Finite Group Databasebeta

← C2×#26561 #6439271

Subgroups — C2×#26561 #6439271

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×C4×Q16C211yesmaximal, characteristic, complemented
3C2×#2421C211yesmaximal, characteristic
4C2×#2362C211yesmaximal, characteristic
5#26561C2116yesmaximal, direct product factor
6C2×#2097C211yesmaximal, characteristic, complemented
7C2×#2056C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic
9C2×#2001C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1928C211yesmaximal, characteristic, complemented
13C2×#1923C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1854C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#485—28no—
24C2×#485C2218yes—
25C2×C4×Q16—28no—
26C2×C4×Q16C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2421—24no—
43#2421C2218yes—
44C2×#400—21no—
45C2×#397—21no—
46#2362—24no—
47#2362C2218yes—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#388—21no—
55C2×#387—21no—
56C2×#387—21no—
57C2×#382C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62#2097C2218yescomplemented
63#2056C2218yescomplemented
64#2036C2218yes—
65#2001C2218yes—
66#1949C2218yescomplemented
67#1942C2218yescomplemented
68#1928C2218yescomplemented
69#1923C2218yes—
70#1865C2218yescomplemented
71#1865C2218yescomplemented
72#1854C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16—24no—
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—42no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—42no—
103C2×C4×Q8—42no—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2314yes—
106C2×#117—22no—
107C2×#117—24no—
108C2×#117—42no—
109C2×#117C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#485—232no—
135C4×Q16—232no—
136C2×#107—24no—
137C2×#107—24no—
138C2×#107—24no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106—24no—
143C2×#106—42no—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—42no—
153C2×#103D418yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×C22⋊C8—24no—
161C2×C22⋊C8—24no—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8C2314yes—
166#400—28no—
167#397—28no—
168#391—22no—
169#391—22no—
170#391C2314yes—
171#391C2314yes—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#388—28no—
177#388—28no—
178#388—28no—
179#387—28no—
180#387—28no—
181#382—22no—
182#382C2314yes—
183#380—22no—
184#380C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—28no—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4D414yes—
189C4×C22⋊C4C2314yes—
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—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25C2×D411yescharacteristic
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×Q8C2×D414yes—
222C22×Q8#4114yes—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C2×Q16—232no—
248C2×Q16—28no—
249C2×Q16—28no—
250C2×Q16—28no—
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×C8C2×D414yes—
262C22×C8#4114yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22⋊Q8—216no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279#117—28no—
280#117—44no—
281#117—48no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
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—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
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—44no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—28no—
347#107—28no—
348#107—28no—
349#106—28no—
350#106—44no—
351C4⋊C8—28no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—44no—
357#103C2×D4116yes—
358C22⋊C8—28no—
359C22⋊C8—28no—
360C4×C8—28no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—28no—
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—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8C22×D414yes—
407C2×Q8C2×#4114yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
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—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
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—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
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#13514yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481Q16—232no—
482C2×C8—28no—
483C2×C8—28no—
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—44no—
495C2×C8—44no—
496C2×C8C22×D414yes—
497C2×C8C2×#4114yes—
498C4⋊C4—216no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4C22×D414yes—
521C4⋊C4C2×#4114yes—
522C22⋊C4—216no—
523C22⋊C4—28no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—48no—
535C42—48no—
536C42—48no—
537C42—48no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23#52512yes—
553C23D4212yes—
554C23#54712yes—
555C23C2×#13512yes—
556C23#57512yes—
557C23#57812yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—24no—
562Q8—24no—
563Q8—24no—
564Q8—42no—
565Q8—42no—
566Q8—48no—
567Q8—48no—
568Q8—42no—
569Q8—42no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
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—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4#52514yes—
627C2×C4#52512yes—
628C2×C4#53914yes—
629C2×C4D4212yes—
630C2×C4#54714yes—
631C2×C4#54712yes—
632C2×C4#55914yes—
633C2×C4C2×#13514yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—24no—
641C8—24no—
642C8—42no—
643C8—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—88no—
652C22#245514yes—
653C22#310214yes—
654C22#312714yes—
655C22#326914yes—
656C22#328514yes—
657C22#334214yes—
658C22#335214yes—
659C22C2×#52511yescharacteristic
660C22C2×D4211yescharacteristic
661C22C2×#54711yescharacteristic
662C22C22×#13511yescharacteristic
663C22C2×#57511yescharacteristic
664C22C2×#57811yescharacteristic
665C22C2×#57811yescharacteristic
666C4—24no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#245514yes—
680C4C2×#52511yescharacteristic
681C4C2×#53911yescharacteristic
682C4C2×#54711yescharacteristic
683C4C2×#55911yescharacteristic
684C2—82no—
685C2#2656118yesdirect product factor
686C2C2×#245511yescharacteristic
687C2C2×#310211yescharacteristic
688C2C2×#312711yescharacteristic
689C2C2×#326911yescharacteristic
690C2C2×#328511yescharacteristic
691C2C2×#334211yescharacteristic
692C2C2×#335211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×Q16C2semidirect
5. #26561C2direct (N × H)
6. C2×#2097C2semidirect
7. C2×#2056C2semidirect
10. C2×#1949C2semidirect
11. C2×#1942C2semidirect
12. C2×#1928C2semidirect
14. C2×#1865C2semidirect
15. C2×#1865C2semidirect
26. C2×C4×Q16C22semidirect
62. #2097C22semidirect
63. #2056C22semidirect
66. #1949C22semidirect
67. #1942C22semidirect
68. #1928C22semidirect
70. #1865C22semidirect
71. #1865C22semidirect
685. C2#26561direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×C4×Q162
C2×#24212
C2×#23622
#265612
C2×#20972
C2×#20562
C2×#20362
C2×#20012
C2×#19492
C2×#19422
C2×#19282
C2×#19232
C2×#18652
C2×#18542