Finite Group Databasebeta

← C2×#25215 #6441593

Subgroups — C2×#25215 #6441593

Order 512.

2332 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×#467C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4C2×#2362C211yesmaximal, characteristic
5#25215C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2072C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2056C211yesmaximal, characteristic, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1927C211yesmaximal, characteristic, complemented
12C2×#1878C211yesmaximal, characteristic, complemented
13C2×#1873C211yesmaximal, characteristic
14C2×#1850C211yesmaximal, characteristic
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1735C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#485—28no—
25C2×#485C2218yes—
26C2×#467—28no—
27C2×#467C2218yes—
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40#2499—24no—
41#2499C2218yes—
42C2×#397—21no—
43C2×#397—21no—
44#2362—24no—
45#2362C2218yes—
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52C2×#387—21no—
53C2×#387—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#2068C2218yescomplemented
63#2056C2218yescomplemented
64#1947C2218yescomplemented
65#1927C2218yescomplemented
66#1878C2218yescomplemented
67#1873C2218yes—
68#1850C2218yes—
69#1805C2218yescomplemented
70#1735C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16—24no—
83C22×Q16—24no—
84C22×Q16—42no—
85C22×Q16C2314yes—
86C23×C8—24no—
87C23×C8—24no—
88C23×C8—42no—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2314yes—
93C23×C8C2311yescharacteristic
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8D418yes—
100C2×C22⋊Q8C2314yes—
101C2×C22⋊Q8C2314yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—24no—
104C2×C4×Q8—42no—
105C2×C4×Q8C2314yes—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—42no—
112C2×#117C2314yes—
113C2×#117C2314yes—
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⋊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⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#485—232no—
139#467—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⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—42no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×C22⋊C8—24no—
163C2×C22⋊C8C2314yes—
164#397—28no—
165#397—28no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391C2314yes—
170#391C2314yes—
171#391C2314yes—
172#389—22no—
173#389C2314yes—
174#388—28no—
175#388—28no—
176#387—28no—
177#387—28no—
178#387—28no—
179#384—22no—
180#384C2314yes—
181#382—22no—
182#382C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—28no—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4D414yes—
187C4×C22⋊C4C2314yes—
188C2×#95—21no—
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×#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×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4C2×D411yescharacteristic
244C23×C4#4114yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C2×Q16—28no—
248C2×Q16—28no—
249C2×Q16—44no—
250C2×Q16—416no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—28no—
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—28no—
264C22×C8—44no—
265C22×C8—416no—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22⋊Q8—216no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—48no—
276C22⋊Q8C2×D4116yes—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—48no—
280#117—28no—
281#117—28no—
282#117—28no—
283#117—44no—
284#117—48no—
285#117—48no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
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—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4#4114yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350#107—28no—
351#107—28no—
352#106—28no—
353C4⋊C8—28no—
354C4⋊C8—44no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—28no—
359#103—28no—
360#103—44no—
361C22⋊C8—28no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24#12114yes—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8C22×D414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
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—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—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4#12114yes—
465C22×C4#12118yes—
466C22×C4#13514yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
482Q16—416no—
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—416no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8—44no—
499C2×C8C2×#4114yes—
500C4⋊C4—216no—
501C4⋊C4—24no—
502C4⋊C4—24no—
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—22no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4C22×D414yes—
523C4⋊C4C2×#4114yes—
524C22⋊C4—216no—
525C22⋊C4—28no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—48no—
537C42—48no—
538C42—48no—
539C42—48no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—28no—
544C23—28no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C2×#12114yes—
554C23C2×#12112yes—
555C23#54612yes—
556C23#54712yes—
557C23C2×#13512yes—
558C23#57512yes—
559C23#57812yes—
560C23#57812yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—42no—
565Q8—48no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—84no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—216no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
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—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4C2×#12114yes—
628C2×C4C2×#12112yes—
629C2×C4#52914yes—
630C2×C4#53914yes—
631C2×C4#54612yes—
632C2×C4#54614yes—
633C2×C4#54712yes—
634C2×C4#54714yes—
635C2×C4C2×#13514yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C8—24no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—416no—
652C22—42no—
653C22—42no—
654C22#242514yes—
655C22#303514yes—
656C22#304414yes—
657C22#330214yes—
658C22#331314yes—
659C22#333614yes—
660C22#334914yes—
661C22C22×#12111yescharacteristic
662C22C2×#54611yescharacteristic
663C22C2×#54711yescharacteristic
664C22C22×#13511yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57811yescharacteristic
667C22C2×#57811yescharacteristic
668C4—24no—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#242514yes—
682C4C2×#52911yescharacteristic
683C4C2×#53911yescharacteristic
684C4C2×#54611yescharacteristic
685C4C2×#54711yescharacteristic
686C2—44no—
687C2#2521518yesdirect product factor
688C2C2×#242511yescharacteristic
689C2C2×#303511yescharacteristic
690C2C2×#304411yescharacteristic
691C2C2×#330211yescharacteristic
692C2C2×#331311yescharacteristic
693C2C2×#333611yescharacteristic
694C2C2×#334911yescharacteristic
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. #25215C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2072C2semidirect
8. C2×#2068C2semidirect
9. C2×#2056C2semidirect
10. C2×#1947C2semidirect
11. C2×#1927C2semidirect
12. C2×#1878C2semidirect
15. C2×#1805C2semidirect
60. #2085C22semidirect
61. #2072C22semidirect
62. #2068C22semidirect
63. #2056C22semidirect
64. #1947C22semidirect
65. #1927C22semidirect
66. #1878C22semidirect
69. #1805C22semidirect
687. C2#25215direct (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×#4672
C2×#24992
C2×#23622
#252152
C2×#20852
C2×#20722
C2×#20682
C2×#20562
C2×#19472
C2×#19272
C2×#18782
C2×#18732
C2×#18502
C2×#18052
C2×#17352