Finite Group Databasebeta

← C2×#29940 #6435637

Subgroups — C2×#29940 #6435637

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3#29940C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2076C211yesmaximal, characteristic
9C2×#2077C211yesmaximal, characteristic
10C2×#2077C211yesmaximal, characteristic
11C2×#2075C211yesmaximal, characteristic
12C2×#1989C211yesmaximal, characteristic, complemented
13C2×#1989C211yesmaximal, characteristic, complemented
14C2×#1949C211yesmaximal, characteristic, complemented
15C2×#1949C211yesmaximal, characteristic, complemented
16C2×#1797C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27#2851—24no—
28#2851C2218yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#387—21no—
32C2×#387—21no—
33C2×#385—21no—
34C2×#385—21no—
35#2283—24no—
36#2283C2218yes—
37C2×#383—21no—
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#381—21no—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×#376—21no—
50#2101C2218yescomplemented
51#2101C2218yescomplemented
52#2085C2218yescomplemented
53#2085C2218yescomplemented
54#2077C2218yes—
55#2077C2218yes—
56#2076C2218yes—
57#2075C2218yes—
58#1989C2218yescomplemented
59#1989C2218yescomplemented
60#1949C2218yescomplemented
61#1949C2218yescomplemented
62#1797C2218yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
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×C4D411yescharacteristic
78C24×C4D411yescharacteristic
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#125—22no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125C2314yes—
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×C42—24no—
105C22×C42—24no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42D414yes—
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C2×#107—42no—
115C2×#107—42no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106—42no—
119C2×#106—42no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8—24no—
123C2×C4⋊C8—24no—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126C2×C22⋊C8—42no—
127C2×C22⋊C8—42no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130#389—22no—
131#389—22no—
132#389C2314yes—
133#389C2314yes—
134#387—28no—
135#387—28no—
136#385—28no—
137#385—28no—
138#383—28no—
139#382—22no—
140#382—22no—
141#382—22no—
142#382C2314yes—
143#382C2314yes—
144#382C2314yes—
145#381—28no—
146#380—22no—
147#380—22no—
148#380—22no—
149#380—22no—
150#380C2314yes—
151#380C2314yes—
152#380C2314yes—
153#380C2314yes—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4C2314yes—
157C4×C4⋊C4C2314yes—
158C4×C22⋊C4—28no—
159#376—28no—
160C2×#95—21no—
161C2×#95—21no—
162C2×#95—21no—
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—
174#340C2314yes—
175#340C2314yes—
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—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D414yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
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—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235#125—28no—
236#125—28no—
237#125—48no—
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—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—216no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D412yes—
299C2×C42C2×D414yes—
300C2×C42C2×D412yes—
301C2×C42#4114yes—
302C2×C42#4114yes—
303C2×C42C2412yes—
304C2×C42C2412yes—
305#107—44no—
306#107—44no—
307#106—44no—
308#106—44no—
309C4⋊C8—28no—
310C4⋊C8—28no—
311C22⋊C8—44no—
312C22⋊C8—44no—
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#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24#13514yes—
334C24C22×D412yes—
335C24C22×D411yescharacteristic
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—28no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
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—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
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×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×C4—42no—
402C22×C4#12314yes—
403C22×C4#12314yes—
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×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414C22×C42−1+412yes—
415C22×C42−1+412yes—
416C22×C4C2511yescharacteristic
417C2×C8—22no—
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×C8—44no—
433C4⋊C4—22no—
434C4⋊C4—22no—
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—48no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C42—28no—
466C42—28no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—48no—
471C42—48no—
472C42#12318yes—
473C42C2×#4114yes—
474C42C2×#4114yes—
475C23—28no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23C2×#12312yes—
489C23#53612yes—
490C23#53712yes—
491C23C2×#13514yes—
492C23C2×#13512yes—
493C23#57512yes—
494C23#57512yes—
495C23#57512yes—
496C23C23×D411yescharacteristic
497C2×C4—28no—
498C2×C4—28no—
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—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
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—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—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4C2×#12314yes—
555C2×C4C2×#12314yes—
556C2×C4C2×#12312yes—
557C2×C4#53612yes—
558C2×C4#53712yes—
559C2×C4#54114yes—
560C2×C4#54114yes—
561C2×C4#54214yes—
562C2×C4#54214yes—
563C2×C4C22×#4111yescharacteristic
564C2×C4C22×#4111yescharacteristic
565C2×C4C2×H(5,𝔽2)11yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C2×C4C2×2−1+411yescharacteristic
568C2×C4C2×2−1+411yescharacteristic
569C8—42no—
570C8—42no—
571C8—42no—
572C8—42no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22#262514yes—
582C22#309114yes—
583C22#309314yes—
584C22#323214yes—
585C22#324014yes—
586C22#325014yes—
587C22#325614yes—
588C22C22×#12311yescharacteristic
589C22C2×#53611yescharacteristic
590C22C2×#53711yescharacteristic
591C22C22×#13511yescharacteristic
592C22C2×#57511yescharacteristic
593C22C2×#57511yescharacteristic
594C22C2×#57511yescharacteristic
595C4—24no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4#262514yes—
609C4C2×#54111yescharacteristic
610C4C2×#54111yescharacteristic
611C4C2×#54211yescharacteristic
612C4C2×#54211yescharacteristic
613C2—44no—
614C2#2994018yesdirect product factor
615C2C2×#262511yescharacteristic
616C2C2×#309111yescharacteristic
617C2C2×#309311yescharacteristic
618C2C2×#323211yescharacteristic
619C2C2×#324011yescharacteristic
620C2C2×#325011yescharacteristic
621C2C2×#325611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #29940C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2101C2semidirect
6. C2×#2085C2semidirect
7. C2×#2085C2semidirect
12. C2×#1989C2semidirect
13. C2×#1989C2semidirect
14. C2×#1949C2semidirect
15. C2×#1949C2semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2085C22semidirect
53. #2085C22semidirect
58. #1989C22semidirect
59. #1989C22semidirect
60. #1949C22semidirect
61. #1949C22semidirect
614. C2#29940direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22832
#299402
C2×#21012
C2×#20852
C2×#20762
C2×#20772
C2×#20752
C2×#19892
C2×#19492
C2×#17972