Finite Group Databasebeta

← C2×#31442 #7504715

Subgroups — C2×#31442 #7504715

Order 512.

1798 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×#2496C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3#31442C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2077C211yesmaximal, characteristic
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1949C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1924C211yesmaximal, characteristic
14C2×#1921C211yesmaximal, characteristic, complemented
15C2×#1852C211yesmaximal, characteristic
16C2×#1823C211yesmaximal, characteristic
17C22×#126—21no—
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×#97C2211yescharacteristic
28#2496—24no—
29#2496C2218yes—
30#2442—24no—
31#2442C2218yes—
32C2×#400—21no—
33C2×#400—21no—
34C2×#398—21no—
35C2×#397—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#385—21no—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#380—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2218yescomplemented
51#2100C2218yescomplemented
52#2088C2218yescomplemented
53#2077C2218yes—
54#2077C2218yes—
55#2059C2218yescomplemented
56#1992C2218yescomplemented
57#1949C2218yescomplemented
58#1944C2218yescomplemented
59#1924C2218yes—
60#1921C2218yescomplemented
61#1852C2218yes—
62#1823C2218yes—
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×C4—21no—
78C24×C4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#126—28no—
84C2×#126—48no—
85C2×#125—24no—
86C2×#125—22no—
87C2×#125—42no—
88C2×#125C2314yes—
89C22×C4⋊C4—24no—
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—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4D411yescharacteristic
114C22×C22⋊C4D411yescharacteristic
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C2×#107—24no—
119C2×#107—42no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106—24no—
123C2×#106—42no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8—24no—
127C2×C4⋊C8C2314yes—
128C2×C22⋊C8—24no—
129C2×C22⋊C8—42no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97—42no—
133C2×#97C2314yes—
134#400—28no—
135#400—28no—
136#398—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#385—28no—
148#384—22no—
149#384C2314yes—
150#382—22no—
151#382C2314yes—
152#380—22no—
153#380—22no—
154#380—22no—
155#380—28no—
156#380C2314yes—
157#380C2314yes—
158#380C2314yes—
159C4×C4⋊C4—22no—
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×#95D411yescharacteristic
174C2×#95D411yescharacteristic
175#340C2314yes—
176#340C2314yes—
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—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×C4—21no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—82no—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2412yes—
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—
232#126—432no—
233#125—28no—
234#125—44no—
235#125—48no—
236C2×C4⋊C4—21no—
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—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4#4114yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2412yes—
308C2×C42C2412yes—
309#107—28no—
310#107—44no—
311#106—28no—
312#106—44no—
313C4⋊C8—28no—
314C22⋊C8—28no—
315C22⋊C8—44no—
316#97—44no—
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—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24AGL(1,ℤ/8)14yes—
336C24C22×D412yes—
337C24C22×D411yescharacteristic
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
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—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
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—22no—
370C22×C4—22no—
371C22×C4—48no—
372C22×C4—42no—
373C22×C4—42no—
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—48no—
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—84no—
401C22×C4—816no—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C22×C4C2511yescharacteristic
415C2×C8—28no—
416C2×C8—22no—
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—44no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8C2×#4114yes—
431C4⋊C4—24no—
432C4⋊C4—22no—
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—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4C2×#4114yes—
464C22⋊C4—28no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—48no—
476C23—28no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23#53612yes—
490C23#54612yes—
491C23#54712yes—
492C23C2×AGL(1,ℤ/8)14yes—
493C23C2×AGL(1,ℤ/8)12yes—
494C23#57512yes—
495C23#57712yes—
496C23#57812yes—
497C23C23×D411yescharacteristic
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
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—22no—
513C2×C4—22no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
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—42no—
547C2×C4—42no—
548C2×C4—816no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4#53612yes—
557C2×C4#53914yes—
558C2×C4#54114yes—
559C2×C4#54612yes—
560C2×C4#54712yes—
561C2×C4#55614yes—
562C2×C4#56314yes—
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—24no—
570C8—42no—
571C8—42no—
572C8—42no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—416no—
580C22—42no—
581C22#268914yes—
582C22#323414yes—
583C22#324214yes—
584C22#330614yes—
585C22#332714yes—
586C22#334314yes—
587C22#335014yes—
588C22C2×#53611yescharacteristic
589C22C2×#54611yescharacteristic
590C22C2×#54711yescharacteristic
591C22C22×AGL(1,ℤ/8)11yescharacteristic
592C22C2×#57511yescharacteristic
593C22C2×#57711yescharacteristic
594C22C2×#57811yescharacteristic
595C4—42no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—42no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4#268914yes—
609C4C2×#53911yescharacteristic
610C4C2×#54111yescharacteristic
611C4C2×#55611yescharacteristic
612C4C2×#56311yescharacteristic
613C2—44no—
614C2#3144218yesdirect product factor
615C2C2×#268911yescharacteristic
616C2C2×#323411yescharacteristic
617C2C2×#324211yescharacteristic
618C2C2×#330611yescharacteristic
619C2C2×#332711yescharacteristic
620C2C2×#334311yescharacteristic
621C2C2×#335011yescharacteristic
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. #31442C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2101C2semidirect
6. C2×#2088C2semidirect
9. C2×#2059C2semidirect
10. C2×#1992C2semidirect
11. C2×#1949C2semidirect
12. C2×#1944C2semidirect
14. C2×#1921C2semidirect
50. #2101C22semidirect
51. #2100C22semidirect
52. #2088C22semidirect
55. #2059C22semidirect
56. #1992C22semidirect
57. #1949C22semidirect
58. #1944C22semidirect
60. #1921C22semidirect
614. C2#31442direct (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×#24422
#314422
C2×#21002
C2×#21012
C2×#20882
C2×#20772
C2×#20592
C2×#19922
C2×#19492
C2×#19442
C2×#19242
C2×#19212
C2×#18522
C2×#18232