Finite Group Databasebeta

← C2×#17368 #6342563

Subgroups — C2×#17368 #6342563

Order 512.

4190 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#2973C212yesmaximal, complemented
3#17368C2116yesmaximal, direct product factor
4C2×#1467C212yesmaximal, complemented
5C2×#1465C212yesmaximal, complemented
6C2×#1464C212yesmaximal, complemented
7C2×#1463C212yesmaximal, complemented
8C2×#1459C212yesmaximal, complemented
9C22×#324C212yesmaximal, complemented
10C23×#41C411yescharacteristic
11C2×#520—28no—
12C2×#520—28no—
13C2×#520C418yes—
14C2×#520C2218yes—
15C22×C4×Q8C411yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C411yescharacteristic
18C22×C4×D4C411yescharacteristic
19C22×C4×D4C411yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#117C411yescharacteristic
24C22×#117C411yescharacteristic
25C22×#117C2211yescharacteristic
26C23×C4⋊C4C411yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29#2973—28no—
30#2973C22116yescomplemented
31C2×#435—28no—
32C2×#432C2214yes—
33C2×#407C2214yes—
34C22×C22⋊C8C2214yes—
35#1467C22116yescomplemented
36#1465C22116yescomplemented
37#1464C22116yescomplemented
38#1463C22116yescomplemented
39#1459C22116yescomplemented
40C2×#324C2214yes—
41C2×#324C2214yes—
42C2×#324C2214yes—
43C2×#324C22116yescomplemented
44C2×#323C2214yes—
45C22×#41—24no—
46C22×#41—24no—
47C22×#41—28no—
48C22×#41—28no—
49C22×#41C2×C418yes—
50C23×Q8C2×C411yescharacteristic
51C23×D4C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C23×D4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2×C411yescharacteristic
56C24×C4C2×C411yescharacteristic
57C24×C4C2×C411yescharacteristic
58C24×C4C2×C411yescharacteristic
59C24×C4C2×C411yescharacteristic
60C24×C4C2×C411yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16—24no—
64C23×C8—24no—
65#520—264no—
66#520—264no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—24no—
71C2×C4×Q8C2×C418yes—
72C2×C4×Q8C2×C414yes—
73C2×C4×Q8C2314yes—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4C2×C418yes—
85C2×C4×D4C2×C418yes—
86C2×C4×D4C2×C418yes—
87C2×C4×D4C2×C414yes—
88C2×C4×D4C2×C414yes—
89C2×C4×D4C2×C414yes—
90C2×C4×D4D418yes—
91C2×C4×D4D418yes—
92C2×C4×D4C2314yes—
93C2×C4×D4C2314yes—
94C2×C4×D4C2314yes—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117C2×C418yes—
102C2×#117C2×C418yes—
103C2×#117C2×C414yes—
104C2×#117C2314yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C411yescharacteristic
113C22×C4⋊C4C2×C411yescharacteristic
114C22×C4⋊C4C2×C418yes—
115C22×C4⋊C4C2×C411yescharacteristic
116C22×C4⋊C4C2×C411yescharacteristic
117C22×C4⋊C4C2×C414yes—
118C22×C4⋊C4C2×C414yes—
119C22×C4⋊C4C2×C411yescharacteristic
120C22×C4⋊C4C2×C411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C22⋊C4C2×C411yescharacteristic
127C22×C22⋊C4C2×C411yescharacteristic
128C22×C22⋊C4C2×C411yescharacteristic
129C22×C22⋊C4C2×C411yescharacteristic
130C22×C22⋊C4C2×C411yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C42C2×C411yescharacteristic
133C22×C42C2×C411yescharacteristic
134C22×C42C2×C411yescharacteristic
135C22×C42C2×C411yescharacteristic
136C22×C42C2×C411yescharacteristic
137C22×C42C2311yescharacteristic
138#435—264no—
139#432—28no—
140#432C23116yes—
141C2×C4⋊C8—24no—
142#407—28no—
143#407C23116yes—
144C2×C22⋊C8—24no—
145C2×C22⋊C8—28no—
146C2×C22⋊C8—24no—
147C2×C22⋊C8C2318yes—
148C2×C22⋊C8C23116yes—
149C2×#97—24no—
150#324C23116yes—
151#324C23116yes—
152#324C23116yes—
153#323C23116yes—
154C25C22×C411yescharacteristic
155C25C22×C411yescharacteristic
156C25C22×C411yescharacteristic
157C2×#41—264no—
158C2×#41—264no—
159C2×#41—232no—
160C2×#41—232no—
161C2×#41—232no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8C22×C414yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4C22⋊C414yes—
177C22×D4C22⋊C414yes—
178C22×D4M1618yescomplemented
179C22×D4C22×C414yes—
180C22×D4C22×C414yes—
181C22×D4C22×C414yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—82no—
202C23×C4—82no—
203C23×C4—82no—
204C23×C4—82no—
205C23×C4—82no—
206C23×C4—82no—
207C23×C4M1618yescomplemented
208C23×C4C22×C414yes—
209C23×C4C22×C414yes—
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C414yes—
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C22×C411yescharacteristic
224C23×C4C22×C411yescharacteristic
225C23×C4C22×C411yescharacteristic
226C23×C4C22×C412yes—
227C23×C4C22×C412yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C2×M16—216no—
236C2×M16—88no—
237C22×C8—28no—
238C22×C8—216no—
239C22×C8—88no—
240C4×Q8—232no—
241C4×Q8—216no—
242C4×Q8—216no—
243C4×D4—232no—
244C4×D4—216no—
245C4×D4—216no—
246C4×D4—216no—
247C4×D4—216no—
248C4×D4—232no—
249C4×D4—232no—
250C4×D4C22⋊C4132yes—
251C4×D4C2×D4132yes—
252#117—232no—
253#117—232no—
254#117—216no—
255#117—216no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—232no—
279C2×C4⋊C4C22⋊C414yes—
280C2×C4⋊C4C22⋊C414yes—
281C2×C4⋊C4C22×C414yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C412yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C414yes—
286C2×C4⋊C4C22×C414yes—
287C2×C4⋊C4C22×C412yes—
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C414yes—
290C2×C4⋊C4C22×C414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4C22⋊C414yes—
303C2×C22⋊C4C22⋊C414yes—
304C2×C22⋊C4C22⋊C414yes—
305C2×C22⋊C4C22⋊C414yes—
306C2×C22⋊C4C22×C414yes—
307C2×C22⋊C4C22×C414yes—
308C2×C22⋊C4C22×C414yes—
309C2×C22⋊C4C22×C412yes—
310C2×C22⋊C4C22×C414yes—
311C2×C22⋊C4C22×C414yes—
312C2×C22⋊C4C2412yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42C22×C414yes—
324C2×C42C22×C414yes—
325C2×C42C22×C414yes—
326C2×C42C22×C414yes—
327C2×C42C22×C414yes—
328C2×C42C22×C412yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2412yes—
332C4⋊C8—216no—
333C22⋊C8—28no—
334C22⋊C8—216no—
335C22⋊C8—216no—
336C22⋊C8C24116yes—
337#97—216no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24C2×C22⋊C412yes—
350C24C2×C22⋊C412yes—
351C24C2×C22⋊C412yes—
352C24C2×C22⋊C412yes—
353C24C2×M1614yes—
354C24C2×M1614yes—
355C24C2×M1614yes—
356C24C23×C411yescharacteristic
357C24C23×C412yes—
358C24C23×C412yes—
359C24C23×C412yes—
360#41—2256no—
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×D4—216no—
367C2×D4—216no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—28no—
379C2×D4C2×C22⋊C4116yes—
380C2×D4C2×M16116yescomplemented
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C412yes—
453C22×C4C2×C22⋊C412yes—
454C22×C4C2×C22⋊C418yes—
455C22×C4C2×C22⋊C418yes—
456C22×C4C2×M1614yes—
457C22×C4C2×M1614yes—
458C22×C4C2×M1614yes—
459C22×C4C2×M16116yescomplemented
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C412yes—
471C22×C4C23×C411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D418yes—
475C22×C4C22×D418yes—
476C22×C4C2511yescharacteristic
477M16—816no—
478C2×C8—216no—
479C2×C8—416no—
480C2×C8—816no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4C2×C22⋊C4116yes—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4C2×C22⋊C4116yes—
500C22⋊C4C2×C22⋊C4116yes—
501C42—28no—
502C42—28no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42C2×C22⋊C418yes—
507C42C22×D418yes—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—48no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23#40718yes—
543C23#40718yes—
544C23C22×C22⋊C414yes—
545C23C22×C22⋊C414yes—
546C23C22×C22⋊C411yescharacteristic
547C23C22×C22⋊C411yescharacteristic
548C23C22×M1612yes—
549C23C22×M1612yes—
550C23C22×M1614yes—
551C23C22×M1614yes—
552C23C22×M1614yes—
553C23C24×C411yescharacteristic
554Q8—232no—
555D4—232no—
556D4—232no—
557D4—232no—
558C2×C4—232no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—232no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4C22×C22⋊C414yes—
615C2×C4C22×C22⋊C414yes—
616C2×C4C22×C22⋊C414yes—
617C2×C4C22×C22⋊C414yes—
618C2×C4C22×C22⋊C412yes—
619C2×C4C22×M1614yes—
620C2×C4C22×M1614yes—
621C2×C4C22×M1614yes—
622C2×C4C23×D412yes—
623C8—416no—
624C22—21no—
625C22—21no—
626C22—24no—
627C22—21no—
628C22—21no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—216no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—44no—
639C22—48no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—48no—
649C22—48no—
650C22C2×#40712yes—
651C22C2×#40718yes—
652C22C2×#40712yes—
653C22#288312yes—
654C22#288912yes—
655C22C23×C22⋊C411yescharacteristic
656C22C23×M1611yescharacteristic
657C22C23×M1611yescharacteristic
658C4—24no—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—24no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C2—22no—
672C2—21no—
673C2—21no—
674C2—28no—
675C2—44no—
676C2—44no—
677C2#1736814yesdirect product factor
678C2C22×#40711yescharacteristic
679C2C2×#288311yescharacteristic
680C2C2×#288911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×#41C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2973C2semidirect
3. #17368C2direct (N × H)
4. C2×#1467C2semidirect
5. C2×#1465C2semidirect
6. C2×#1464C2semidirect
7. C2×#1463C2semidirect
8. C2×#1459C2semidirect
9. C22×#324C2semidirect
30. #2973C22semidirect
35. #1467C22semidirect
36. #1465C22semidirect
37. #1464C22semidirect
38. #1463C22semidirect
39. #1459C22semidirect
43. C2×#324C22semidirect
178. C22×D4M16semidirect
207. C23×C4M16semidirect
380. C2×D4C2×M16semidirect
459. C22×C4C2×M16semidirect
677. C2#17368direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#29732
#173682
C2×#14672
C2×#14652
C2×#14642
C2×#14632
C2×#14592
C22×#3242