Finite Group Databasebeta

← C2×#31312 #7504328

Subgroups — C2×#31312 #7504328

Order 512.

2396 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3#31312C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2043C211yesmaximal, characteristic, complemented
7C2×#2043C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic
9C2×#2035C211yesmaximal, characteristic
10C2×#1981C211yesmaximal, characteristic, complemented
11C2×#1960C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1928C211yesmaximal, characteristic, complemented
14C2×#1924C211yesmaximal, characteristic
15C2×#1787C211yesmaximal, characteristic
16C2×#1787C211yesmaximal, characteristic
17C24×Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×C22⋊C8C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×#97C2211yescharacteristic
39#2494—24no—
40#2494C2218yes—
41#2434—24no—
42#2434C2218yes—
43C2×#401—21no—
44C2×#397—21no—
45C2×#395—21no—
46C2×#395—21no—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#387—21no—
53C2×#385—21no—
54C2×#385—21no—
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2097C2218yescomplemented
61#2066C2218yescomplemented
62#2043C2218yescomplemented
63#2043C2218yescomplemented
64#2036C2218yes—
65#2035C2218yes—
66#1981C2218yescomplemented
67#1960C2218yescomplemented
68#1935C2218yescomplemented
69#1928C2218yescomplemented
70#1924C2218yes—
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×C22⋊Q8—22no—
104C2×C22⋊Q8—42no—
105C2×C22⋊Q8—42no—
106C2×C22⋊Q8C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—42no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×C22⋊C8—42no—
158C2×C22⋊C8—42no—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97—42no—
162C2×#97C2314yes—
163#401—28no—
164#397—28no—
165#395—28no—
166#395—28no—
167#391—22no—
168#391—22no—
169#391C2314yes—
170#391C2314yes—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#388—28no—
176#387—28no—
177#385—28no—
178#385—28no—
179#382—22no—
180#382C2314yes—
181#380—22no—
182#380—22no—
183#380C2314yes—
184#380C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4D414yes—
188C4×C22⋊C4C2314yes—
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×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25C2×D411yescharacteristic
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—28no—
214C22×Q8—28no—
215C22×Q8—22no—
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—44no—
230C22×Q8—44no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8—44no—
235C22×Q8—48no—
236C22×Q8—48no—
237C22×Q8—44no—
238C22×Q8—48no—
239C22×Q8—416no—
240C22×Q8—416no—
241C22×Q8—416no—
242C22×Q8—416no—
243C22×Q8#4114yes—
244C22×Q8#4114yes—
245C22×Q8C2412yes—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—24no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C22×C8—22no—
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×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—28no—
291C4⋊Q8—28no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—48no—
295C4⋊Q8—48no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—48no—
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—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—82no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353#107—44no—
354#106—44no—
355C4⋊C8—44no—
356#103—28no—
357#103—28no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370#97—44no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D414yes—
383#95C2×D414yes—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24#13714yes—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
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—48no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—416no—
419C2×Q8—416no—
420C2×Q8—416no—
421C2×Q8—464no—
422C2×Q8—416no—
423C2×Q8—432no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8C2×#4114yes—
440C2×Q8C2×#4114yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
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—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
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—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—432no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
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#13714yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C4C2511yescharacteristic
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
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—44no—
531C2×C8—44no—
532C2×C8—44no—
533C2×C8—44no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—48no—
555C4⋊C4—84no—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—84no—
561C42—24no—
562C42—24no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—84no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23#53512yes—
582C23#54612yes—
583C23#54612yes—
584C23C2×#13714yes—
585C23C2×#13712yes—
586C23C2×#13712yes—
587C23#57712yes—
588C23#57812yes—
589C23C23×D411yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—24no—
593Q8—24no—
594Q8—432no—
595Q8—42no—
596Q8—42no—
597Q8—48no—
598Q8—48no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—84no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—432no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—48no—
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—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4#53512yes—
664C2×C4#53814yes—
665C2×C4#53814yes—
666C2×C4#54612yes—
667C2×C4#54612yes—
668C2×C4#55114yes—
669C2×C4#56014yes—
670C2×C4C2×#13714yes—
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—416no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22#268514yes—
690C22#321814yes—
691C22#322514yes—
692C22#331214yes—
693C22#332014yes—
694C22#332014yes—
695C22#332814yes—
696C22C2×#53511yescharacteristic
697C22C2×#54611yescharacteristic
698C22C2×#54611yescharacteristic
699C22C22×#13711yescharacteristic
700C22C22×#13711yescharacteristic
701C22C2×#57711yescharacteristic
702C22C2×#57811yescharacteristic
703C4—24no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#268514yes—
717C4C2×#53811yescharacteristic
718C4C2×#53811yescharacteristic
719C4C2×#55111yescharacteristic
720C4C2×#56011yescharacteristic
721C2—44no—
722C2#3131218yesdirect product factor
723C2C2×#268511yescharacteristic
724C2C2×#321811yescharacteristic
725C2C2×#322511yescharacteristic
726C2C2×#331211yescharacteristic
727C2C2×#332011yescharacteristic
728C2C2×#332011yescharacteristic
729C2C2×#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. #31312C2direct (N × H)
4. C2×#2097C2semidirect
5. C2×#2066C2semidirect
6. C2×#2043C2semidirect
7. C2×#2043C2semidirect
10. C2×#1981C2semidirect
11. C2×#1960C2semidirect
12. C2×#1935C2semidirect
13. C2×#1928C2semidirect
60. #2097C22semidirect
61. #2066C22semidirect
62. #2043C22semidirect
63. #2043C22semidirect
66. #1981C22semidirect
67. #1960C22semidirect
68. #1935C22semidirect
69. #1928C22semidirect
722. C2#31312direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#24342
#313122
C2×#20972
C2×#20662
C2×#20432
C2×#20362
C2×#20352
C2×#19812
C2×#19602
C2×#19352
C2×#19282
C2×#19242
C2×#17872