Finite Group Databasebeta

← C2×#25233 #6441713

Subgroups — C2×#25233 #6441713

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C22×C8×D4C211yesmaximal, characteristic
3C2×#2485C211yesmaximal, characteristic
4C2×#2415C211yesmaximal, characteristic
5#25233C2116yesmaximal, direct product factor
6C2×#2086C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2035C211yesmaximal, characteristic, complemented
10C2×#1941C211yesmaximal, characteristic, complemented
11C2×#1933C211yesmaximal, characteristic, complemented
12C2×#1892C211yesmaximal, characteristic, complemented
13C2×#1892C211yesmaximal, characteristic, complemented
14C2×#1854C211yesmaximal, characteristic, complemented
15C2×#1808C211yesmaximal, characteristic, complemented
16C2×#1735C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#485—28no—
25C2×#485C2218yescomplemented
26C2×C8×D4—28no—
27C2×C8×D4C2218yes—
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2485—24no—
41#2485C2218yes—
42#2415—24no—
43#2415C2218yes—
44C2×#397—21no—
45C2×#397—21no—
46C2×#394—21no—
47C2×#394—21no—
48C2×#391C2211yescharacteristic
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55C2×#387—21no—
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×#380C2211yescharacteristic
60#2086C2218yescomplemented
61#2077C2218yescomplemented
62#2068C2218yescomplemented
63#2035C2218yescomplemented
64#1941C2218yescomplemented
65#1933C2218yescomplemented
66#1892C2218yescomplemented
67#1892C2218yescomplemented
68#1854C2218yescomplemented
69#1808C2218yescomplemented
70#1735C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8—24no—
84C23×C8—24no—
85C23×C8—24no—
86C23×C8—24no—
87C23×C8—42no—
88C23×C8—42no—
89C23×C8C2311yescharacteristic
90C23×C8C2314yes—
91C23×C8C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—42no—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4D414yes—
106C2×C4×D4C2314yes—
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⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4D411yescharacteristic
137C22×C22⋊C4D411yescharacteristic
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#485—232no—
142C8×D4—232no—
143C2×#107—24no—
144C2×#107—42no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106—24no—
148C2×#106—24no—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8—24no—
152C2×C4⋊C8C2314yes—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—42no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×C22⋊C8—24no—
162C2×C22⋊C8D418yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C4×C8—24no—
166C2×C4×C8C2314yes—
167#397—28no—
168#397—28no—
169#394—28no—
170#394—28no—
171#391—22no—
172#391C2314yes—
173#389—28no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#388—28no—
181#388—28no—
182#387—28no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380C2314yes—
188#380C2314yes—
189#380C2314yes—
190#380C2314yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208C25C2×D411yescharacteristic
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4C2×D414yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D414yes—
246C23×C4C2×D411yescharacteristic
247C23×C4#4114yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—28no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—28no—
262C22×C8—28no—
263C22×C8—28no—
264C22×C8—44no—
265C22×C8—44no—
266C22×C8—416no—
267C22×C8—416no—
268C22×C8C2×D414yes—
269C22×C8C2×D414yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—44no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C4×D4—216no—
282C4×D4—28no—
283C4×D4—28no—
284#117—28no—
285#117—28no—
286#117—28no—
287#117—44no—
288#117—48no—
289#117—48no—
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—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—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4#4114yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—82no—
347C2×C22⋊C4—82no—
348C2×C22⋊C4C2×D414yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358#107—28no—
359#107—44no—
360#106—28no—
361#106—28no—
362C4⋊C8—28no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—44no—
367C22⋊C8—28no—
368C22⋊C8C2×D4116yes—
369C4×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#95—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24#12114yes—
392C24#13514yes—
393C24C22×D412yes—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×D4—232no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
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—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
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×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495C2×C8—28no—
496C2×C8—28no—
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—416no—
508C2×C8—416no—
509C2×C8—44no—
510C2×C8C22×D414yes—
511C2×C8C22×D414yes—
512C2×C8C2×#4114yes—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
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—22no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—84no—
539C4⋊C4C2×#4114yes—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—48no—
554C42—48no—
555C23—28no—
556C23—22no—
557C23—28no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—28no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23C2×#12114yes—
582C23C2×#12112yes—
583C23#54612yes—
584C23#54712yes—
585C23C2×#13514yes—
586C23C2×#13512yes—
587C23C2×#13512yes—
588C23#57712yes—
589C23#57812yes—
590C23C23×D411yescharacteristic
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—84no—
596D4—232no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—216no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
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—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—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4C2×#12114yes—
654C2×C4C2×#12112yes—
655C2×C4#52914yes—
656C2×C4#53914yes—
657C2×C4#54612yes—
658C2×C4#54614yes—
659C2×C4#54712yes—
660C2×C4#54714yes—
661C2×C4C2×#13514yes—
662C2×C4C23×D411yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C8—24no—
669C8—24no—
670C8—24no—
671C8—42no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—416no—
684C22—416no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22#242514yes—
689C22#305014yes—
690C22#306314yes—
691C22#330114yes—
692C22#330214yes—
693C22#333614yes—
694C22#334014yes—
695C22C22×#12111yescharacteristic
696C22C2×#54611yescharacteristic
697C22C2×#54711yescharacteristic
698C22C22×#13511yescharacteristic
699C22C22×#13511yescharacteristic
700C22C2×#57711yescharacteristic
701C22C2×#57811yescharacteristic
702C4—24no—
703C4—24no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#242514yes—
715C4C2×#52911yescharacteristic
716C4C2×#53911yescharacteristic
717C4C2×#54611yescharacteristic
718C4C2×#54711yescharacteristic
719C2—44no—
720C2—44no—
721C2#2523318yesdirect product factor
722C2C2×#242511yescharacteristic
723C2C2×#305011yescharacteristic
724C2C2×#306311yescharacteristic
725C2C2×#330111yescharacteristic
726C2C2×#330211yescharacteristic
727C2C2×#333611yescharacteristic
728C2C2×#334011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#485C2semidirect
5. #25233C2direct (N × H)
6. C2×#2086C2semidirect
7. C2×#2077C2semidirect
8. C2×#2068C2semidirect
9. C2×#2035C2semidirect
10. C2×#1941C2semidirect
11. C2×#1933C2semidirect
12. C2×#1892C2semidirect
13. C2×#1892C2semidirect
14. C2×#1854C2semidirect
15. C2×#1808C2semidirect
16. C2×#1735C2semidirect
25. C2×#485C22semidirect
60. #2086C22semidirect
61. #2077C22semidirect
62. #2068C22semidirect
63. #2035C22semidirect
64. #1941C22semidirect
65. #1933C22semidirect
66. #1892C22semidirect
67. #1892C22semidirect
68. #1854C22semidirect
69. #1808C22semidirect
70. #1735C22semidirect
721. C2#25233direct (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×C8×D42
C2×#24852
C2×#24152
#252332
C2×#20862
C2×#20772
C2×#20682
C2×#20352
C2×#19412
C2×#19332
C2×#18922
C2×#18542
C2×#18082
C2×#17352