Finite Group Databasebeta

← C2×#33054 #6629474

Subgroups — C2×#33054 #6629474

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic, complemented
2C22×#405C211yesmaximal, characteristic, complemented
3C2×#2620C211yesmaximal, characteristic
4C2×#2628C211yesmaximal, characteristic, complemented
5#33054C2116yesmaximal, direct product factor
6C2×#1990C211yesmaximal, characteristic, complemented
7C2×#1958C211yesmaximal, 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×#1836C211yesmaximal, characteristic, complemented
17C23×M16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#489—28no—
25C2×#489C2218yescomplemented
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#2628—24no—
41#2628C2218yescomplemented
42#2620—24no—
43#2620C2218yes—
44C2×#400—21no—
45C2×#399—21no—
46C2×#396—21no—
47C2×#396—21no—
48C2×#395—21no—
49C2×#394—21no—
50C2×#394—21no—
51C2×#393—21no—
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×#380C2211yescharacteristic
60#1990C2218yescomplemented
61#1958C2218yescomplemented
62#1936C2218yescomplemented
63#1935C2218yescomplemented
64#1934C2218yescomplemented
65#1933C2218yescomplemented
66#1895C2218yes—
67#1894C2218yescomplemented
68#1893C2218yes—
69#1892C2218yescomplemented
70#1836C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16—24no—
85C22×M16—24no—
86C22×M16—24no—
87C22×M16C2314yes—
88C23×C8—24no—
89C23×C8—24no—
90C23×C8—24no—
91C23×C8C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2314yes—
100C2×#124—22no—
101C2×#124—24no—
102C2×#124—24no—
103C2×#124C2314yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8C2314yes—
108C2×#121—22no—
109C2×#121—24no—
110C2×#121—24no—
111C2×#121C2314yes—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117D414yes—
115C2×#117C2314yes—
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⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
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⋊C4D411yescharacteristic
139C22×C22⋊C4D411yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#489—232no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102—24no—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159#405—232no—
160C2×#97—24no—
161C2×#97D418yescomplemented
162C2×#97C2314yes—
163C2×#97C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8—24no—
166C2×C4×C8C2314yes—
167C2×C4×C8C2314yes—
168#400—28no—
169#399—28no—
170#396—28no—
171#396—28no—
172#395—28no—
173#394—28no—
174#394—28no—
175#393—28no—
176#391—22no—
177#391C2314yes—
178#390—22no—
179#390C2314yes—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#384—22no—
185#384C2314yes—
186#382—22no—
187#382C2314yes—
188#380—22no—
189#380—22no—
190#380C2314yes—
191#380C2314yes—
192C2×#95—21no—
193C2×#95—21no—
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—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—44no—
219C22×Q8—82no—
220C22×Q8—82no—
221C22×Q8C2412yes—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—44no—
231C22×D4—82no—
232C22×D4—82no—
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4#4114yes—
252C23×C4#4114yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C2×M16—232no—
256C2×M16—28no—
257C2×M16—28no—
258C2×M16—28no—
259C22×C8—232no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—28no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—28no—
270C22×C8—28no—
271C22×C8C2×D414yes—
272C22×C8C2×D414yes—
273C22×C8#4114yes—
274C22×C8#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281C4⋊Q8—48no—
282#124—28no—
283#124—28no—
284#124—48no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—48no—
288#121—28no—
289#121—28no—
290#121—48no—
291#117—216no—
292#117—28no—
293#117—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
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—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
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⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
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—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4—82no—
351C2×C22⋊C4—82no—
352C2×C22⋊C4C2×D414yes—
353C2×C22⋊C4C2×D414yes—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D414yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—28no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—28no—
370#97—28no—
371#97C2×D4116yescomplemented
372C4×C8—28no—
373C4×C8—28no—
374#95—24no—
375#95—24no—
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—48no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×Q8—816no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—816no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—24no—
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—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
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—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4#12114yes—
484C22×C4#12114yes—
485C22×C4#12414yes—
486C22×C4#12418yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2511yescharacteristic
502M16—232no—
503C2×C8—232no—
504C2×C8—28no—
505C2×C8—28no—
506C2×C8—28no—
507C2×C8—28no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C22×D414yes—
517C2×C8C22×D414yes—
518C2×C8C2×#4114yes—
519C2×C8C2×#4114yes—
520C4⋊C4—28no—
521C4⋊C4—28no—
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—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—48no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42#12118yes—
560C42C22×D414yes—
561C42C2×#4114yes—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23C2×#12112yes—
590C23C2×#12414yes—
591C23C2×#12412yes—
592C23#53212yes—
593C23#57612yes—
594C23#57712yes—
595C23#57712yes—
596C23#57812yes—
597C23C23×D411yescharacteristic
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—816no—
603Q8—84no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—816no—
609D4—84no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—216no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—44no—
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—42no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4C2×#12114yes—
661C2×C4C2×#12114yes—
662C2×C4C2×#12112yes—
663C2×C4#52514yes—
664C2×C4#52514yes—
665C2×C4C2×#12414yes—
666C2×C4C2×#12412yes—
667C2×C4#52914yes—
668C2×C4#52914yes—
669C2×C4#53212yes—
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C8—24no—
677C8—24no—
678C8—24no—
679C8—24no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—42no—
684C22—42no—
685C22—416no—
686C22—42no—
687C22—42no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—168no—
696C22#285014yes—
697C22#306214yes—
698C22#306314yes—
699C22#313614yes—
700C22#313714yes—
701C22#317314yes—
702C22#317514yes—
703C22C22×#12111yescharacteristic
704C22C22×#12411yescharacteristic
705C22C2×#53211yescharacteristic
706C22C2×#57611yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57711yescharacteristic
709C22C2×#57811yescharacteristic
710C4—24no—
711C4—24no—
712C4—24no—
713C4—42no—
714C4—42no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#285014yes—
723C4C2×#52511yescharacteristic
724C4C2×#52511yescharacteristic
725C4C2×#52911yescharacteristic
726C4C2×#52911yescharacteristic
727C2—44no—
728C2—162no—
729C2#3305418yesdirect product factor
730C2C2×#285011yescharacteristic
731C2C2×#306211yescharacteristic
732C2C2×#306311yescharacteristic
733C2C2×#313611yescharacteristic
734C2C2×#313711yescharacteristic
735C2C2×#317311yescharacteristic
736C2C2×#317511yescharacteristic
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×#489C2semidirect
2. C22×#405C2semidirect
4. C2×#2628C2semidirect
5. #33054C2direct (N × H)
6. C2×#1990C2semidirect
7. C2×#1958C2semidirect
8. C2×#1936C2semidirect
9. C2×#1935C2semidirect
10. C2×#1934C2semidirect
11. C2×#1933C2semidirect
13. C2×#1894C2semidirect
15. C2×#1892C2semidirect
16. C2×#1836C2semidirect
25. C2×#489C22semidirect
35. C2×#405C22semidirect
41. #2628C22semidirect
60. #1990C22semidirect
61. #1958C22semidirect
62. #1936C22semidirect
63. #1935C22semidirect
64. #1934C22semidirect
65. #1933C22semidirect
67. #1894C22semidirect
69. #1892C22semidirect
70. #1836C22semidirect
161. C2×#97D4semidirect
371. #97C2×D4semidirect
729. C2#33054direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4052
C2×#26202
C2×#26282
#330542
C2×#19902
C2×#19582
C2×#19362
C2×#19352
C2×#19342
C2×#19332
C2×#18952
C2×#18942
C2×#18932
C2×#18922
C2×#18362