23×C412yes—
494C22×C4C23×C414yes—
495C22×C4C22×D414yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C22×D414yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501M16—48no—
502M16—48no—
503C2×C8—28no—
504C2×C8—28no—
505C2×C8—48no—
506C2×C8—84no—
507C2×C8—88no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4C2×C22⋊C414yes—
523C4⋊C4C2×C22⋊C414yes—
524C4⋊C4C2×C22⋊C418yes—
525C4⋊C4C2×C22⋊C418yes—
526C4⋊C4C23×C412yes—
527C4⋊C4C23×C412yes—
528C4⋊C4C23×C414yes—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—22no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—22no—
537C22⋊C4—24no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4C2×C22⋊C418yes—
543C22⋊C4C2×C22⋊C418yes—
544C22⋊C4C2×C22⋊C418yes—
545C42—28no—
546C42—24no—
547C42—28no—
548C42—24no—
549C42C2×C22⋊C414yes—
550C42C2×C22⋊C414yes—
551C42C2×C22⋊C412yes—
552C42C2×C22⋊C412yes—
553C42C2×C22⋊C412yes—
554C42C2×C22⋊C414yes—
555C42C23×C411yescharacteristic
556C42C23×C412yes—
557C42C23×C414yes—
558C42C22×D414yes—
559C42C22×D412yes—
560C42C22×D411yescharacteristic
561C42C22×D414yes—
562C42C22×D411yescharacteristic
563C42C22×D412yes—
564C42C2511yescharacteristic
565C23—24no—
566C23—22no—
567C23—24no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—48no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—48no—
579C23—42no—
580C23—84no—
581C23—816no—
582C23#41614yes—
583C23C22×C22⋊C414yes—
584C23C22×C22⋊C412yes—
585C23C22×C22⋊C411yescharacteristic
586Q8—28no—
587Q8—28no—
588Q8—44no—
589Q8—84no—
590D4—24no—
591D4—216no—
592D4—28no—
593D4—22no—
594D4—24no—
595D4—48no—
596D4—42no—
597D4—416no—
598D4—88no—
599D4#56814yes—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—416no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—88no—
634C2×C4#41614yes—
635C2×C4#41618yes—
636C2×C4C22×C22⋊C414yes—
637C2×C4C22×C22⋊C414yes—
638C2×C4C22×C22⋊C411yescharacteristic
639C2×C4C22×C22⋊C411yescharacteristic
640C2×C4C22×C22⋊C412yes—
641C2×C4C22×C22⋊C414yes—
642C2×C4C22×C22⋊C412yes—
643C2×C4C22×C22⋊C411yescharacteristic
644C2×C4C22×C22⋊C412yes—
645C2×C4C2×#13514yes—
646C2×C4C2×#13514yes—
647C2×C4C2×#13514yes—
648C2×C4C24×C411yescharacteristic
649C2×C4C23×D411yescharacteristic
650C2×C4C23×D411yescharacteristic
651C8—48no—
652C8—48no—
653C22—21no—
654C22—22no—
655C22—21no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—44no—
662C22—48no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—84no—
669C22—816no—
670C22—84no—
671C22C2×#41612yes—
672C22C2×#41614yes—
673C22C23×C22⋊C411yescharacteristic
674C4—24no—
675C4—21no—
676C4—22no—
677C4—24no—
678C4—21no—
679C4—21no—
680C4—24no—
681C4—22no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4C2×#41612yes—
687C4C2×#41614yes—
688C4C2×#56812yes—
689C4C22×#13512yes—
690C2—22no—
691C2—24no—
692C2—44no—
693C2—44no—
694C2—82no—
695C2#6006411yescharacteristic
696C2C22×#41611yescharacteristic
697C2#6009311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radicalC2×#3475C2
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. #60276C2semidirect
3. #60125C2semidirect
4. #17508C2semidirect
5. #17487C2semidirect
6. #17486C2semidirect
7. #17443C2semidirect
8. C2×#1486C2semidirect
34. C8×#41C22semidirect
36. #2925C22semidirect
38. #2920C22semidirect
40. C2×#431C22semidirect
44. #1494C22semidirect
45. #1493C22semidirect
46. #1488C22semidirect
47. #1487C22semidirect
50. #1486C22semidirect
51. #1482C22semidirect
52. #1481C22semidirect
53. #1476C22semidirect
54. C2×D4⋊C8C22semidirect
170. #432C23semidirect
178. #431C23semidirect
179. #431C23semidirect
195. C2×C4×C8C23semidirect
196. Q8⋊C8C23semidirect

Maximal subgroups

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

subgroupindex
C2×#34752
#602762
#601252
#175082
#174872
#174862
#174432
C2×#14862