Finite Group Databasebeta

← C2×#25287 #6448015

Subgroups — C2×#25287 #6448015

Order 512.

2268 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3C2×#2545C211yesmaximal, characteristic
4C2×#2369C211yesmaximal, characteristic, complemented
5#25287C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2043C211yesmaximal, characteristic, complemented
9C2×#2042C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic
11C2×#1983C211yesmaximal, characteristic
12C2×#1981C211yesmaximal, characteristic, complemented
13C2×#1940C211yesmaximal, characteristic, complemented
14C2×#1926C211yesmaximal, characteristic
15C2×#1927C211yesmaximal, characteristic, complemented
16C2×#1763C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#489—28no—
23C2×#489C2218yes—
24C2×#475—28no—
25C2×#475C2218yescomplemented
26C22×#106C2211yescharacteristic
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×#97C2211yescharacteristic
44#2545—24no—
45#2545C2218yes—
46C2×#396—21no—
47C2×#396—21no—
48#2369—24no—
49#2369C2218yescomplemented
50C2×#391—21no—
51C2×#391—21no—
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389—21no—
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#388—21no—
58C2×#388—21no—
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63C2×C4×C4⋊C4C2211yescharacteristic
64#2066C2218yescomplemented
65#2065C2218yes—
66#2043C2218yescomplemented
67#2042C2218yes—
68#1983C2218yes—
69#1983C2218yes—
70#1981C2218yescomplemented
71#1940C2218yescomplemented
72#1927C2218yescomplemented
73#1926C2218yes—
74#1763C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—42no—
92C2×C4⋊Q8D418yes—
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2314yes—
95C2×#124—24no—
96C2×#124—22no—
97C2×#124—22no—
98C2×#124—24no—
99C2×#124—24no—
100C2×#124—42no—
101C2×#124C2314yes—
102C2×#124C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8C2314yes—
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#489—232no—
134#475—232no—
135C2×#106—24no—
136C2×#106—24no—
137C2×#106C2314yes—
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×#97—24no—
170C2×#97C2314yes—
171#396—28no—
172#396—28no—
173#391—22no—
174#391—28no—
175#391—28no—
176#391C2314yes—
177#390—22no—
178#390C2314yes—
179#389—28no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#388—28no—
185#388—28no—
186C4×C4⋊C4—28no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
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—28no—
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—21no—
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—22no—
223C22×Q8—22no—
224C22×Q8—82no—
225C22×Q8—82no—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
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×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—28no—
268C4⋊Q8—216no—
269C4⋊Q8—28no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8C2×D4116yes—
273#124—28no—
274#124—28no—
275#124—28no—
276#124—44no—
277#124—48no—
278#124—48no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—48no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
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—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42—82no—
341C2×C42—82no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D414yes—
346C2×C42#4114yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350#106—28no—
351#106—28no—
352C4⋊C8—28no—
353C4⋊C8—44no—
354C4⋊C8—44no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—44no—
367#97—28no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22×D411yescharacteristic
386C2×Q8—216no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—816no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×Q8C2×#4114yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—44no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
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—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—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4#12114yes—
473C22×C4#12114yes—
474C22×C4#13714yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489C2×C8—22no—
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—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8—44no—
505C4⋊C4—216no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—28no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22×D414yes—
526C4⋊C4C22×D414yes—
527C4⋊C4C2×#4114yes—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—48no—
541C42—48no—
542C42—84no—
543C42—84no—
544C42#12118yes—
545C42C22×D414yes—
546C42C2×#4114yes—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23C2×#12112yes—
562C23#54612yes—
563C23#54812yes—
564C23C2×#13712yes—
565C23#57512yes—
566C23#57712yes—
567C23#57712yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—48no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—816no—
579Q8—84no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—84no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
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—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12112yes—
637C2×C4#52914yes—
638C2×C4#54214yes—
639C2×C4#54612yes—
640C2×C4#54812yes—
641C2×C4#54814yes—
642C2×C4#55014yes—
643C2×C4C2×#13714yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—168no—
662C22#242614yes—
663C22#303714yes—
664C22#304214yes—
665C22#331614yes—
666C22#332114yes—
667C22#336914yes—
668C22#337614yes—
669C22C22×#12111yescharacteristic
670C22C2×#54611yescharacteristic
671C22C2×#54811yescharacteristic
672C22C22×#13711yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57711yescharacteristic
676C4—24no—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#242614yes—
690C4C2×#52911yescharacteristic
691C4C2×#54211yescharacteristic
692C4C2×#54811yescharacteristic
693C4C2×#55011yescharacteristic
694C2—162no—
695C2#2528718yesdirect product factor
696C2C2×#242611yescharacteristic
697C2C2×#303711yescharacteristic
698C2C2×#304211yescharacteristic
699C2C2×#331611yescharacteristic
700C2C2×#332111yescharacteristic
701C2C2×#336911yescharacteristic
702C2C2×#337611yescharacteristic
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×#475C2semidirect
4. C2×#2369C2semidirect
5. #25287C2direct (N × H)
6. C2×#2066C2semidirect
8. C2×#2043C2semidirect
12. C2×#1981C2semidirect
13. C2×#1940C2semidirect
15. C2×#1927C2semidirect
16. C2×#1763C2semidirect
25. C2×#475C22semidirect
49. #2369C22semidirect
64. #2066C22semidirect
66. #2043C22semidirect
70. #1981C22semidirect
71. #1940C22semidirect
72. #1927C22semidirect
74. #1763C22semidirect
695. C2#25287direct (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×#4752
C2×#25452
C2×#23692
#252872
C2×#20662
C2×#20652
C2×#20432
C2×#20422
C2×#19832
C2×#19812
C2×#19402
C2×#19262
C2×#19272
C2×#17632