Finite Group Databasebeta

← C2×#28276 #6465995

Subgroups — C2×#28276 #6465995

Order 512.

2532 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic, complemented
2C22×#442C211yesmaximal, characteristic
3C2×#2538C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic, complemented
5#28276C2116yesmaximal, direct product factor
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2065C211yesmaximal, characteristic, complemented
10C2×#1972C211yesmaximal, characteristic
11C2×#1970C211yesmaximal, characteristic
12C2×#1943C211yesmaximal, characteristic, complemented
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#446—28no—
23C2×#446C2218yescomplemented
24C2×#442—28no—
25C2×#442C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×#97C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2580—24no—
44#2580C2218yescomplemented
45#2538—24no—
46#2538C2218yescomplemented
47C2×#399—21no—
48C2×#396—21no—
49C2×#396—21no—
50C2×#393—21no—
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#385—21no—
60C2×#385—21no—
61C2×#384C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#380—21no—
64#2067C2218yescomplemented
65#2067C2218yescomplemented
66#2065C2218yescomplemented
67#2065C2218yescomplemented
68#1972C2218yes—
69#1970C2218yes—
70#1943C2218yescomplemented
71#1941C2218yescomplemented
72#1847C2218yescomplemented
73#1804C2218yescomplemented
74#1802C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8D411yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C22×D8—24no—
85C22×D8—24no—
86C22×D8Q818yescomplemented
87C22×D8C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—48no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D414yes—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
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×C22⋊C4—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#446—232no—
135#442—232no—
136C2×#107—24no—
137C2×#107—42no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106—42no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102—42no—
155C2×#102—42no—
156C2×#102—42no—
157C2×#102—42no—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#97—24no—
165C2×#97D418yes—
166C2×#97C2314yes—
167C2×#97C2314yes—
168C2×C4×C8—24no—
169C2×C4×C8C2314yes—
170#399—28no—
171#396—28no—
172#396—28no—
173#393—28no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#390C2314yes—
181#390C2314yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#385—28no—
191#385—28no—
192#384—22no—
193#384C2314yes—
194#382—22no—
195#382C2314yes—
196#380—28no—
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—
207C25—21no—
208C25—21no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8C2×D414yes—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2×Q814yes—
226C22×D4C2×Q814yes—
227C22×D4C2412yes—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×D8—232no—
251C2×D8—28no—
252C2×D8—28no—
253C2×D8C2×Q8116yescomplemented
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8C2×D414yes—
263C22×C8C2×D414yes—
264C22×C8C2×Q814yes—
265C22×C8#4114yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22⋊Q8—432no—
271C4×Q8—216no—
272C4×Q8—28no—
273C4×Q8—28no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—48no—
279C4×D4—48no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
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—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
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×C22⋊C4—82no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D414yes—
350C2×C42#4114yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—28no—
355#107—44no—
356#106—28no—
357#106—28no—
358#106—44no—
359C4⋊C8—28no—
360C4⋊C8—28no—
361C4⋊C8—28no—
362#102—28no—
363#102—28no—
364#102—44no—
365#102—44no—
366#102—44no—
367#102—44no—
368#97—28no—
369#97C2×D4116yes—
370C4×C8—28no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D411yescharacteristic
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—44no—
397C2×Q8—416no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—28no—
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—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4C22×Q814yes—
423C2×D4C22×Q814yes—
424C22×C4—22no—
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—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—22no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
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—48no—
466C22×C4—48no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—816no—
470C22×C4—816no—
471C22×C4#12114yes—
472C22×C4#12114yes—
473C22×C4AGL(1,ℤ/8)14yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488D8—232no—
489C2×C8—28no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—44no—
499C2×C8—44no—
500C2×C8C22⋊Q8116yescomplemented
501C2×C8C22×D414yes—
502C2×C8C22×D414yes—
503C2×C8C22×Q814yes—
504C2×C8C2×#4114yes—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—48no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42#12118yes—
549C42C22×D414yes—
550C42C2×#4114yes—
551C23—24no—
552C23—24no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—48no—
562C23—48no—
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—
579C23C2×#12112yes—
580C23D4×Q812yes—
581C23#55012yes—
582C23C2×AGL(1,ℤ/8)12yes—
583C23#57512yes—
584C23#57612yes—
585C23#57712yes—
586C23C23×D411yescharacteristic
587Q8—416no—
588D4—24no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—48no—
597D4—48no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
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—816no—
638C2×C4—816no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4C2×#12114yes—
646C2×C4C2×#12114yes—
647C2×C4C2×#12112yes—
648C2×C4C2×C22⋊Q814yes—
649C2×C4#53714yes—
650C2×C4D4×Q812yes—
651C2×C4#55012yes—
652C2×C4#55414yes—
653C2×C4#55614yes—
654C2×C4C2×AGL(1,ℤ/8)14yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C23×Q811yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—24no—
662C8—42no—
663C8—42no—
664C8C2×C22⋊Q818yescomplemented
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—816no—
676C22—816no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22#253814yes—
682C22#305514yes—
683C22#306214yes—
684C22#338414yes—
685C22#338914yes—
686C22#339914yes—
687C22#340114yes—
688C22C22×#12111yescharacteristic
689C22C2×D4×Q811yescharacteristic
690C22C2×#55011yescharacteristic
691C22C22×AGL(1,ℤ/8)11yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57611yescharacteristic
694C22C2×#57711yescharacteristic
695C4—24no—
696C4—24no—
697C4—24no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#253814yes—
708C4C22×C22⋊Q811yescharacteristic
709C4C2×#53711yescharacteristic
710C4C2×#55411yescharacteristic
711C4C2×#55611yescharacteristic
712C2—84no—
713C2—84no—
714C2#2827618yesdirect product factor
715C2C2×#253811yescharacteristic
716C2C2×#305511yescharacteristic
717C2C2×#306211yescharacteristic
718C2C2×#338411yescharacteristic
719C2C2×#338911yescharacteristic
720C2C2×#339911yescharacteristic
721C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#446C2semidirect
3. C2×#2538C2semidirect
4. C2×#2580C2semidirect
5. #28276C2direct (N × H)
6. C2×#2067C2semidirect
7. C2×#2067C2semidirect
8. C2×#2065C2semidirect
9. C2×#2065C2semidirect
12. C2×#1943C2semidirect
13. C2×#1941C2semidirect
14. C2×#1847C2semidirect
15. C2×#1804C2semidirect
16. C2×#1802C2semidirect
23. C2×#446C22semidirect
44. #2580C22semidirect
46. #2538C22semidirect
64. #2067C22semidirect
65. #2067C22semidirect
66. #2065C22semidirect
67. #2065C22semidirect
70. #1943C22semidirect
71. #1941C22semidirect
72. #1847C22semidirect
73. #1804C22semidirect
74. #1802C22semidirect
86. C22×D8Q8semidirect
253. C2×D8C2×Q8semidirect
500. C2×C8C22⋊Q8semidirect
664. C8C2×C22⋊Q8semidirect
714. C2#28276direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4462
C22×#4422
C2×#25382
C2×#25802
#282762
C2×#20672
C2×#20652
C2×#19722
C2×#19702
C2×#19432
C2×#19412
C2×#18472
C2×#18042
C2×#18022