Finite Group Databasebeta

← C2×#27842 #6457733

Subgroups — C2×#27842 #6457733

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2496C211yesmaximal, characteristic
3#27842C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2076C211yesmaximal, characteristic
7C2×#2077C211yesmaximal, characteristic
8C2×#1943C211yesmaximal, characteristic, complemented
9C2×#1942C211yesmaximal, characteristic, complemented
10C2×#1919C211yesmaximal, characteristic, complemented
11C2×#1918C211yesmaximal, characteristic, complemented
12C2×#1847C211yesmaximal, characteristic, complemented
13C2×#1809C211yesmaximal, characteristic
14C2×#1809C211yesmaximal, characteristic
15C2×C4×#106C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C22×#126—21no—
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2496—24no—
29#2496C2218yes—
30#2494—24no—
31#2494C2218yes—
32C2×#401—21no—
33C2×#400—21no—
34C2×#397—21no—
35C2×#397—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388—21no—
41C2×#387—21no—
42C2×#387—21no—
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2100C2218yescomplemented
54#2077C2218yes—
55#2076C2218yes—
56#1943C2218yescomplemented
57#1942C2218yescomplemented
58#1919C2218yescomplemented
59#1918C2218yescomplemented
60#1847C2218yescomplemented
61#1809C2218yes—
62#1809C2218yes—
63C4×#106C2218yescomplemented
64#1753C2218yes—
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
78C22×M16—24no—
79C22×M16—24no—
80C22×M16—24no—
81C22×M16C2314yes—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#126—28no—
87C2×#126—48no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
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×C22⋊C8—24no—
125C2×C22⋊C8C2314yes—
126C2×#97—24no—
127C2×#97C2314yes—
128C2×C4×C8—24no—
129C2×C4×C8C2314yes—
130#401—28no—
131#400—28no—
132#397—28no—
133#397—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#387—28no—
144#387—28no—
145#384—22no—
146#384C2314yes—
147#382—22no—
148#382C2314yes—
149#380—22no—
150#380—22no—
151#380—22no—
152#380—22no—
153#380C2314yes—
154#380C2314yes—
155#380C2314yes—
156#380C2314yes—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4C2314yes—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4D414yes—
162C4×C22⋊C4C2314yes—
163C2×#95—21no—
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×#95—21no—
175C2×#95D411yescharacteristic
176C2×#95D411yescharacteristic
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25C2×D411yescharacteristic
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×M16—28no—
213C2×M16—28no—
214C2×M16—28no—
215C2×M16—416no—
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#4114yes—
225C22×C8#4114yes—
226C22×C8#4114yes—
227C22×C8#4114yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232#126—432no—
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—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—82no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—82no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2412yes—
300C2×C42C2412yes—
301#107—28no—
302#107—28no—
303#107—28no—
304#106—28no—
305#106—28no—
306#106—28no—
307C22⋊C8—28no—
308#97—28no—
309C4×C8—28no—
310#95—24no—
311#95—24no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
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—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
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—42no—
392C22×C4AGL(1,ℤ/8)14yes—
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
406M16—416no—
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—44no—
417C2×C8—44no—
418C2×C8#126116yescomplemented
419C2×C8C2×#4114yes—
420C2×C8C2×#4114yes—
421C2×C8C2×#4114yes—
422C2×C8C2×#4114yes—
423C4⋊C4—22no—
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—22no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—84no—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—48no—
466C42—84no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23#53212yes—
482C23#54712yes—
483C23#54712yes—
484C23C2×AGL(1,ℤ/8)12yes—
485C23#57512yes—
486C23#57712yes—
487C23#57812yes—
488C23C23×D411yescharacteristic
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
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—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4C2×#12614yes—
546C2×C4#53212yes—
547C2×C4#54114yes—
548C2×C4#54712yes—
549C2×C4#54712yes—
550C2×C4#55214yes—
551C2×C4#55614yes—
552C2×C4C2×AGL(1,ℤ/8)14yes—
553C2×C4C22×#4111yescharacteristic
554C2×C4C22×#4111yescharacteristic
555C2×C4C22×#4111yescharacteristic
556C2×C4C22×#4111yescharacteristic
557C2×C4C2×2−1+411yescharacteristic
558C2×C4C2×2−1+411yescharacteristic
559C8—24no—
560C8—42no—
561C8—42no—
562C8C2×#12618yescomplemented
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—88no—
571C22#249614yes—
572C22#317114yes—
573C22#317814yes—
574C22#334514yes—
575C22#334914yes—
576C22#335114yes—
577C22#335314yes—
578C22C2×#53211yescharacteristic
579C22C2×#54711yescharacteristic
580C22C2×#54711yescharacteristic
581C22C22×AGL(1,ℤ/8)11yescharacteristic
582C22C2×#57511yescharacteristic
583C22C2×#57711yescharacteristic
584C22C2×#57811yescharacteristic
585C4—24no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#249614yes—
599C4C22×#12611yescharacteristic
600C4C2×#54111yescharacteristic
601C4C2×#55211yescharacteristic
602C4C2×#55611yescharacteristic
603C2—82no—
604C2#2784218yesdirect product factor
605C2C2×#249611yescharacteristic
606C2C2×#317111yescharacteristic
607C2C2×#317811yescharacteristic
608C2C2×#334511yescharacteristic
609C2C2×#334911yescharacteristic
610C2C2×#335111yescharacteristic
611C2C2×#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. #27842C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2101C2semidirect
8. C2×#1943C2semidirect
9. C2×#1942C2semidirect
10. C2×#1919C2semidirect
11. C2×#1918C2semidirect
12. C2×#1847C2semidirect
15. C2×C4×#106C2semidirect
52. #2101C22semidirect
53. #2100C22semidirect
56. #1943C22semidirect
57. #1942C22semidirect
58. #1919C22semidirect
59. #1918C22semidirect
60. #1847C22semidirect
63. C4×#106C22semidirect
418. C2×C8#126semidirect
562. C8C2×#126semidirect
604. C2#27842direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#24962
#278422
C2×#21002
C2×#21012
C2×#20762
C2×#20772
C2×#19432
C2×#19422
C2×#19192
C2×#19182
C2×#18472
C2×#18092
C2×C4×#1062
C2×#17532