Finite Group Databasebeta

← C2×#25283 #6448005

Subgroups — C2×#25283 #6448005

Order 512.

2230 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic, complemented
3C2×#2545C211yesmaximal, characteristic
4C2×#2370C211yesmaximal, characteristic, complemented
5#25283C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2043C211yesmaximal, characteristic, complemented
9C2×#2042C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic
11C2×#1983C211yesmaximal, characteristic
12C2×#1981C211yesmaximal, characteristic, complemented
13C2×#1939C211yesmaximal, characteristic, complemented
14C2×#1927C211yesmaximal, characteristic, complemented
15C2×#1925C211yesmaximal, characteristic
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#488—28no—
23C2×#488C2218yes—
24C2×#479—28no—
25C2×#479C2218yescomplemented
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×C4×C8C2211yescharacteristic
44#2545—24no—
45#2545C2218yes—
46C2×#396—21no—
47C2×#396—21no—
48#2370—24no—
49#2370C2218yescomplemented
50C2×#391C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389—21no—
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#388—21no—
56C2×#388—21no—
57C2×#387—21no—
58C2×#387—21no—
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63C2×C4×C4⋊C4C2211yescharacteristic
64#2068C2218yescomplemented
65#2065C2218yes—
66#2043C2218yescomplemented
67#2042C2218yes—
68#1983C2218yes—
69#1983C2218yes—
70#1981C2218yescomplemented
71#1939C2218yescomplemented
72#1927C2218yescomplemented
73#1925C2218yes—
74C4×#103C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8C2314yes—
90C2×#125—22no—
91C2×#125—24no—
92C2×#125D418yes—
93C2×#125C2314yes—
94C2×#124—24no—
95C2×#124—22no—
96C2×#124—22no—
97C2×#124—24no—
98C2×#124—24no—
99C2×#124—42no—
100C2×#124C2314yes—
101C2×#124C2314yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8C2314yes—
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⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#488—232no—
134#479—232no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—42no—
141C2×C4⋊C8—42no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—42no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102—24no—
163C2×#102—24no—
164C2×#102—42no—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C4×C8—24no—
170C2×C4×C8C2314yes—
171#396—28no—
172#396—28no—
173#391—22no—
174#391C2314yes—
175#390—22no—
176#390C2314yes—
177#389—28no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#388—28no—
183#388—28no—
184#387—28no—
185#387—28no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—28no—
190C4×C4⋊C4D414yes—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2314yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8#4114yes—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266#125—216no—
267#125—28no—
268#125C2×D4116yes—
269#124—28no—
270#124—28no—
271#124—28no—
272#124—44no—
273#124—48no—
274#124—48no—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—48no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
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—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4#4114yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—82no—
339C2×C42—82no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D414yes—
344C2×C42#4114yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—28no—
349#106—28no—
350C4⋊C8—28no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—28no—
359#103—28no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—28no—
364#102—44no—
365C4×C8—28no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8C2×#4114yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
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—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
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—42no—
466C22×C4#12114yes—
467C22×C4#12114yes—
468C22×C4#13514yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8—44no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—84no—
520C4⋊C4C22×D414yes—
521C4⋊C4C22×D414yes—
522C4⋊C4C2×#4114yes—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—48no—
536C42—48no—
537C42—84no—
538C42—84no—
539C42#12118yes—
540C42C22×D414yes—
541C42C2×#4114yes—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23C2×#12112yes—
557C23#54612yes—
558C23#54812yes—
559C23C2×#13512yes—
560C23#57512yes—
561C23#57712yes—
562C23#57812yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—24no—
566Q8—48no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—84no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—84no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4C2×#12114yes—
629C2×C4C2×#12114yes—
630C2×C4C2×#12112yes—
631C2×C4#52914yes—
632C2×C4#54214yes—
633C2×C4#54612yes—
634C2×C4#54812yes—
635C2×C4#54814yes—
636C2×C4#55014yes—
637C2×C4C2×#13514yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#242614yes—
657C22#303514yes—
658C22#304614yes—
659C22#330014yes—
660C22#332614yes—
661C22#336014yes—
662C22#337714yes—
663C22C22×#12111yescharacteristic
664C22C2×#54611yescharacteristic
665C22C2×#54811yescharacteristic
666C22C22×#13511yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57711yescharacteristic
669C22C2×#57811yescharacteristic
670C4—24no—
671C4—24no—
672C4—24no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#242614yes—
684C4C2×#52911yescharacteristic
685C4C2×#54211yescharacteristic
686C4C2×#54811yescharacteristic
687C4C2×#55011yescharacteristic
688C2—162no—
689C2#2528318yesdirect product factor
690C2C2×#242611yescharacteristic
691C2C2×#303511yescharacteristic
692C2C2×#304611yescharacteristic
693C2C2×#330011yescharacteristic
694C2C2×#332611yescharacteristic
695C2C2×#336011yescharacteristic
696C2C2×#337711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#479C2semidirect
4. C2×#2370C2semidirect
5. #25283C2direct (N × H)
6. C2×#2068C2semidirect
8. C2×#2043C2semidirect
12. C2×#1981C2semidirect
13. C2×#1939C2semidirect
14. C2×#1927C2semidirect
16. C2×C4×#103C2semidirect
25. C2×#479C22semidirect
49. #2370C22semidirect
64. #2068C22semidirect
66. #2043C22semidirect
70. #1981C22semidirect
71. #1939C22semidirect
72. #1927C22semidirect
74. C4×#103C22semidirect
689. C2#25283direct (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×#4792
C2×#25452
C2×#23702
#252832
C2×#20682
C2×#20652
C2×#20432
C2×#20422
C2×#19832
C2×#19812
C2×#19392
C2×#19272
C2×#19252
C2×C4×#1032