Finite Group Databasebeta

← C2×#31288 #7504310

Subgroups — C2×#31288 #7504310

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3#31288C2116yesmaximal, direct product factor
4C2×#2096C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2043C211yesmaximal, characteristic, complemented
7C2×#2043C211yesmaximal, characteristic, complemented
8C2×#2035C211yesmaximal, characteristic
9C2×#2035C211yesmaximal, characteristic
10C2×#1981C211yesmaximal, characteristic, complemented
11C2×#1956C211yesmaximal, characteristic, complemented
12C2×#1933C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1923C211yesmaximal, characteristic
15C2×#1787C211yesmaximal, characteristic
16C2×#1787C211yesmaximal, characteristic
17C24×Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126—21no—
21C22×C22⋊Q8C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2496—24no—
41#2496C2218yes—
42#2434—24no—
43#2434C2218yes—
44C2×#400—21no—
45C2×#397—21no—
46C2×#395—21no—
47C2×#395—21no—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#387—21no—
53C2×#385—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2096C2218yescomplemented
61#2066C2218yescomplemented
62#2043C2218yescomplemented
63#2043C2218yescomplemented
64#2035C2218yes—
65#2035C2218yes—
66#1981C2218yescomplemented
67#1956C2218yescomplemented
68#1933C2218yescomplemented
69#1927C2218yescomplemented
70#1923C2218yes—
71#1787C2218yes—
72#1787C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8—22no—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—42no—
82C23×Q8—42no—
83C23×Q8—42no—
84C23×Q8—42no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2314yes—
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2314yes—
103C2×#126—28no—
104C2×#126—48no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—42no—
107C2×C22⋊Q8—42no—
108C2×C22⋊Q8C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#106—42no—
132C2×#106—42no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×#103—24no—
138C2×#103—24no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103—42no—
147C2×#103—42no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×C22⋊C8—42no—
162C2×C22⋊C8—42no—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C4×C8—42no—
166C2×C4×C8C2314yes—
167#400—28no—
168#397—28no—
169#395—28no—
170#395—28no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#391C2314yes—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#387—28no—
180#385—28no—
181#385—28no—
182#384—22no—
183#384C2314yes—
184#380—22no—
185#380—22no—
186#380C2314yes—
187#380C2314yes—
188C4×C4⋊C4—22no—
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×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207C25C2×D411yescharacteristic
208C22×Q8—21no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—28no—
215C22×Q8—28no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—44no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—44no—
239C22×Q8—48no—
240C22×Q8—416no—
241C22×Q8—416no—
242C22×Q8—416no—
243C22×Q8—416no—
244C22×Q8#4114yes—
245C22×Q8#4114yes—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—24no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4—44no—
267C23×C4C2×D412yes—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—28no—
292C4⋊Q8—28no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—48no—
296C4⋊Q8—48no—
297#126—432no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—48no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4SD1618yes—
347C2×C22⋊C4C2×D414yes—
348C2×C22⋊C4C2×D414yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359#106—44no—
360#106—44no—
361C4⋊C8—44no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374C22⋊C8—44no—
375C22⋊C8—44no—
376C4×C8—44no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95C2×D414yes—
387#95C2×D414yes—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24C2×SD1614yes—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—416no—
423C2×Q8—416no—
424C2×Q8—416no—
425C2×Q8—464no—
426C2×Q8—416no—
427C2×Q8—432no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8C2×#4114yes—
444C2×Q8C2×#4114yes—
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—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—432no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4C2×SD1614yes—
507C22×C4C2×SD1614yes—
508C22×C4#13714yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C4H(5,𝔽2)12yes—
521C22×C4C2511yescharacteristic
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—44no—
535C2×C8—44no—
536C2×C8—44no—
537C2×C8—44no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—48no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—48no—
559C4⋊C4—48no—
560C4⋊C4—48no—
561C4⋊C4—48no—
562C22⋊C4—28no—
563C22⋊C4—48no—
564C22⋊C4—48no—
565C22⋊C4—48no—
566C22⋊C4C2×SD16116yes—
567C42—24no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—48no—
574C23—22no—
575C23—28no—
576C23—22no—
577C23—22no—
578C23—28no—
579C23—28no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23#53512yes—
588C23#54612yes—
589C23#54612yes—
590C23C22×SD1614yes—
591C23C22×SD1612yes—
592C23C2×#13712yes—
593C23#57812yes—
594C23#57812yes—
595C23C23×D411yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—24no—
599Q8—24no—
600Q8—432no—
601Q8—42no—
602Q8—42no—
603Q8—48no—
604Q8—48no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—84no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—432no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—48no—
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—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4#53512yes—
668C2×C4#53814yes—
669C2×C4#53814yes—
670C2×C4#54612yes—
671C2×C4#54612yes—
672C2×C4#55114yes—
673C2×C4#56014yes—
674C2×C4C22×SD1614yes—
675C2×C4C22×SD1614yes—
676C2×C4C2×#13714yes—
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C8—42no—
684C8—42no—
685C8—42no—
686C8—42no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—416no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22#268514yes—
696C22#320614yes—
697C22#322314yes—
698C22#329814yes—
699C22#329814yes—
700C22#331214yes—
701C22#332814yes—
702C22C2×#53511yescharacteristic
703C22C2×#54611yescharacteristic
704C22C2×#54611yescharacteristic
705C22C23×SD1611yescharacteristic
706C22C22×#13711yescharacteristic
707C22C2×#57811yescharacteristic
708C22C2×#57811yescharacteristic
709C4—24no—
710C4—24no—
711C4—24no—
712C4—42no—
713C4—44no—
714C4—44no—
715C4—42no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#268514yes—
723C4C2×#53811yescharacteristic
724C4C2×#53811yescharacteristic
725C4C2×#55111yescharacteristic
726C4C2×#56011yescharacteristic
727C2—44no—
728C2#3128818yesdirect product factor
729C2C2×#268511yescharacteristic
730C2C2×#320611yescharacteristic
731C2C2×#322311yescharacteristic
732C2C2×#329811yescharacteristic
733C2C2×#329811yescharacteristic
734C2C2×#331211yescharacteristic
735C2C2×#332811yescharacteristic
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
3. #31288C2direct (N × H)
4. C2×#2096C2semidirect
5. C2×#2066C2semidirect
6. C2×#2043C2semidirect
7. C2×#2043C2semidirect
10. C2×#1981C2semidirect
11. C2×#1956C2semidirect
12. C2×#1933C2semidirect
13. C2×#1927C2semidirect
60. #2096C22semidirect
61. #2066C22semidirect
62. #2043C22semidirect
63. #2043C22semidirect
66. #1981C22semidirect
67. #1956C22semidirect
68. #1933C22semidirect
69. #1927C22semidirect
728. C2#31288direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#24342
#312882
C2×#20962
C2×#20662
C2×#20432
C2×#20352
C2×#19812
C2×#19562
C2×#19332
C2×#19272
C2×#19232
C2×#17872