Finite Group Databasebeta

← C2×#25213 #6441591

Subgroups — C2×#25213 #6441591

Order 512.

2462 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C22×#465C211yesmaximal, characteristic
3C2×#2500C211yesmaximal, characteristic
4C2×#2362C211yesmaximal, characteristic, complemented
5#25213C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2072C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2055C211yesmaximal, characteristic, complemented
10C2×#1948C211yesmaximal, characteristic, complemented
11C2×#1925C211yesmaximal, characteristic, complemented
12C2×#1880C211yesmaximal, characteristic, complemented
13C2×#1875C211yesmaximal, characteristic
14C2×#1850C211yesmaximal, characteristic, complemented
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1735C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#485—28no—
25C2×#485C2218yescomplemented
26C2×#465—28no—
27C2×#465C2218yes—
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40#2500—24no—
41#2500C2218yes—
42C2×#394—21no—
43C2×#394—21no—
44#2362—24no—
45#2362C2218yescomplemented
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52C2×#388—21no—
53C2×#388—21no—
54C2×#387—21no—
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×C4×C4⋊C4—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2085C2218yescomplemented
61#2072C2218yescomplemented
62#2065C2218yescomplemented
63#2055C2218yescomplemented
64#1948C2218yescomplemented
65#1925C2218yescomplemented
66#1880C2218yescomplemented
67#1875C2218yes—
68#1850C2218yescomplemented
69#1805C2218yescomplemented
70#1735C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C23×C8—24no—
88C23×C8—24no—
89C23×C8—42no—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2314yes—
94C23×C8C2311yescharacteristic
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8D418yes—
98C2×C22⋊Q8C2314yes—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#121C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—24no—
105C2×C4×Q8—42no—
106C2×C4×Q8C2314yes—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—42no—
113C2×#117C2314yes—
114C2×#117C2314yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42—21no—
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#485—232no—
143#465—232no—
144C2×#107—24no—
145C2×#107C2314yes—
146C2×#106—24no—
147C2×#106—24no—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8—24no—
151C2×C4⋊C8—42no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102—24no—
161C2×#102—24no—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8—24no—
167C2×C22⋊C8C2314yes—
168#394—28no—
169#394—28no—
170#391—22no—
171#391C2314yes—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2314yes—
176#389—22no—
177#389C2314yes—
178#388—28no—
179#388—28no—
180#388—28no—
181#388—28no—
182#387—28no—
183#384—22no—
184#384C2314yes—
185#382—22no—
186#382C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—28no—
189C4×C4⋊C4C2314yes—
190C4×C22⋊C4D414yes—
191C4×C22⋊C4C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8C2×D414yes—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—82no—
228C22×D4—82no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D414yes—
247C23×C4C2×D411yescharacteristic
248C23×C4#4114yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—416no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—28no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—28no—
268C22×C8—44no—
269C22×C8—416no—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22⋊Q8—216no—
276C22⋊Q8—28no—
277C22⋊Q8C2×D4116yes—
278#121—28no—
279#121—28no—
280#121—48no—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—48no—
284#117—28no—
285#117—28no—
286#117—28no—
287#117—44no—
288#117—48no—
289#117—48no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
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—22no—
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—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
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—42no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42—44no—
354C2×C42—44no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—28no—
361#106—28no—
362#106—28no—
363C4⋊C8—28no—
364C4⋊C8—44no—
365#103—28no—
366#103—28no—
367#103—28no—
368#102—28no—
369#102—28no—
370#102—44no—
371C22⋊C8—28no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D414yes—
379#95C2×D414yes—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24#12114yes—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8C22×D414yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—816no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
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—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
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#12114yes—
479C22×C4#12118yes—
480C22×C4#13514yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496SD16—416no—
497C2×C8—28no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—416no—
510C2×C8—44no—
511C2×C8—44no—
512C2×C8—44no—
513C2×C8C2×#4114yes—
514C4⋊C4—216no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4C22×D414yes—
536C4⋊C4C2×#4114yes—
537C22⋊C4—216no—
538C22⋊C4—28no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—48no—
554C42—48no—
555C42—48no—
556C42—48no—
557C23—28no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—28no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23C2×#12114yes—
585C23C2×#12112yes—
586C23#54612yes—
587C23#54712yes—
588C23C2×#13512yes—
589C23#57512yes—
590C23#57712yes—
591C23#57712yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—42no—
596Q8—48no—
597Q8—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—816no—
603D4—84no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—216no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
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—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4C2×#12114yes—
656C2×C4C2×#12112yes—
657C2×C4#52914yes—
658C2×C4#53914yes—
659C2×C4#54612yes—
660C2×C4#54614yes—
661C2×C4#54712yes—
662C2×C4#54714yes—
663C2×C4C2×#13514yes—
664C2×C4C23×D411yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C8—24no—
671C8—42no—
672C8—42no—
673C8—42no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—42no—
678C22—42no—
679C22—416no—
680C22—42no—
681C22—42no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—168no—
690C22#242514yes—
691C22#303514yes—
692C22#304314yes—
693C22#330314yes—
694C22#332114yes—
695C22#333714yes—
696C22#335014yes—
697C22C22×#12111yescharacteristic
698C22C2×#54611yescharacteristic
699C22C2×#54711yescharacteristic
700C22C22×#13511yescharacteristic
701C22C2×#57511yescharacteristic
702C22C2×#57711yescharacteristic
703C22C2×#57711yescharacteristic
704C4—24no—
705C4—24no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—42no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#242514yes—
717C4C2×#52911yescharacteristic
718C4C2×#53911yescharacteristic
719C4C2×#54611yescharacteristic
720C4C2×#54711yescharacteristic
721C2—44no—
722C2—162no—
723C2#2521318yesdirect product factor
724C2C2×#242511yescharacteristic
725C2C2×#303511yescharacteristic
726C2C2×#304311yescharacteristic
727C2C2×#330311yescharacteristic
728C2C2×#332111yescharacteristic
729C2C2×#333711yescharacteristic
730C2C2×#335011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#485C2semidirect
4. C2×#2362C2semidirect
5. #25213C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2072C2semidirect
8. C2×#2065C2semidirect
9. C2×#2055C2semidirect
10. C2×#1948C2semidirect
11. C2×#1925C2semidirect
12. C2×#1880C2semidirect
14. C2×#1850C2semidirect
15. C2×#1805C2semidirect
16. C2×#1735C2semidirect
25. C2×#485C22semidirect
45. #2362C22semidirect
60. #2085C22semidirect
61. #2072C22semidirect
62. #2065C22semidirect
63. #2055C22semidirect
64. #1948C22semidirect
65. #1925C22semidirect
66. #1880C22semidirect
68. #1850C22semidirect
69. #1805C22semidirect
70. #1735C22semidirect
723. C2#25213direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4652
C2×#25002
C2×#23622
#252132
C2×#20852
C2×#20722
C2×#20652
C2×#20552
C2×#19482
C2×#19252
C2×#18802
C2×#18752
C2×#18502
C2×#18052
C2×#17352