Finite Group Databasebeta

← C2×#30008 #6433332

Subgroups — C2×#30008 #6433332

Order 512.

2390 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#30008C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic, complemented
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic, complemented
9C2×#2034C211yesmaximal, characteristic
10C2×#2035C211yesmaximal, characteristic, complemented
11C2×#2000C211yesmaximal, characteristic
12C2×#1927C211yesmaximal, characteristic, complemented
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1797C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4—21no—
21C22×C4×D4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36#2855—24no—
37#2855C2218yescomplemented
38C4×#121—24no—
39C4×#121C2218yes—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#387—21no—
48C2×#387—21no—
49C2×#386—21no—
50C2×#385—21no—
51C2×#381—21no—
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4—21no—
56#2088C2218yescomplemented
57#2077C2218yescomplemented
58#2066C2218yescomplemented
59#2065C2218yescomplemented
60#2055C2218yescomplemented
61#2035C2218yescomplemented
62#2034C2218yes—
63#2000C2218yes—
64#1927C2218yescomplemented
65#1920C2218yescomplemented
66#1877C2218yescomplemented
67#1864C2218yescomplemented
68#1797C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16—42no—
87C22×SD16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—42no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8C2314yes—
96C2×#121—24no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121C2314yes—
100C2×C4×D4—28no—
101C2×C4×D4—22no—
102C2×C4×D4—42no—
103C2×C4×D4—42no—
104C2×C4×D4—48no—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42—42no—
125C22×C42—42no—
126C22×C42D414yes—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C2×#106—24no—
133C2×#106—42no—
134C2×#106—42no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8—42no—
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8—42no—
155C2×C22⋊C8—42no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#391—22no—
159#391—22no—
160#391C2314yes—
161#391C2314yes—
162#390—22no—
163#390C2314yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#388—28no—
171#387—28no—
172#387—28no—
173#386—28no—
174#385—28no—
175#381—28no—
176#380—22no—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4—28no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8C2412yes—
211C22×D4—28no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4#4114yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4SD1618yescomplemented
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D414yes—
253C23×C4C2×D414yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×SD16—416no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279#121—216no—
280#121—28no—
281#121—28no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—432no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
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—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—48no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#106—28no—
350#106—44no—
351#106—44no—
352C4⋊C8—44no—
353C4⋊C8—44no—
354#103—28no—
355#103—44no—
356#103—44no—
357#102—28no—
358#102—44no—
359#102—44no—
360C22⋊C8—44no—
361C22⋊C8—44no—
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—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C2×SD1614yes—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×D4—216no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—432no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4C2×#4114yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
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—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
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—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—48no—
483C22×C4—816no—
484C22×C4C2×SD1614yes—
485C22×C4C2×SD1614yes—
486C22×C4C2×SD16116yescomplemented
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500SD16—416no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8—44no—
515C2×C8—44no—
516C2×C8—44no—
517C4⋊C4—24no—
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—22no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—48no—
538C4⋊C4C2×#4114yes—
539C22⋊C4—216no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—44no—
553C42—44no—
554C42—48no—
555C23—28no—
556C23—22no—
557C23—28no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23#52512yes—
583C23#53412yes—
584C23#53712yes—
585C23C22×SD1614yes—
586C23C22×SD1612yes—
587C23#57512yes—
588C23#57512yes—
589C23#57512yes—
590C23C23×D411yescharacteristic
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—84no—
596D4—24no—
597D4—24no—
598D4—432no—
599D4—48no—
600D4—42no—
601D4—42no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—416no—
619C2×C4—416no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
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—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—816no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4#52512yes—
657C2×C4#53412yes—
658C2×C4#53712yes—
659C2×C4#53814yes—
660C2×C4#53914yes—
661C2×C4#54014yes—
662C2×C4#54214yes—
663C2×C4C22×SD1614yes—
664C2×C4C22×SD1614yes—
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—42no—
672C8—42no—
673C8—42no—
674C8—42no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—416no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—816no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22#263014yes—
692C22#310114yes—
693C22#311114yes—
694C22#318714yes—
695C22#319714yes—
696C22#324614yes—
697C22#325614yes—
698C22C2×#52511yescharacteristic
699C22C2×#53411yescharacteristic
700C22C2×#53711yescharacteristic
701C22C23×SD1611yescharacteristic
702C22C2×#57511yescharacteristic
703C22C2×#57511yescharacteristic
704C22C2×#57511yescharacteristic
705C4—24no—
706C4—24no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4—42no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#263014yes—
718C4C2×#53811yescharacteristic
719C4C2×#53911yescharacteristic
720C4C2×#54011yescharacteristic
721C4C2×#54211yescharacteristic
722C2—44no—
723C2—84no—
724C2#3000818yesdirect product factor
725C2C2×#263011yescharacteristic
726C2C2×#310111yescharacteristic
727C2C2×#311111yescharacteristic
728C2C2×#318711yescharacteristic
729C2C2×#319711yescharacteristic
730C2C2×#324611yescharacteristic
731C2C2×#325611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2855C2semidirect
3. #30008C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2077C2semidirect
6. C2×#2066C2semidirect
7. C2×#2065C2semidirect
8. C2×#2055C2semidirect
10. C2×#2035C2semidirect
12. C2×#1927C2semidirect
13. C2×#1920C2semidirect
14. C2×#1877C2semidirect
15. C2×#1864C2semidirect
16. C2×#1797C2semidirect
37. #2855C22semidirect
56. #2088C22semidirect
57. #2077C22semidirect
58. #2066C22semidirect
59. #2065C22semidirect
60. #2055C22semidirect
61. #2035C22semidirect
64. #1927C22semidirect
65. #1920C22semidirect
66. #1877C22semidirect
67. #1864C22semidirect
68. #1797C22semidirect
245. C23×C4SD16semidirect
486. C22×C4C2×SD16semidirect
724. C2#30008direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28552
C2×C4×#1212
#300082
C2×#20882
C2×#20772
C2×#20662
C2×#20652
C2×#20552
C2×#20342
C2×#20352
C2×#20002
C2×#19272
C2×#19202
C2×#18772
C2×#18642
C2×#17972