Finite Group Databasebeta

← C2×#27165 #6450807

Subgroups — C2×#27165 #6450807

Order 512.

2060 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#27165C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2036C211yesmaximal, characteristic
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1942C211yesmaximal, characteristic, complemented
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1923C211yesmaximal, characteristic
12C2×#1918C211yesmaximal, characteristic, complemented
13C2×#1872C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1852C211yesmaximal, characteristic
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117—21no—
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2421—24no—
34#2421C2218yes—
35C2×#400—21no—
36C2×#397—21no—
37#2344—24no—
38#2344C2218yes—
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#387—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×#382C2211yescharacteristic
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2077C2218yes—
56#2068C2218yescomplemented
57#2036C2218yes—
58#1949C2218yescomplemented
59#1942C2218yescomplemented
60#1928C2218yescomplemented
61#1923C2218yes—
62#1918C2218yescomplemented
63#1872C2218yes—
64#1865C2218yescomplemented
65#1852C2218yes—
66C4×#103C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—42no—
87C2×C22⋊Q8C2314yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—24no—
90C2×C4×Q8—42no—
91C2×C4×Q8C2314yes—
92C2×#117—28no—
93C2×#117—48no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C2×#107—24no—
122C2×#107—24no—
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×C4⋊C8—24no—
130C2×C4⋊C8C2314yes—
131C2×#103—24no—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×C22⋊C8—24no—
140C2×C22⋊C8—42no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8—42no—
144C2×C4×C8C2314yes—
145#400—28no—
146#397—28no—
147#391—22no—
148#391C2314yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#388—28no—
156#387—28no—
157#387—28no—
158#387—28no—
159#382—22no—
160#382C2314yes—
161#381—28no—
162#380—22no—
163#380—22no—
164#380C2314yes—
165#380C2314yes—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C22⋊C4D414yes—
171C4×C22⋊C4C2314yes—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25C2×D411yescharacteristic
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8#4114yes—
203C22×Q8C2412yes—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8#4114yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22⋊Q8—28no—
247C22⋊Q8—44no—
248C22⋊Q8—48no—
249C4×Q8—28no—
250C4×Q8—44no—
251C4×Q8—48no—
252#117—432no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4#4114yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319C2×C42C2412yes—
320#107—28no—
321#107—28no—
322#107—44no—
323#106—44no—
324C4⋊C8—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—28no—
329C22⋊C8—28no—
330C22⋊C8—44no—
331C4×C8—44no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95C2×D414yes—
344#95C2×D414yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24#13514yes—
352C24C22×D412yes—
353C24C22×D411yescharacteristic
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—44no—
366C2×Q8C2×#4114yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—432no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4AGL(1,ℤ/8)14yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C22×C4C2511yescharacteristic
444C2×C8—28no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—44no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8C2×#4114yes—
460C4⋊C4—216no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4C2×#4114yes—
486C4⋊C4C2×#4114yes—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—48no—
499C42—48no—
500C42—48no—
501C23—28no—
502C23—28no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23#52512yes—
515C23#54612yes—
516C23#54712yes—
517C23C2×#13514yes—
518C23C2×#13512yes—
519C23C2×AGL(1,ℤ/8)12yes—
520C23#57812yes—
521C23#57812yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—42no—
526Q8—42no—
527Q8—48no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—432no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#52512yes—
586C2×C4#52914yes—
587C2×C4#54214yes—
588C2×C4#54612yes—
589C2×C4#54712yes—
590C2×C4#55214yes—
591C2×C4#55314yes—
592C2×C4C2×AGL(1,ℤ/8)14yes—
593C2×C4C22×#4111yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—24no—
600C8—42no—
601C8—42no—
602C8—42no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—416no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22#247014yes—
612C22#310514yes—
613C22#312714yes—
614C22#330014yes—
615C22#330814yes—
616C22#333614yes—
617C22#335314yes—
618C22C2×#52511yescharacteristic
619C22C2×#54611yescharacteristic
620C22C2×#54711yescharacteristic
621C22C22×#13511yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C2×#57811yescharacteristic
624C22C2×#57811yescharacteristic
625C4—24no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#247014yes—
639C4C2×#52911yescharacteristic
640C4C2×#54211yescharacteristic
641C4C2×#55211yescharacteristic
642C4C2×#55311yescharacteristic
643C2—44no—
644C2#2716518yesdirect product factor
645C2C2×#247011yescharacteristic
646C2C2×#310511yescharacteristic
647C2C2×#312711yescharacteristic
648C2C2×#330011yescharacteristic
649C2C2×#330811yescharacteristic
650C2C2×#333611yescharacteristic
651C2C2×#335311yescharacteristic
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
3. #27165C2direct (N × H)
4. C2×#2101C2semidirect
6. C2×#2068C2semidirect
8. C2×#1949C2semidirect
9. C2×#1942C2semidirect
10. C2×#1928C2semidirect
12. C2×#1918C2semidirect
14. C2×#1865C2semidirect
16. C2×C4×#103C2semidirect
54. #2101C22semidirect
56. #2068C22semidirect
58. #1949C22semidirect
59. #1942C22semidirect
60. #1928C22semidirect
62. #1918C22semidirect
64. #1865C22semidirect
66. C4×#103C22semidirect
644. C2#27165direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24212
C2×#23442
#271652
C2×#21012
C2×#20772
C2×#20682
C2×#20362
C2×#19492
C2×#19422
C2×#19282
C2×#19232
C2×#19182
C2×#18722
C2×#18652
C2×#18522
C2×C4×#1032