Finite Group Databasebeta

← C2×#30813 #7501941

Subgroups — C2×#30813 #7501941

Order 512.

2524 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#486C211yesmaximal, characteristic
2C22×#476C211yesmaximal, characteristic, complemented
3C2×#2551C211yesmaximal, characteristic, complemented
4C2×#2445C211yesmaximal, characteristic
5#30813C2116yesmaximal, direct product factor
6C2×#2094C211yesmaximal, characteristic, complemented
7C2×#2071C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#2041C211yesmaximal, characteristic
11C2×#2023C211yesmaximal, characteristic, complemented
12C2×#1941C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic, complemented
14C2×#1926C211yesmaximal, characteristic, complemented
15C2×#1916C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#127C2211yescharacteristic
21C22×#123—21no—
22C22×C4×D4C2211yescharacteristic
23C2×#486—28no—
24C2×#486C2218yes—
25C2×#476—28no—
26C2×#476C2218yescomplemented
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2551—24no—
42#2551C2218yescomplemented
43#2445—24no—
44#2445C2218yes—
45C2×#395—21no—
46C2×#395—21no—
47C2×#394—21no—
48C2×#394—21no—
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389—21no—
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#386—21no—
57C2×#384C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2094C2218yescomplemented
61#2071C2218yescomplemented
62#2065C2218yescomplemented
63#2058C2218yescomplemented
64#2041C2218yes—
65#2023C2218yescomplemented
66#1941C2218yescomplemented
67#1926C2218yescomplemented
68#1926C2218yescomplemented
69#1916C2218yescomplemented
70#1864C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×D4—21no—
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—48no—
98C2×C4⋊Q8C2314yes—
99C2×C4⋊Q8C2314yes—
100C2×#127—24no—
101C2×#127—22no—
102C2×#127—42no—
103C2×#127C2314yes—
104C2×#123—28no—
105C2×#123—48no—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—42no—
109C2×C4×D4C2314yes—
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⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#486—232no—
137#476—232no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106—24no—
141C2×#106—42no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102—42no—
155C2×#102D418yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8C2314yes—
166#395—28no—
167#395—28no—
168#394—28no—
169#394—28no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#390C2314yes—
176#389—28no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#386—28no—
184#384—22no—
185#384C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D414yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2314yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95D411yescharacteristic
195C2×#95D411yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203#336C2314yes—
204C25—21no—
205C25—21no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—82no—
213C22×Q8—82no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—82no—
234C22×D4—82no—
235C22×D4C2×D414yes—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
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—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8C2×D414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—48no—
276C4⋊Q8—48no—
277C4⋊Q8—432no—
278#127—28no—
279#127—44no—
280#127—48no—
281#123—432no—
282C4×D4—28no—
283C4×D4—44no—
284C4×D4—48no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
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—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4SD1618yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4#4114yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4—82no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355#106—28no—
356#106—28no—
357#106—28no—
358#106—44no—
359C4⋊C8—28no—
360#102—28no—
361#102—28no—
362#102—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—44no—
367#102C2×D4116yes—
368C4×C8—28no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22×D411yescharacteristic
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—84no—
397C2×Q8—84no—
398C2×Q8—816no—
399C2×Q8—816no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—816no—
432C2×D4—816no—
433C2×D4C22×D414yes—
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—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—48no—
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—48no—
474C22×C4—84no—
475C22×C4—816no—
476C22×C4C2×SD1614yes—
477C22×C4C2×SD1614yes—
478C22×C4C2×SD1614yes—
479C22×C4AGL(1,ℤ/8)14yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C42−1+412yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
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—44no—
507C2×C8—44no—
508C2×C8—44no—
509C2×C8C22×D414yes—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—22no—
514C4⋊C4—22no—
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—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—84no—
534C4⋊C4C2×SD16116yes—
535C4⋊C4C22×D414yes—
536C4⋊C4C2×#4114yes—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—24no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—48no—
554C23—24no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23#53612yes—
583C23D4212yes—
584C23#54812yes—
585C23C22×SD1612yes—
586C23C2×AGL(1,ℤ/8)12yes—
587C23#57612yes—
588C23#57712yes—
589C23C23×D411yescharacteristic
590Q8—816no—
591Q8—816no—
592D4—24no—
593D4—24no—
594D4—48no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—816no—
602D4—816no—
603D4—84no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—48no—
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—816no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4#53612yes—
650C2×C4#53614yes—
651C2×C4#54214yes—
652C2×C4D4212yes—
653C2×C4#54714yes—
654C2×C4#54812yes—
655C2×C4#56314yes—
656C2×C4C22×SD1614yes—
657C2×C4C22×SD1614yes—
658C2×C4C22×SD1614yes—
659C2×C4C2×AGL(1,ℤ/8)14yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×2−1+411yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C8—24no—
667C8—42no—
668C8—42no—
669C8—42no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—168no—
686C22#267414yes—
687C22#322714yes—
688C22#324114yes—
689C22#326414yes—
690C22D4×SD1614yes—
691C22#336214yes—
692C22#337314yes—
693C22C2×#53611yescharacteristic
694C22C2×D4211yescharacteristic
695C22C2×#54811yescharacteristic
696C22C23×SD1611yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C2×#57611yescharacteristic
699C22C2×#57711yescharacteristic
700C4—24no—
701C4—24no—
702C4—24no—
703C4—24no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—42no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#267414yes—
713C4C2×#53611yescharacteristic
714C4C2×#54211yescharacteristic
715C4C2×#54711yescharacteristic
716C4C2×#56311yescharacteristic
717C2—84no—
718C2—162no—
719C2#3081318yesdirect product factor
720C2C2×#267411yescharacteristic
721C2C2×#322711yescharacteristic
722C2C2×#324111yescharacteristic
723C2C2×#326411yescharacteristic
724C2C2×D4×SD1611yescharacteristic
725C2C2×#336211yescharacteristic
726C2C2×#337311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#476C2semidirect
3. C2×#2551C2semidirect
5. #30813C2direct (N × H)
6. C2×#2094C2semidirect
7. C2×#2071C2semidirect
8. C2×#2065C2semidirect
9. C2×#2058C2semidirect
11. C2×#2023C2semidirect
12. C2×#1941C2semidirect
13. C2×#1926C2semidirect
14. C2×#1926C2semidirect
15. C2×#1916C2semidirect
16. C2×#1864C2semidirect
26. C2×#476C22semidirect
42. #2551C22semidirect
60. #2094C22semidirect
61. #2071C22semidirect
62. #2065C22semidirect
63. #2058C22semidirect
65. #2023C22semidirect
66. #1941C22semidirect
67. #1926C22semidirect
68. #1926C22semidirect
69. #1916C22semidirect
70. #1864C22semidirect
719. C2#30813direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4862
C22×#4762
C2×#25512
C2×#24452
#308132
C2×#20942
C2×#20712
C2×#20652
C2×#20582
C2×#20412
C2×#20232
C2×#19412
C2×#19262
C2×#19162
C2×#18642