Finite Group Databasebeta

← C2×#27168 #6450806

Subgroups — C2×#27168 #6450806

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3#27168C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2076C211yesmaximal, characteristic
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2035C211yesmaximal, characteristic
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1941C211yesmaximal, characteristic, complemented
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1923C211yesmaximal, characteristic
12C2×#1918C211yesmaximal, characteristic, complemented
13C2×#1873C211yesmaximal, characteristic
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2421—24no—
33#2421C2218yes—
34C2×#400—21no—
35C2×#397—21no—
36#2346—24no—
37#2346C2218yes—
38C2×#391C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#388—21no—
45C2×#387—21no—
46C2×#382—21no—
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×#376—21no—
54#2101C2218yescomplemented
55#2076C2218yes—
56#2066C2218yescomplemented
57#2035C2218yes—
58#1949C2218yescomplemented
59#1941C2218yescomplemented
60#1928C2218yescomplemented
61#1923C2218yes—
62#1918C2218yescomplemented
63#1873C2218yes—
64#1863C2218yescomplemented
65#1854C2218yes—
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×C4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—42no—
86C2×C22⋊Q8C2314yes—
87C2×C4×Q8—22no—
88C2×C4×Q8—24no—
89C2×C4×Q8—42no—
90C2×C4×Q8C2314yes—
91C2×#117—22no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117C2314yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D411yescharacteristic
112C22×C22⋊C4D411yescharacteristic
113C22×C42—21no—
114C22×C42—21no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—24no—
120C2×#107—42no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—24no—
124C2×#106—42no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8—24no—
128C2×C4⋊C8C2314yes—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103—24no—
132C2×#103—24no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8—24no—
138C2×C22⋊C8—42no—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8—42no—
142C2×C4×C8C2314yes—
143#400—28no—
144#397—28no—
145#391—22no—
146#391C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#388—28no—
154#388—28no—
155#388—28no—
156#387—28no—
157#382—22no—
158#382—28no—
159#382C2314yes—
160#380—22no—
161#380—22no—
162#380C2314yes—
163#380C2314yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4D414yes—
169C4×C22⋊C4C2314yes—
170#376—28no—
171C2×#95—21no—
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×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8#4114yes—
202C22×Q8C2412yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8#4114yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22⋊Q8—28no—
242C22⋊Q8—44no—
243C22⋊Q8—48no—
244C4×Q8—28no—
245C4×Q8—44no—
246C4×Q8—48no—
247#117—216no—
248#117—28no—
249#117—28no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4#4114yes—
280C2×C4⋊C4#4114yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—82no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316#107—28no—
317#107—44no—
318#106—28no—
319#106—44no—
320C4⋊C8—28no—
321#103—28no—
322#103—28no—
323#103—28no—
324#103—28no—
325C22⋊C8—28no—
326C22⋊C8—44no—
327C4×C8—44no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95C2×D414yes—
340#95C2×D414yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24C22×D412yes—
349C24C22×D411yescharacteristic
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—44no—
362C2×Q8C2×#4114yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4#13514yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C22×C4C2511yescharacteristic
436C2×C8—28no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8—44no—
451C2×C8C2×#4114yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4C2×#4114yes—
477C4⋊C4C2×#4114yes—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C42—216no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—48no—
492C42—48no—
493C42—48no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23#52512yes—
509C23#54612yes—
510C23#54712yes—
511C23C2×#13512yes—
512C23#57512yes—
513C23#57712yes—
514C23#57812yes—
515C23C23×D411yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—42no—
519Q8—42no—
520Q8—48no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4#52512yes—
578C2×C4#52914yes—
579C2×C4#54214yes—
580C2×C4#54612yes—
581C2×C4#54712yes—
582C2×C4#55214yes—
583C2×C4#55314yes—
584C2×C4C2×#13514yes—
585C2×C4C22×#4111yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C8—24no—
592C8—42no—
593C8—42no—
594C8—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—88no—
603C22#247014yes—
604C22#310214yes—
605C22#312014yes—
606C22#330014yes—
607C22#332114yes—
608C22#334014yes—
609C22#335214yes—
610C22C2×#52511yescharacteristic
611C22C2×#54611yescharacteristic
612C22C2×#54711yescharacteristic
613C22C22×#13511yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57711yescharacteristic
616C22C2×#57811yescharacteristic
617C4—24no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#247014yes—
631C4C2×#52911yescharacteristic
632C4C2×#54211yescharacteristic
633C4C2×#55211yescharacteristic
634C4C2×#55311yescharacteristic
635C2—82no—
636C2#2716818yesdirect product factor
637C2C2×#247011yescharacteristic
638C2C2×#310211yescharacteristic
639C2C2×#312011yescharacteristic
640C2C2×#330011yescharacteristic
641C2C2×#332111yescharacteristic
642C2C2×#334011yescharacteristic
643C2C2×#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
3. #27168C2direct (N × H)
4. C2×#2101C2semidirect
6. C2×#2066C2semidirect
8. C2×#1949C2semidirect
9. C2×#1941C2semidirect
10. C2×#1928C2semidirect
12. C2×#1918C2semidirect
14. C2×#1863C2semidirect
16. C2×C4×#103C2semidirect
54. #2101C22semidirect
56. #2066C22semidirect
58. #1949C22semidirect
59. #1941C22semidirect
60. #1928C22semidirect
62. #1918C22semidirect
64. #1863C22semidirect
66. C4×#103C22semidirect
636. C2#27168direct (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×#23462
#271682
C2×#21012
C2×#20762
C2×#20662
C2×#20352
C2×#19492
C2×#19412
C2×#19282
C2×#19232
C2×#19182
C2×#18732
C2×#18632
C2×#18542
C2×C4×#1032