Finite Group Databasebeta

← C2×#25309 #6448102

Subgroups — C2×#25309 #6448102

Order 512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#463C211yesmaximal, characteristic
2C22×#446C211yesmaximal, characteristic, complemented
3C2×#2538C211yesmaximal, characteristic, complemented
4C2×#2592C211yesmaximal, characteristic, complemented
5#25309C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic, complemented
9C2×#2054C211yesmaximal, characteristic
10C2×#1972C211yesmaximal, characteristic
11C2×#1968C211yesmaximal, characteristic, complemented
12C2×#1959C211yesmaximal, characteristic, complemented
13C2×#1948C211yesmaximal, characteristic, complemented
14C2×#1935C211yesmaximal, characteristic, complemented
15C2×#1934C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#463—28no—
24C2×#463C2218yes—
25C2×#446—28no—
26C2×#446C2218yescomplemented
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×#97C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2592—24no—
44#2592C2218yescomplemented
45#2538—24no—
46#2538C2218yescomplemented
47C2×#397—21no—
48C2×#396—21no—
49C2×#396—21no—
50C2×#393—21no—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#390C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#388—21no—
60C2×#388—21no—
61C2×#384C2211yescharacteristic
62C2×#380—21no—
63C2×C4×C4⋊C4C2211yescharacteristic
64#2066C2218yescomplemented
65#2065C2218yescomplemented
66#2055C2218yescomplemented
67#2054C2218yes—
68#1972C2218yes—
69#1968C2218yescomplemented
70#1959C2218yescomplemented
71#1948C2218yescomplemented
72#1935C2218yescomplemented
73#1934C2218yescomplemented
74#1762C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8D411yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—42no—
87C22×SD16C2314yes—
88C22×D8—24no—
89C22×D8—24no—
90C22×D8—42no—
91C22×D8C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×#124—22no—
97C2×#124—24no—
98C2×#124—24no—
99C2×#124C2314yes—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—48no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8D414yes—
105C2×C4×Q8C2314yes—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#463—232no—
138#446—232no—
139C2×#106—42no—
140C2×#106—42no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8—24no—
145C2×C4⋊C8D418yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—42no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102—24no—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—42no—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#97—24no—
166C2×#97—24no—
167C2×#97C2314yes—
168C2×#97C2314yes—
169C2×C4×C8—24no—
170C2×C4×C8C2314yes—
171#397—28no—
172#396—28no—
173#396—28no—
174#393—28no—
175#391—22no—
176#391—22no—
177#391C2314yes—
178#391C2314yes—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390C2314yes—
184#390C2314yes—
185#390C2314yes—
186#390C2314yes—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#388—28no—
192#388—28no—
193#384—22no—
194#384C2314yes—
195#380—28no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4C2314yes—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209C25—21no—
210C25—21no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—42no—
221C22×Q8C2×D414yes—
222C22×Q8C2412yes—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4#4114yes—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—416no—
260C2×D8—28no—
261C2×D8—28no—
262C2×D8—44no—
263C2×D8—416no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8C2×D414yes—
274C22×C8C2×D414yes—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280#124—28no—
281#124—28no—
282#124—48no—
283C22⋊Q8—432no—
284C4×Q8—216no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—48no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
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—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4—82no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—82no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D414yes—
356C2×C42#4114yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#106—44no—
361#106—44no—
362C4⋊C8—28no—
363C4⋊C8—28no—
364C4⋊C8C2×D4116yes—
365#103—28no—
366#103—28no—
367#103—44no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372#102—44no—
373#97—28no—
374#97—28no—
375C4×C8—28no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24C22×D411yescharacteristic
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—416no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4C2×#4114yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—48no—
485C22×C4—816no—
486C22×C4#12114yes—
487C22×C4#12114yes—
488C22×C4AGL(1,ℤ/8)14yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C42−1+412yes—
502C22×C4C2511yescharacteristic
503SD16—416no—
504D8—416no—
505C2×C8—28no—
506C2×C8—28no—
507C2×C8—28no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—44no—
518C2×C8C22×D414yes—
519C2×C8C22×D414yes—
520C2×C8C2×#4114yes—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—84no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—84no—
561C42#12118yes—
562C42C22×D414yes—
563C42C2×#4114yes—
564C23—24no—
565C23—48no—
566C23—48no—
567C23—42no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23C2×#12112yes—
593C23#54812yes—
594C23#55012yes—
595C23C2×AGL(1,ℤ/8)12yes—
596C23#57512yes—
597C23#57712yes—
598C23#57712yes—
599C23C23×D411yescharacteristic
600Q8—416no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—84no—
606D4—24no—
607D4—24no—
608D4—42no—
609D4—48no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—84no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—816no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4C2×#12114yes—
663C2×C4C2×#12114yes—
664C2×C4C2×#12112yes—
665C2×C4#52914yes—
666C2×C4#54214yes—
667C2×C4#54614yes—
668C2×C4#54812yes—
669C2×C4#54814yes—
670C2×C4#55012yes—
671C2×C4C2×AGL(1,ℤ/8)14yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×2−1+411yescharacteristic
678C8—24no—
679C8—24no—
680C8—24no—
681C8—42no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—816no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—168no—
698C22#242614yes—
699C22#305514yes—
700C22#306014yes—
701C22#337314yes—
702C22#337614yes—
703C22#339714yes—
704C22#340114yes—
705C22C22×#12111yescharacteristic
706C22C2×#54811yescharacteristic
707C22C2×#55011yescharacteristic
708C22C22×AGL(1,ℤ/8)11yescharacteristic
709C22C2×#57511yescharacteristic
710C22C2×#57711yescharacteristic
711C22C2×#57711yescharacteristic
712C4—24no—
713C4—24no—
714C4—24no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4#242614yes—
725C4C2×#52911yescharacteristic
726C4C2×#54211yescharacteristic
727C4C2×#54611yescharacteristic
728C4C2×#54811yescharacteristic
729C2—84no—
730C2—162no—
731C2#2530918yesdirect product factor
732C2C2×#242611yescharacteristic
733C2C2×#305511yescharacteristic
734C2C2×#306011yescharacteristic
735C2C2×#337311yescharacteristic
736C2C2×#337611yescharacteristic
737C2C2×#339711yescharacteristic
738C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#446C2semidirect
3. C2×#2538C2semidirect
4. C2×#2592C2semidirect
5. #25309C2direct (N × H)
6. C2×#2066C2semidirect
7. C2×#2065C2semidirect
8. C2×#2055C2semidirect
11. C2×#1968C2semidirect
12. C2×#1959C2semidirect
13. C2×#1948C2semidirect
14. C2×#1935C2semidirect
15. C2×#1934C2semidirect
16. C2×#1762C2semidirect
26. C2×#446C22semidirect
44. #2592C22semidirect
46. #2538C22semidirect
64. #2066C22semidirect
65. #2065C22semidirect
66. #2055C22semidirect
69. #1968C22semidirect
70. #1959C22semidirect
71. #1948C22semidirect
72. #1935C22semidirect
73. #1934C22semidirect
74. #1762C22semidirect
731. C2#25309direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4632
C22×#4462
C2×#25382
C2×#25922
#253092
C2×#20662
C2×#20652
C2×#20552
C2×#20542
C2×#19722
C2×#19682
C2×#19592
C2×#19482
C2×#19352
C2×#19342
C2×#17622