Finite Group Databasebeta

← C2×#59642 #7666241

Subgroups — C2×#59642 #7666241

Order 512.

6022 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59642C2132yesmaximal, direct product factor
2C2×#2807C214yesmaximal, complemented
3C2×#2582C212yesmaximal, complemented
4C2×#2568C211yesmaximal, characteristic, complemented
5C2×#2542C211yesmaximal, characteristic
6C2×#2570C212yesmaximal, complemented
7C2×#2580C212yesmaximal
8C2×#2589C214yesmaximal, complemented
9C2×#2492C212yesmaximal, complemented
10C2×#2502C212yesmaximal, complemented
11C2×#2423C212yesmaximal, complemented
12C2×#2367C212yesmaximal
13C2×#2360C212yesmaximal, complemented
14C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
15C2×D4×C4⋊C4C211yesmaximal, characteristic
16C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
17C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
18C2×C4×#124C211yesmaximal, characteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2212yes—
27C22×C4×Q8C2212yes—
28C22×C4×D4C2211yescharacteristic
29C22×C4×D4C2212yes—
30C23×C4⋊C4C2212yes—
31#2807C22164yescomplemented
32#2589C22164yescomplemented
33#2582C22132yescomplemented
34#2580C22132yes—
35#2570C22132yescomplemented
36#2568C22116yescomplemented
37#2542C22116yes—
38#2502C22132yescomplemented
39#2492C22132yescomplemented
40#2423C22132yescomplemented
41C2×#400C2214yes—
42C2×#400C2214yes—
43C2×#399C2214yes—
44C2×#399C2214yes—
45C2×#398C2214yes—
46C2×#398C2214yes—
47C2×#398C2214yes—
48C2×#393C2214yes—
49#2367C22132yes—
50#2360C22132yescomplemented
51Q8×C4⋊C4C22116yescomplemented
52D4×C4⋊C4C22116yes—
53C4×C4⋊Q8C22116yescomplemented
54C4×C4⋊Q8C22116yescomplemented
55C4×#124C22116yes—
56C2×#391C2212yes—
57C2×#391C2214yes—
58C2×#391C2212yes—
59C2×#391C2212yes—
60C2×#391C2212yes—
61C2×#391C2211yescharacteristic
62C2×#391C2212yes—
63C2×#390C2211yescharacteristic
64C2×#390C2212yes—
65C2×#389C2214yes—
66C2×#389C2212yes—
67C2×#389C2212yes—
68C2×#389C2214yes—
69C2×#389C2212yes—
70C2×#389C2212yes—
71C2×#389C2212yes—
72C2×#389C2211yescharacteristic
73C2×#389C2211yescharacteristic
74C2×#388C2212yes—
75C2×#388C2214yes—
76C2×#388C2212yes—
77C2×#388C2214yes—
78C2×#387C2214yes—
79C2×#387C2214yes—
80C2×#385C2214yes—
81C2×#385C2214yes—
82C2×#385C2214yes—
83C2×#385C2214yes—
84C2×#384C2211yescharacteristic
85C2×#384C2212yes—
86C2×#384C2212yes—
87C2×#384C2211yescharacteristic
88C2×#382C2212yes—
89C2×#382C2211yescharacteristic
90C2×#380C2212yes—
91C2×C4×C4⋊C4C2212yes—
92C2×C4×C4⋊C4C2211yescharacteristic
93C2×C4×C4⋊C4C2212yes—
94C2×C4×C4⋊C4C2211yescharacteristic
95C2×C4×C4⋊C4C2212yes—
96C2×C4×C4⋊C4C2212yes—
97C2×C4×C22⋊C4C2214yes—
98C2×C4×C22⋊C4C2212yes—
99C2×C4×C22⋊C4C2212yes—
100C2×C43C2211yescharacteristic
101C23×Q8C2312yes—
102C23×Q8C2312yes—
103C23×Q8C2311yescharacteristic
104C23×D4C2311yescharacteristic
105C24×C4C2312yes—
106C24×C4C2314yes—
107C2×C4⋊Q8—28no—
108C2×C4⋊Q8—28no—
109C2×C4⋊Q8C2318yes—
110C2×C4⋊Q8C2318yes—
111C2×#124Q8116yescomplemented
112C2×#124C2318yes—
113C2×C22⋊Q8—216no—
114C2×C22⋊Q8—48no—
115C2×C22⋊Q8C23116yes—
116C2×C22⋊Q8C23116yes—
117C2×C4×Q8—28no—
118C2×C4×Q8—48no—
119C2×C4×Q8—48no—
120C2×C4×Q8—48no—
121C2×C4×Q8C2318yes—
122C2×C4×Q8C23116yes—
123C2×C4×Q8C23116yes—
124C2×C4×Q8C23116yes—
125C2×C4×D4—28no—
126C2×C4×D4—48no—
127C2×C4×D4C2318yes—
128C2×C4×D4C23116yes—
129C22×C4⋊C4—48no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C23116yes—
143C22×C4⋊C4C2314yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2314yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2311yescharacteristic
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2314yes—
158C22×C22⋊C4C2314yes—
159C22×C22⋊C4C2312yes—
160C22×C22⋊C4C2312yes—
161C22×C22⋊C4C2312yes—
162C22×C42C2312yes—
163C22×C42C2311yescharacteristic
164C22×C42C2311yescharacteristic
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2311yescharacteristic
168C22×C42C2311yescharacteristic
169C22×C42C2314yes—
170C22×C42C2314yes—
171C22×C42C2311yescharacteristic
172C22×C42C2311yescharacteristic
173C22×C42C2311yescharacteristic
174#400C23132yes—
175#400C23132yes—
176#399C23132yes—
177#399C23132yes—
178#398C23132yes—
179#398C23132yes—
180#398C23132yes—
181#393C23132yes—
182#391C23116yes—
183#391C23116yes—
184#391C23116yes—
185#391C23116yes—
186#391C23116yes—
187#391C2318yes—
188#391C23132yes—
189#390C2318yes—
190#390C23116yes—
191#389C23116yes—
192#389C2318yes—
193#389C2318yes—
194#389C23116yes—
195#389C23116yes—
196#389C23116yes—
197#389C23116yes—
198#389C23132yes—
199#389C23132yes—
200#388C23116yes—
201#388C23132yes—
202#388C23132yes—
203#388C23116yes—
204#387C23132yes—
205#387C23132yes—
206#385C23132yes—
207#385C23132yes—
208#385C23132yes—
209#385C23132yes—
210#384C2318yes—
211#384C2318yes—
212#384C23116yes—
213#384C23116yes—
214#382C23116yes—
215#382C2318yes—
216#380C23116yes—
217C4×C4⋊C4C2318yes—
218C4×C4⋊C4C23116yes—
219C4×C4⋊C4C23116yes—
220C4×C4⋊C4C23116yes—
221C4×C4⋊C4C2318yes—
222C4×C4⋊C4C23116yes—
223C4×C22⋊C4C23132yes—
224C4×C22⋊C4C23116yes—
225C4×C22⋊C4C23116yes—
226C2×#95C2314yes—
227C2×#95C2314yes—
228C2×#95C2314yes—
229C2×#95C2312yes—
230C2×#95C2312yes—
231C2×#95C2312yes—
232C2×#95C2314yes—
233C2×#95C2314yes—
234C2×#95C2314yes—
235C2×#95C2314yes—
236C2×#95C2314yes—
237C2×#95C2312yes—
238C2×#95C2312yes—
239C2×#95C2312yes—
240C2×#95C2312yes—
241C2×#95C2312yes—
242C43C2318yes—
243C25C2412yes—
244C22×Q8—24no—
245C22×Q8—28no—
246C22×Q8—28no—
247C22×Q8—28no—
248C22×Q8—24no—
249C22×Q8—28no—
250C22×Q8—28no—
251C22×Q8—44no—
252C22×Q8C2×Q818yes—
253C22×Q8C2418yes—
254C22×Q8C2418yes—
255C22×Q8C2414yes—
256C22×D4—24no—
257C22×D4—24no—
258C22×D4C2×Q818yes—
259C22×D4C2414yes—
260C23×C4—28no—
261C23×C4—28no—
262C23×C4—28no—
263C23×C4—48no—
264C23×C4—48no—
265C23×C4—48no—
266C23×C4C2418yes—
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C23×C4C2414yes—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2414yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C23×C4C2412yes—
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2412yes—
280C23×C4C2411yescharacteristic
281C23×C4C24116yes—
282C23×C4C2411yescharacteristic
283C23×C4C2411yescharacteristic
284C4⋊Q8—232no—
285C4⋊Q8—232no—
286#124C2×Q8164yescomplemented
287C22⋊Q8—264no—
288C22⋊Q8—432no—
289C4×Q8—232no—
290C4×Q8—432no—
291C4×Q8—432no—
292C4×Q8—432no—
293C4×D4—232no—
294C4×D4—432no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—216no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—432no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4#4118yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2418yes—
321C2×C4⋊C4C2418yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2418yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C24116yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2418yes—
329C2×C4⋊C4C2418yes—
330C2×C4⋊C4C2418yes—
331C2×C4⋊C4C24116yes—
332C2×C4⋊C4C2418yes—
333C2×C4⋊C4C24116yes—
334C2×C4⋊C4C2418yes—
335C2×C4⋊C4C24116yes—
336C2×C4⋊C4C2418yes—
337C2×C4⋊C4C24116yes—
338C2×C4⋊C4C2418yes—
339C2×C4⋊C4C2418yes—
340C2×C4⋊C4C24116yes—
341C2×C4⋊C4C2414yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—216no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4C2×Q8116yes—
349C2×C22⋊C4C2×Q8116yes—
350C2×C22⋊C4C2418yes—
351C2×C22⋊C4C24116yes—
352C2×C22⋊C4C2418yes—
353C2×C22⋊C4C24116yes—
354C2×C22⋊C4C24116yes—
355C2×C22⋊C4C2418yes—
356C2×C22⋊C4C2418yes—
357C2×C22⋊C4C2418yes—
358C2×C42—28no—
359C2×C42—24no—
360C2×C42—28no—
361C2×C42—24no—
362C2×C42—28no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—48no—
366C2×C42—48no—
367C2×C42—42no—
368C2×C42C2×Q818yes—
369C2×C42#4118yes—
370C2×C42C2418yes—
371C2×C42C2414yes—
372C2×C42C2414yes—
373C2×C42C2418yes—
374C2×C42C2418yes—
375C2×C42C2414yes—
376C2×C42C2414yes—
377C2×C42C24116yes—
378C2×C42C24116yes—
379C2×C42C2414yes—
380C2×C42C2414yes—
381C2×C42C2414yes—
382#95C2418yes—
383#95C2418yes—
384#95C2418yes—
385#95C24116yes—
386#95C2418yes—
387#95C2418yes—
388#95C2418yes—
389#95C24116yes—
390#95C24116yes—
391#95C24116yes—
392#95C2418yes—
393#95C2418yes—
394#95C24116yes—
395#95C24116yes—
396#95C24116yes—
397#95C24116yes—
398C24—24no—
399C24—24no—
400C24—24no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24C22×Q818yes—
405C24C2514yes—
406C24C2511yescharacteristic
407C2×Q8—232no—
408C2×Q8—28no—
409C2×Q8—216no—
410C2×Q8—216no—
411C2×Q8—216no—
412C2×Q8—28no—
413C2×Q8—216no—
414C2×Q8—216no—
415C2×Q8—48no—
416C2×Q8—432no—
417C2×Q8—432no—
418C2×Q8C22×Q8116yes—
419C2×D4—232no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4C22×Q8116yes—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—24no—
459C22×C4—216no—
460C22×C4—216no—
461C22×C4—216no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—432no—
466C22×C4—832no—
467C22×C4C22×Q814yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q818yes—
470C22×C4C22×Q814yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C42−1+414yes—
479C22×C4C2514yes—
480C22×C4C2514yes—
481C22×C4C2518yes—
482C22×C4C2514yes—
483C22×C4C2514yes—
484C22×C4C2518yes—
485C22×C4C2512yes—
486C22×C4C2514yes—
487C22×C4C2512yes—
488C22×C4C2512yes—
489C22×C4C2514yes—
490C22×C4C2512yes—
491C22×C4C2514yes—
492C22×C4C2512yes—
493C22×C4C2512yes—
494C22×C4C2512yes—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—232no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—416no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—416no—
514C4⋊C4—48no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4C2×#41116yes—
518C22⋊C4—216no—
519C22⋊C4—232no—
520C22⋊C4—216no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4C22×Q8132yes—
525C22⋊C4C22×Q8132yes—
526C42—216no—
527C42—28no—
528C42—216no—
529C42—28no—
530C42—216no—
531C42—28no—
532C42—28no—
533C42—44no—
534C42—416no—
535C42—416no—
536C42C22×Q8116yes—
537C42C2×#41116yes—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—216no—
545C23—216no—
546C23—216no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23C23×Q818yes—
552C23C23×Q812yes—
553C23C22×#4112yes—
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+412yes—
558C23#58512yes—
559C23C2611yescharacteristic
560Q8—232no—
561Q8—432no—
562Q8—432no—
563D4—232no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—216no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—216no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—48no—
607C2×C4—416no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—416no—
613C2×C4—48no—
614C2×C4—432no—
615C2×C4—832no—
616C2×C4#554116yes—
617C2×C4#559116yes—
618C2×C4C23×Q814yes—
619C2×C4C23×Q814yes—
620C2×C4C23×Q818yes—
621C2×C4C23×Q814yes—
622C2×C4C22×#4114yes—
623C2×C4C22×#4118yes—
624C2×C4C22×#4114yes—
625C2×C4C22×#4114yes—
626C2×C4C22×#4114yes—
627C2×C4C2×H(5,𝔽2)14yes—
628C2×C4C2×H(5,𝔽2)14yes—
629C2×C4C2×2−1+414yes—
630C22—28no—
631C22—28no—
632C22—28no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—432no—
638C22C2×#55414yes—
639C22Q8×#4114yes—
640C22C2×#55914yes—
641C22#350314yes—
642C22#350714yes—
643C22#352014yes—
644C22#354614yes—
645C22C24×Q811yescharacteristic
646C22C23×#4111yescharacteristic
647C22C23×#4111yescharacteristic
648C22C22×H(5,𝔽2)11yescharacteristic
649C22C22×H(5,𝔽2)11yescharacteristic
650C22C22×2−1+411yescharacteristic
651C22C2×#58511yescharacteristic
652C4—28no—
653C4—24no—
654C4—24no—
655C4—24no—
656C4—28no—
657C4—24no—
658C4—42no—
659C4—44no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—48no—
666C4C2×#55418yes—
667C4C2×#55918yes—
668C2—48no—
669C2#5964218yesdirect product factor
670C2C22×#55411yescharacteristic
671C2C22×#55911yescharacteristic
672C2C2×Q8×#4111yescharacteristic
673C2C2×#354611yescharacteristic
674C2C2×#352011yescharacteristic
675C2C2×#350711yescharacteristic
676C2C2×#350311yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59642C2direct (N × H)
2. C2×#2807C2semidirect
3. C2×#2582C2semidirect
4. C2×#2568C2semidirect
6. C2×#2570C2semidirect
8. C2×#2589C2semidirect
9. C2×#2492C2semidirect
10. C2×#2502C2semidirect
11. C2×#2423C2semidirect
13. C2×#2360C2semidirect
14. C2×Q8×C4⋊C4C2semidirect
16. C2×C4×C4⋊Q8C2semidirect
17. C2×C4×C4⋊Q8C2semidirect
31. #2807C22semidirect
32. #2589C22semidirect
33. #2582C22semidirect
35. #2570C22semidirect
36. #2568C22semidirect
38. #2502C22semidirect
39. #2492C22semidirect
40. #2423C22semidirect
50. #2360C22semidirect
51. Q8×C4⋊C4C22semidirect
53. C4×C4⋊Q8C22semidirect
54. C4×C4⋊Q8C22semidirect
111. C2×#124Q8semidirect
286. #124C2×Q8semidirect
669. C2#59642direct (N × H)

Maximal subgroups

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

subgroupindex
#596422
C2×#28072
C2×#25822
C2×#25682
C2×#25422
C2×#25702
C2×#25802
C2×#25892
C2×#24922
C2×#25022
C2×#24232
C2×#23672
C2×#23602
C2×Q8×C4⋊C42
C2×D4×C4⋊C42
C2×C4×C4⋊Q82
C2×C4×#1242