Finite Group Databasebeta

← C2×#28285 #6466033

Subgroups — C2×#28285 #6466033

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3C2×C4×#125C211yesmaximal, characteristic, complemented
4C2×C4×#124C211yesmaximal, characteristic
5#28285C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic
11C2×#1983C211yesmaximal, characteristic
12C2×#1949C211yesmaximal, characteristic, complemented
13C2×#1949C211yesmaximal, characteristic, complemented
14C2×#1842C211yesmaximal, characteristic, complemented
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#479—28no—
22C2×#479C2218yescomplemented
23C2×#478—28no—
24C2×#478C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C4×#125—24no—
44C4×#125C2218yescomplemented
45C4×#124—24no—
46C4×#124C2218yes—
47C2×#391C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#387—21no—
54C2×#387—21no—
55C2×#382C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×C4×C4⋊C4—21no—
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4—21no—
62C2×C4×C22⋊C4—21no—
63C2×C43C2211yescharacteristic
64#2068C2218yescomplemented
65#2067C2218yes—
66#2066C2218yescomplemented
67#2065C2218yes—
68#1983C2218yes—
69#1983C2218yes—
70#1949C2218yescomplemented
71#1949C2218yescomplemented
72#1842C2218yescomplemented
73#1805C2218yescomplemented
74#1804C2218yes—
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#125—22no—
87C2×#125—24no—
88C2×#125D418yes—
89C2×#125C2314yes—
90C2×#124—22no—
91C2×#124—24no—
92C2×#124Q818yes—
93C2×#124C2314yes—
94C2×C4×Q8—22no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#479—232no—
129#478—232no—
130C2×#107—24no—
131C2×#107—24no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8—24no—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—24no—
141C2×C4⋊C8—24no—
142C2×C4⋊C8—42no—
143C2×C4⋊C8—42no—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—42no—
153C2×#103—42no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—42no—
161C2×#102—42no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166#391—22no—
167#391C2314yes—
168#390—22no—
169#390C2314yes—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#388—28no—
175#388—28no—
176#387—28no—
177#387—28no—
178#382—22no—
179#382—22no—
180#382C2314yes—
181#382C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—28no—
185C4×C4⋊C4—28no—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—28no—
189C4×C22⋊C4—28no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C43D414yes—
201C43C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206C25—21no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8C2×Q814yes—
214C22×Q8C2412yes—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Q814yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#125—216no—
258#125—28no—
259#125C2×D4116yes—
260#124—216no—
261#124—28no—
262#124C2×Q8116yes—
263C4×Q8—28no—
264C4×Q8—28no—
265C4×Q8—48no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D412yes—
331C2×C42C2×D414yes—
332C2×C42C2×D412yes—
333C2×C42C2×Q814yes—
334C2×C42#4114yes—
335C2×C42#4114yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—28no—
340#107—28no—
341#106—28no—
342#106—28no—
343C4⋊C8—28no—
344C4⋊C8—28no—
345C4⋊C8—28no—
346C4⋊C8—28no—
347C4⋊C8—44no—
348C4⋊C8—44no—
349#103—28no—
350#103—28no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8C22×Q814yes—
389C2×D4—24no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4C22×Q814yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—48no—
446C22×C4—816no—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4C22⋊Q814yes—
450C22×C4C22⋊Q814yes—
451C22×C4#13514yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×Q812yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—44no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—48no—
503C4⋊C4C22×D414yes—
504C4⋊C4C22×D414yes—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C42—24no—
511C42—28no—
512C42—22no—
513C42—22no—
514C42—44no—
515C42—44no—
516C42—44no—
517C42—48no—
518C42—48no—
519C42—48no—
520C42#12118yes—
521C42C22⋊Q818yes—
522C42C22×D414yes—
523C42C22×Q814yes—
524C42C2×#4114yes—
525C42C2×#4114yes—
526C23—24no—
527C23—48no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23C2×#12112yes—
541C23C2×C22⋊Q812yes—
542C23#53712yes—
543C23C2×#13512yes—
544C23#57512yes—
545C23#57512yes—
546C23#57512yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—42no—
551Q8—42no—
552Q8—48no—
553D4—24no—
554D4—24no—
555D4—42no—
556D4—42no—
557D4—48no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4C2×#12112yes—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q814yes—
611C2×C4C2×C22⋊Q812yes—
612C2×C4#53712yes—
613C2×C4D4×Q814yes—
614C2×C4#55014yes—
615C2×C4#55414yes—
616C2×C4#55614yes—
617C2×C4C2×#13514yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C8—42no—
625C8—42no—
626C8—42no—
627C8—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22#253814yes—
637C22#303514yes—
638C22#304014yes—
639C22#307614yes—
640C22#308014yes—
641C22#325014yes—
642C22#325614yes—
643C22C22×#12111yescharacteristic
644C22C22×C22⋊Q811yescharacteristic
645C22C2×#53711yescharacteristic
646C22C22×#13511yescharacteristic
647C22C2×#57511yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57511yescharacteristic
650C4—24no—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—24no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#253814yes—
664C4C2×D4×Q811yescharacteristic
665C4C2×#55011yescharacteristic
666C4C2×#55411yescharacteristic
667C4C2×#55611yescharacteristic
668C2—84no—
669C2#2828518yesdirect product factor
670C2C2×#253811yescharacteristic
671C2C2×#303511yescharacteristic
672C2C2×#304011yescharacteristic
673C2C2×#307611yescharacteristic
674C2C2×#308011yescharacteristic
675C2C2×#325011yescharacteristic
676C2C2×#325611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#479C2semidirect
3. C2×C4×#125C2semidirect
5. #28285C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2066C2semidirect
12. C2×#1949C2semidirect
13. C2×#1949C2semidirect
14. C2×#1842C2semidirect
15. C2×#1805C2semidirect
22. C2×#479C22semidirect
44. C4×#125C22semidirect
64. #2068C22semidirect
66. #2066C22semidirect
70. #1949C22semidirect
71. #1949C22semidirect
72. #1842C22semidirect
73. #1805C22semidirect
669. C2#28285direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4782
C2×C4×#1252
C2×C4×#1242
#282852
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#19832
C2×#19492
C2×#18422
C2×#18052
C2×#18042