Finite Group Databasebeta

← C2×#27469 #6450720

Subgroups — C2×#27469 #6450720

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3#27469C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2083C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic
7C2×#2075C211yesmaximal, characteristic
8C2×#1948C211yesmaximal, characteristic, complemented
9C2×#1942C211yesmaximal, characteristic, complemented
10C2×#1923C211yesmaximal, characteristic
11C2×#1921C211yesmaximal, characteristic, complemented
12C2×#1920C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1854C211yesmaximal, characteristic
15C2×#1854C211yesmaximal, characteristic
16C2×C4×#106C211yesmaximal, characteristic, complemented
17C22×#117C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2497—24no—
29#2497C2218yes—
30C2×#400—21no—
31C2×#396—21no—
32#2346—24no—
33#2346C2218yes—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388—21no—
39C2×#388—21no—
40C2×#388—21no—
41C2×#387—21no—
42C2×#384C2211yescharacteristic
43C2×#382—21no—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×#376—21no—
52#2088C2218yescomplemented
53#2083C2218yescomplemented
54#2077C2218yes—
55#2075C2218yes—
56#1948C2218yescomplemented
57#1942C2218yescomplemented
58#1923C2218yes—
59#1921C2218yescomplemented
60#1920C2218yescomplemented
61#1918C2218yescomplemented
62#1854C2218yes—
63#1854C2218yes—
64C4×#106C2218yescomplemented
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×C4C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#117—22no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117C2314yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4D411yescharacteristic
105C22×C22⋊C4D411yescharacteristic
106C22×C42—21no—
107C22×C42—21no—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C2×#107—24no—
113C2×#107—24no—
114C2×#107—24no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106—24no—
119C2×#106—24no—
120C2×#106—24no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8—24no—
125C2×C4⋊C8C2314yes—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—24no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C4×C8—24no—
131C2×C4×C8C2314yes—
132#400—28no—
133#396—28no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389C2314yes—
139#389C2314yes—
140#389C2314yes—
141#389C2314yes—
142#388—28no—
143#388—28no—
144#388—28no—
145#387—28no—
146#384—22no—
147#384C2314yes—
148#382—28no—
149#380—22no—
150#380—22no—
151#380—22no—
152#380C2314yes—
153#380C2314yes—
154#380C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4C2314yes—
159C4×C4⋊C4C2314yes—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4D414yes—
162C4×C22⋊C4C2314yes—
163#376—28no—
164C2×#95—21no—
165C2×#95—21no—
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×#95D411yescharacteristic
175C2×#95D411yescharacteristic
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188C25C2×D411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8#4114yes—
222C22×C8#4114yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227#117—216no—
228#117—28no—
229#117—28no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
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—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4#4114yes—
265C2×C4⋊C4#4114yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4—82no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—82no—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2412yes—
301C2×C42C2412yes—
302#107—28no—
303#107—28no—
304#107—28no—
305#106—28no—
306#106—28no—
307#106—28no—
308C4⋊C8—28no—
309C22⋊C8—28no—
310C22⋊C8—28no—
311C4×C8—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C22×D412yes—
332C24C22×D411yescharacteristic
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
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#13514yes—
392C22×C4#13714yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C42−1+412yes—
404C22×C42−1+412yes—
405C22×C4C2511yescharacteristic
406C2×C8—28no—
407C2×C8—28no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8C2×#4114yes—
421C2×C8C2×#4114yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4C2×#4114yes—
451C4⋊C4C2×#4114yes—
452C22⋊C4—216no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C42—216no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—48no—
466C42—48no—
467C42—84no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23#52512yes—
483C23#54712yes—
484C23#54712yes—
485C23C2×#13512yes—
486C23C2×#13712yes—
487C23#57612yes—
488C23#57712yes—
489C23C23×D411yescharacteristic
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—42no—
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—42no—
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—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4#52512yes—
546C2×C4#52914yes—
547C2×C4#54114yes—
548C2×C4#54712yes—
549C2×C4#54712yes—
550C2×C4#55214yes—
551C2×C4#55614yes—
552C2×C4C2×#13514yes—
553C2×C4C2×#13714yes—
554C2×C4C22×#4111yescharacteristic
555C2×C4C22×#4111yescharacteristic
556C2×C4C22×#4111yescharacteristic
557C2×C4C22×#4111yescharacteristic
558C2×C4C2×2−1+411yescharacteristic
559C2×C4C2×2−1+411yescharacteristic
560C8—24no—
561C8—24no—
562C8—42no—
563C8—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—88no—
572C22#247814yes—
573C22#310714yes—
574C22#311914yes—
575C22#333714yes—
576C22#334114yes—
577C22#334414yes—
578C22#335714yes—
579C22C2×#52511yescharacteristic
580C22C2×#54711yescharacteristic
581C22C2×#54711yescharacteristic
582C22C22×#13511yescharacteristic
583C22C22×#13711yescharacteristic
584C22C2×#57611yescharacteristic
585C22C2×#57711yescharacteristic
586C4—24no—
587C4—24no—
588C4—42no—
589C4—42no—
590C4—44no—
591C4—44no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4#247814yes—
600C4C2×#52911yescharacteristic
601C4C2×#54111yescharacteristic
602C4C2×#55211yescharacteristic
603C4C2×#55611yescharacteristic
604C2—82no—
605C2#2746918yesdirect product factor
606C2C2×#247811yescharacteristic
607C2C2×#310711yescharacteristic
608C2C2×#311911yescharacteristic
609C2C2×#333711yescharacteristic
610C2C2×#334111yescharacteristic
611C2C2×#334411yescharacteristic
612C2C2×#335711yescharacteristic
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. #27469C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2083C2semidirect
8. C2×#1948C2semidirect
9. C2×#1942C2semidirect
11. C2×#1921C2semidirect
12. C2×#1920C2semidirect
13. C2×#1918C2semidirect
16. C2×C4×#106C2semidirect
52. #2088C22semidirect
53. #2083C22semidirect
56. #1948C22semidirect
57. #1942C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1918C22semidirect
64. C4×#106C22semidirect
605. C2#27469direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#23462
#274692
C2×#20882
C2×#20832
C2×#20772
C2×#20752
C2×#19482
C2×#19422
C2×#19232
C2×#19212
C2×#19202
C2×#19182
C2×#18542
C2×C4×#1062