Finite Group Databasebeta

← C2×#29784 #6433260

Subgroups — C2×#29784 #6433260

Order 512.

2532 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#468C211yesmaximal, characteristic
2C22×#423C211yesmaximal, characteristic, complemented
3C2×#2855C211yesmaximal, characteristic, complemented
4C2×#2854C211yesmaximal, characteristic
5#29784C2116yesmaximal, direct product factor
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2066C211yesmaximal, characteristic, complemented
10C2×#2067C211yesmaximal, characteristic, complemented
11C2×#2065C211yesmaximal, characteristic, complemented
12C2×#2055C211yesmaximal, characteristic, complemented
13C2×#2055C211yesmaximal, characteristic, complemented
14C2×#1895C211yesmaximal, characteristic
15C2×#1894C211yesmaximal, characteristic, complemented
16C2×#1809C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C23×M16C2211yescharacteristic
20C22×#123—21no—
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C2×#468—28no—
25C2×#468C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#423—28no—
30C2×#423C2218yescomplemented
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41#2855—24no—
42#2855C2218yescomplemented
43#2854—24no—
44#2854C2218yes—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55C2×#387—21no—
56C2×#387—21no—
57C2×#386—21no—
58C2×#385—21no—
59C2×#381—21no—
60C2×#380C2211yescharacteristic
61C2×#380C2211yescharacteristic
62#2087C2218yescomplemented
63#2086C2218yescomplemented
64#2068C2218yescomplemented
65#2067C2218yescomplemented
66#2066C2218yescomplemented
67#2065C2218yescomplemented
68#2055C2218yescomplemented
69#2055C2218yescomplemented
70#1895C2218yes—
71#1894C2218yescomplemented
72#1809C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C22×SD16—24no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C22×M16—24no—
90C22×M16—42no—
91C22×M16—42no—
92C22×M16—42no—
93C22×M16—42no—
94C22×M16D418yes—
95C22×M16C2314yes—
96C22×M16C2314yes—
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C2×#123—28no—
102C2×#123—48no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2314yes—
107C2×#121—22no—
108C2×#121—24no—
109C2×#121—24no—
110C2×#121C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—24no—
131C22×C42—24no—
132C22×C42D414yes—
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#468—232no—
139C2×#107—24no—
140C2×#107—42no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106—42no—
144C2×#106C2314yes—
145#423—232no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8—24no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102—24no—
161C2×#102—42no—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166#391—22no—
167#391—22no—
168#391C2314yes—
169#391C2314yes—
170#390—22no—
171#390—22no—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#388—28no—
183#388—28no—
184#387—28no—
185#387—28no—
186#386—28no—
187#385—28no—
188#381—28no—
189#380—22no—
190#380—22no—
191#380C2314yes—
192#380C2314yes—
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×#95—21no—
202C2×#95—21no—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209C25—21no—
210C25C2×D411yescharacteristic
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8C2412yes—
219C22×D4—24no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4C2412yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D414yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4#4114yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—416no—
264C2×M16—28no—
265C2×M16—416no—
266C2×M16—44no—
267C2×M16—44no—
268C2×M16—44no—
269C2×M16—44no—
270C2×M16—416no—
271C2×M16C2×D4116yes—
272C22×C8—22no—
273C22×C8—22no—
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×C8C2×D414yes—
283C22×C8C2×D414yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288#123—432no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—48no—
292#121—28no—
293#121—28no—
294#121—48no—
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—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
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—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—216no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—28no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—28no—
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—28no—
361#107—44no—
362#106—44no—
363C4⋊C8—28no—
364C4⋊C8—28no—
365C4⋊C8—28no—
366C4⋊C8—28no—
367#103—28no—
368#103—44no—
369#103—44no—
370#102—28no—
371#102—44no—
372#102—44no—
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#95—24no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24#12114yes—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—816no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—416no—
466C22×C4—416no—
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—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4#12114yes—
489C22×C4#12118yes—
490C22×C4AGL(1,ℤ/8)14yes—
491C22×C4#13714yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C42−1+412yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507SD16—416no—
508M16—416no—
509M16—416no—
510C2×C8—28no—
511C2×C8—28no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—44no—
523C2×C8—44no—
524C2×C8C22×D414yes—
525C2×C8C22×D414yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—48no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C42—28no—
558C42—24no—
559C42—24no—
560C42—28no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C23—28no—
566C23—22no—
567C23—22no—
568C23—28no—
569C23—28no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23C2×#12114yes—
593C23C2×#12112yes—
594C23#53712yes—
595C23#53712yes—
596C23C2×AGL(1,ℤ/8)12yes—
597C23C2×#13712yes—
598C23#57512yes—
599C23#57512yes—
600C23C23×D411yescharacteristic
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—84no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—816no—
611D4—84no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—216no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—416no—
633C2×C4—416no—
634C2×C4—44no—
635C2×C4—44no—
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—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—84no—
664C2×C4C2×#12114yes—
665C2×C4C2×#12112yes—
666C2×C4#53414yes—
667C2×C4#53614yes—
668C2×C4#53712yes—
669C2×C4#53712yes—
670C2×C4#53914yes—
671C2×C4#53914yes—
672C2×C4C2×AGL(1,ℤ/8)14yes—
673C2×C4C2×#13714yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×H(5,𝔽2)11yescharacteristic
678C2×C4C2×2−1+411yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C8—24no—
681C8—24no—
682C8—42no—
683C8—42no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—416no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—168no—
700C22#261814yes—
701C22#305514yes—
702C22#305614yes—
703C22#325114yes—
704C22#325214yes—
705C22#325414yes—
706C22#325614yes—
707C22C22×#12111yescharacteristic
708C22C2×#53711yescharacteristic
709C22C2×#53711yescharacteristic
710C22C22×AGL(1,ℤ/8)11yescharacteristic
711C22C22×#13711yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57511yescharacteristic
714C4—24no—
715C4—24no—
716C4—24no—
717C4—42no—
718C4—42no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4#261814yes—
727C4C2×#53411yescharacteristic
728C4C2×#53611yescharacteristic
729C4C2×#53911yescharacteristic
730C4C2×#53911yescharacteristic
731C2—44no—
732C2—162no—
733C2#2978418yesdirect product factor
734C2C2×#261811yescharacteristic
735C2C2×#305511yescharacteristic
736C2C2×#305611yescharacteristic
737C2C2×#325111yescharacteristic
738C2C2×#325211yescharacteristic
739C2C2×#325411yescharacteristic
740C2C2×#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
2. C22×#423C2semidirect
3. C2×#2855C2semidirect
5. #29784C2direct (N × H)
6. C2×#2087C2semidirect
7. C2×#2086C2semidirect
8. C2×#2068C2semidirect
9. C2×#2066C2semidirect
10. C2×#2067C2semidirect
11. C2×#2065C2semidirect
12. C2×#2055C2semidirect
13. C2×#2055C2semidirect
15. C2×#1894C2semidirect
16. C2×#1809C2semidirect
30. C2×#423C22semidirect
42. #2855C22semidirect
62. #2087C22semidirect
63. #2086C22semidirect
64. #2068C22semidirect
65. #2067C22semidirect
66. #2066C22semidirect
67. #2065C22semidirect
68. #2055C22semidirect
69. #2055C22semidirect
71. #1894C22semidirect
72. #1809C22semidirect
733. C2#29784direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4682
C22×#4232
C2×#28552
C2×#28542
#297842
C2×#20872
C2×#20862
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#20552
C2×#18952
C2×#18942
C2×#18092