Finite Group Databasebeta

← C2×#27360 #6450271

Subgroups — C2×#27360 #6450271

Order 512.

1962 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2507C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#27360C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2084C211yesmaximal, characteristic, complemented
6C2×#2071C211yesmaximal, characteristic
7C2×#2065C211yesmaximal, characteristic
8C2×#1948C211yesmaximal, characteristic, complemented
9C2×#1934C211yesmaximal, characteristic
10C2×#1926C211yesmaximal, characteristic
11C2×#1920C211yesmaximal, characteristic, complemented
12C2×#1920C211yesmaximal, characteristic, complemented
13C2×#1916C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic
15C2×#1864C211yesmaximal, characteristic
16C2×C4×#106C211yesmaximal, characteristic, complemented
17C22×C4×D4C2211yescharacteristic
18C22×#117—21no—
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2507—24no—
30#2507C2218yescomplemented
31C2×#398—21no—
32C2×#398—21no—
33#2354—24no—
34#2354C2218yes—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#389—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#387—21no—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2218yescomplemented
53#2084C2218yescomplemented
54#2071C2218yes—
55#2065C2218yes—
56#1948C2218yescomplemented
57#1934C2218yes—
58#1926C2218yes—
59#1920C2218yescomplemented
60#1920C2218yescomplemented
61#1916C2218yescomplemented
62#1864C2218yes—
63#1864C2218yes—
64C4×#106C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4C2314yes—
87C2×#117—28no—
88C2×#117—48no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C2×#106—24no—
118C2×#106—24no—
119C2×#106—24no—
120C2×#106—24no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8—24no—
126C2×C4⋊C8C2314yes—
127C2×#102—24no—
128C2×#102—24no—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C4×C8—24no—
136C2×C4×C8C2314yes—
137#398—28no—
138#398—28no—
139#390—22no—
140#390—22no—
141#390C2314yes—
142#390C2314yes—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—28no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#388—28no—
153#388—28no—
154#387—28no—
155#384—22no—
156#384—22no—
157#384C2314yes—
158#384C2314yes—
159#380—28no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4D414yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95D411yescharacteristic
176C2×#95D411yescharacteristic
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4#4114yes—
197C22×D4C2412yes—
198C23×C4—28no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—24no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2411yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8#4114yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C4×D4—216no—
233C4×D4—28no—
234C4×D4—28no—
235#117—432no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4SD1618yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4#4114yes—
276C2×C4⋊C4#4114yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4—82no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—82no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2412yes—
310C2×C42C2412yes—
311C2×C42C2412yes—
312#106—28no—
313#106—28no—
314#106—28no—
315#106—28no—
316C4⋊C8—28no—
317#102—28no—
318#102—28no—
319#102—28no—
320#102—28no—
321C4×C8—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—24no—
332C24—22no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24C22×D411yescharacteristic
340C2×D4—24no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4C2×#4114yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—216no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
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—432no—
379C22×C4—48no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C2×SD1614yes—
401C22×C4C2×SD1614yes—
402C22×C4C2×SD1614yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C42−1+412yes—
415C22×C42−1+412yes—
416C22×C4C2511yescharacteristic
417C2×C8—28no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8C2×#4114yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4C2×SD16116yes—
462C4⋊C4C2×#4114yes—
463C4⋊C4C2×#4114yes—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—48no—
479C42—48no—
480C42—84no—
481C23—24no—
482C23—48no—
483C23—48no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23#52512yes—
496C23#54712yes—
497C23#54812yes—
498C23C22×SD1612yes—
499C23#57512yes—
500C23#57612yes—
501C23#57712yes—
502C23C23×D411yescharacteristic
503D4—24no—
504D4—24no—
505D4—42no—
506D4—42no—
507D4—48no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—432no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4#52512yes—
559C2×C4#52914yes—
560C2×C4#54114yes—
561C2×C4#54712yes—
562C2×C4#54812yes—
563C2×C4#55214yes—
564C2×C4#55214yes—
565C2×C4C22×SD1614yes—
566C2×C4C22×SD1614yes—
567C2×C4C22×SD1614yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C8—24no—
575C8—42no—
576C8—42no—
577C8—42no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—816no—
586C22#247514yes—
587C22#310114yes—
588C22#311914yes—
589C22#333514yes—
590C22#334714yes—
591C22#336214yes—
592C22#337614yes—
593C22C2×#52511yescharacteristic
594C22C2×#54711yescharacteristic
595C22C2×#54811yescharacteristic
596C22C23×SD1611yescharacteristic
597C22C2×#57511yescharacteristic
598C22C2×#57611yescharacteristic
599C22C2×#57711yescharacteristic
600C4—24no—
601C4—24no—
602C4—24no—
603C4—42no—
604C4—42no—
605C4—44no—
606C4—44no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4#247514yes—
614C4C2×#52911yescharacteristic
615C4C2×#54111yescharacteristic
616C4C2×#55211yescharacteristic
617C4C2×#55211yescharacteristic
618C2—84no—
619C2#2736018yesdirect product factor
620C2C2×#247511yescharacteristic
621C2C2×#310111yescharacteristic
622C2C2×#311911yescharacteristic
623C2C2×#333511yescharacteristic
624C2C2×#334711yescharacteristic
625C2C2×#336211yescharacteristic
626C2C2×#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
1. C2×#2507C2semidirect
3. #27360C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2084C2semidirect
8. C2×#1948C2semidirect
11. C2×#1920C2semidirect
12. C2×#1920C2semidirect
13. C2×#1916C2semidirect
16. C2×C4×#106C2semidirect
30. #2507C22semidirect
52. #2099C22semidirect
53. #2084C22semidirect
56. #1948C22semidirect
59. #1920C22semidirect
60. #1920C22semidirect
61. #1916C22semidirect
64. C4×#106C22semidirect
619. C2#27360direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25072
C2×#23542
#273602
C2×#20992
C2×#20842
C2×#20712
C2×#20652
C2×#19482
C2×#19342
C2×#19262
C2×#19202
C2×#19162
C2×#18642
C2×C4×#1062