Finite Group Databasebeta

← C2×#27454 #6450713

Subgroups — C2×#27454 #6450713

Order 512.

2006 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#27454C2116yesmaximal, direct product factor
4C2×#2086C211yesmaximal, characteristic, complemented
5C2×#2083C211yesmaximal, characteristic, complemented
6C2×#2075C211yesmaximal, characteristic
7C2×#2075C211yesmaximal, characteristic
8C2×#1947C211yesmaximal, characteristic, complemented
9C2×#1942C211yesmaximal, characteristic, complemented
10C2×#1923C211yesmaximal, characteristic
11C2×#1918C211yesmaximal, characteristic, complemented
12C2×#1918C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1852C211yesmaximal, characteristic
15C2×#1852C211yesmaximal, characteristic
16C2×C4×#107C211yesmaximal, characteristic, complemented
17C22×#126—21no—
18C22×#117—21no—
19C23×C4⋊C4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2496—24no—
31#2496C2218yes—
32C2×#400—21no—
33C2×#397—21no—
34#2344—24no—
35#2344C2218yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#387—21no—
42C2×#387—21no—
43C2×#384C2211yescharacteristic
44C2×#381—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2086C2218yescomplemented
53#2083C2218yescomplemented
54#2075C2218yes—
55#2075C2218yes—
56#1947C2218yescomplemented
57#1942C2218yescomplemented
58#1923C2218yes—
59#1918C2218yescomplemented
60#1918C2218yescomplemented
61#1918C2218yescomplemented
62#1852C2218yes—
63#1852C2218yes—
64C4×#107C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#126—28no—
84C2×#126—48no—
85C2×#117—28no—
86C2×#117—48no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
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⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
110C22×C22⋊C4D411yescharacteristic
111C22×C42—21no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C2×#107—24no—
117C2×#107—24no—
118C2×#107—24no—
119C2×#107—24no—
120C2×#107—24no—
121C2×#107—24no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×C4⋊C8—24no—
129C2×C4⋊C8C2314yes—
130C2×C22⋊C8—24no—
131C2×C22⋊C8—24no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8—24no—
135C2×C4×C8C2314yes—
136#400—28no—
137#397—28no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389C2314yes—
143#389C2314yes—
144#389C2314yes—
145#389C2314yes—
146#387—28no—
147#387—28no—
148#387—28no—
149#384—22no—
150#384C2314yes—
151#381—28no—
152#380—22no—
153#380—22no—
154#380—22no—
155#380C2314yes—
156#380C2314yes—
157#380C2314yes—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C22⋊C4D414yes—
165C4×C22⋊C4C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95D411yescharacteristic
177C2×#95D411yescharacteristic
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25C2×D411yescharacteristic
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
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×C8#4114yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C22×C8C2412yes—
233#126—432no—
234#117—432no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
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—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4#4114yes—
275C2×C4⋊C4#4114yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4D818yes—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2412yes—
307C2×C42C2412yes—
308#107—28no—
309#107—28no—
310#107—28no—
311#107—28no—
312#107—28no—
313#107—28no—
314C4⋊C8—28no—
315C22⋊C8—28no—
316C22⋊C8—28no—
317C4×C8—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95C2×D414yes—
329#95C2×D414yes—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24C2×D814yes—
337C24C22×D412yes—
338C24C22×D411yescharacteristic
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—216no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—432no—
373C22×C4—48no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
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×C4—42no—
401C22×C4C2×D814yes—
402C22×C4C2×D814yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C42−1+412yes—
414C22×C42−1+412yes—
415C22×C4C2511yescharacteristic
416C2×C8—28no—
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—44no—
429C2×C8—44no—
430C2×C8C2×#4114yes—
431C2×C8C2×#4114yes—
432C4⋊C4—216no—
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—22no—
448C4⋊C4—22no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4C2×#4114yes—
462C4⋊C4C2×#4114yes—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4C2×D8116yes—
468C42—24no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—48no—
475C42—48no—
476C42—48no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23#52512yes—
491C23#54712yes—
492C23#54712yes—
493C23C22×D814yes—
494C23C22×D812yes—
495C23#57512yes—
496C23#57812yes—
497C23#57812yes—
498C23C23×D411yescharacteristic
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—432no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—48no—
520C2×C4—48no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
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—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4#52512yes—
556C2×C4#52914yes—
557C2×C4#54114yes—
558C2×C4#54712yes—
559C2×C4#54712yes—
560C2×C4#55214yes—
561C2×C4#55614yes—
562C2×C4C22×D814yes—
563C2×C4C22×D814yes—
564C2×C4C22×#4111yescharacteristic
565C2×C4C22×#4111yescharacteristic
566C2×C4C22×#4111yescharacteristic
567C2×C4C22×#4111yescharacteristic
568C2×C4C2×2−1+411yescharacteristic
569C2×C4C2×2−1+411yescharacteristic
570C8—24no—
571C8—24no—
572C8—42no—
573C8—42no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—416no—
580C22—42no—
581C22—42no—
582C22#247814yes—
583C22#310014yes—
584C22#312714yes—
585C22#333414yes—
586C22#333414yes—
587C22#334914yes—
588C22#335214yes—
589C22C2×#52511yescharacteristic
590C22C2×#54711yescharacteristic
591C22C2×#54711yescharacteristic
592C22C23×D811yescharacteristic
593C22C2×#57511yescharacteristic
594C22C2×#57811yescharacteristic
595C22C2×#57811yescharacteristic
596C4—24no—
597C4—24no—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—44no—
602C4—42no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#247814yes—
610C4C2×#52911yescharacteristic
611C4C2×#54111yescharacteristic
612C4C2×#55211yescharacteristic
613C4C2×#55611yescharacteristic
614C2—44no—
615C2#2745418yesdirect product factor
616C2C2×#247811yescharacteristic
617C2C2×#310011yescharacteristic
618C2C2×#312711yescharacteristic
619C2C2×#333411yescharacteristic
620C2C2×#333411yescharacteristic
621C2C2×#334911yescharacteristic
622C2C2×#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. #27454C2direct (N × H)
4. C2×#2086C2semidirect
5. C2×#2083C2semidirect
8. C2×#1947C2semidirect
9. C2×#1942C2semidirect
11. C2×#1918C2semidirect
12. C2×#1918C2semidirect
13. C2×#1918C2semidirect
16. C2×C4×#107C2semidirect
52. #2086C22semidirect
53. #2083C22semidirect
56. #1947C22semidirect
57. #1942C22semidirect
59. #1918C22semidirect
60. #1918C22semidirect
61. #1918C22semidirect
64. C4×#107C22semidirect
615. C2#27454direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#23442
#274542
C2×#20862
C2×#20832
C2×#20752
C2×#19472
C2×#19422
C2×#19232
C2×#19182
C2×#18522
C2×C4×#1072