Finite Group Databasebeta

← C2×#33047 #6629472

Subgroups — C2×#33047 #6629472

Order 512.

2520 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic, complemented
2C22×#405C211yesmaximal, characteristic, complemented
3C2×#2629C211yesmaximal, characteristic, complemented
4C2×#2620C211yesmaximal, characteristic
5#33047C2116yesmaximal, direct product factor
6C2×#1992C211yesmaximal, characteristic, complemented
7C2×#1959C211yesmaximal, characteristic, complemented
8C2×#1936C211yesmaximal, characteristic, complemented
9C2×#1935C211yesmaximal, characteristic, complemented
10C2×#1934C211yesmaximal, characteristic, complemented
11C2×#1933C211yesmaximal, characteristic, complemented
12C2×#1895C211yesmaximal, characteristic
13C2×#1894C211yesmaximal, characteristic, complemented
14C2×#1893C211yesmaximal, characteristic
15C2×#1892C211yesmaximal, characteristic, complemented
16C2×#1834C211yesmaximal, characteristic, complemented
17C23×M16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#488—28no—
25C2×#488C2218yescomplemented
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C2×#405—28no—
35C2×#405C2218yescomplemented
36C22×#97C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2629—24no—
41#2629C2218yescomplemented
42#2620—24no—
43#2620C2218yes—
44C2×#398—21no—
45C2×#398—21no—
46C2×#396—21no—
47C2×#396—21no—
48C2×#396—21no—
49C2×#396—21no—
50C2×#394—21no—
51C2×#394—21no—
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×#380C2211yescharacteristic
60#1992C2218yescomplemented
61#1959C2218yescomplemented
62#1936C2218yescomplemented
63#1935C2218yescomplemented
64#1934C2218yescomplemented
65#1933C2218yescomplemented
66#1895C2218yes—
67#1894C2218yescomplemented
68#1893C2218yes—
69#1892C2218yescomplemented
70#1834C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16—24no—
84C22×M16—24no—
85C22×M16—24no—
86C22×M16C2314yes—
87C23×C8—24no—
88C23×C8—24no—
89C23×C8—24no—
90C23×C8C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#125—22no—
96C2×#125—24no—
97C2×#125—24no—
98C2×#125C2314yes—
99C2×#124—22no—
100C2×#124—24no—
101C2×#124—24no—
102C2×#124C2314yes—
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—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117D414yes—
114C2×#117C2314yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4D411yescharacteristic
141C22×C22⋊C4D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#488—232no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102—24no—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102—24no—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161#405—232no—
162C2×#97—24no—
163C2×#97—24no—
164C2×#97C2314yes—
165C2×#97C2314yes—
166C2×C4×C8—24no—
167C2×C4×C8D418yescomplemented
168C2×C4×C8C2314yes—
169C2×C4×C8C2314yes—
170#398—28no—
171#398—28no—
172#396—28no—
173#396—28no—
174#396—28no—
175#396—28no—
176#394—28no—
177#394—28no—
178#391—22no—
179#391C2314yes—
180#390—22no—
181#390C2314yes—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#389C2314yes—
186#384—22no—
187#384—22no—
188#384C2314yes—
189#384C2314yes—
190#380—22no—
191#380—22no—
192#380C2314yes—
193#380C2314yes—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208C25—21no—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—82no—
228C22×D4—82no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4#4114yes—
248C23×C4#4114yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—232no—
252C2×M16—28no—
253C2×M16—28no—
254C2×M16—28no—
255C22×C8—232no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—28no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—28no—
266C22×C8—28no—
267C22×C8C2×D414yes—
268C22×C8C2×D414yes—
269C22×C8#4114yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#125—28no—
276#125—28no—
277#125—48no—
278#124—28no—
279#124—28no—
280#124—48no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—48no—
284#121—28no—
285#121—28no—
286#121—48no—
287#117—216no—
288#117—28no—
289#117—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
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—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
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—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—82no—
351C2×C22⋊C4—82no—
352C2×C22⋊C4—82no—
353C2×C22⋊C4—82no—
354C2×C22⋊C4C2×D414yes—
355C2×C22⋊C4C2×D414yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D414yes—
361C2×C42#4114yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372#97—28no—
373#97—28no—
374C4×C8—28no—
375C4×C8C2×D4116yescomplemented
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24#12414yes—
397C24C22×D412yes—
398C24C22×D411yescharacteristic
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—816no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
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—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—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—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4#12114yes—
480C22×C4#12114yes—
481C22×C4#12414yes—
482C22×C4#12418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2511yescharacteristic
498M16—232no—
499C2×C8—232no—
500C2×C8—28no—
501C2×C8—28no—
502C2×C8—28no—
503C2×C8—28no—
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×C8C22×D414yes—
513C2×C8C22×D414yes—
514C2×C8C2×#4114yes—
515C2×C8C2×#4114yes—
516C4⋊C4—28no—
517C4⋊C4—28no—
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—22no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42#12118yes—
559C42C22×D414yes—
560C42C2×#4114yes—
561C23—28no—
562C23—28no—
563C23—28no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23C2×#12112yes—
589C23C2×#12414yes—
590C23C2×#12412yes—
591C23#53212yes—
592C23#57612yes—
593C23#57712yes—
594C23#57712yes—
595C23#57812yes—
596C23C23×D411yescharacteristic
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—84no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—816no—
607D4—84no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—216no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—44no—
629C2×C4—44no—
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—42no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4C2×#12114yes—
659C2×C4C2×#12114yes—
660C2×C4C2×#12112yes—
661C2×C4#52514yes—
662C2×C4#52514yes—
663C2×C4C2×#12414yes—
664C2×C4C2×#12412yes—
665C2×C4#52914yes—
666C2×C4#52914yes—
667C2×C4#53212yes—
668C2×C4C23×D411yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C8—24no—
675C8—24no—
676C8—24no—
677C8—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—416no—
684C22—42no—
685C22—42no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—168no—
694C22#285014yes—
695C22#306014yes—
696C22#306114yes—
697C22#313614yes—
698C22#313714yes—
699C22#317814yes—
700C22#317914yes—
701C22C22×#12111yescharacteristic
702C22C22×#12411yescharacteristic
703C22C2×#53211yescharacteristic
704C22C2×#57611yescharacteristic
705C22C2×#57711yescharacteristic
706C22C2×#57711yescharacteristic
707C22C2×#57811yescharacteristic
708C4—24no—
709C4—24no—
710C4—24no—
711C4—42no—
712C4—42no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#285014yes—
721C4C2×#52511yescharacteristic
722C4C2×#52511yescharacteristic
723C4C2×#52911yescharacteristic
724C4C2×#52911yescharacteristic
725C2—44no—
726C2—162no—
727C2#3304718yesdirect product factor
728C2C2×#285011yescharacteristic
729C2C2×#306011yescharacteristic
730C2C2×#306111yescharacteristic
731C2C2×#313611yescharacteristic
732C2C2×#313711yescharacteristic
733C2C2×#317811yescharacteristic
734C2C2×#317911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#488C2semidirect
2. C22×#405C2semidirect
3. C2×#2629C2semidirect
5. #33047C2direct (N × H)
6. C2×#1992C2semidirect
7. C2×#1959C2semidirect
8. C2×#1936C2semidirect
9. C2×#1935C2semidirect
10. C2×#1934C2semidirect
11. C2×#1933C2semidirect
13. C2×#1894C2semidirect
15. C2×#1892C2semidirect
16. C2×#1834C2semidirect
25. C2×#488C22semidirect
35. C2×#405C22semidirect
41. #2629C22semidirect
60. #1992C22semidirect
61. #1959C22semidirect
62. #1936C22semidirect
63. #1935C22semidirect
64. #1934C22semidirect
65. #1933C22semidirect
67. #1894C22semidirect
69. #1892C22semidirect
70. #1834C22semidirect
167. C2×C4×C8D4semidirect
375. C4×C8C2×D4semidirect
727. C2#33047direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4882
C22×#4052
C2×#26292
C2×#26202
#330472
C2×#19922
C2×#19592
C2×#19362
C2×#19352
C2×#19342
C2×#19332
C2×#18952
C2×#18942
C2×#18932
C2×#18922
C2×#18342