Finite Group Databasebeta

← C2×#25206 #6441600

Subgroups — C2×#25206 #6441600

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C22×#467C211yesmaximal, characteristic
3C2×#2502C211yesmaximal, characteristic
4C2×#2360C211yesmaximal, characteristic
5#25206C2116yesmaximal, direct product factor
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2072C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2056C211yesmaximal, characteristic, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1928C211yesmaximal, characteristic, complemented
12C2×#1878C211yesmaximal, characteristic, complemented
13C2×#1872C211yesmaximal, characteristic
14C2×#1850C211yesmaximal, characteristic
15C2×#1803C211yesmaximal, characteristic, complemented
16C22×#340C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C2×#484—28no—
25C2×#484C2218yes—
26C2×#467—28no—
27C2×#467C2218yes—
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40#2502—24no—
41#2502C2218yes—
42C2×#398—21no—
43C2×#398—21no—
44#2360—24no—
45#2360C2218yes—
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#387—21no—
52C2×#385—21no—
53C2×#385—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4—21no—
59C2×C4×C22⋊C4C2211yescharacteristic
60#2083C2218yescomplemented
61#2072C2218yescomplemented
62#2068C2218yescomplemented
63#2056C2218yescomplemented
64#1947C2218yescomplemented
65#1928C2218yescomplemented
66#1878C2218yescomplemented
67#1872C2218yes—
68#1850C2218yes—
69#1803C2218yescomplemented
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2218yes—
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4—21no—
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16—42no—
87C22×Q16C2314yes—
88C23×C8—24no—
89C23×C8—24no—
90C23×C8—42no—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2314yes—
95C23×C8C2311yescharacteristic
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8D418yes—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—24no—
106C2×C4×Q8—42no—
107C2×C4×Q8C2314yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—42no—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#484—232no—
141#467—232no—
142C2×#107—24no—
143C2×#107—24no—
144C2×#107—24no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×C4⋊C8—24no—
149C2×C4⋊C8—42no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103—42no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×C22⋊C8—24no—
165C2×C22⋊C8C2314yes—
166#398—28no—
167#398—28no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#391C2314yes—
174#389—22no—
175#389—28no—
176#389C2314yes—
177#387—28no—
178#385—28no—
179#385—28no—
180#385—28no—
181#384—22no—
182#384—22no—
183#384C2314yes—
184#384C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—28no—
188C4×C22⋊C4D414yes—
189C4×C22⋊C4C2314yes—
190C2×#95—21no—
191C2×#95—21no—
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—
208C25C2×D411yescharacteristic
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8C2×D414yes—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D414yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4#4114yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—416no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—28no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—28no—
272C22×C8—44no—
273C22×C8—416no—
274C22×C8#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22⋊Q8—216no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—48no—
284C22⋊Q8C2×D4116yes—
285C4×Q8—28no—
286C4×Q8—44no—
287C4×Q8—48no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4#4114yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4Q1618yescomplemented
344C2×C22⋊C4C2×D414yes—
345C2×C22⋊C4C2×D414yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42—82no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358#107—28no—
359#107—28no—
360#107—28no—
361C4⋊C8—28no—
362C4⋊C8—44no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—44no—
369C22⋊C8—28no—
370#95—24no—
371#95—24no—
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#12114yes—
385C24C2×Q1614yes—
386C24#13714yes—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8C22×D414yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—48no—
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—48no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—48no—
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—84no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—816no—
476C22×C4—816no—
477C22×C4#12114yes—
478C22×C4#12118yes—
479C22×C4C2×Q1614yes—
480C22×C4C2×Q1614yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
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
496Q16—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—48no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4C22×D414yes—
541C4⋊C4C2×#4114yes—
542C22⋊C4—216no—
543C22⋊C4—28no—
544C22⋊C4—48no—
545C22⋊C4C2×Q16116yescomplemented
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—48no—
553C42—84no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—22no—
558C23—22no—
559C23—28no—
560C23—28no—
561C23—28no—
562C23—22no—
563C23—22no—
564C23#462116yes—
565C23C2×#12114yes—
566C23C2×#12112yes—
567C23#54612yes—
568C23#54712yes—
569C23C22×Q1614yes—
570C23C22×Q1612yes—
571C23C2×#13714yes—
572C23C2×#13712yes—
573C23#57612yes—
574C23#57712yes—
575C23C23×D411yescharacteristic
576Q8—24no—
577Q8—24no—
578Q8—42no—
579Q8—48no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—84no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—216no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—816no—
636C2×C4—816no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12112yes—
644C2×C4#52914yes—
645C2×C4#53914yes—
646C2×C4#54612yes—
647C2×C4#54614yes—
648C2×C4#54712yes—
649C2×C4#54714yes—
650C2×C4C22×Q1614yes—
651C2×C4C22×Q1614yes—
652C2×C4C23×D411yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C8—24no—
659C8—42no—
660C8—42no—
661C8—42no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—232no—
667C22—24no—
668C22—24no—
669C22#242514yes—
670C22C2×#46218yes—
671C22C2×#46214yes—
672C22#304514yes—
673C22#329914yes—
674C22#331414yes—
675C22#333814yes—
676C22#334414yes—
677C22C22×#12111yescharacteristic
678C22C2×#54611yescharacteristic
679C22C2×#54711yescharacteristic
680C22C23×Q1611yescharacteristic
681C22C22×#13711yescharacteristic
682C22C2×#57611yescharacteristic
683C22C2×#57711yescharacteristic
684C4—24no—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4#242514yes—
698C4C2×#52911yescharacteristic
699C4C2×#53911yescharacteristic
700C4C2×#54611yescharacteristic
701C4C2×#54711yescharacteristic
702C2—28no—
703C2#2520618yesdirect product factor
704C2C2×#242511yescharacteristic
705C2C22×#46211yescharacteristic
706C2C2×#304511yescharacteristic
707C2C2×#329911yescharacteristic
708C2C2×#331411yescharacteristic
709C2C2×#333811yescharacteristic
710C2C2×#334411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #25206C2direct (N × H)
6. C2×#2083C2semidirect
7. C2×#2072C2semidirect
8. C2×#2068C2semidirect
9. C2×#2056C2semidirect
10. C2×#1947C2semidirect
11. C2×#1928C2semidirect
12. C2×#1878C2semidirect
15. C2×#1803C2semidirect
60. #2083C22semidirect
61. #2072C22semidirect
62. #2068C22semidirect
63. #2056C22semidirect
64. #1947C22semidirect
65. #1928C22semidirect
66. #1878C22semidirect
69. #1803C22semidirect
343. C2×C22⋊C4Q16semidirect
545. C22⋊C4C2×Q16semidirect
703. C2#25206direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4672
C2×#25022
C2×#23602
#252062
C2×#20832
C2×#20722
C2×#20682
C2×#20562
C2×#19472
C2×#19282
C2×#18782
C2×#18722
C2×#18502
C2×#18032
C22×#3402