Finite Group Databasebeta

← C2×#24825 #6445542

Subgroups — C2×#24825 #6445542

Order 512.

2602 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C22×#440C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic
5#24825C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2083C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2066C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1940C211yesmaximal, characteristic, complemented
12C2×#1877C211yesmaximal, characteristic
13C2×#1877C211yesmaximal, characteristic
14C2×#1803C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×#1802C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4—21no—
24C2×#478—28no—
25C2×#478C2218yes—
26C2×#440—28no—
27C2×#440C2218yes—
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41#2565—24no—
42#2565C2218yescomplemented
43C2×#398—21no—
44C2×#398—21no—
45D4×C4⋊C4—24no—
46D4×C4⋊C4C2218yes—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#387—21no—
52C2×#387—21no—
53C2×#385—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#380—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60#2085C2218yescomplemented
61#2083C2218yescomplemented
62#2068C2218yescomplemented
63#2066C2218yescomplemented
64#1992C2218yescomplemented
65#1940C2218yescomplemented
66#1877C2218yes—
67#1877C2218yes—
68#1804C2218yes—
69#1803C2218yescomplemented
70#1802C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4—21no—
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16Q818yes—
87C22×SD16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×#125—22no—
93C2×#125—24no—
94C2×#125—24no—
95C2×#125C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8C2314yes—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—24no—
106C2×C4×D4—42no—
107C2×C4×D4—42no—
108C2×C4×D4D418yes—
109C2×C4×D4C2314yes—
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—28no—
122C22×C4⋊C4—48no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#478—232no—
139#440—232no—
140C2×#107—24no—
141C2×#107—24no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—42no—
148C2×C4⋊C8—42no—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—42no—
154C2×#103—42no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—24no—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#97—24no—
165C2×#97C2314yes—
166#398—28no—
167#398—28no—
168#391—22no—
169#391—22no—
170#391C2314yes—
171#391C2314yes—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#387—28no—
177#387—28no—
178#385—28no—
179#385—28no—
180#384—22no—
181#384—22no—
182#384C2314yes—
183#384C2314yes—
184#382—22no—
185#382C2314yes—
186#380—28no—
187C4×C4⋊C4D414yes—
188C4×C4⋊C4C2314yes—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95D411yescharacteristic
198C2×#95D411yescharacteristic
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207C25—21no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8C2×Q814yes—
215C22×Q8C2412yes—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4C2×D414yes—
221C22×D4C2×Q814yes—
222C22×D4#4114yes—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—232no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16C2×Q8116yes—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8C2×Q814yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270#125—28no—
271#125—28no—
272#125—48no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—48no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—216no—
279C4×D4—28no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—48no—
284C4×D4C2×D4116yes—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—432no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—44no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D414yes—
348C2×C42#4114yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—28no—
353#107—28no—
354#106—28no—
355C4⋊C8—28no—
356C4⋊C8—44no—
357C4⋊C8—44no—
358#103—28no—
359#103—44no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—28no—
364#97—28no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—22no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8C22×Q814yes—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4C22⋊Q8116yes—
407C2×D4C22×D414yes—
408C2×D4C22×Q814yes—
409C2×D4C2×#4114yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—216no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—48no—
436C22×C4—432no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—816no—
464C22×C4—816no—
465C22×C4#12114yes—
466C22×C4#12114yes—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
482SD16—232no—
483C2×C8—28no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8C22×Q814yes—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—84no—
527C4⋊C4C22×D414yes—
528C22⋊C4—216no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—48no—
538C42#12118yes—
539C42C22×D414yes—
540C42C2×#4114yes—
541C23—24no—
542C23—216no—
543C23—24no—
544C23—24no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C2×#12112yes—
556C23#54612yes—
557C23D4×Q812yes—
558C23C2×AGL(1,ℤ/8)12yes—
559C23#57512yes—
560C23#57712yes—
561C23#57812yes—
562C23C23×D411yescharacteristic
563Q8—24no—
564Q8—24no—
565Q8—48no—
566Q8—42no—
567Q8—42no—
568D4—216no—
569D4—24no—
570D4—24no—
571D4—24no—
572D4C2×C22⋊Q818yes—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—432no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
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—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—816no—
619C2×C4—816no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4C2×#12114yes—
626C2×C4C2×#12114yes—
627C2×C4C2×#12112yes—
628C2×C4C2×C22⋊Q814yes—
629C2×C4#53714yes—
630C2×C4#54612yes—
631C2×C4D4×Q812yes—
632C2×C4#55214yes—
633C2×C4#55414yes—
634C2×C4C2×AGL(1,ℤ/8)14yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C23×Q811yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C8—24no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—28no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—432no—
652C22—44no—
653C22#241714yes—
654C22#303614yes—
655C22#304614yes—
656C22#331014yes—
657C22#331514yes—
658C22#338614yes—
659C22#338814yes—
660C22C22×#12111yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×D4×Q811yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57711yescharacteristic
666C22C2×#57811yescharacteristic
667C4—24no—
668C4—24no—
669C4—24no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#241714yes—
681C4C22×C22⋊Q811yescharacteristic
682C4C2×#53711yescharacteristic
683C4C2×#55211yescharacteristic
684C4C2×#55411yescharacteristic
685C2—48no—
686C2#2482518yesdirect product factor
687C2C2×#241711yescharacteristic
688C2C2×#303611yescharacteristic
689C2C2×#304611yescharacteristic
690C2C2×#331011yescharacteristic
691C2C2×#331511yescharacteristic
692C2C2×#338611yescharacteristic
693C2C2×#338811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2565C2semidirect
5. #24825C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2083C2semidirect
8. C2×#2068C2semidirect
9. C2×#2066C2semidirect
10. C2×#1992C2semidirect
11. C2×#1940C2semidirect
14. C2×#1803C2semidirect
42. #2565C22semidirect
60. #2085C22semidirect
61. #2083C22semidirect
62. #2068C22semidirect
63. #2066C22semidirect
64. #1992C22semidirect
65. #1940C22semidirect
69. #1803C22semidirect
686. C2#24825direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C22×#4402
C2×#25652
C2×D4×C4⋊C42
#248252
C2×#20852
C2×#20832
C2×#20682
C2×#20662
C2×#19922
C2×#19402
C2×#18772
C2×#18032
C2×#18042
C2×#18022